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data/OlymMATH-EN-EASY.jsonl
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{"problem": "Given a non-negative integer sequence $\\{a_n\\}$ satisfying $a_1 = 2016$, $a_{n+1} \\le \\sqrt{a_n}$, and if the number of terms is at least 2, then any two terms in the sequence are not equal. Find the number of such sequences $\\{a_n\\}$.", "answer": "948", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-0-EN"}
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{"problem": "Given that $AB$ is a diameter of circle $\\odot C$ with radius $2$, circle $\\odot D$ is internally tangent to circle $\\odot C$ at point $A$, circle $\\odot E$ is internally tangent to circle $\\odot C$, externally tangent to circle $\\odot D$, and tangent to line segment $AB$ at point $F$. If the radius of circle $\\odot D$ is $3$ times the radius of circle $\\odot E$, find the radius of circle $\\odot D$.", "answer": "4\\sqrt{15}-14", "subject": "Geometry", "unique_id": "OlymMATH-EASY-1-EN"}
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{"problem": "Calculate the value of $\\sqrt{9+8\\cos 20^{\\circ }}-\\sec 20^{\\circ }$.", "answer": "3", "subject": "Algebra", "unique_id": "OlymMATH-EASY-2-EN"}
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{"problem": "A sphere is circumscribed around tetrahedron $ABCD$, and another sphere with radius $1$ is tangent to plane $ABC$. The two spheres are internally tangent at point $D$. If $AD=3$, $\\cos \\angle BAC=\\frac{4}{5}$, $\\cos \\angle BAD=\\cos \\angle CAD=\\frac{1}{\\sqrt{2}}$, find the volume of tetrahedron $ABCD$.", "answer": "\\frac{18}{5}", "subject": "Geometry", "unique_id": "OlymMATH-EASY-3-EN"}
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{"problem": "Find the minimum value of $f(x) = \\sum_{i=1}^{2017} i|x-i|$ when $x \\in [1, 2017]$.", "answer": "801730806", "subject": "Algebra", "unique_id": "OlymMATH-EASY-4-EN"}
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{"problem": "In a triangle, the three interior angles form an arithmetic sequence. The difference between the longest and shortest sides is 4 times the height to the third side. Find how much larger the largest interior angle is than the smallest interior angle. (Express the answer using inverse trigonometric functions)", "answer": "\\pi -\\arccos \\frac{1}{8}", "subject": "Geometry", "unique_id": "OlymMATH-EASY-5-EN"}
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{"problem": "What is the distance between the foci of the quadratic curve $(3x+4y-13)(7x-24y+3)=200$?", "answer": "2\\sqrt{10}", "subject": "Geometry", "unique_id": "OlymMATH-EASY-6-EN"}
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{"problem": "In a cube, any two vertices determine a line. Find how many pairs of lines are perpendicular and skew (non-intersecting) to each other.", "answer": "78", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-7-EN"}
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{"problem": "Find the range of the function $f(x)=\\frac{(x-x^3)(1-6x^2+x^4)}{(1+x^2)^4}$.", "answer": "\\left[ -\\frac{1}{8}, \\frac{1}{8} \\right]", "subject": "Algebra", "unique_id": "OlymMATH-EASY-8-EN"}
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{"problem": "Let the set of positive integers $A = \\{a_1, a_2, \\dots, a_{1000}\\}$, where $a_1 < a_2 < \\dots < a_{1000} \\le 2017$. If for any $1 \\le i, j \\le 1000$, whenever $i+j \\in A$, we have $a_i + a_j \\in A$, find the number of sets $A$ that satisfy this condition.", "answer": "2^{17}", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-9-EN"}
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{"problem": "Given $x, y \\in \\mathbf{R}$, for any $n \\in \\mathbf{Z}_{+}$, $nx+\\frac{1}{n}y\\geq 1$. Find the minimum value of $41x+2y$.", "answer": "9", "subject": "Algebra", "unique_id": "OlymMATH-EASY-10-EN"}
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{"problem": "Let $T$ be the set consisting of all positive divisors of $2020^{100}$. The set $S$ satisfies: (1) $S$ is a subset of $T$; (2) No element in $S$ is a multiple of another element in $S$. Find the maximum number of elements in $S$.", "answer": "10201", "subject": "Number Theory", "unique_id": "OlymMATH-EASY-11-EN"}
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{"problem": "Find the maximum number of right angles among all interior angles of a simple 300-sided polygon (without self-intersections) in a plane.", "answer": "201", "subject": "Geometry", "unique_id": "OlymMATH-EASY-12-EN"}
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{"problem": "Let $x$, $y$, $z$ be complex numbers satisfying $x^2 + y^2 + z^2 = xy + yz + zx$, $|x+y+z| = 21$, $|x-y| = 2\\sqrt{3}$, $|x| = 3\\sqrt{3}$. Find the value of $|y|^2 + |z|^2$.", "answer": "132", "subject": "Algebra", "unique_id": "OlymMATH-EASY-13-EN"}
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{"problem": "Define an \"operation\" as replacing a known positive integer $n$ with a randomly chosen non-negative integer less than it (with equal probability for each number). Find the probability that when performing multiple operations to transform $2019$ into $0$, the numbers $10$, $100$, and $1000$ all appear during the process.", "answer": "\\frac{1}{1112111}", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-14-EN"}
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{"problem": "In the Cartesian coordinate system, consider the set of points $\\{(m, n) | m, n \\in \\mathbf{Z}_{+}, 1 \\leqslant m, n \\leqslant 6\\}$. Each point is colored either red or blue. Find the number of different coloring schemes such that each unit square has exactly two red vertices.", "answer": "126", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-15-EN"}
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{"problem": "Given that circle $\\odot O$ has equation $x^2 + y^2 = 4$, circle $\\odot M$ has equation $(x - 5\\cos\\theta)^2 + (y - 5\\sin\\theta)^2 = 1 (\\theta \\in \\mathbf{R})$. From any point $P$ on circle $\\odot M$, draw two tangent lines $PE$ and $PF$ to circle $\\odot O$, with points of tangency $E$ and $F$ respectively. Find the minimum value of $\\overrightarrow{PE} \\cdot \\overrightarrow{PF}$.", "answer": "6", "subject": "Geometry", "unique_id": "OlymMATH-EASY-16-EN"}
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{"problem": "Given an integer sequence $\\{a_{i,j}\\}$ ($i, j \\in \\mathbf{N}$), where $a_{1,n} = n^n$ ($n \\in \\mathbf{Z}_{+}$), $a_{i,j} = a_{i-1,j} + a_{i-1,j+1}$ ($i, j \\geqslant 1$). Find the units digit of $a_{128,1}$.", "answer": "4", "subject": "Algebra", "unique_id": "OlymMATH-EASY-17-EN"}
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{"problem": "There are $100$ distinct points and $n$ distinct lines $l_1, l_2, \\dots, l_n$ on a plane. Let $a_k$ denote the number of points that line $l_k$ passes through. If $a_1 + a_2 + \\dots + a_n = 250$, find the minimum possible value of $n$.", "answer": "21", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-18-EN"}
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{"problem": "Let $k = -\\frac{1}{2} + \\frac{\\sqrt{3}}{2}i$. In the complex plane, the vertices of triangle $\\triangle ABC$ correspond to complex numbers $z_1$, $z_2$, $z_3$ satisfying $z_1 + kz_2 + k^2(2z_3 - z_1) = 0$. Find the radian measure of the smallest interior angle of this triangle.", "answer": "\\frac{\\pi}{6}", "subject": "Algebra", "unique_id": "OlymMATH-EASY-19-EN"}
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{"problem": "Suppose 40 people vote anonymously, each with one ballot. Each person can vote for one or two candidates among three candidates. There are no invalid ballots. Find the number of different possible voting outcomes.", "answer": "45961", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-20-EN"}
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{"problem": "In an $n \\times n$ square grid, there are $(n+1)^2$ intersection points. The number of squares (which can be tilted) with vertices at these intersection points is denoted as $a_n$. It is known that when $n=2$, $a_2 = 6$, and when $n=3$, $a_3 = 20$. Find the value of $a_{26}$.", "answer": "44226", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-21-EN"}
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{"problem": "An ellipse $\\frac{x^{2}}{8}+\\frac{y^{2}}{4}=1$, a line passing through point $F(2,0)$ intersects the ellipse at points $A$ and $B$, point $C$ is on the line $x=4$. If $\\triangle ABC$ is an equilateral triangle, find the area of $\\triangle ABC$.", "answer": "\\frac{72\\sqrt{3}}{25}", "subject": "Geometry", "unique_id": "OlymMATH-EASY-22-EN"}
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{"problem": "The ordered positive integer array $(a_1, a_2, \\dots, a_{23})$ satisfies: (1) $a_1 < a_2 < \\dots < a_{23} = 50$; (2) any three numbers in the array can form the three sides of a triangle. Find the number of arrays that satisfy these conditions.", "answer": "2576", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-23-EN"}
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{"problem": "Given that the cubic equation $x^{3}-x^{2}-5x-1=0$ has three distinct roots $x_{1}$, $x_{2}$, $x_{3}$. Find the value of $\\left({x}_{1}^{2}-4x_{1}x_{2}+{x}_{2}^{2}\\right)\\left({x}_{2}^{2}-4x_{2}x_{3}+{x}_{3}^{2}\\right)\\left({x}_{3}^{2}-4x_{3}x_{1}+{x}_{1}^{2}\\right)$.", "answer": "444", "subject": "Algebra", "unique_id": "OlymMATH-EASY-24-EN"}
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{"problem": "Let $\\triangle ABC$ be an acute triangle where the lengths of the sides opposite to the angles are $a$, $b$, and $c$ respectively. If $2a^{2}=2b^{2}+c^{2}$, find the minimum value of $\\tan A+\\tan B+\\tan C$.", "answer": "6", "subject": "Algebra", "unique_id": "OlymMATH-EASY-25-EN"}
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{"problem": "In the tetrahedron $ABCD$, $DA=DB=DC=1$, and $DA$, $DB$, $DC$ are perpendicular to each other. Find the length of the curve formed by points on the surface of the tetrahedron that are at a distance of $\\frac{2\\sqrt{3}}{3}$ from point $A$.", "answer": "\\frac{\\sqrt{3}\\pi}{2}", "subject": "Geometry", "unique_id": "OlymMATH-EASY-26-EN"}
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{"problem": "Consider all three-element subsets $\\{a, b, c\\}$ of the set $\\{1, 2, \\cdots, 81\\}$, where $a < b < c$. We call $b$ the middle element. Find the sum $S$ of all middle elements.", "answer": "3498120", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-27-EN"}
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{"problem": "In a cube $ABCD-A_{1}B_{1}C_{1}D_{1}$ with edge length $1$, $M$ and $N$ are the midpoints of edges $C_{1}D_{1}$ and $B_{1}C_{1}$ respectively. Find the area of the cross-section formed when plane $AMN$ intersects this cube.", "answer": "\\frac{7\\sqrt{17}}{24}", "subject": "Geometry", "unique_id": "OlymMATH-EASY-28-EN"}
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{"problem": "In the Cartesian coordinate system $xOy$, $F$ is the focus of the parabola $\\Gamma: y^2 = 2px(p>0)$. Point $B$ is on the $x$-axis and to the right of point $F$. Point $A$ is on $\\Gamma$, and $|AF|=|BF|$. The lines $AF$ and $AB$ intersect $\\Gamma$ at second points $M$ and $N$ respectively. If $\\angle AMN=90^\\circ$, find the slope of line $AF$.", "answer": "\\sqrt{3}", "subject": "Geometry", "unique_id": "OlymMATH-EASY-29-EN"}
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{"problem": "Given that the function $f\\colon \\mathbf{R}\\rightarrow \\mathbf{R}$ satisfies $f(x^2)+f(y^2)=f^2(x+y)-2xy$ for all $x$, $y\\in \\mathbf{R}$. Let $S={\\sum}_{n=-2020}^{2020}f(n)$. Find how many possible values $S$ can take.", "answer": "2041211", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-30-EN"}
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| 32 |
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{"problem": "Given that the lateral edge length of a regular triangular pyramid is 1, and the dihedral angle between the side face and the base face is $45^{\\circ}$. Find the volume of the circumscribed sphere of this regular triangular pyramid.", "answer": "\\frac{5\\sqrt{5}\\pi}{6}", "subject": "Geometry", "unique_id": "OlymMATH-EASY-31-EN"}
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{"problem": "Given $x_i \\in \\mathbf{R} (1 \\leqslant i \\leqslant 2020)$, and $x_{1010} = 1$. Find the minimum value of ${\\sum}_{i,j=1}^{2020} \\min\\{i,j\\}x_i x_j$.", "answer": "\\frac{1}{2}", "subject": "Algebra", "unique_id": "OlymMATH-EASY-32-EN"}
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{"problem": "A digital clock displays hours, minutes, and seconds using two digits each (such as 10:09:18). Between 05:00:00 and 22:59:59 on the same day, what is the probability that all six digits on the clock face are different?", "answer": "\\frac{16}{135}", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-33-EN"}
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{"problem": "Given the set $M = \\{1, 2, \\dots, 2020\\}$. Now each number in $M$ is colored either red, yellow, or blue, and each color exists. Let $S_1 = \\{(x, y, z) \\in M^3 \\mid x, y, z \\text{ are the same color}, 2020 \\mid (x+y+z)\\}$, $S_2 = \\{(x, y, z) \\in M^3 \\mid x, y, z \\text{ are pairwise different colors}, 2020 \\mid (x+y+z)\\}$. Find the minimum value of $2|S_1| - |S_2|$.", "answer": "2", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-34-EN"}
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{"problem": "Let $O$ be the incenter of $\\triangle ABC$, $AB=3$, $AC=4$, $BC=5$, $\\overrightarrow{OP}=x\\overrightarrow{OA}+y\\overrightarrow{OB}+z\\overrightarrow{OC}$, $0 \\leqslant x, y, z \\leqslant 1$. Find the area of the plane region covered by the trajectory of the moving point $P$.", "answer": "12", "subject": "Geometry", "unique_id": "OlymMATH-EASY-35-EN"}
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{"problem": "Let $\\{a, b, c, d\\}$ be a subset of $\\{1, 2, \\cdots, 17\\}$. If $17|(a - b + c - d)$, then $\\{a, b, c, d\\}$ is called a \"good subset\". Find the number of good subsets.", "answer": "476", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-36-EN"}
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{"problem": "In the quadrangular pyramid $P-ABCD$, $\\overrightarrow{DC}=3\\overrightarrow{AB}$. A plane passing through line $AB$ divides the quadrangular pyramid into two parts of equal volume. Let $E$ be the point where this plane intersects edge $PC$. Find the value of $\\frac{PE}{PC}$.", "answer": "\\frac{2}{3}", "subject": "Geometry", "unique_id": "OlymMATH-EASY-37-EN"}
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{"problem": "In the cube $ABCD-EFGH$, $M$ is the midpoint of edge $GH$. Plane $AFM$ divides the cube into two parts with volumes $V_1$ and $V_2$ ($V_1 \\leqslant V_2$). Find the value of $\\frac{V_1}{V_2}$.", "answer": "\\frac{7}{17}", "subject": "Geometry", "unique_id": "OlymMATH-EASY-38-EN"}
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{"problem": "Given that set $S$ contains all integers between 1 and $2^{40}$ whose binary representation has exactly two 1's and the rest are 0's. Find the probability that a randomly selected number from $S$ is divisible by 9.", "answer": "\\frac{133}{780}", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-39-EN"}
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{"problem": "Given an ellipse $\\frac{x^2}{4} + \\frac{y^2}{3} = 1$, $F_1$ and $F_2$ are its left and right foci, respectively. A moving line $l$ is tangent to this ellipse. The symmetric point of the right focus $F_2$ with respect to the line $l$ is $P(m, n)$, $S = |3m + 4n - 24|$. Find the range of values for $S$.", "answer": "[7, 47]", "subject": "Geometry", "unique_id": "OlymMATH-EASY-40-EN"}
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{"problem": "In the regular triangular pyramid $P-ABC$, $AP = 3$, $AB = 4$, $D$ is a point on line $BC$, and the angle between face $APD$ and line $BC$ is $45^\\circ$. Find the length of segment $PD$.", "answer": "\\frac{\\sqrt{89}}{3}", "subject": "Geometry", "unique_id": "OlymMATH-EASY-41-EN"}
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{"problem": "In a Cartesian coordinate system, four points are fixed: $A(0,0)$, $B(2,0)$, $C(4,2)$, $D(4,4)$. Two ants crawl from point $A$ to point $D$ and from point $B$ to point $C$ respectively. The ants can only move in the positive direction of the coordinate axes, and can only change direction at integer points. Find the number of path pairs such that the two ants never meet.", "answer": "195", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-42-EN"}
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{"problem": "Let $h_n$ represent the number of regions that a convex polygon with $n+2$ sides is divided into by its diagonals. Assume that no three diagonals intersect at the same point, and define $h_0 = 0$. Find the value of $h_{26}$.", "answer": "20826", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-43-EN"}
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{"problem": "A line $l$ passing through point $P(2,1)$ intersects the positive $x$-axis and the positive $y$-axis at points $A$ and $B$ respectively. $O$ is the origin of the coordinate system. Find the $y$-intercept of line $l$ when the perimeter of triangle $\\triangle AOB$ is minimum.", "answer": "\\frac{5}{2}", "subject": "Geometry", "unique_id": "OlymMATH-EASY-44-EN"}
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{"problem": "Define a function $f(x)$ on the set $\\{x\\in \\mathbb{Z}_{+} | 1\\leqslant x\\leqslant 12\\}$ satisfying $|f(x+1)-f(x)|=1$ ($x=1, 2, \\dots, 11$), and $f(1)$, $f(6)$, $f(12)$ form a geometric sequence. If $f(1)=1$, find the number of different functions $f(x)$ that satisfy these conditions.", "answer": "355", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-45-EN"}
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{"problem": "There are three colors of small balls in a box: red, yellow, and blue. There are $12$ red balls, $18$ yellow balls, and $30$ blue balls. Each time, one ball is taken out from the box until all balls are taken out. Find the probability that the red balls are the first to be completely taken out.", "answer": "\\frac{18}{35}", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-46-EN"}
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{"problem": "Find the sum of squares of all distinct real roots of the equation $x^8 - 14x^4 - 8x^3 - x^2 + 1 = 0$ with respect to $x$.", "answer": "8", "subject": "Algebra", "unique_id": "OlymMATH-EASY-47-EN"}
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{"problem": "Let $z_1, z_2, \\dots, z_7$ be the seven distinct complex roots of $z^7 = -1$. Find the value of $\\sum_{j=1}^7 \\frac{1}{|1 - z_j|^2}$.", "answer": "\\frac{49}{4}", "subject": "Algebra", "unique_id": "OlymMATH-EASY-48-EN"}
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| 50 |
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{"problem": "Given a hyperbola $\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1$ ($a> 0$, $b> 0$) with left and right foci $F_{1}$ and $F_{2}$ respectively. A line passing through $F_{2}$ intersects the right branch of the hyperbola at points $A$ and $B$. If $\\left| AF_{2}\\right| =3\\left| F_{2}B\\right| $ and $\\left| AF_{1}\\right| =\\left| AB\\right| $, find the eccentricity of the hyperbola.", "answer": "2", "subject": "Geometry", "unique_id": "OlymMATH-EASY-49-EN"}
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{"problem": "Let $a$, $b$, $c$ be positive real numbers. Find the minimum value of $\\frac{(a+b+c)(a^2+3b^2+15c^2)}{abc}$.", "answer": "36", "subject": "Algebra", "unique_id": "OlymMATH-EASY-50-EN"}
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{"problem": "A frisbee toy is a circular disc divided into 20 sectors by 20 rays emanating from the center, with each sector colored either red or blue (only the front side is colored), and any two opposite sectors are colored differently. If frisbee toys that are the same after rotation are considered identical, how many different frisbee toys are there in total? (Answer with a specific number.)", "answer": "52", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-51-EN"}
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{"problem": "Ten points are given on a plane, with no three points collinear. Four line segments are drawn, each connecting two points on the plane. These line segments are chosen randomly, and each line segment has the same probability of being selected. Find the probability that three of these line segments form a triangle with three of the given ten points as vertices.", "answer": "\\frac{16}{473}", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-52-EN"}
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{"problem": "Find the smallest positive real number $c$ such that for any positive integer $n(n \\geqslant 2)$ and positive real numbers $a_1 \\leqslant a_2 \\leqslant \\cdots \\leqslant a_n$, we have $\\sum_{k=2}^{n}\\frac{a_{k}-2\\sqrt{a_{k-1}(a_{k}-a_{k-1})}}{c^{k}}\\geqslant \\frac{a_{n}}{nc^{n}}-\\frac{a_{1}}{c}$.", "answer": "2", "subject": "Algebra", "unique_id": "OlymMATH-EASY-53-EN"}
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{"problem": "From $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$, select $7$ different numbers to form a sequence $a_1, a_2, \\cdots, a_7$, such that the sum of any $4$ adjacent terms is a multiple of $3$. Find the number of such sequences.", "answer": "3024", "subject": "Number Theory", "unique_id": "OlymMATH-EASY-54-EN"}
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| 56 |
+
{"problem": "For a positive integer $n$, denote the sum of its digits as $s(n)$ and the product of its digits as $p(n)$. If $s(n)+p(n)=n$, then $n$ is called a coincidental number. Find the sum of all coincidental numbers.", "answer": "531", "subject": "Number Theory", "unique_id": "OlymMATH-EASY-55-EN"}
|
| 57 |
+
{"problem": "Nine consecutive positive integers are arranged in ascending order as a sequence $a_1<\\cdots<a_9$. If $a_1+a_3+a_5+a_7+a_9$ is a perfect square, and $a_2+a_4+a_6+a_8$ is a perfect cube, find the minimum value of the sum of these nine positive integers.", "answer": "18000", "subject": "Number Theory", "unique_id": "OlymMATH-EASY-56-EN"}
|
| 58 |
+
{"problem": "If the number of 1's in the binary representation of $n$ is greater than the number of 0's, then the positive integer $n$ is called a good number. Find the number of good numbers not exceeding $2017$.", "answer": "1169", "subject": "Number Theory", "unique_id": "OlymMATH-EASY-57-EN"}
|
| 59 |
+
{"problem": "Consider each permutation of $1, 2, \\cdots, 8$ as an eight-digit number. Find the number of such eight-digit numbers that are divisible by $11$.", "answer": "4608", "subject": "Number Theory", "unique_id": "OlymMATH-EASY-58-EN"}
|
| 60 |
+
{"problem": "Find the value of the integer $n$ that satisfies $133^5 + 110^5 + 84^5 + 27^5 = n^5$.", "answer": "144", "subject": "Number Theory", "unique_id": "OlymMATH-EASY-59-EN"}
|
| 61 |
+
{"problem": "Choose a set of numbers from $1, 2, \\cdots, 2018$ such that for any two numbers in the set, their sum cannot be divided by their difference. Find the maximum possible size of such a set.", "answer": "673", "subject": "Number Theory", "unique_id": "OlymMATH-EASY-60-EN"}
|
| 62 |
+
{"problem": "Find the number of 2012-digit even numbers consisting of digits 0, 1, and 2, where each digit appears at least once.", "answer": "4\\times 3^{2010}-5\\times 2^{2010}+1", "subject": "Number Theory", "unique_id": "OlymMATH-EASY-61-EN"}
|
| 63 |
+
{"problem": "For a positive integer $n$, let $\\varphi(n)$ denote the number of positive integers not exceeding $n$ and coprime to $n$. Let $f(n)$ denote the smallest positive integer greater than $n$ that is not coprime to $n$. If $f(n) = m$ and $\\varphi(m) = n$, then $(m, n)$ is called a lucky pair. Consider the set $S$ of all lucky pairs, find the value of $\\sum_{(m, n)\\in S} (m + n)$.", "answer": "6", "subject": "Number Theory", "unique_id": "OlymMATH-EASY-62-EN"}
|
| 64 |
+
{"problem": "If a positive integer $a$ satisfies: there exists a prime number $p$ such that $a^2+p$ is also a perfect square, then $a$ is called a good number. Find the number of good numbers in the set $M=\\{1, 2, \\cdots, 100\\}$.", "answer": "45", "subject": "Number Theory", "unique_id": "OlymMATH-EASY-63-EN"}
|
| 65 |
+
{"problem": "Given the set $A = \\{1, 2, \\cdots, 2019\\}$, a mapping $f: A \\rightarrow A$ satisfies that for any $k \\in A$, we have $f(k) \\leqslant k$, and the image has exactly $2018$ different values. Find how many such mappings $f$ exist.", "answer": "2^{2019} - 2020", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-64-EN"}
|
| 66 |
+
{"problem": "Given a sequence $\\{a_n\\}$ that satisfies $a_{n+1} + (-1)^n a_n = 2n - 1$, and the sum of the first $2019$ terms of the sequence $\\{a_n - n\\}$ is $2019$, find the value of $a_{2020}$.", "answer": "1", "subject": "Algebra", "unique_id": "OlymMATH-EASY-65-EN"}
|
| 67 |
+
{"problem": "Given the set $\\{1, 2, \\cdots, 30\\}$, a three-element subset is called \"interesting\" if the product of its three elements is a multiple of $8$. Find how many interesting subsets of $\\{1, 2, \\cdots, 30\\}$ there are.", "answer": "1925", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-66-EN"}
|
| 68 |
+
{"problem": "Given a regular tetrahedron $ABCD$ with edge length $1$, $M$ is the midpoint of $AC$, and $P$ is on the line segment $DM$. Find the minimum value of $AP+BP$.", "answer": "\\sqrt{1+\\frac{\\sqrt{6}}{3}}", "subject": "Geometry", "unique_id": "OlymMATH-EASY-67-EN"}
|
| 69 |
+
{"problem": "Let $(a_1, a_2, \\cdots, a_{2022})$ be a circular arrangement of integers $1, 2, \\ldots, 2022$ in clockwise order. If $\\sum_{i=1}^{2022} |a_i - a_{i+1}| = 4042$ ($a_{2023} = a_1$), find the number of circular arrangements that satisfy this condition.", "answer": "2^{2020}", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-68-EN"}
|
| 70 |
+
{"problem": "Given a hyperbola $\\Gamma: \\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$, $F$ is its left focus. The line $y = kx$ intersects the left and right branches of $\\Gamma$ at points $A$ and $B$ respectively, satisfying $FA \\perp AB$ and $\\angle ABF = \\angle AFO$ ($O$ is the origin). Find the eccentricity of $\\Gamma$.", "answer": "\\frac{3\\sqrt{2}+\\sqrt{6}}{2}", "subject": "Geometry", "unique_id": "OlymMATH-EASY-69-EN"}
|
| 71 |
+
{"problem": "Given that the distances from a point $P$ in space to the vertices $A$ and $B$ of a regular tetrahedron $ABCD$ are $2$ and $3$ respectively. Find the maximum distance from point $P$ to the line $CD$ when the edge length of the regular tetrahedron varies.", "answer": "\\frac{5\\sqrt{3}}{2}", "subject": "Geometry", "unique_id": "OlymMATH-EASY-70-EN"}
|
| 72 |
+
{"problem": "Given that two vertices of an equilateral triangle are on the parabola $y^2 = 4x$, and the third vertex is on the directrix of the parabola, and the distance from the center of the triangle to the directrix equals $\\frac{1}{9}$ of the perimeter. Find the area of the triangle.", "answer": "36\\sqrt{3}", "subject": "Geometry", "unique_id": "OlymMATH-EASY-71-EN"}
|
| 73 |
+
{"problem": "Given that the right focus of the ellipse $\\frac{x^{2}}{9}+\\frac{y^{2}}{5}=1$ is $F$, $P$ is a point on the ellipse, and point $A\\left(0,2\\sqrt{3}\\right)$. Find the area of $\\triangle APF$ when the perimeter of $\\triangle APF$ is at its maximum.", "answer": "\\frac{21\\sqrt{3}}{4}", "subject": "Geometry", "unique_id": "OlymMATH-EASY-72-EN"}
|
| 74 |
+
{"problem": "Let complex numbers $z_{1}=-\\sqrt{3}-\\mathrm{i}$, $z_{2}=3+\\sqrt{3}\\mathrm{i}$, $z=\\left(2+\\cos \\theta \\right)+\\mathrm{i}\\sin \\theta$. Find the minimum value of $\\left| z-z_{1}\\right| +\\left| z-z_{2}\\right| $.", "answer": "2+2\\sqrt{3}", "subject": "Algebra", "unique_id": "OlymMATH-EASY-73-EN"}
|
| 75 |
+
{"problem": "Let $A = \\{1, 2, \\cdots, 6\\}$, and function $f: A \\rightarrow A$. Define $p(f) = f(1) \\cdots f(6)$. Find the number of functions such that $p(f) | 36$.", "answer": "580", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-74-EN"}
|
| 76 |
+
{"problem": "Given the parabola $C: y^2 = 4x$ with focus $F$, the symmetric point of $F$ with respect to the origin is $M$, and $\\odot M$ is a circle with radius $1$. Line $l$ passes through a point $A$ on $\\odot M$ (different from the origin), and is tangent to the parabola $C$ at point $T$. Find the maximum value of $\\frac{|FA|}{|FT|}$.", "answer": "\\frac{1 + \\sqrt{5}}{2}", "subject": "Geometry", "unique_id": "OlymMATH-EASY-75-EN"}
|
| 77 |
+
{"problem": "Let $x_{i} \\geq 0 (i = 1, 2, \\cdots, 6)$, and satisfy $\\begin{cases} x_{1} + x_{2} + \\cdots + x_{6} = 1, \\\\ x_{1} x_{3} x_{5} + x_{2} x_{4} x_{6} \\geq \\frac{1}{540} \\end{cases}$. Find the maximum value of $x_{1} x_{2} x_{3} + x_{2} x_{3} x_{4} + x_{3} x_{4} x_{5} + x_{4} x_{5} x_{6} + x_{5} x_{6} x_{1} + x_{6} x_{1} x_{2}$.", "answer": "\\frac{19}{540}", "subject": "Algebra", "unique_id": "OlymMATH-EASY-76-EN"}
|
| 78 |
+
{"problem": "Given real numbers $a_1, a_2, \\cdots, a_{224}$ such that for any $i = 1, 2, \\cdots, 224$, we have $i \\leqslant a_i \\leqslant 2i$. Find the minimum value of $\\frac{(\\sum_{i=1}^{224} i a_i)^2}{\\sum_{i=1}^{224} a_i^2}$.", "answer": "\\frac{10057600}{3}", "subject": "Algebra", "unique_id": "OlymMATH-EASY-77-EN"}
|
| 79 |
+
{"problem": "Color each cell in a $5 \\times 5$ grid with one of five colors, such that the number of cells of each color is the same. If two adjacent cells have different colors, their common edge is called a \"dividing edge\". Find the minimum number of dividing edges.", "answer": "16", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-78-EN"}
|
| 80 |
+
{"problem": "If 3 points are randomly selected from the vertices of a regular $17$-sided polygon, what is the probability that these points form an acute triangle?", "answer": "\\frac{3}{10}", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-79-EN"}
|
| 81 |
+
{"problem": "Given that the function $f(x) = 10x^2 + mx + n$ ($m, n \\in \\mathbf{Z}$) has two distinct real roots in the interval $(1, 3)$. Find the maximum possible value of $f(1)f(3)$.", "answer": "99", "subject": "Algebra", "unique_id": "OlymMATH-EASY-80-EN"}
|
| 82 |
+
{"problem": "In the Cartesian coordinate system $xOy$, a circle with center at the origin $C$ and radius $1$ has a tangent line $l$ that intersects the $x$-axis at point $N$ and the $y$-axis at point $M$. Point $A(3,4)$, and $\\overrightarrow{AC}=x\\overrightarrow{AM}+y\\overrightarrow{AN}$. Let point $P(x,y)$.\n\nFind the minimum value of $9x^{2}+16y^{2}$.", "answer": "4", "subject": "Geometry", "unique_id": "OlymMATH-EASY-81-EN"}
|
| 83 |
+
{"problem": "Given $x, y, z \\in \\mathbf{R}_{+}$ and $x+y+z=1$, find the maximum value of $x+\\sqrt{2xy}+3\\sqrt[3]{xyz}$.", "answer": "2", "subject": "Algebra", "unique_id": "OlymMATH-EASY-82-EN"}
|
| 84 |
+
{"problem": "Given three points $A$, $B$, $C$ on the ellipse $\\frac{x^{2}}{4}+y^{2}=1$, the line $BC$ with negative slope intersects the $y$-axis at point $M$. If the origin $O$ is the centroid of $\\triangle ABC$, and the ratio of the areas of $\\triangle BMA$ to $\\triangle CMO$ is $3:2$. The possible slopes of line $BC$ form a set $S$. Find the sum of squares of all elements in $S$.", "answer": "\\frac{41}{6}", "subject": "Geometry", "unique_id": "OlymMATH-EASY-83-EN"}
|
| 85 |
+
{"problem": "In triangle $\\triangle ABC$, $z = \\frac{\\sqrt{65}}{5} \\sin \\frac{A+B}{2} + i \\cos \\frac{A-B}{2}$, $|z| = \\frac{3\\sqrt{5}}{5}$. Find the maximum value of $\\angle C$.", "answer": "\\pi - \\arctan \\frac{12}{5}", "subject": "Algebra", "unique_id": "OlymMATH-EASY-84-EN"}
|
| 86 |
+
{"problem": "In triangle $\\triangle ABC$, the sides opposite to angles $\\angle A$, $\\angle B$, $\\angle C$ are $a$, $b$, $c$ respectively. If $\\angle A = 39^{\\circ}$ and $(a^2 - b^2)(a^2 + ac - b^2) = b^2c^2$, find the value of $\\angle C$.", "answer": "115^{\\circ}", "subject": "Geometry", "unique_id": "OlymMATH-EASY-85-EN"}
|
| 87 |
+
{"problem": "Through the point $P(1,\\frac{1}{2})$ inside the ellipse $\\frac{x^{2}}{6}+\\frac{y^{2}}{3}=1$, draw a line that does not pass through the origin, intersecting the ellipse at points $A$ and $B$. Find the maximum value of the area of triangle $\\triangle OAB$.", "answer": "\\frac{3\\sqrt{6}}{4}", "subject": "Geometry", "unique_id": "OlymMATH-EASY-86-EN"}
|
| 88 |
+
{"problem": "Nine cards numbered $1, 2, \\dots, 9$ are randomly arranged in a row. If the first card (from the left) has the number $k$, then reversing the order of the first $k$ cards is considered one operation. The game stops when an operation cannot be performed (i.e., when the first card has the number 1). If an arrangement cannot be operated on, and it is obtained by exactly one other arrangement after one operation, then this arrangement is called a \"secondary terminating arrangement\". Among all possible arrangements, find the probability that a secondary terminating arrangement occurs.", "answer": "\\frac{103}{2520}", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-87-EN"}
|
| 89 |
+
{"problem": "Let $a_{i}(i\\in \\mathbb{Z}_{+},i\\leqslant 2020)$ be non-negative real numbers, and $\\sum_{i=1}^{2020}a_{i}=1$. Find the maximum value of $\\sum_{\\substack{i\\neq j\\\\i|j}}a_{i}a_{j}$.", "answer": "\\frac{5}{11}", "subject": "Algebra", "unique_id": "OlymMATH-EASY-88-EN"}
|
| 90 |
+
{"problem": "Given the ellipse $C: \\frac{x^{2}}{25}+\\frac{y^{2}}{9}=1$, and a moving circle $\\Gamma: x^{2}+y^{2}=r^{2}(3< r< 5)$. If $M$ is a point on the ellipse $C$, $N$ is a point on the moving circle $\\Gamma$, and the line $MN$ is tangent to both the ellipse $C$ and the moving circle $\\Gamma$, find the maximum value of the distance $|MN|$ between points $M$ and $N$.", "answer": "2", "subject": "Geometry", "unique_id": "OlymMATH-EASY-89-EN"}
|
| 91 |
+
{"problem": "Let $a$, $b$, $c$ be distinct non-zero real numbers satisfying: the equations $ax^3+bx+c=0$, $bx^3+cx+a=0$, $cx^3+ax+b=0$ have a common root, and among these three equations, there exist two equations with no imaginary roots. Find the minimum value of $\\frac{a}{b}+\\frac{b}{c}+\\frac{c}{a}$.", "answer": "\\frac{17}{12}", "subject": "Algebra", "unique_id": "OlymMATH-EASY-90-EN"}
|
| 92 |
+
{"problem": "In the sequence $\\{u_n\\} (n \\in \\mathbb{Z}_{+})$, $u_1 = 2$, $u_2 = 8$, $u_{n+1} = 4u_n - u_{n-1} (n \\geqslant 2)$. Find $\\sum_{n=1}^{\\infty} \\operatorname{arccot} u_n^2$.", "answer": "\\frac{\\pi}{12}", "subject": "Algebra", "unique_id": "OlymMATH-EASY-91-EN"}
|
| 93 |
+
{"problem": "Let $f(x): [0, 1] \\rightarrow \\mathbb{R}$, satisfying: (1) $f(\\frac{x}{3}) = \\frac{1}{2}f(x)$; (2) $f(1-x) = 1 - f(x)$; (3) $f(x) = \\frac{1}{2} (x \\in [\\frac{1}{3}, \\frac{2}{3}])$. If $n=2023$, find the value of $S_n = \\sum_{\\substack{1 \\leqslant k \\leqslant 3^n \\\\ k \\text{\\ is\\ odd}}} f(\\frac{k}{3^n})$.", "answer": "\\frac{3^{2023} + 3}{4}", "subject": "Algebra", "unique_id": "OlymMATH-EASY-92-EN"}
|
| 94 |
+
{"problem": "Parabola $\\Gamma: x^2 = 4y$, a line $l$ with slope $1$ intersects the parabola $\\Gamma$ at points $A$ and $B$. Tangent lines to the parabola $\\Gamma$ are drawn at points $A$ and $B$, intersecting at point $M$. $F$ is the focus of the parabola $\\Gamma$. Let $S_1$, $S_2$, and $S_3$ be the areas of triangles $\\triangle AFM$, $\\triangle BFM$, and $\\triangle ABM$ respectively. Find the minimum value of $\\frac{S_1 S_2}{S_3}$.", "answer": "\\sqrt[4]{\\frac{64}{27}}", "subject": "Geometry", "unique_id": "OlymMATH-EASY-93-EN"}
|
| 95 |
+
{"problem": "Let ellipse $C: \\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{20}=1\\left(a> 2\\sqrt{5}\\right)$ have its left focus at $F$. It is known that there exists a line $l$ passing through point $P\\left(1,1\\right)$ intersecting the ellipse at points $A$ and $B$, and $M$ is the midpoint of $AB$, such that $\\left| FM\\right|$ is the geometric mean of $\\left| FA\\right|$ and $\\left| FB\\right|$. Find the minimum positive integer value of $a$.", "answer": "7", "subject": "Geometry", "unique_id": "OlymMATH-EASY-94-EN"}
|
| 96 |
+
{"problem": "Given that $A$, $B$, $C$ are the three interior angles of $\\triangle ABC$, vector $\\boldsymbol{\\alpha} = \\left( \\cos \\frac{A-B}{2}, \\sqrt{3} \\sin \\frac{A+B}{2} \\right)$, and $|\\boldsymbol{\\alpha}| = \\sqrt{2}$. If when angle $C$ is at its maximum, there exists a moving point $M$ such that $|MA|$, $|AB|$, $|MB|$ form an arithmetic sequence, find the maximum value of $\\frac{|MC|}{|AB|}$.", "answer": "\\frac{2\\sqrt{3}+\\sqrt{2}}{4}", "subject": "Geometry", "unique_id": "OlymMATH-EASY-95-EN"}
|
| 97 |
+
{"problem": "Let $p=2017$ be a prime number. Let set $A$ consist of numbers from the set $\\{1,3,5,\\cdots,p-2\\}$ that are quadratic residues modulo $p$, and let set $B$ consist of numbers from this set that are not quadratic residues modulo $p$. Find the value of $(\\sum_{a\\in A}\\cos \\frac{a\\pi}{p})^{2}+(\\sum_{b\\in B}\\cos \\frac{b\\pi}{p})^{2}$.", "answer": "\\frac{1009}{4}", "subject": "Number Theory", "unique_id": "OlymMATH-EASY-96-EN"}
|
| 98 |
+
{"problem": "Through the right focus $F_{2}(c,0)$ of the ellipse $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1(a>b>0)$, draw a line $l$ that intersects the ellipse at points $P$ and $Q$. On the circle $x^{2}+y^{2}=b^{2}$, find a point $M$, and connect $MP$ and $MQ$. We denote the maximum area of the triangle $\\triangle MPQ$ as $F(a, b)$. Find $F(3, 2\\sqrt{2}) + F(2, 1)$.", "answer": "\\frac{19\\sqrt{2}+11}{3}", "subject": "Geometry", "unique_id": "OlymMATH-EASY-97-EN"}
|
| 99 |
+
{"problem": "Let the set $S$ consist of all integer solutions to the equation $2^x + 3^y = z^2$. Find $\\sum_{(x, y, z)\\in S}(x + y + z^2)$.", "answer": "96", "subject": "Number Theory", "unique_id": "OlymMATH-EASY-98-EN"}
|
| 100 |
+
{"problem": "A mathematics competition consists of $6$ problems, each worth $7$ points for a correct answer and $0$ points for an incorrect answer or no answer. After the competition, a participating team obtained a total score of $161$ points. When analyzing the scores, it was found that any two contestants from this team had at most two correctly solved problems in common, and there were no three contestants who all correctly solved the same two problems. Find the minimum number of contestants on this team.", "answer": "7", "subject": "Combinatorics", "unique_id": "OlymMATH-EASY-99-EN"}
|
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| 1 |
+
{"problem": "Let $a, b, c \\in \\mathbb{R}$, $a^3 b + b^3 c + c^3 a = 3$, find the minimum value of the expression $f(a, b, c) = (\\sum a^4)^4 + 1000 \\sum a^2 b^2$.", "answer": "2625", "subject": "Algebra", "unique_id": "OlymMATH-HARD-0-EN"}
|
| 2 |
+
{"problem": "If the distances from the eight vertices of a cube to a certain plane are $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$ respectively, consider all possible edge lengths of this cube. Assuming the possible edge lengths form a set $S$, find the sum of squares of all elements in $S$.", "answer": "210", "subject": "Geometry", "unique_id": "OlymMATH-HARD-1-EN"}
|
| 3 |
+
{"problem": "For $i = 1, 2, \\cdots, n$, we have $x_i < 1$, and $| x_1 | + | x_2 | + \\cdots + | x_n | = 19 + | x_1 + x_2 + \\cdots + x_n |$. Find the minimum value of the positive integer $n$.", "answer": "11", "subject": "Algebra", "unique_id": "OlymMATH-HARD-2-EN"}
|
| 4 |
+
{"problem": "Find the minimum number of cubes (which can be suspended in air) needed so that all three views (front, top, and side) are $3 \\times 3$ grids.", "answer": "8", "subject": "Geometry", "unique_id": "OlymMATH-HARD-3-EN"}
|
| 5 |
+
{"problem": "Let $x$, $y$, $z$ be positive real numbers. Find the minimum value of $f(x, y, z) = \\frac{(2 + 5y)(3x + z)(x + 3y)(2z + 5)}{xyz}$.", "answer": "241 + 44\\sqrt{30}", "subject": "Algebra", "unique_id": "OlymMATH-HARD-4-EN"}
|
| 6 |
+
{"problem": "Given the ellipse $x^{2} / 4 + y^{2} = 1$, $N_{1}(-1, 0)$, $N_{2}(1, 0)$, $M(3, 0)$, a line passing through $M$ intersects the ellipse at two points $P$ and $Q$. Connect $N_{1}P$ and $N_{2}Q$ to get the intersection point $R$. It can be proven that the locus of $R$ forms a conic section. Find its eccentricity.", "answer": "\\frac{\\sqrt{51}}{6}", "subject": "Geometry", "unique_id": "OlymMATH-HARD-5-EN"}
|
| 7 |
+
{"problem": "Nine small balls numbered $1, 2, \\dots, 9$ are randomly placed at nine equally spaced points on a circle, with one ball at each point. Let $S$ be the sum of the absolute differences between the numbers of all adjacent balls on the circle. Find the probability of the arrangement that minimizes the value of $S$. Note: If one arrangement can coincide with another after rotation or mirror reflection, then they are considered the same arrangement.", "answer": "\\frac{1}{315}", "subject": "Combinatorics", "unique_id": "OlymMATH-HARD-6-EN"}
|
| 8 |
+
{"problem": "A triangle with sides of length $10$, $12$, $14$ is folded along its three medians to form a tetrahedron. Find the diameter of the circumscribed sphere of the tetrahedron.", "answer": "\\sqrt{55}", "subject": "Geometry", "unique_id": "OlymMATH-HARD-7-EN"}
|
| 9 |
+
{"problem": "For $k$, $n \\in \\mathbf{Z}_{+}$, consider a finite sequence $\\{a_{k}\\}$ with $n$ terms, where $a_{k} \\leqslant m$, $a_{k}$, $m \\in \\mathbf{Z}_{+}$. If $m=2025$, determine the maximum value of $n$ such that the sequence $\\{a_{k}\\}$ satisfies: (1) for any term $a_{k}$, if $a_{k-1}$ and $a_{k+1}$ exist, then $a_{k-1} \\neq a_{k+1}$; (2) there do not exist positive integers $i_{1} < i_{2} < i_{3} < i_{4}$ such that $a_{{i_{1}}} = a_{{i_{3}}} \\neq a_{{i_{2}}} = a_{{i_{4}}}$.", "answer": "8098", "subject": "Combinatorics", "unique_id": "OlymMATH-HARD-8-EN"}
|
| 10 |
+
{"problem": "Given an ellipse $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1(a>b>0)$ with left focus $F$, $P(x_{0},y_{0})$ is a point on the ellipse, where $x_{0}>0$. Draw a tangent line from point $P$ to the circle $x^{2}+y^{2}=b^{2}$, which intersects the ellipse at a second point $Q$. Let $I$ be the incenter of triangle $\\triangle PFQ$, and $\\angle PFQ=2\\alpha$. If $a^2=\\sqrt{3}, b^2=\\sqrt{2}$, find the value of $|FI| \\cos \\alpha$.", "answer": "\\sqrt[4]{3}", "subject": "Geometry", "unique_id": "OlymMATH-HARD-9-EN"}
|
| 11 |
+
{"problem": "Arrange the ten digits from 0 to 9 into a ten-digit number without repetition and with a non-zero first digit. Find the number of such ten-digit numbers that are divisible by 99.", "answer": "285120", "subject": "Number Theory", "unique_id": "OlymMATH-HARD-10-EN"}
|
| 12 |
+
{"problem": "Let the sum of $n$ distinct positive integers $a_1, a_2, \\dots, a_n$ be $2000$. Denote $A = \\max\\{a_1, a_2, \\dots, a_n\\}$. Find the minimum value of $A+n$. ($n$ is not given in advance)", "answer": "110", "subject": "Number Theory", "unique_id": "OlymMATH-HARD-11-EN"}
|
| 13 |
+
{"problem": "Given a positive integer $n=2024$. Find the maximum value of the integer $M$ such that for any positive integers $a_{1}, a_{2}, \\ldots, a_{n}$, we have $[\\sqrt{a_{1}}]+[\\sqrt{a_{2}}]+\\cdots +[\\sqrt{a_{n}}]\\geqslant [\\sqrt{a_{1}+a_{2}+\\cdots +a_{n}+M\\min \\{a_{1},a_{2},\\cdots ,a_{n}\\}}]$, where $[x]$ denotes the greatest integer not exceeding the real number $x$.", "answer": "1364850", "subject": "Algebra", "unique_id": "OlymMATH-HARD-12-EN"}
|
| 14 |
+
{"problem": "Given $m> 0$, the equation $(mx-3+\\sqrt{2})^{2}-\\sqrt{x+m}=0$ in $x$ has exactly two distinct real roots in the interval $[0,1]$. Find the range of values of the real number $m$.", "answer": "[3,193-132\\sqrt{2}]", "subject": "Algebra", "unique_id": "OlymMATH-HARD-13-EN"}
|
| 15 |
+
{"problem": "A class has 25 students. The teacher wants to prepare $N$ candies for a competition and distribute them according to grades (equal scores receive equal numbers of candies, lower scores receive fewer candies, which can be 0). Find the minimum value of $N$ such that regardless of how many questions are in the competition and how students answer the questions, the candies can be distributed in this way.", "answer": "600", "subject": "Combinatorics", "unique_id": "OlymMATH-HARD-14-EN"}
|
| 16 |
+
{"problem": "Let $n$ be a positive integer, and set $T_n$ be a subset of the set $A_n=\\{k \\mid k \\in \\mathbf{Z}_{+}, \\text{ and } k \\leqslant n\\}$, such that the difference between any two numbers in $T_n$ is not equal to 4 or 7. If the maximum number of elements in $T_n$ is denoted as $f_n$ (for example, $f_1=1$, $f_2=2$), find the value of $\\sum_{n=1}^{2023}f_n$.", "answer": "932604", "subject": "Combinatorics", "unique_id": "OlymMATH-HARD-15-EN"}
|
| 17 |
+
{"problem": "Find the largest positive integer $n \\le 2025$ such that there exists a strictly increasing sequence of positive integers $a_1 < a_2 < \\cdots < a_n$, where all sums $a_i + a_j (1 \\le i < j \\le n)$ are distinct, and in modulo 4, each remainder appears the same number of times.", "answer": "1296", "subject": "Number Theory", "unique_id": "OlymMATH-HARD-16-EN"}
|
| 18 |
+
{"problem": "Let $[x]$ denote the greatest integer not exceeding the real number $x$. The sequence $\\{x_n\\}$ satisfies: $x_1 = 1$, $x_{n+1} = 4x_n + [\\sqrt{11}x_n]$. Find the units digit of $x_{2021}$.", "answer": "9", "subject": "Number Theory", "unique_id": "OlymMATH-HARD-17-EN"}
|
| 19 |
+
{"problem": "Given two regular triangular pyramids $P-ABC$ and $Q-ABC$ inscribed in the same unit sphere $O$, with the two vertices $P$ and $Q$ on opposite sides of the base $ABC$. Let the plane angles of the dihedral angles $P-AB-C$ and $Q-AB-C$ be $\\alpha$ and $\\beta$ respectively. Find the value of $AB \\tan(\\alpha + \\beta)$.", "answer": "-\\frac{4\\sqrt{3}}{3}", "subject": "Geometry", "unique_id": "OlymMATH-HARD-18-EN"}
|
| 20 |
+
{"problem": "In $\\triangle ABC$, $AB = AC$, $\\angle BAC = 30^\\circ$. On side $AB$, take five equal division points $T_1$, $T_2$, $T_3$, $T_4$, with points $A$, $T_1$, $T_2$, $T_3$, $T_4$, $B$ arranged in sequence. Let $\\theta_k = \\angle BT_k C$ ($k = 1, 2, 3, 4$). Find the value of $\\tan A \\cdot \\tan \\theta_1 + \\sum_{k=1}^3 \\tan \\theta_k \\cdot \\tan \\theta_{k+1} - \\tan \\theta_4 \\cdot \\tan B$.", "answer": "-5 - \\frac{10 \\sqrt{3}}{3}", "subject": "Algebra", "unique_id": "OlymMATH-HARD-19-EN"}
|
| 21 |
+
{"problem": "Given a $2022 \\times 2022$ grid. Each cell in the grid is filled with one of the four colors $A$, $B$, $C$, $D$. If every $2 \\times 2$ square in the grid contains all four colors, find how many different perfect grids there are.", "answer": "12 \\times 2^{2022} - 24", "subject": "Combinatorics", "unique_id": "OlymMATH-HARD-20-EN"}
|
| 22 |
+
{"problem": "Let positive integers $a$, $b$, $c$, $d$ satisfy $a < b < c < d$, and any three distinct numbers among them can form an obtuse triangle with these three numbers as the side lengths. Find the minimum value of $d$.", "answer": "14", "subject": "Number Theory", "unique_id": "OlymMATH-HARD-21-EN"}
|
| 23 |
+
{"problem": "Let function $f(x)=\\sin^4 \\omega x - \\sin \\omega x \\cdot \\cos \\omega x + \\cos^4 \\omega x (\\omega > 0)$. If there exist $a, b \\in [0, \\pi]$ such that $f(a) + f(b) = \\frac{9}{4}$, find the minimum value of $\\omega$.", "answer": "\\frac{7}{12}", "subject": "Algebra", "unique_id": "OlymMATH-HARD-22-EN"}
|
| 24 |
+
{"problem": "Given a $3\\times 2025$ grid, an ant starts from the bottom-left cell and can move to any adjacent cell that shares an edge. If the ant visits every cell of the grid exactly once and finally reaches the top-right corner, how many different paths are possible?", "answer": "2^2023", "subject": "Combinatorics", "unique_id": "OlymMATH-HARD-23-EN"}
|
| 25 |
+
{"problem": "Given that the left and right foci of the hyperbola $x^2 - \\frac{y^2}{3} = 1$ are $F_1$ and $F_2$, a line passing through $F_2$ intersects the right branch of the hyperbola at points $A$ and $B$. Find the range of values for the sum of the radii of the incircles of triangles $\\triangle AF_1F_2$ and $\\triangle BF_1F_2$.", "answer": "\\left[2, \\frac{4}{3}\\sqrt{3}\\right)", "subject": "Geometry", "unique_id": "OlymMATH-HARD-24-EN"}
|
| 26 |
+
{"problem": "For any positive integer $n$, $\\tau(n)$ represents the number of positive divisors of $n$, and $\\varphi(n)$ represents the number of positive integers that are less than $n$ and coprime to $n$. If a positive integer $n$ satisfies that one of $n$, $\\tau(n)$, $\\varphi(n)$ is the arithmetic mean of the other two, then $n$ is called a good number. Find how many good numbers exist.", "answer": "4", "subject": "Number Theory", "unique_id": "OlymMATH-HARD-25-EN"}
|
| 27 |
+
{"problem": "Let $a, b, c$ be positive rational numbers such that $a+1/b, b+1/c, c+1/a$ are all integers. The set of all possible values of $a + b + c$ forms a set $S$. Find the product of all elements in $S$.", "answer": "\\frac{21}{2}", "subject": "Number Theory", "unique_id": "OlymMATH-HARD-26-EN"}
|
| 28 |
+
{"problem": "Given an ellipse $C: x^{2} / a^{2}+y^{2} / b^{2}=1$ $(a>b>0)$ with eccentricity $e=4 / 5$, let $P$ be any point on the ellipse different from the left and right vertices $A$ and $B$ on the major axis, $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse respectively, and $\\angle APB=2 \\alpha$, $\\angle F_{1} P F_{2}=2 \\beta$. Find the minimum value of $\\tan \\beta \\cdot \\tan 2 \\alpha$.", "answer": "-\\frac{5}{2}", "subject": "Geometry", "unique_id": "OlymMATH-HARD-27-EN"}
|
| 29 |
+
{"problem": "Given $\\frac{by}{z}+\\frac{cz}{y}=a$, $\\frac{cz}{x}+\\frac{ax}{z}=b$, $\\frac{ax}{y}+\\frac{by}{x}=c$, and $abc=1$, find the value of $a^{3}+b^{3}+c^{3}$.", "answer": "5", "subject": "Algebra", "unique_id": "OlymMATH-HARD-28-EN"}
|
| 30 |
+
{"problem": "In rectangle $ABCD$, $AB=2$, $AD=4$, point $E$ is on segment $AD$, and $AE=3$. Now fold triangle $\\triangle ABE$ along $BE$ and fold triangle $\\triangle DCE$ along $CE$, so that point $D$ falls on segment $AE$. Find the cosine value of the dihedral angle $D-EC-B$.", "answer": "\\frac{7}{8}", "subject": "Geometry", "unique_id": "OlymMATH-HARD-29-EN"}
|
| 31 |
+
{"problem": "Let $f(x) = || \\cdots || x^{10} - 2^{2007}| - 2^{2006}| - \\cdots - 2^2| - 2| $. Find the value of $f(2007)$.", "answer": "1", "subject": "Algebra", "unique_id": "OlymMATH-HARD-30-EN"}
|
| 32 |
+
{"problem": "A regular tetrahedron $ABCD$ has its edges colored with six different colors, with each edge colored with only one color and edges sharing a vertex must have different colors. Find the probability that all edges have different colors.", "answer": "\\frac{3}{17}", "subject": "Combinatorics", "unique_id": "OlymMATH-HARD-31-EN"}
|
| 33 |
+
{"problem": "For a cube $ABCD-A_{1}B_{1}C_{1}D_{1}$, place the numbers $1, 2, \\cdots, 8$ at the eight vertices of the cube, with the requirement that the sum of any three numbers on each face is not less than $10$. Find the number of different ways to place the numbers.", "answer": "480", "subject": "Combinatorics", "unique_id": "OlymMATH-HARD-32-EN"}
|
| 34 |
+
{"problem": "Given 2024 points on a straight line. Now randomly pair all points into 1012 pairs, connecting them into 1012 line segments. Find the probability that there exists a line segment that intersects with all the other 1011 line segments.", "answer": "\\frac{2}{3}", "subject": "Combinatorics", "unique_id": "OlymMATH-HARD-33-EN"}
|
| 35 |
+
{"problem": "Five tennis players participate in a round-robin tournament (exactly one match between any two players), and there are no draws. In each of these ten matches, both players have a $50\\%$ probability of winning, and the results of each match are independent. Find the probability that during the entire tournament, there exist four distinct players $P_1$, $P_2$, $P_3$, $P_4$, such that $P_1$ defeats $P_2$, $P_2$ defeats $P_3$, $P_3$ defeats $P_4$, and $P_4$ defeats $P_1$.", "answer": "\\frac{49}{64}", "subject": "Combinatorics", "unique_id": "OlymMATH-HARD-34-EN"}
|
| 36 |
+
{"problem": "In the ellipse $\\Gamma: \\frac{x^{2}}{2019} + \\frac{y^{2}}{2018} = 1$, $F$ is the left focus. Line $l$ passing through the right focus intersects the left directrix of ellipse $\\Gamma$ and the ellipse $\\Gamma$ at points $C$, $A$, and $B$, respectively. If $\\angle FAB = 40^{\\circ}$ and $\\angle FBA = 10^{\\circ}$, find the value of $\\angle FCA$.", "answer": "15^{\\circ}", "subject": "Geometry", "unique_id": "OlymMATH-HARD-35-EN"}
|
| 37 |
+
{"problem": "Initially, a zookeeper places a carrot with mass $a$ and a rabbit in the top-left cell of a $20 \\times 20$ grid. Next, if the rabbit and the carrot are in the same cell, it will eat $\\frac{1}{20}a$ mass of the carrot, and then the zookeeper will place the remaining carrot in one of the cells (possibly the current cell) with equal probability; otherwise, the rabbit will move to an adjacent cell (two cells are adjacent if and only if they share a common edge), and this movement will shorten the distance between it and the carrot. Find the expected number of moves the rabbit makes before eating the entire carrot.", "answer": "\\frac{2318}{5}", "subject": "Combinatorics", "unique_id": "OlymMATH-HARD-36-EN"}
|
| 38 |
+
{"problem": "There is an $n \\times n$ ($n \\geqslant 2$, $n \\in \\mathbb{Z}_{+}$) grid, where each $1 \\times 1$ cell is called a unit cell. In each unit cell, either one chess piece is placed or nothing is placed. If after placing all the chess pieces, it is found that for any unit cell, there must be a chess piece in some unit cell adjacent to it (i.e., a unit cell different from this unit cell and sharing at least one common vertex with this unit cell), then the total number of chess pieces placed is called an \"$n$-good number\". For each $n \\geqslant 2(n \\in \\mathbb{Z}_{+})$, let $f(n)$ be the minimum of all $n$-good numbers. If the constant $c$ satisfies: for all $n \\geqslant 2(n \\in \\mathbb{Z}_{+})$, $f(n) \\geqslant cn^{2}$ holds, find the maximum value of $c$.", "answer": "\\frac{1}{7}", "subject": "Combinatorics", "unique_id": "OlymMATH-HARD-37-EN"}
|
| 39 |
+
{"problem": "Given a line segment $x+y=1$ ($x\\geqslant 0$, $y\\geqslant 0$) with $2020$ points on it. Find the smallest positive integer $k$, such that for any such $2020$ points, there always exists a way to divide these $2020$ points into two groups, where in one group the sum of y-coordinates is not greater than $k$, and in the other group the sum of x-coordinates is not greater than $k$ (these $2020$ points may coincide).", "answer": "506", "subject": "Combinatorics", "unique_id": "OlymMATH-HARD-38-EN"}
|
| 40 |
+
{"problem": "Let set $A = \\{1, 2, \\cdots, 5\\}$, and the set consisting of all subsets of set $A$ is called the power set of $A$, denoted as $2^A$. A mapping $f: 2^A \\rightarrow A$ is called a \"perfect mapping\" if for any $X, Y \\in 2^A$, we have $f(X \\cap Y) = \\min\\{f(X), f(Y)\\}$. Find the number of perfect mappings.", "answer": "4425", "subject": "Combinatorics", "unique_id": "OlymMATH-HARD-39-EN"}
|
| 41 |
+
{"problem": "Given a regular polygon where each side and diagonal is colored with one of $2018$ colors, and not all sides and diagonals have the same color. If there are no \"two-colored triangles\" (i.e., triangles whose three sides are colored with exactly two colors) in the regular polygon, then the coloring of the polygon is called \"harmonious\". Find the largest positive integer $N$ such that there exists a harmonious coloring of a regular $N$-sided polygon.", "answer": "2017^2", "subject": "Combinatorics", "unique_id": "OlymMATH-HARD-40-EN"}
|
| 42 |
+
{"problem": "Define a function $f: \\mathbb{Z} \\rightarrow \\mathbb{Z}$ such that for any $x, y \\in \\mathbb{Z}$, we have $f(x^2 - 3y^2) + f(x^2 + y^2) = 2(x+y)f(x-y)$. If $n > 0$, then $f(n) > 0$, and $f(2015)f(2016)$ is a perfect square. Find the minimum value of $f(1) + f(2)$.", "answer": "246", "subject": "Number Theory", "unique_id": "OlymMATH-HARD-41-EN"}
|
| 43 |
+
{"problem": "Let the set $X=\\{1,2,\\cdots,2022\\}$. A family of sets $\\mathcal{F}$ consists of several distinct subsets of $X$, satisfying: for any $F\\in \\mathcal{F}$, we have $|F| \\geqslant 800$; and for any $x\\in X$, there are at least $800$ sets $F\\in \\mathcal{F}$ such that $x\\in F$. Find the smallest positive integer $m$ such that there must exist $m$ sets in $\\mathcal{F}$ whose union is $X$.", "answer": "1222", "subject": "Combinatorics", "unique_id": "OlymMATH-HARD-42-EN"}
|
| 44 |
+
{"problem": "Given that in a Cartesian coordinate system, the trajectory of point $P(x, y)$ satisfies the system of equations\n$\\begin{cases}\na^{2}x-axy-y=0, \\\\\na^{2}y+axy+x=0,\n\\end{cases}$.\nPoints $A(1,t)$ and $B(s,2)$ are centrally symmetric with respect to the origin. Find the minimum value of $\\overrightarrow{AP} \\cdot \\overrightarrow{BP}$.", "answer": "6\\sqrt{3}-5", "subject": "Geometry", "unique_id": "OlymMATH-HARD-43-EN"}
|
| 45 |
+
{"problem": "Given that the cross-section $\\alpha$ that forms a $60^\\circ$ angle with the base of cylinder $OO'$ intersects the lateral surface of the cylinder to form an elliptical plane figure. Spheres $C_1$ and $C_2$ are located on opposite sides of the cross-section $\\alpha$, and they are tangent to the lateral surface of the cylinder, one base, and the cross-section $\\alpha$ respectively. Let the volumes of spheres $C_1$, $C_2$, and cylinder $OO'$ be $V_1$, $V_2$, and $V$ respectively. Find the value of $\\frac{V_1+V_2}{V}$.", "answer": "\\frac{4}{9}", "subject": "Geometry", "unique_id": "OlymMATH-HARD-44-EN"}
|
| 46 |
+
{"problem": "The $64$ cells of an $8 \\times 8$ grid are numbered from $1, 2, \\cdots, 64$, such that for all $1 \\le i \\le 63$, the two cells numbered $i$ and $i+1$ share a common edge. Find the maximum possible sum of the numbers in the eight cells along the main diagonal.", "answer": "432", "subject": "Combinatorics", "unique_id": "OlymMATH-HARD-45-EN"}
|
| 47 |
+
{"problem": "In a $101 \\times 101$ grid, each cell is filled with a number from the set $\\{1, 2, \\cdots, 101^2\\}$, and each number in the set is used exactly once. The left and right boundaries of the grid are considered the same line, and the top and bottom boundaries are also considered the same line (i.e., it is a torus). If no matter how we fill the grid, there always exist two adjacent cells (cells sharing an edge) such that the difference between the two numbers filled in is not less than $M$, find the maximum value of $M$.", "answer": "201", "subject": "Combinatorics", "unique_id": "OlymMATH-HARD-46-EN"}
|
| 48 |
+
{"problem": "There are two chess pieces each of red, green, white, and blue (identical except for color). Now, seven pieces are selected to be embedded at the vertices of a regular hexagonal pyramid, with one piece at each vertex. Find the number of different embedding methods.", "answer": "424", "subject": "Combinatorics", "unique_id": "OlymMATH-HARD-47-EN"}
|
| 49 |
+
{"problem": "For a parabola $y^2=2px$, consider a right triangle $\\mathrm{Rt}\\triangle ABC$ inscribed in it, with the hypotenuse $BC \\perp x$-axis at point $M$. Extend $MA$ to point $D$ such that circle $\\odot N$ with diameter $AD$ is tangent to the $x$-axis at point $E$. Connect $BE$, which intersects the parabola at point $F$. If the area of quadrilateral $AFBC$ is $8p^2$, points $A$ and $F$ do not coincide, and $p^2=\\sqrt{2}$, find the area of triangle $\\triangle ACD$.", "answer": "\\frac{15\\sqrt{2}}{2}", "subject": "Geometry", "unique_id": "OlymMATH-HARD-48-EN"}
|
| 50 |
+
{"problem": "Given the set of integers $A = \\{1, 2, \\cdots, 100\\}$. Let the function $f: A \\rightarrow A$ satisfy: (1) for any $1 \\leqslant i \\leqslant 99$, we have $|f(i) - f(i+1)| \\leqslant 1$; (2) for any $1 \\leqslant i \\leqslant 100$, we have $f(f(i)) = 100$. Find the minimum possible value of $\\sum_{i=1}^{100} f(i)$.", "answer": "8350", "subject": "Combinatorics", "unique_id": "OlymMATH-HARD-49-EN"}
|
| 51 |
+
{"problem": "Let set $A = \\{0, 1, \\cdots, 2018\\}$. If $x, y, z \\in A$, and $x^2 + y^2 - z^2 = 2019^2$, find the sum of the maximum and minimum values of $x + y + z$.", "answer": "7962", "subject": "Number Theory", "unique_id": "OlymMATH-HARD-50-EN"}
|
| 52 |
+
{"problem": "If the inequality $2\\sin^2 C + \\sin A \\cdot \\sin B > k \\sin B \\cdot \\sin C$ holds for any triangle $\\triangle ABC$, find the maximum value of the real number $k$.", "answer": "2\\sqrt{2}-1", "subject": "Geometry", "unique_id": "OlymMATH-HARD-51-EN"}
|
| 53 |
+
{"problem": "Find the minimum real number $a$ such that for all positive integers $n$ and real numbers $0 = x_0 < x_1 < \\cdots < x_n$ satisfying\n$$a \\sum_{k=1}^{n} \\frac{\\sqrt{(k+1)^3}}{\\sqrt{x_k^2 - x_{k-1}^2}} \\geq \\sum_{k=1}^{n} \\frac{k^2 + 3k + 3}{x_k}.$$", "answer": "\\frac{16\\sqrt{2}}{9}", "subject": "Algebra", "unique_id": "OlymMATH-HARD-52-EN"}
|
| 54 |
+
{"problem": "Given non-zero non-collinear vectors $\\overrightarrow{OA}$ and $\\overrightarrow{OB}$. Let $\\overrightarrow{OC} = \\frac{1}{1+r} \\overrightarrow{OA} + \\frac{r}{1+r} \\overrightarrow{OB}$. Define the set of points $M = \\{K \\mid \\frac{\\overrightarrow{KA} \\cdot \\overrightarrow{KC}}{|\\overrightarrow{KA}|} = \\frac{\\overrightarrow{KB} \\cdot \\overrightarrow{KC}}{|\\overrightarrow{KB}|} \\}$. When $K_1$, $K_2 \\in M$, if for any $r \\geq 2$, the inequality $|\\overrightarrow{K_1 K_2}| \\leq c |\\overrightarrow{AB}|$ always holds, find the minimum value of the real number $c$.", "answer": "\\frac{4}{3}", "subject": "Geometry", "unique_id": "OlymMATH-HARD-53-EN"}
|
| 55 |
+
{"problem": "In the plane region $M = \\{(x, y) | 0 \\le y \\le 2 - x, 0 \\le x \\le 2 \\}$, $k$ points are chosen arbitrarily. It is always possible to divide these $k$ points into two groups $A$ and $B$, such that the sum of the x-coordinates of all points in group $A$ does not exceed $6$, and the sum of the y-coordinates of all points in group $B$ does not exceed $6$. Find the maximum value of the positive integer $k$.", "answer": "11", "subject": "Combinatorics", "unique_id": "OlymMATH-HARD-54-EN"}
|
| 56 |
+
{"problem": "Let the three roots of the equation $4^{1-2x} + \\log_2 x = 0$ be $x_1, x_2, x_3$ ($x_1 < x_2 < x_3$). Find the value of $\\frac{\\log_2 x_2}{x_1 x_2 x_3}$.", "answer": "-32", "subject": "Algebra", "unique_id": "OlymMATH-HARD-55-EN"}
|
| 57 |
+
{"problem": "Find the maximum value of $C \\in \\mathbf{R}_{+}$ such that from any real sequence $a_{1}, a_{2}, \\ldots, a_{2022}$, it is possible to select some terms that simultaneously satisfy the following conditions: (1) no three consecutive terms are all selected; (2) at least one of any three consecutive terms is selected; (3) the absolute value of the sum of the selected terms is not less than $C(|a_{1}| + |a_{2}| + \\cdots + |a_{2022}|)$.", "answer": "\\frac{1}{6}", "subject": "Combinatorics", "unique_id": "OlymMATH-HARD-56-EN"}
|
| 58 |
+
{"problem": "Given that for any real number $x$, the inequality $f(x) = 1 - a \\cos x - b \\sin x - A \\cos 2x - B \\sin 2x \\ge 0$ holds. Find the maximum value of $(A^2 + B^2)(a^2 + b^2)$.", "answer": "2", "subject": "Algebra", "unique_id": "OlymMATH-HARD-57-EN"}
|
| 59 |
+
{"problem": "Given several numbers in the interval $[0, 1]$ (which can be the same), their sum does not exceed $S$. Find the maximum value of $S$ such that it is always possible to divide these numbers into two groups, where the sum of numbers in each group does not exceed $11$.", "answer": "\\frac{253}{12}", "subject": "Combinatorics", "unique_id": "OlymMATH-HARD-58-EN"}
|
| 60 |
+
{"problem": "Given $\\begin{cases} \\sin \\alpha = \\sin(\\alpha + \\beta + \\gamma) + 1, \\\\ \\sin \\beta = 3\\sin(\\alpha + \\beta + \\gamma) + 2, \\\\ \\sin \\gamma = 5\\sin(\\alpha + \\beta + \\gamma) + 3. \\end{cases}$ Find the product of all possible values of $\\sin \\alpha \\cdot \\sin \\beta \\cdot \\sin \\gamma$.", "answer": "\\frac{3}{512}", "subject": "Algebra", "unique_id": "OlymMATH-HARD-59-EN"}
|
| 61 |
+
{"problem": "Let $n \\in \\mathbf{Z}_{+}$, $n \\geqslant 2$, $a_{1}, a_{2}, \\cdots, a_{n} \\in \\mathbf{R}$, and $a_{1} + a_{2} + \\cdots + a_{n} = 1$. Define $b_{k} = \\sqrt{1 - \\frac{1}{16^{k}}} \\sqrt{a_{1}^{2} + a_{2}^{2} + \\cdots + a_{k}^{2}}$ $(1 \\leqslant k \\leqslant n)$. Find the minimum value of $b_{1} + b_{2} + \\cdots + b_{n-1} + \\frac{4}{3} b_{n}$.", "answer": "\\frac{\\sqrt{15}}{3}", "subject": "Algebra", "unique_id": "OlymMATH-HARD-60-EN"}
|
| 62 |
+
{"problem": "Define a tetrahedron with equal skew edges as an isosceles tetrahedron. Let the isosceles tetrahedron $DBMN$ have circumscribed sphere radius $R$, and the circumscribed circle radius of triangle $\\triangle BMN$ be $r$. Given that $DB=MN=a$, $DM=BN=b$, $DN=BM=c$, find the range of values for $\\frac{r}{R}$.", "answer": "\\left[\\frac{2\\sqrt{2}}{3},1\\right)", "subject": "Geometry", "unique_id": "OlymMATH-HARD-61-EN"}
|
| 63 |
+
{"problem": "Given $n = \\overline{d_1 d_2 \\cdots d_{2017}}$, where $d_i \\in \\{1, 3, 5, 7, 9\\}$ $(i = 1, 2, \\cdots, 2017)$, and $\\sum_{i=1}^{1009} d_i d_{i+1} \\equiv 1 \\pmod{4}$, $\\sum_{i=1010}^{2016} d_i d_{i+1} \\equiv 1 \\pmod{4}$. Find the number of values of $n$ that satisfy these conditions.", "answer": "6 \\times 5^{2015}", "subject": "Number Theory", "unique_id": "OlymMATH-HARD-62-EN"}
|
| 64 |
+
{"problem": "Let $x\\in (0,1)$, $\\frac{1}{x}\\notin \\mathbf{Z}$, $a_{n}=\\frac{nx}{(1-x)(1-2x)\\cdots (1-nx)}$, where $n=1, 2, {\\ldots}$. We call $x$ a \"good number\" if and only if $x$ makes the sequence $\\{a_{n}\\}$ defined above satisfy $a_{1}+a_{2}+\\cdots +a_{10}> -1$ and $a_{1}a_{2}\\cdots a_{10}> 0$. Find the sum of the lengths of all intervals on the number line corresponding to all good numbers.", "answer": "\\frac{61}{210}", "subject": "Algebra", "unique_id": "OlymMATH-HARD-63-EN"}
|
| 65 |
+
{"problem": "Given $a>0$, $b\\in \\mathbf{R}$. If $|ax^3-bx^2+ax|\\leqslant bx^4+(a+2b)x^2+b$ holds for any $x\\in [\\frac{1}{2},2]$, find the range of values for $\\frac{b}{a}$.", "answer": "\\left[\\frac{\\sqrt{2}-1}{2},+\\infty \\right)", "subject": "Algebra", "unique_id": "OlymMATH-HARD-64-EN"}
|
| 66 |
+
{"problem": "Given that $P$ is a point on the edge $AB$ of the cube $ABCD-A_1B_1C_1D_1$, and the angle between line $A_1B$ and plane $B_1CP$ is $60^\\circ$. Find the tangent value of the dihedral angle $A_1-B_1P-C$.", "answer": "-\\sqrt{5}", "subject": "Geometry", "unique_id": "OlymMATH-HARD-65-EN"}
|
| 67 |
+
{"problem": "For $x \\in [0, 2\\pi]$, find the maximum value of the function $f(x) = \\sqrt{4\\cos^2x + 4\\sqrt{6}\\cos x + 6} + \\sqrt{4\\cos^2x - 8\\sqrt{6}\\cos x + 4\\sqrt{2}\\sin x + 22}$.", "answer": "2(\\sqrt{6}+\\sqrt{2})", "subject": "Algebra", "unique_id": "OlymMATH-HARD-66-EN"}
|
| 68 |
+
{"problem": "Find all prime numbers $p$ such that $p^2 - 87p + 729$ is a perfect cube.", "answer": "2011", "subject": "Number Theory", "unique_id": "OlymMATH-HARD-67-EN"}
|
| 69 |
+
{"problem": "For any positive real numbers $a_1, a_2, \\cdots, a_5$, if $\\sum_{i=1}^{5}\\frac{a_i}{\\sqrt{a_i^2+2^{i-1}a_{i+1}a_{i+2}}}\\geqslant \\lambda$, find the maximum value of $\\lambda$.", "answer": "1", "subject": "Algebra", "unique_id": "OlymMATH-HARD-68-EN"}
|
| 70 |
+
{"problem": "If real numbers $x$, $y$ satisfy the condition $x^2 - y^2 = 4$, find the range of values for $\\frac{1}{x^2} - \\frac{y}{x}$.", "answer": "\\left[-1, \\frac{5}{4}\\right]", "subject": "Algebra", "unique_id": "OlymMATH-HARD-69-EN"}
|
| 71 |
+
{"problem": "Find the number of sets of positive integer solutions to the equation $\\arctan \\frac{1}{m} + \\arctan \\frac{1}{n} + \\arctan \\frac{1}{p} = \\frac{\\pi}{4}$.", "answer": "15", "subject": "Number Theory", "unique_id": "OlymMATH-HARD-70-EN"}
|
| 72 |
+
{"problem": "In triangle $\\triangle ABC$, the inscribed circle is tangent to sides $AB$ and $AC$ at points $E$ and $F$ respectively. $AD$ is the altitude from vertex $A$ to side $BC$, and $AE+AF=AD$. Find the range of values for $\\sin \\frac{A}{2}$.", "answer": "\\left[\\frac{3}{5},\\frac{\\sqrt{2}}{2}\\right)", "subject": "Geometry", "unique_id": "OlymMATH-HARD-71-EN"}
|
| 73 |
+
{"problem": "Given the function $f(x) = a(|\\sin x| + |\\cos x|) - 3\\sin 2x - 7$, where $a$ is a real parameter. Consider the ordered pairs $(a, n)$ ($n \\in \\mathbf{Z}_{+}$) such that the function $y = f(x)$ has exactly $2019$ zeros in the interval $(0, n\\pi)$. All such ordered pairs form a set $S$. Find $\\sum_{(a_0, n_0)\\in S} (a_0^2+n_0)$.", "answer": "4650", "subject": "Algebra", "unique_id": "OlymMATH-HARD-72-EN"}
|
| 74 |
+
{"problem": "For a regular tetrahedron $ABCD$, $M$ and $N$ are the midpoints of edges $AB$ and $AC$ respectively, $P$ and $Q$ are the centroids of faces $ACD$ and $ABD$ respectively. Find the angle between $MP$ and $NQ$.", "answer": "\\arccos \\frac{7}{18}", "subject": "Geometry", "unique_id": "OlymMATH-HARD-73-EN"}
|
| 75 |
+
{"problem": "In space, there are four points $A$, $B$, $C$, $D$ satisfying $AB = BC = CD$. If $\\angle ABC = \\angle BCD = \\angle CDA = 36^{\\circ}$, find the sum of all possible values of the angle formed by lines $AC$ and $BD$.", "answer": "126^{\\circ}", "subject": "Geometry", "unique_id": "OlymMATH-HARD-74-EN"}
|
| 76 |
+
{"problem": "For any 2016 complex numbers $z_{1}, z_{2}, \\cdots, z_{2016}$, we have $\\sum_{k=1}^{2016} | z_{k} |^{2} \\geq \\lambda \\min_{1 \\leq k \\leq 2016} \\{ | z_{k+1} - z_{k} |^{2} \\}$, where $z_{2017} = z_{1}$. Find the maximum value of $\\lambda$.", "answer": "504", "subject": "Algebra", "unique_id": "OlymMATH-HARD-75-EN"}
|
| 77 |
+
{"problem": "Let $\\triangle ABC$ be an inscribed triangle of the ellipse $\\Gamma: \\frac{x^2}{4} + y^2 = 1$, where $A$ is the intersection point of the ellipse $\\Gamma$ with the positive x-axis, and the product of the slopes of lines $AB$ and $AC$ is $-\\frac{1}{4}$. If $G$ is the centroid of $\\triangle ABC$, find the range of values for $|GA| + |GB| + |GC|$.", "answer": "\\left[\\frac{2\\sqrt{13}+4}{3}, \\frac{16}{3}\\right)", "subject": "Geometry", "unique_id": "OlymMATH-HARD-76-EN"}
|
| 78 |
+
{"problem": "Given that $O$ is the origin, $F$ is the right focus of the ellipse $C: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$, a line $l$ passing through point $F$ intersects the ellipse $C$ at points $A$ and $B$, and points $P$ and $Q$ on the ellipse satisfy $\\overrightarrow{OP} + \\overrightarrow{OA} + \\overrightarrow{OB} = \\overrightarrow{OP} + \\overrightarrow{OQ} = \\mathbf{0}$ and points $P$, $A$, $Q$, $B$ are concyclic. Find the eccentricity of ellipse $C$.", "answer": "\\frac{\\sqrt{2}}{2}", "subject": "Geometry", "unique_id": "OlymMATH-HARD-77-EN"}
|
| 79 |
+
{"problem": "Given $P(x) = x^8 + 3x^7 + 6x^6 + 10x^5 + 15x^4 + 21x^3 + 28x^2 + 36x + 45$, $z = \\cos \\frac{2\\pi}{11} + i\\sin \\frac{2\\pi}{11}$. Find the value of $P(z)P(z^2)\\cdots P(z^{10})$.", "answer": "11^8 (5^{11} - 4^{11})", "subject": "Algebra", "unique_id": "OlymMATH-HARD-78-EN"}
|
| 80 |
+
{"problem": "Given a parabola $C_{1}: x^{2}=y$, a circle $C_{2}: x^{2}+(y-4)^{2}=1$, and $P$, $A$, $B$ are three distinct points on the parabola $C_{1}$, where point $P$ is different from the origin. It is known that the lines $PA$ and $PB$ are both tangent to the circle $C_{2}$, and $|PA|=|PB|$. Find the y-coordinate of point $P$.", "answer": "\\frac{23}{5}", "subject": "Geometry", "unique_id": "OlymMATH-HARD-79-EN"}
|
| 81 |
+
{"problem": "Let $n = 108$, and $x_1, x_2, \\dots, x_n$ be a sequence of $n$ positive numbers satisfying $0 < x_1 \\leqslant x_2 \\leqslant \\cdots \\leqslant x_n$ and $x_1 + x_2 \\leqslant x_n$. Find the minimum value of $\\left(x_1 + x_2 + \\cdots + x_n\\right)\\left(\\frac{1}{x_1} + \\frac{1}{x_2} + \\cdots + \\frac{1}{x_n}\\right)$.", "answer": "210\\sqrt{10}+11035", "subject": "Algebra", "unique_id": "OlymMATH-HARD-80-EN"}
|
| 82 |
+
{"problem": "Through vertex $A$ of a regular tetrahedron $ABCD$, create a cross-section in the shape of an isosceles triangle, such that the angle between this cross-section and face $BCD$ is $75 ^{\\circ}$. Find how many such cross-sections exist.", "answer": "18", "subject": "Geometry", "unique_id": "OlymMATH-HARD-81-EN"}
|
| 83 |
+
{"problem": "Let the sequence $\\{a_n\\}$ satisfy $a_0=0$, $a_{n+1}=\\frac{8}{5}a_n+\\frac{6}{5}\\sqrt{4^n-a_n^2}\\left(n\\in\\mathbb{N}\\right)$. Find the decimal part of $\\sum_{k=0}^{2005} a_k$ (expressed as a decimal).", "answer": "0.84", "subject": "Algebra", "unique_id": "OlymMATH-HARD-82-EN"}
|
| 84 |
+
{"problem": "In a rectangular coordinate system, $A(-1, 0), B(1, 0), C(0, 1)$. If there exists a parameter $a$ such that the line $l:y=ax+b$ divides the triangle $\\triangle ABC$ into two parts of equal area, find the range of values for $b$.", "answer": "\\left[1-\\frac{1}{\\sqrt{2}}, \\frac{1}{2}\\right)", "subject": "Geometry", "unique_id": "OlymMATH-HARD-83-EN"}
|
| 85 |
+
{"problem": "$a_1, a_2, \\cdots, a_{2016}$ is a permutation of $1, 2, \\cdots, 2016$, and satisfies $2017 | (a_1 a_2 + a_2 a_3 + \\cdots + a_{2015} a_{2016})$. There are $K$ such permutations, find the remainder when $K$ is divided by $4066272$.", "answer": "2016", "subject": "Number Theory", "unique_id": "OlymMATH-HARD-84-EN"}
|
| 86 |
+
{"problem": "Given that $n$ is a positive integer not exceeding 2021, and satisfying $\\left( \\left[ \\sqrt{n} \\right]^2 + 1 \\right) | \\left( n^2 + 1 \\right)$, find the number of such $n$.", "answer": "47", "subject": "Number Theory", "unique_id": "OlymMATH-HARD-85-EN"}
|
| 87 |
+
{"problem": "Find the number of ordered pairs of positive integers $(m,k)$ that satisfy the following conditions, where $3 \\leqslant k \\leqslant 12$ and $2 \\leqslant m \\leqslant 20$. Additionally, when $\\frac{1}{k}$ is represented as a repeating decimal in base $m$, the digits in the repeating portion are all distinct, and by deleting the first few digits of the decimal part, we can obtain the base $m$ repeating decimal representations of $\\frac{2}{k}, \\cdots, \\frac{k-1}{k}$.", "answer": "21", "subject": "Number Theory", "unique_id": "OlymMATH-HARD-86-EN"}
|
| 88 |
+
{"problem": "Given that a positive integer $n$ satisfies: in any consecutive $n$ positive integers, it is always possible to select two numbers $a$, $b$ ($a \\neq b$), and there exists a positive integer $k$, such that $210|(a^k-b^k)$. Find the minimum value of $n$ that satisfies this condition.", "answer": "9", "subject": "Number Theory", "unique_id": "OlymMATH-HARD-87-EN"}
|
| 89 |
+
{"problem": "A tetrahedron $ABCD$ has vertices $A, B, C, D$. $M_1, \\cdots, M_6$ are the midpoints of the six edges. If 4 points are selected randomly from these 10 points, find the probability that they are not coplanar.", "answer": "\\frac{37}{70}", "subject": "Combinatorics", "unique_id": "OlymMATH-HARD-88-EN"}
|
| 90 |
+
{"problem": "Let $a_1, a_2, a_3, a_4, a_5 \\in [0, 1]$, find the maximum value of $\\prod_{1 \\le i < j \\le 5} |a_i - a_j|$.", "answer": "\\frac{3\\sqrt{21}}{38416}", "subject": "Algebra", "unique_id": "OlymMATH-HARD-89-EN"}
|
| 91 |
+
{"problem": "Find the remainder of $\\sum_{k=0}^{1234}\\binom{2016\\times 1234}{2016k}$ modulo $2017^2$ (provide the value in the range $[0, 2017^2)$).", "answer": "1581330", "subject": "Number Theory", "unique_id": "OlymMATH-HARD-90-EN"}
|
| 92 |
+
{"problem": "Write out all positive integers from $1$ to $10000$ from left to right, then delete those numbers that are divisible by $5$ or $7$, and form a new number by connecting the remaining numbers in a row. Find the remainder when this new number is divided by $11$ (give the value in the range $[0, 11)$).", "answer": "8", "subject": "Number Theory", "unique_id": "OlymMATH-HARD-91-EN"}
|
| 93 |
+
{"problem": "Denote a decimal number of the form $0.a_1 a_2^{(k)} \\cdots a_n^{(k)} \\cdots$ as $A(a_1, k)$, where the digit $a_1$ can be any natural number from $1$ to $9$. When $a_1$ is given, $a_2^{(k)}$ equals the ones digit of the product $ka_1$, and $a_n^{(k)}$ equals the ones digit of the product $ka_{n-1}^{(k)}$, where $n=3, 4, \\cdots$. Find the value of $\\sum_{k=1}^9 \\sum_{a_1=1}^9 A(a_1, k)$.", "answer": "\\frac{401}{9}", "subject": "Number Theory", "unique_id": "OlymMATH-HARD-92-EN"}
|
| 94 |
+
{"problem": "Let $S\\subset \\{1, 2, \\cdots, 100\\}$ be a set. It is known that for any two distinct elements $a, b$ in $S$, there exists a positive integer $k$ and two distinct elements $c, d$ in $S$ (which may equal $a$ or $b$), such that $c < d$ and $a + b = c^k d$. Find the maximum number of elements in set $S$.", "answer": "48", "subject": "Number Theory", "unique_id": "OlymMATH-HARD-93-EN"}
|
| 95 |
+
{"problem": "Let the sequence $\\{a_n\\}$ satisfy: (1) $a_1$ is a perfect square number (2) For any positive integer $n$, $a_{n + 1}$ is the smallest positive integer such that $2^na_1+2^{n-1}a_2+\\cdots+2a_n+a_{n+1}$ is a perfect square number. If there exists a positive integer $s$ such that $a_s = a_{s + 1} = t$, find the minimum possible value of $t$.", "answer": "31", "subject": "Number Theory", "unique_id": "OlymMATH-HARD-94-EN"}
|
| 96 |
+
{"problem": "Given positive integers $x_1, x_2, \\cdots, x_{2005}$ satisfying $\\sum_{i = 1} ^ {2005} x_i = 432972$, find the maximum value of $\\sum_{i = 1} ^ {2005} \\gcd(x_i, x_{i+1}, x_{i+2})$, where the indices are taken modulo $2005$.", "answer": "432756", "subject": "Number Theory", "unique_id": "OlymMATH-HARD-95-EN"}
|
| 97 |
+
{"problem": "Find the smallest integer $m\\ge 2017$ such that for any integers $a_1, a_2, \\cdots, a_{m}$, there exist $1 < i_1 < i_2 < \\cdots < i_{2017} \\le m$ and $\\varepsilon_1, \\varepsilon_2, \\cdots, \\varepsilon_{2017} \\in \\{-1, 1\\}$, such that $\\sum_{j=1}^{2017}\\varepsilon_j a_{i_j}$ is divisible by $2017$.", "answer": "2027", "subject": "Number Theory", "unique_id": "OlymMATH-HARD-96-EN"}
|
| 98 |
+
{"problem": "A positive integer is called a \"good number\" if it can be represented as the sum of squares of pairwise differences of $1893$ integers. Find the smallest positive integer $a$ that is not a perfect square, such that multiplying any good number by $a$ still yields a good number.", "answer": "43", "subject": "Number Theory", "unique_id": "OlymMATH-HARD-97-EN"}
|
| 99 |
+
{"problem": "Let $a_1, a_2, \\cdots, a_{20}$ be $20$ distinct positive integers, and the set $\\{a_i + a_j | 1 \\le i, j \\le 20\\}$ contains $201$ distinct elements. Find the minimum possible number of distinct elements in the set $\\{|a_i - a_j| | 1 \\le i, j \\le 20\\}$.", "answer": "100", "subject": "Combinatorics", "unique_id": "OlymMATH-HARD-98-EN"}
|
| 100 |
+
{"problem": "Find the number of positive integers $t$ not exceeding $2009$ such that for all natural numbers $n$, $\\sum_{k = 0}^n \\binom{2n+1}{2k+1} t^k$ is coprime to $2009$.", "answer": "980", "subject": "Number Theory", "unique_id": "OlymMATH-HARD-99-EN"}
|
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| 1 |
+
{"problem": "已知非负整数数列 $\\{a_n\\}$ 满足 $a_1 = 2016$, $a_{n+1} \\le \\sqrt{a_n}$, 且若项数不少于 2, 则其中任意两项均不相等. 求这样的数列 $\\{a_n\\}$ 的个数.", "answer": "948", "subject": "组合", "unique_id": "OlymMATH-EASY-0-ZH"}
|
| 2 |
+
{"problem": "已知 $AB$ 是半径为 $2$ 的 $\\odot C$ 的一条直径, $\\odot D$ 与 $\\odot C$ 内切于点 $A$, $\\odot E$ 与 $\\odot C$ 内切, 与 $\\odot D$ 外切, 且与 $AB$ 切于点 $F$. 若 $\\odot D$ 的半径是 $\\odot E$ 半径的 $3$ 倍, 求 $\\odot D$ 的半径.", "answer": "4\\sqrt{15}-14", "subject": "几何", "unique_id": "OlymMATH-EASY-1-ZH"}
|
| 3 |
+
{"problem": "计算 $\\sqrt{9+8\\cos 20^{\\circ }}-\\sec 20^{\\circ }$ 的值.", "answer": "3", "subject": "代数", "unique_id": "OlymMATH-EASY-2-ZH"}
|
| 4 |
+
{"problem": "一个球外接于四面体 $ABCD$, 另一个半径为 $1$ 的球与平面 $ABC$ 相切, 且两球内切于点 $D$. 若 $AD=3$, $\\cos \\angle BAC=\\frac{4}{5}$, $\\cos \\angle BAD=\\cos \\angle CAD=\\frac{1}{\\sqrt{2}}$, 求四面体 $ABCD$ 的体积.", "answer": "\\frac{18}{5}", "subject": "几何", "unique_id": "OlymMATH-EASY-3-ZH"}
|
| 5 |
+
{"problem": "求当 $x \\in [1, 2017]$ 时, $f(x) = \\sum_{i=1}^{2017} i|x-i|$ 的最小值.", "answer": "801730806", "subject": "代数", "unique_id": "OlymMATH-EASY-4-ZH"}
|
| 6 |
+
{"problem": "已知三内角成等差数列的三角形的最长, 最短两边之差为第三边上的高的 4 倍, 求最大内角比最小内角大多少. (用反三角函数表示)", "answer": "\\pi -\\arccos \\frac{1}{8}", "subject": "几何", "unique_id": "OlymMATH-EASY-5-ZH"}
|
| 7 |
+
{"problem": "二次曲线 $(3x+4y-13)(7x-24y+3)=200$ 的焦点之间的距离为多少?", "answer": "2\\sqrt{10}", "subject": "几何", "unique_id": "OlymMATH-EASY-6-ZH"}
|
| 8 |
+
{"problem": "在正方体中, 任两个顶点确定一条直线, 求这些直线中垂直且异面的直线共有多少对.", "answer": "78", "subject": "组合", "unique_id": "OlymMATH-EASY-7-ZH"}
|
| 9 |
+
{"problem": "求函数 $f(x)=\\frac{(x-x^3)(1-6x^2+x^4)}{(1+x^2)^4}$ 的值域.", "answer": "\\left[ -\\frac{1}{8}, \\frac{1}{8} \\right]", "subject": "代数", "unique_id": "OlymMATH-EASY-8-ZH"}
|
| 10 |
+
{"problem": "设正整数集合 $A = \\{a_1, a_2, \\dots, a_{1000}\\}$, 其中 $a_1 < a_2 < \\dots < a_{1000} \\le 2017$. 若对于任意的 $1 \\le i, j \\le 1000$, $i+j \\in A$ 均有 $a_i + a_j \\in A$, 求满足条件的集合 $A$ 的个数.", "answer": "2^{17}", "subject": "组合", "unique_id": "OlymMATH-EASY-9-ZH"}
|
| 11 |
+
{"problem": "已知 $x, y \\in \\mathbf{R}$, 对于任意的 $n \\in \\mathbf{Z}_{+}$, $nx+\\frac{1}{n}y\\geq 1$. 求 $41x+2y$ 的最小值.", "answer": "9", "subject": "代数", "unique_id": "OlymMATH-EASY-10-ZH"}
|
| 12 |
+
{"problem": "设 $T$ 是由 $2020^{100}$ 的所有正约数组成的集合, 集合 $S$ 满足：（1）$S$ 为 $T$ 的子集；（2）$S$ 中任何一个元素均不为 $S$ 中另一个元素的倍数. 求 $S$ 中元素个数的最大值.", "answer": "10201", "subject": "数论", "unique_id": "OlymMATH-EASY-11-ZH"}
|
| 13 |
+
{"problem": "求平面上的简单 300 边形（不自交）的所有内角中直角个数的最大值.", "answer": "201", "subject": "几何", "unique_id": "OlymMATH-EASY-12-ZH"}
|
| 14 |
+
{"problem": "设 $x$, $y$, $z$ 为复数, 且满足 $x^2 + y^2 + z^2 = xy + yz + zx$, $|x+y+z| = 21$, $|x-y| = 2\\sqrt{3}$, $|x| = 3\\sqrt{3}$. 求 $|y|^2 + |z|^2$ 的值.", "answer": "132", "subject": "代数", "unique_id": "OlymMATH-EASY-13-ZH"}
|
| 15 |
+
{"problem": "设一个“操作”为将已知的正整数 $n$ 随机换为一个比其小的非负整数（每个数概率相同）, 求将 $2019$ 进行若干次操作得到 $0$, 操作过程中 $10$, $100$, $1000$ 均出现的概率.", "answer": "\\frac{1}{1112111}", "subject": "组合", "unique_id": "OlymMATH-EASY-14-ZH"}
|
| 16 |
+
{"problem": "将平面直角坐标系中点集 $\\{(m, n) | m, n \\in \\mathbf{Z}_{+}, 1 \\leqslant m, n \\leqslant 6\\}$ 中的点染成红色或蓝色. 求每个单位正方形恰有两个红色顶点的不同染色方案数.", "answer": "126", "subject": "组合", "unique_id": "OlymMATH-EASY-15-ZH"}
|
| 17 |
+
{"problem": "已知 $\\odot O$ 的方程为 $x^2 + y^2 = 4$, $\\odot M$ 的方程为 $(x - 5\\cos\\theta)^2 + (y - 5\\sin\\theta)^2 = 1 (\\theta \\in \\mathbf{R})$. 过 $\\odot M$ 上任意一点 $P$ 作 $\\odot O$ 的两条切线 $PE$, $PF$, 切点分别为 $E$, $F$. 求 $\\overrightarrow{PE} \\cdot \\overrightarrow{PF}$ 的最小值.", "answer": "6", "subject": "几何", "unique_id": "OlymMATH-EASY-16-ZH"}
|
| 18 |
+
{"problem": "已知整数数列 $\\{a_{i,j}\\}$（$i, j \\in \\mathbf{N}$）, 其中, $a_{1,n} = n^n$（$n \\in \\mathbf{Z}_{+}$）, $a_{i,j} = a_{i-1,j} + a_{i-1,j+1}$（$i, j \\geqslant 1$）. 求 $a_{128,1}$ 所取值的个位数字.", "answer": "4", "subject": "代数", "unique_id": "OlymMATH-EASY-17-ZH"}
|
| 19 |
+
{"problem": "平面上有 $100$ 个不同的点和 $n$ 条不同的直线 $l_1, l_2, \\dots, l_n$, 记直线 $l_k$ 经过的点数为 $a_k$. 若 $a_1 + a_2 + \\dots + a_n = 250$, 求 $n$ 的最小可能值.", "answer": "21", "subject": "组合", "unique_id": "OlymMATH-EASY-18-ZH"}
|
| 20 |
+
{"problem": "设 $k = -\\frac{1}{2} + \\frac{\\sqrt{3}}{2}i$, 复平面上 $\\triangle ABC$ 的顶点对应的复数 $z_1$, $z_2$, $z_3$ 满足 $z_1 + kz_2 + k^2(2z_3 - z_1) = 0$. 求该三角形的最小内角的弧度数.", "answer": "\\frac{\\pi}{6}", "subject": "代数", "unique_id": "OlymMATH-EASY-19-ZH"}
|
| 21 |
+
{"problem": "设 $40$ 人匿名投票, 每人一张选票, 可投选三位候选人中的一人或两人, 无废票, 求不同的开票结果数.", "answer": "45961", "subject": "组合", "unique_id": "OlymMATH-EASY-20-ZH"}
|
| 22 |
+
{"problem": "在 $n \\times n$ 的正方形网格中, 共有 $(n+1)^2$ 个交点, 以这些交点为顶点的正方形（可以倾斜）个数记为 $a_n$. 已知当 $n=2$ 时, $a_2 = 6$, 当 $n=3$ 时, $a_3 = 20$. 求 $a_26$ 的值.", "answer": "44226", "subject": "组合", "unique_id": "OlymMATH-EASY-21-ZH"}
|
| 23 |
+
{"problem": "椭圆 $\\frac{x^{2}}{8}+\\frac{y^{2}}{4}=1$, 过点 $F(2,0)$ 的直线与椭圆交于 $A$, $B$ 两点, 点 $C$ 在直线 $x=4$ 上. 若 $\\triangle ABC$ 为正三角形, 求 $\\triangle ABC$ 的面积.", "answer": "\\frac{72\\sqrt{3}}{25}", "subject": "几何", "unique_id": "OlymMATH-EASY-22-ZH"}
|
| 24 |
+
{"problem": "有序正整数组 $(a_1, a_2, \\dots, a_{23})$ 满足: (1) $a_1 < a_2 < \\dots < a_{23} = 50$; (2) 数组中任意三个数可组成三角形的三边. 求满足条件的数组个数.", "answer": "2576", "subject": "组合", "unique_id": "OlymMATH-EASY-23-ZH"}
|
| 25 |
+
{"problem": "已知一元三次方程 $x^{3}-x^{2}-5x-1=0$ 的三个不同的根为 $x_{1}$, $x_{2}$, $x_{3}$. 求 $\\left({x}_{1}^{2}-4x_{1}x_{2}+{x}_{2}^{2}\\right)\\left({x}_{2}^{2}-4x_{2}x_{3}+{x}_{3}^{2}\\right)\\left({x}_{3}^{2}-4x_{3}x_{1}+{x}_{1}^{2}\\right)$ 的值.", "answer": "444", "subject": "代数", "unique_id": "OlymMATH-EASY-24-ZH"}
|
| 26 |
+
{"problem": "设锐角 $\\triangle ABC$ 内角对边长分别为 $a$, $b$, $c$. 若 $2a^{2}=2b^{2}+c^{2}$, 求 $\\tan A+\\tan B+\\tan C$ 的最小值.", "answer": "6", "subject": "代数", "unique_id": "OlymMATH-EASY-25-ZH"}
|
| 27 |
+
{"problem": "在四面体 $ABCD$ 中, $DA=DB=DC=1$, 且 $DA$, $DB$, $DC$ 两两垂直. 求在该四面体表面上与点 $A$ 距离为 $\\frac{2\\sqrt{3}}{3}$ 的点形成的曲线的长度.", "answer": "\\frac{\\sqrt{3}\\pi}{2}", "subject": "几何", "unique_id": "OlymMATH-EASY-26-ZH"}
|
| 28 |
+
{"problem": "考虑集合 $\\{1, 2, \\cdots, 81\\}$ 的所有三元子集 $\\{a, b, c\\}$, 其中 $a < b < c$, 称 $b$ 为中间元素. 求所有的中间元素之和 $S$.", "answer": "3498120", "subject": "组合", "unique_id": "OlymMATH-EASY-27-ZH"}
|
| 29 |
+
{"problem": "在棱长为 $1$ 的正方体 $ABCD-A_{1}B_{1}C_{1}D_{1}$ 中, $M$, $N$ 分别为棱 $C_{1}D_{1}$, $B_{1}C_{1}$ 的中点. 求平面 $AMN$ 截此正方体所得截面的面积.", "answer": "\\frac{7\\sqrt{17}}{24}", "subject": "几何", "unique_id": "OlymMATH-EASY-28-ZH"}
|
| 30 |
+
{"problem": "在平面直角坐标系 $xOy$ 中, $F$ 为抛物线 $\\Gamma: y^2 = 2px(p>0)$ 的焦点. 点 $B$ 在 $x$ 轴上, 且在点 $F$ 的右侧. 点 $A$ 在 $\\Gamma$ 上, 且 $|AF|=|BF|$. 直线 $AF$, $AB$ 与 $\\Gamma$ 的第二个交点分别为 $M$, $N$. 若 $\\angle AMN=90^\\circ$, 求直线 $AF$ 的斜率.", "answer": "\\sqrt{3}", "subject": "几何", "unique_id": "OlymMATH-EASY-29-ZH"}
|
| 31 |
+
{"problem": "已知函数 $f\\colon \\mathbf{R}\\rightarrow \\mathbf{R}$ 满足对于任意的 $x$, $y\\in \\mathbf{R}$, 均有 $f(x^2)+f(y^2)=f^2(x+y)-2xy$. 设 $S={\\sum}_{n=-2020}^{2020}f(n)$. 求 $S$ 共有多少种可能取值.", "answer": "2041211", "subject": "组合", "unique_id": "OlymMATH-EASY-30-ZH"}
|
| 32 |
+
{"problem": "已知正三棱锥的侧棱长为 1, 侧面与底面所成二面角的大小为 $45^{\\circ}$. 求该正三棱锥的外接球的体积.", "answer": "\\frac{5\\sqrt{5}\\pi}{6}", "subject": "几何", "unique_id": "OlymMATH-EASY-31-ZH"}
|
| 33 |
+
{"problem": "已知 $x_i \\in \\mathbf{R} (1 \\leqslant i \\leqslant 2020)$, 且 $x_{1010} = 1$. 求 ${\\sum}_{i,j=1}^{2020} \\min\\{i,j\\}x_i x_j$ 的最小值.", "answer": "\\frac{1}{2}", "subject": "代数", "unique_id": "OlymMATH-EASY-32-ZH"}
|
| 34 |
+
{"problem": "数字钟分别用两个数字显示小时, 分, 秒（如 10:09:18）, 在同一天的 05:00:00-22:59:59 之间, 求钟面上的六个数字全不相同的概率.", "answer": "\\frac{16}{135}", "subject": "组合", "unique_id": "OlymMATH-EASY-33-ZH"}
|
| 35 |
+
{"problem": "已知集合 $M = \\{1, 2, \\dots, 2020\\}$. 现将 $M$ 中的每个数染为红, 黄, 蓝三色之一, 且每种颜色均存在. 设 $S_1 = \\{(x, y, z) \\in M^3 \\mid x, y, z \\text{ 同色}, 2020 \\mid (x+y+z)\\}$, $S_2 = \\{(x, y, z) \\in M^3 \\mid x, y, z \\text{ 两两异色}, 2020 \\mid (x+y+z)\\}$. 求 $2|S_1| - |S_2|$ 的最小值.", "answer": "2", "subject": "组合", "unique_id": "OlymMATH-EASY-34-ZH"}
|
| 36 |
+
{"problem": "设 $O$ 为 $\\triangle ABC$ 的内心, $AB=3$, $AC=4$, $BC=5$, $\\overrightarrow{OP}=x\\overrightarrow{OA}+y\\overrightarrow{OB}+z\\overrightarrow{OC}$, $0 \\leqslant x, y, z \\leqslant 1$. 求动点 $P$ 的轨迹所覆盖的平面区域的面积.", "answer": "12", "subject": "几何", "unique_id": "OlymMATH-EASY-35-ZH"}
|
| 37 |
+
{"problem": "设$\\{a, b, c, d\\}$ 是 $\\{1, 2, \\cdots, 17\\}$ 的子集. 若 $17|(a - b + c - d)$, 则称 $\\{a, b, c, d\\}$ 为 ``好子集''. 求好子集的个数.", "answer": "476", "subject": "组合", "unique_id": "OlymMATH-EASY-36-ZH"}
|
| 38 |
+
{"problem": "在四棱锥 $P-ABCD$ 中, $\\overrightarrow{DC}=3\\overrightarrow{AB}$, 过直线 $AB$ 的平面将四棱锥截成体积相等的两个部分, 设该平面与棱 $PC$ 交于点 $E$. 求 $\\frac{PE}{PC}$ 的值.", "answer": "\\frac{2}{3}", "subject": "几何", "unique_id": "OlymMATH-EASY-37-ZH"}
|
| 39 |
+
{"problem": "在正方体 $ABCD-EFGH$ 中, $M$ 为棱 $GH$ 的中点, 平面 $AFM$ 将正方体分割成体积分别为 $V_1$, $V_2$ ($V_1 \\leqslant V_2$) 的两部分, 求 $\\frac{V_1}{V_2}$ 的值.", "answer": "\\frac{7}{17}", "subject": "几何", "unique_id": "OlymMATH-EASY-38-ZH"}
|
| 40 |
+
{"problem": "已知集合 $S$ 包含所有介于 1 至 $2^{40}$ 的整数, 它们的二进制表示中恰有两个 1, 其余为 0. 求从 $S$ 中随机取出一个数, 其可被 9 整除的概率.", "answer": "\\frac{133}{780}", "subject": "组合", "unique_id": "OlymMATH-EASY-39-ZH"}
|
| 41 |
+
{"problem": "已知椭圆 $\\frac{x^2}{4} + \\frac{y^2}{3} = 1$, $F_1$, $F_2$ 为其左, 右焦点, 动直线 $l$ 为此椭圆的切线, 右焦点 $F_2$ 关于直线 $l$ 的对称点为 $P(m, n)$, $S = |3m + 4n - 24|$. 求 $S$ 的取值范围.", "answer": "[7, 47]", "subject": "几何", "unique_id": "OlymMATH-EASY-40-ZH"}
|
| 42 |
+
{"problem": "在正三棱锥 $P-ABC$ 中, $AP = 3$, $AB = 4$, $D$ 是直线 $BC$ 上一点, 且面 $APD$ 与直线 $BC$ 的夹角大小为 $45^\\circ$. 求线段 $PD$ 的长度.", "answer": "\\frac{\\sqrt{89}}{3}", "subject": "几何", "unique_id": "OlymMATH-EASY-41-ZH"}
|
| 43 |
+
{"problem": "在平面直角坐标系内取定四个点, $A(0,0)$, $B(2,0)$, $C(4,2)$, $D(4,4)$. 有两只蚂蚁分别从点 $A$ 爬到点 $D$ 和从点 $B$ 爬到点 $C$, 爬行方向为坐标轴正方向, 且只能在整点处改变方向. 求满足两只蚂蚁不相遇的路径数.", "answer": "195", "subject": "组合", "unique_id": "OlymMATH-EASY-42-ZH"}
|
| 44 |
+
{"problem": "令 $h_n$ 表示具有 $n+2$ 条边的凸边形区域被其对角线所分成的区域数. 假设没有三条对角线共点, 定义 $h_0 = 0$. 求 $h_{26}$ 的值.", "answer": "20826", "subject": "组合", "unique_id": "OlymMATH-EASY-43-ZH"}
|
| 45 |
+
{"problem": "过点 $P(2,1)$ 的直线 $l$ 分别与 $x$ 轴正半轴, $y$ 轴正半轴交于 $A$, $B$ 两点, $O$ 为坐标原点. 求当 $\\triangle AOB$ 的周长最小时直线 $l$ 在 $y$ 轴上的截距.", "answer": "\\frac{5}{2}", "subject": "几何", "unique_id": "OlymMATH-EASY-44-ZH"}
|
| 46 |
+
{"problem": "定义在集合 $\\{x\\in \\mathbb{Z}_{+} | 1\\leqslant x\\leqslant 12\\}$ 上的函数 $f(x)$ 满足 $|f(x+1)-f(x)|=1$（$x=1, 2, \\dots, 11$）, 且 $f(1)$, $f(6)$, $f(12)$ 成等比数列. 若 $f(1)=1$, 求满足条件的不同函数 $f(x)$ 的个数.", "answer": "355", "subject": "组合", "unique_id": "OlymMATH-EASY-45-ZH"}
|
| 47 |
+
{"problem": "盒子里有红, 黄, 蓝三种颜色的小球, 其中红球有 $12$ 个, 黄球有 $18$ 个, 蓝球有 $30$ 个. 每次从盒子中取出一个球, 直至取完. 求红球最先被取完的概率.", "answer": "\\frac{18}{35}", "subject": "组合", "unique_id": "OlymMATH-EASY-46-ZH"}
|
| 48 |
+
{"problem": "求关于 $x$ 的方程 $x^8 - 14x^4 - 8x^3 - x^2 + 1 = 0$ 的所有两两不同的实数根的平方和.", "answer": "8", "subject": "代数", "unique_id": "OlymMATH-EASY-47-ZH"}
|
| 49 |
+
{"problem": "设 $z^7 = -1$ 的七个两两不同的复数根为 $z_1, z_2, \\dots, z_7$, 求 $\\sum_{j=1}^7 \\frac{1}{|1 - z_j|^2}$ 的值.", "answer": "\\frac{49}{4}", "subject": "代数", "unique_id": "OlymMATH-EASY-48-ZH"}
|
| 50 |
+
{"problem": "已知双曲线 $\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1$（$a> 0$, $b> 0$）的左, 右焦点分别为 $F_{1}$, $F_{2}$, 过 $F_{2}$ 的直线交右支于 $A$, $B$ 两点. 若 $\\left| AF_{2}\\right| =3\\left| F_{2}B\\right| $, $\\left| AF_{1}\\right| =\\left| AB\\right| $, 求该双曲线的离心率.", "answer": "2", "subject": "几何", "unique_id": "OlymMATH-EASY-49-ZH"}
|
| 51 |
+
{"problem": "设 $a$, $b$, $c$ 是正实数, 求 $\\frac{(a+b+c)(a^2+3b^2+15c^2)}{abc}$ 的最小值.", "answer": "36", "subject": "代数", "unique_id": "OlymMATH-EASY-50-ZH"}
|
| 52 |
+
{"problem": "一个飞盘玩具是指一个圆盘被 $20$ 条从圆心出发的射线等分为 $20$ 个扇形, 每个扇形被染为红色或蓝色（只有正面染色）, 且相对的两个扇形都不同色. 若旋转后相同的飞盘玩具视为同一种, 求不同的飞盘玩具共有多少种.（用具体数字作答）", "answer": "52", "subject": "组合", "unique_id": "OlymMATH-EASY-51-ZH"}
|
| 53 |
+
{"problem": "平面上给出十个点, 任何三点都不共线, 作四条线段, 每条线段联结平面上的两个点. 这些线段是任选的, 且这些线段都有相同的被选的可能性. 求这些线段中的某三条线段构成以给定十个点中三点为顶点的三角形的概率.", "answer": "\\frac{16}{473}", "subject": "组合", "unique_id": "OlymMATH-EASY-52-ZH"}
|
| 54 |
+
{"problem": "求最小的正实数 $c$, 使��对任意正整数 $n(n \\geqslant 2)$ 及正实数 $a_1 \\leqslant a_2 \\leqslant \\cdots \\leqslant a_n$, 都有 $\\sum_{k=2}^{n}\\frac{a_{k}-2\\sqrt{a_{k-1}(a_{k}-a_{k-1})}}{c^{k}}\\geqslant \\frac{a_{n}}{nc^{n}}-\\frac{a_{1}}{c}$.", "answer": "2", "subject": "代数", "unique_id": "OlymMATH-EASY-53-ZH"}
|
| 55 |
+
{"problem": "从 $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$ 中选出 $7$ 个不同的数排成一个数列 $a_1, a_2, \\cdots, a_7$, 使得其任意相邻 $4$ 项之和是 $3$ 的倍数, 求这样的数列的个数.", "answer": "3024", "subject": "数论", "unique_id": "OlymMATH-EASY-54-ZH"}
|
| 56 |
+
{"problem": "对于正整数 $n$, 将其各位数字之和记为 $s(n)$, 各位数字之积记为 $p(n)$, 若成立 $s(n)+p(n)=n$, 就称 $n$ 为巧合数. 求所有巧合数的和.", "answer": "531", "subject": "数论", "unique_id": "OlymMATH-EASY-55-ZH"}
|
| 57 |
+
{"problem": "九个连续正整数从小到大排成一个数列 $a_1<\\cdots<a_9$. 若 $a_1+a_3+a_5+a_7+a_9$ 为一个平方数, $a_2+a_4+a_6+a_8$ 为一个立方数, 求这九个正整数之和的最小值.", "answer": "18000", "subject": "数论", "unique_id": "OlymMATH-EASY-56-ZH"}
|
| 58 |
+
{"problem": "如果 $n$ 的二进制表示中 $1$ 的个数多于 $0$ 的个数, 则称正整数 $n$ 为好数. 求不超过 $2017$ 的好数的个数.", "answer": "1169", "subject": "数论", "unique_id": "OlymMATH-EASY-57-ZH"}
|
| 59 |
+
{"problem": "将 $1, 2, \\cdots, 8$ 的每一个全排列看做一个八位数, 求其中是 $11$ 的倍数的八位数的个数.", "answer": "4608", "subject": "数论", "unique_id": "OlymMATH-EASY-58-ZH"}
|
| 60 |
+
{"problem": "求满足 $133^5 + 110^5 + 84^5 + 27^5 = n^5$ 的整数 $n$ 的值.", "answer": "144", "subject": "数论", "unique_id": "OlymMATH-EASY-59-ZH"}
|
| 61 |
+
{"problem": "在 $1, 2, \\cdots, 2018$ 中任取一组数, 使得其中任意两数之和不能被其差整除. 求最多能取的数的个数.", "answer": "673", "subject": "数论", "unique_id": "OlymMATH-EASY-60-ZH"}
|
| 62 |
+
{"problem": "求正整数中, 由 $0, 1, 2$ 组成, 每个数字至少出现 $1$ 次的 $2012$ 位偶数的个数.", "answer": "4\\times 3^{2010}-5\\times 2^{2010}+1", "subject": "数论", "unique_id": "OlymMATH-EASY-61-ZH"}
|
| 63 |
+
{"problem": "对正整数 $n$, 用 $\\varphi(n)$ 表示不超过 $n$ 且与 $n$ 互质的正整数的个数. $f(n)$ 表示大于 $n$ 且与 $n$ 不互质的最小正整数. 若 $f(n) = m$ 且 $\\varphi(m) = n$, 则称 $(m, n)$ 为幸运数对. 考虑所有的幸运数对组成的集合 $S$, 求 $\\sum_{(m, n)\\in S} (m + n)$ 的值.", "answer": "6", "subject": "数论", "unique_id": "OlymMATH-EASY-62-ZH"}
|
| 64 |
+
{"problem": "若正整数 $a$ 满足: 存在素数 $p$, 使得 $a^2+p$ 也为平方数, 则称 $a$ 为好数. 求在集合 $M=\\{1, 2, \\cdots, 100\\}$ 中好数的个数.", "answer": "45", "subject": "数论", "unique_id": "OlymMATH-EASY-63-ZH"}
|
| 65 |
+
{"problem": "已知集合 $A = \\{1, 2, \\cdots, 2019\\}$, 映射 $f: A \\rightarrow A$ 满足对于任意的 $k \\in A$, 均有 $f(k) \\leqslant k$, 且像恰有 $2018$ 个不同的值. 求满足条件的映射 $f$ 共有多少个.", "answer": "2^{2019} - 2020", "subject": "组合", "unique_id": "OlymMATH-EASY-64-ZH"}
|
| 66 |
+
{"problem": "已知数列 $\\{a_n\\}$ 满足 $a_{n+1} + (-1)^n a_n = 2n - 1$, 且数列 $\\{a_n - n\\}$ 的前 $2019$ 项之和为 $2019$, 求 $a_{2020}$ 的值.", "answer": "1", "subject": "代数", "unique_id": "OlymMATH-EASY-65-ZH"}
|
| 67 |
+
{"problem": "已知集合为 $\\{1, 2, \\cdots, 30\\}$ 的三元子集, 若其三个元素的乘积为 $8$ 的倍数, 则称其为``有趣的''. 求 $\\{1, 2, \\cdots, 30\\}$ 有多少个有趣的集合.", "answer": "1925", "subject": "组合", "unique_id": "OlymMATH-EASY-66-ZH"}
|
| 68 |
+
{"problem": "已知棱长为 $1$ 的正四面体 $ABCD$, $M$ 为 $AC$ 的中点, $P$ 在线段 $DM$ 上. 求 $AP+BP$ 的最小值.", "answer": "\\sqrt{1+\\frac{\\sqrt{6}}{3}}", "subject": "几何", "unique_id": "OlymMATH-EASY-67-ZH"}
|
| 69 |
+
{"problem": "记 $(a_1, a_2, \\cdots, a_{2022})$ 为整数 $1, 2, \\ldots, 2022$ 的一个顺时针方向的圆排列. 若 $\\sum_{i=1}^{2022} |a_i - a_{i+1}| = 4042$ ($a_{2023} = a_1$), 求满足条件的圆排列有多少个.", "answer": "2^{2020}", "subject": "组合", "unique_id": "OlymMATH-EASY-68-ZH"}
|
| 70 |
+
{"problem": "已知双曲线 $\\Gamma: \\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$, $F$ 为其左焦点. 直线 $y = kx$ 分别与 $\\Gamma$ 的左、右两支的交点为 $A$、$B$, 且满足 $FA \\perp AB$, $\\angle ABF = \\angle AFO$ ($O$ 为原点). 求 $\\Gamma$ 的离心率.", "answer": "\\frac{3\\sqrt{2}+\\sqrt{6}}{2}", "subject": "几何", "unique_id": "OlymMATH-EASY-69-ZH"}
|
| 71 |
+
{"problem": "已知空间一点 $P$ 到正四面体 $ABCD$ 的顶点 $A$, $B$ 的距离分别为 $2$, $3$. 求当正四面体的棱长位置变化时, 点 $P$ 到 $CD$ 所在直线的最大距离.", "answer": "\\frac{5\\sqrt{3}}{2}", "subject": "几何", "unique_id": "OlymMATH-EASY-70-ZH"}
|
| 72 |
+
{"problem": "已知等边三角形的两个顶点在抛物线 $y^2 = 4x$ 上, 第三个顶点在抛物线的准线上, 且三角形的中心到该准线的距离等于周长的 $\\frac{1}{9}$. 求三角形的面积.", "answer": "36\\sqrt{3}", "subject": "几何", "unique_id": "OlymMATH-EASY-71-ZH"}
|
| 73 |
+
{"problem": "已知椭圆 $\\frac{x^{2}}{9}+\\frac{y^{2}}{5}=1$ 的右焦点为 $F$, $P$ 为椭圆上一点, 点 $A\\left(0,2\\sqrt{3}\\right)$. 求 $\\triangle APF$ 的周长最大时, $\\triangle APF$ 的面积.", "answer": "\\frac{21\\sqrt{3}}{4}", "subject": "几何", "unique_id": "OlymMATH-EASY-72-ZH"}
|
| 74 |
+
{"problem": "设复数 $z_{1}=-\\sqrt{3}-\\mathrm{i}$, $z_{2}=3+\\sqrt{3}\\mathrm{i}$, $z=\\left(2+\\cos \\theta \\right)+\\mathrm{i}\\sin \\theta$. 求 $\\left| z-z_{1}\\right| +\\left| z-z_{2}\\right| $ 的最小值.", "answer": "2+2\\sqrt{3}", "subject": "代数", "unique_id": "OlymMATH-EASY-73-ZH"}
|
| 75 |
+
{"problem": "设 $A = \\{1, 2, \\cdots, 6\\}$, 函数 $f: A \\rightarrow A$. 记 $p(f) = f(1) \\cdots f(6)$. 求使得 $p(f) | 36$ 的函数共有多少个.", "answer": "580", "subject": "组合", "unique_id": "OlymMATH-EASY-74-ZH"}
|
| 76 |
+
{"problem": "已知抛物线 $C: y^2 = 4x$ 的焦点 $F$, $F$ 关于原点的对称点是 $M$, $\\odot M$ 是半径为 $1$ 的圆. 直线 $l$ 经过 $\\odot M$ 上一点 $A$ (异于原点), 且与抛物线 $C$ 切于点 $T$. 求 $\\frac{|FA|}{|FT|}$ 的最大值.", "answer": "\\frac{1 + \\sqrt{5}}{2}", "subject": "几何", "unique_id": "OlymMATH-EASY-75-ZH"}
|
| 77 |
+
{"problem": "设 $x_{i} \\geq 0 (i = 1, 2, \\cdots, 6)$, 且满足 $\\begin{cases} x_{1} + x_{2} + \\cdots + x_{6} = 1, \\\\ x_{1} x_{3} x_{5} + x_{2} x_{4} x_{6} \\geq \\frac{1}{540} \\end{cases}$. 求 $x_{1} x_{2} x_{3} + x_{2} x_{3} x_{4} + x_{3} x_{4} x_{5} + x_{4} x_{5} x_{6} + x_{5} x_{6} x_{1} + x_{6} x_{1} x_{2}$ 的最大值.", "answer": "\\frac{19}{540}", "subject": "代数", "unique_id": "OlymMATH-EASY-76-ZH"}
|
| 78 |
+
{"problem": "已知实数 $a_1, a_2, \\cdots, a_{224}$ 满足对任意 $i = 1, 2, \\cdots, 224$, 均有 $i \\leqslant a_i \\leqslant 2i$. 求 $\\frac{(\\sum_{i=1}^{224} i a_i)^2}{\\sum_{i=1}^{224} a_i^2}$ 的最小值.", "answer": "\\frac{10057600}{3}", "subject": "代数", "unique_id": "OlymMATH-EASY-77-ZH"}
|
| 79 |
+
{"problem": "将一个 $5 \\times 5$ 方格表中每个格染五种颜色之一, 使得每种颜色的格的个数相同. 若相邻两格的颜色不同, 则称它们的公共边为``分割边''. 求分割边条数的最小值.", "answer": "16", "subject": "组合", "unique_id": "OlymMATH-EASY-78-ZH"}
|
| 80 |
+
{"problem": "从正 $17$ 边形的顶点中随机取出 $3$ 点, 求所取点形成锐角三角形的概率.", "answer": "\\frac{3}{10}", "subject": "组合", "unique_id": "OlymMATH-EASY-79-ZH"}
|
| 81 |
+
{"problem": "已知函数 $f(x) = 10x^2 + mx + n$ ($m, n \\in \\mathbf{Z}$) 在区间 $(1, 3)$ 上有两个不同的实数根. 求 $f(1)f(3)$ 的最大可能值.", "answer": "99", "subject": "代数", "unique_id": "OlymMATH-EASY-80-ZH"}
|
| 82 |
+
{"problem": "在平面直角坐标系 $xOy$ 中, 圆心在坐标原点 $C$、半径为 $1$ 的圆的切线 $l$ 与 $x$ 轴交于点 $N$、与 $y$ 轴交于点 $M$, 点 $A(3,4)$, 且 $\\overrightarrow{AC}=x\\overrightarrow{AM}+y\\overrightarrow{AN}$. 设点 $P(x,y)$.\n\n求 $9x^{2}+16y^{2}$ 的最小值.", "answer": "4", "subject": "几何", "unique_id": "OlymMATH-EASY-81-ZH"}
|
| 83 |
+
{"problem": "已知 $x, y, z \\in \\mathbf{R}_{+}$ 且 $x+y+z=1$, 求 $x+\\sqrt{2xy}+3\\sqrt[3]{xyz}$ 的最大值.", "answer": "2", "subject": "代数", "unique_id": "OlymMATH-EASY-82-ZH"}
|
| 84 |
+
{"problem": "已知椭圆 $\\frac{x^{2}}{4}+y^{2}=1$ 上的三点 $A$, $B$, $C$, 斜率为负数的直线 $BC$ 与 $y$ 轴交于点 $M$. 若原点 $O$ 为 $\\triangle ABC$ 的重心, 且 $\\triangle BMA$ 与 $\\triangle CMO$ 的面积之比为 $3:2$. 直线 $BC$ 可能的斜率组成了集合 $S$, 求 $S$ 中所有元素的平方和.", "answer": "\\frac{41}{6}", "subject": "几何", "unique_id": "OlymMATH-EASY-83-ZH"}
|
| 85 |
+
{"problem": "在 $\\triangle ABC$ 中, $z = \\frac{\\sqrt{65}}{5} \\sin \\frac{A+B}{2} + i \\cos \\frac{A-B}{2}$, $|z| = \\frac{3\\sqrt{5}}{5}$. 求 $\\angle C$ 的最大值.", "answer": "\\pi - \\arctan \\frac{12}{5}", "subject": "代数", "unique_id": "OlymMATH-EASY-84-ZH"}
|
| 86 |
+
{"problem": "在 $\\triangle ABC$ 中, $\\angle A$, $\\angle B$, $\\angle C$ 所对的边分别为 $a$, $b$, $c$. 若 $\\angle A = 39^{\\circ}$, $(a^2 - b^2)(a^2 + ac - b^2) = b^2c^2$, 求 $\\angle C$ 的值.", "answer": "115^{\\circ}", "subject": "几何", "unique_id": "OlymMATH-EASY-85-ZH"}
|
| 87 |
+
{"problem": "过椭圆 $\\frac{x^{2}}{6}+\\frac{y^{2}}{3}=1$ 内的点 $P(1,\\frac{1}{2})$ 作一条不过原点的直线, 与椭圆交于 $A$, $B$ 两点. 求 $\\triangle OAB$ 的面积的最大值.", "answer": "\\frac{3\\sqrt{6}}{4}", "subject": "几何", "unique_id": "OlymMATH-EASY-86-ZH"}
|
| 88 |
+
{"problem": "将写有 $1, 2, \\dots, 9$ 的九张卡片随机地排成一行, 若第一张卡片（左起）上的标数为 $k$, 则将前 $k$ 张卡片逆序排过来称为一次操作, 无法操作时（即第一张卡片上的标数为 1）游戏停止. 若一个排列无法操作, 且恰由唯一的另一个排列经过一次操作得到, 则此排列称为``二次终止排列''. 在所有可能的排列中, 求二次终止排列出现的概率.", "answer": "\\frac{103}{2520}", "subject": "组合", "unique_id": "OlymMATH-EASY-87-ZH"}
|
| 89 |
+
{"problem": "设 $a_{i}(i\\in \\mathbb{Z}_{+},i\\leqslant 2020)$ 为非负实数, 且 $\\sum_{i=1}^{2020}a_{i}=1$. 求 $\\sum_{\\substack{i\\neq j\\\\i|j}}a_{i}a_{j}$ 的最大值.", "answer": "\\frac{5}{11}", "subject": "代数", "unique_id": "OlymMATH-EASY-88-ZH"}
|
| 90 |
+
{"problem": "已知椭圆 $C: \\frac{x^{2}}{25}+\\frac{y^{2}}{9}=1$, 动圆 $\\Gamma: x^{2}+y^{2}=r^{2}(3< r< 5)$. 若 $M$ 为椭圆 $C$ 上的点, $N$ 为动圆 $\\Gamma$ 上的点, 且直线 $MN$ 与椭圆 $C$, 动圆 $\\Gamma$ 均相切, 求 $M, N$ 两点的距离 $|MN|$ 的最大值.", "answer": "2", "subject": "几何", "unique_id": "OlymMATH-EASY-89-ZH"}
|
| 91 |
+
{"problem": "设 $a$, $b$, $c$ 为互不相同的非零实数, 满足：方程 $ax^3+bx+c=0$, $bx^3+cx+a=0$, $cx^3+ax+b=0$ 有公共根, 且这三个方程中存在两个方程无虚根. 求 $\\frac{a}{b}+\\frac{b}{c}+\\frac{c}{a}$ 的最小值.", "answer": "\\frac{17}{12}", "subject": "代数", "unique_id": "OlymMATH-EASY-90-ZH"}
|
| 92 |
+
{"problem": "在数列 $\\{u_n\\} (n \\in \\mathbb{Z}_{+})$ 中, $u_1 = 2$, $u_2 = 8$, $u_{n+1} = 4u_n - u_{n-1} (n \\geqslant 2)$. 求 $\\sum_{n=1}^{\\infty} \\operatorname{arccot} u_n^2$.", "answer": "\\frac{\\pi}{12}", "subject": "代数", "unique_id": "OlymMATH-EASY-91-ZH"}
|
| 93 |
+
{"problem": "设 $f(x): [0, 1] \\rightarrow \\mathbb{R}$, 满足：(1) $f(\\frac{x}{3}) = \\frac{1}{2}f(x)$；(2) $f(1-x) = 1 - f(x)$；(3) $f(x) = \\frac{1}{2} (x \\in [\\frac{1}{3}, \\frac{2}{3}])$. 若 $n=2023$, 求 $S_n = \\sum_{\\substack{1 \\leqslant k \\leqslant 3^n \\\\ k \\text{是奇数}}} f(\\frac{k}{3^n})$ 的值.", "answer": "\\frac{3^{2023} + 3}{4}", "subject": "代数", "unique_id": "OlymMATH-EASY-92-ZH"}
|
| 94 |
+
{"problem": "抛物线 $\\Gamma: x^2 = 4y$, 斜率为 $1$ 的直线 $l$ 与抛物线 $\\Gamma$ 交于 $A$, $B$ 两点, 过点 $A$, $B$ 分别作抛物线 $\\Gamma$ 的切线, 交于点 $M$, $F$ 为抛物线 $\\Gamma$ 的焦点. 记 $\\triangle AFM$, $\\triangle BFM$, $\\triangle ABM$ 的面积分别为 $S_1$, $S_2$, $S_3$. 求 $\\frac{S_1 S_2}{S_3}$ 的最小值.", "answer": "\\sqrt[4]{\\frac{64}{27}}", "subject": "几何", "unique_id": "OlymMATH-EASY-93-ZH"}
|
| 95 |
+
{"problem": "设椭圆 $C: \\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{20}=1\\left(a> 2\\sqrt{5}\\right)$ 的左焦点为 $F$. 已知存在过点 $P\\left(1,1\\right)$ 的直线 $l$ 与椭圆交于点 $A$, $B$, $M$ 是 $AB$ 的中点, 使得 $\\left| FM\\right|$ 是 $\\left| FA\\right|$ 与 $\\left| FB\\right|$ 的等比中项. 求 $a$ 的最小正整数值.", "answer": "7", "subject": "几何", "unique_id": "OlymMATH-EASY-94-ZH"}
|
| 96 |
+
{"problem": "已知 $A$, $B$, $C$ 是 $\\triangle ABC$ 的三个内角, 向量 $\\boldsymbol{\\alpha} = \\left( \\cos \\frac{A-B}{2}, \\sqrt{3} \\sin \\frac{A+B}{2} \\right)$, 且 $|\\boldsymbol{\\alpha}| = \\sqrt{2}$. 若当角 $C$ 最大时, 存在动点 $M$ 使得 $|MA|$, $|AB|$, $|MB|$ 成等差数列, 求 $\\frac{|MC|}{|AB|}$ 的最大值.", "answer": "\\frac{2\\sqrt{3}+\\sqrt{2}}{4}", "subject": "几何", "unique_id": "OlymMATH-EASY-95-ZH"}
|
| 97 |
+
{"problem": "设 $p=2017$ 为素数, 集合 $\\{1,3,5,\\cdots,p-2\\}$ 中是模 $p$ 的二次剩余的数组成集合 $A$, 不是模 $p$ 的二次剩余的数组成集合 $B$. 求 $(\\sum_{a\\in A}\\cos \\frac{a\\pi}{p})^{2}+(\\sum_{b\\in B}\\cos \\frac{b\\pi}{p})^{2}$ 的值.", "answer": "\\frac{1009}{4}", "subject": "数论", "unique_id": "OlymMATH-EASY-96-ZH"}
|
| 98 |
+
{"problem": "过椭圆 $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1(a>b>0)$ 的右焦点 $F_{2}(c,0)$ 作直线 $l$, 交椭圆于点 $P$, $Q$. 在圆 $x^{2}+y^{2}=b^{2}$ 上找一点 $M$, 联结 $MP$, $MQ$. 我们将 $\\triangle MPQ$ 面积的最大值记作 $F(a, b)$. 求 $F(3, 2\\sqrt{2}) + F(2, 1)$.", "answer": "\\frac{19\\sqrt{2}+11}{3}", "subject": "几何", "unique_id": "OlymMATH-EASY-97-ZH"}
|
| 99 |
+
{"problem": "设方程 $2^x + 3^y = z^2$ 的全体整数解构成集合 $S$, 求 $\\sum_{(x, y, z)\\in S}(x + y + z^2)$.", "answer": "96", "subject": "数论", "unique_id": "OlymMATH-EASY-98-ZH"}
|
| 100 |
+
{"problem": "一次数学竞赛共 $6$ 道题, 每题答对得 $7$ 分, 答错或不答得 $0$ 分. 赛后某参赛代表队获团体总分 $161$ 分, 且统计分数时发现, 该队任两名选手至多答对两道相同的题目, 没有三名选手都答对两道相同的题目. 求该队选手至少有多少人.", "answer": "7", "subject": "组合", "unique_id": "OlymMATH-EASY-99-ZH"}
|
data/OlymMATH-ZH-HARD.jsonl
ADDED
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| 1 |
+
{"problem": "设 $a, b, c \\in \\mathbb{R}$, $a^3 b + b^3 c + c^3 a = 3$, 求下式的最小值 $f(a, b, c) = (\\sum a^4)^4 + 1000 \\sum a^2 b^2$.", "answer": "2625", "subject": "代数", "unique_id": "OlymMATH-HARD-0-ZH"}
|
| 2 |
+
{"problem": "若一个正方体的八个顶点到某平面的距离分别为 $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, 考虑这个正方体的所有可能的棱长. 假设可能的棱长构成了集合 $S$, 求 $S$ 中所有元素的平方和.", "answer": "210", "subject": "几何", "unique_id": "OlymMATH-HARD-1-ZH"}
|
| 3 |
+
{"problem": "对于 $i = 1, 2, \\cdots, n$, 都有 $x_i < 1$, 且 $| x_1 | + | x_2 | + \\cdots + | x_n | = 19 + | x_1 + x_2 + \\cdots + x_n |$, 求正整数 $n$ 的最小值.", "answer": "11", "subject": "代数", "unique_id": "OlymMATH-HARD-2-ZH"}
|
| 4 |
+
{"problem": "求最少需要几个正方体 (可悬空), 使其三视图均为 $3 \\times 3$ 的方格.", "answer": "8", "subject": "几何", "unique_id": "OlymMATH-HARD-3-ZH"}
|
| 5 |
+
{"problem": "设 $x$, $y$, $z$ 都是正实数, 求下式的最小值 $f(x, y, z) = \\frac{(2 + 5y)(3x + z)(x + 3y)(2z + 5)}{xyz}$.", "answer": "241 + 44\\sqrt{30}", "subject": "代数", "unique_id": "OlymMATH-HARD-4-ZH"}
|
| 6 |
+
{"problem": "已知椭圆 $x^{2} / 4 + y^{2} = 1$, $N_{1}(-1, 0)$, $N_{2}(1, 0)$, $M(3, 0)$, 过 $M$ 的直线交椭圆于 $P$, $Q$ 两点, 连 $N_{1}P$, $N_{2}Q$, 得交点 $R$. 可以证明 $R$ 的轨迹形成一条二次曲线, 求它的离心率.", "answer": "\\frac{\\sqrt{51}}{6}", "subject": "几何", "unique_id": "OlymMATH-HARD-5-ZH"}
|
| 7 |
+
{"problem": "将编号为 $1, 2, \\dots, 9$ 的九个小球随机放置在圆周的九个等分点上, 每个等分点上各有一个小球. 设圆周上所有相邻两球号码之差的绝对值之和为 $S$. 求使 $S$ 达到最小值的放法的概率. 注：如果某种放法经旋转或镜面反射后可与另一种放法重合, 则认为是相同的放法.", "answer": "\\frac{1}{315}", "subject": "组合", "unique_id": "OlymMATH-HARD-6-ZH"}
|
| 8 |
+
{"problem": "将边长为 $10$, $12$, $14$ 的三角形沿三条中位线折起来围成四面体. 求四面体的外接球直径的值.", "answer": "\\sqrt{55}", "subject": "几何", "unique_id": "OlymMATH-HARD-7-ZH"}
|
| 9 |
+
{"problem": "对于 $k$, $n \\in \\mathbf{Z}_{+}$, 设有限数列 $\\{a_{k}\\}$ 的项数为 $n$, 其中, $a_{k} \\leqslant m$, $a_{k}$, $m \\in \\mathbf{Z}_{+}$. 若 $m=2025$, 试确定项数 $n$ 的最大值, 使得数列 $\\{a_{k}\\}$ 满足：(1) 对任意一项 $a_{k}$, 若存在 $a_{k-1}$ 与 $a_{k+1}$, 则 $a_{k-1} \\neq a_{k+1}$；(2) 不存在正整数 $i_{1} < i_{2} < i_{3} < i_{4}$, 使得 $a_{{i_{1}}} = a_{{i_{3}}} \\neq a_{{i_{2}}} = a_{{i_{4}}}$.", "answer": "8098", "subject": "组合", "unique_id": "OlymMATH-HARD-8-ZH"}
|
| 10 |
+
{"problem": "已知椭圆 $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1(a>b>0)$ 的左焦点为 $F$, $P(x_{0},y_{0})$ 为椭圆上的一点, 且 $x_{0}>0$. 过点 $P$ 作圆 $x^{2}+y^{2}=b^{2}$ 的切线, 与椭圆的第二个交点为 $Q$. 设 $I$ 为 $\\triangle PFQ$ 的内心, $\\angle PFQ=2\\alpha$. 若 $a^2=\\sqrt{3}, b^2=\\sqrt{2}$, 求 $|FI| \\cos \\alpha$ 的值.", "answer": "\\sqrt[4]{3}", "subject": "几何", "unique_id": "OlymMATH-HARD-9-ZH"}
|
| 11 |
+
{"problem": "将 0 至 9 这十个数字排列成首位不为零的不重复的十位数, 求能被 99 整除的十位数的个数.", "answer": "285120", "subject": "数论", "unique_id": "OlymMATH-HARD-10-ZH"}
|
| 12 |
+
{"problem": "设 $n$ 个互异的正整数 $a_1, a_2, \\dots, a_n$ 之和为 $2000$. 记 $A = \\max\\{a_1, a_2, \\dots, a_n\\}$. 求 $A+n$ 的最小值. （$n$ 不是事先给定的）", "answer": "110", "subject": "数论", "unique_id": "OlymMATH-HARD-11-ZH"}
|
| 13 |
+
{"problem": "给定正整数 $n=2024$. 求最大的整数 $M$ 的值, 使得对任意正整数 $a_{1}, a_{2}, \\ldots, a_{n}$, 均有 $[\\sqrt{a_{1}}]+[\\sqrt{a_{2}}]+\\cdots +[\\sqrt{a_{n}}]\\geqslant [\\sqrt{a_{1}+a_{2}+\\cdots +a_{n}+M\\min \\{a_{1},a_{2},\\cdots ,a_{n}\\}}]$, 其中, $[x]$ 表示不超过实数 $x$ 的最大整数.", "answer": "1364850", "subject": "代数", "unique_id": "OlymMATH-HARD-12-ZH"}
|
| 14 |
+
{"problem": "已知 $m> 0$, 关于 $x$ 的方程 $(mx-3+\\sqrt{2})^{2}-\\sqrt{x+m}=0$ 在区间 $[0,1]$ 上恰有两个不同的实根. 求实数 $m$ 的取值范围.", "answer": "[3,193-132\\sqrt{2}]", "subject": "代数", "unique_id": "OlymMATH-HARD-13-ZH"}
|
| 15 |
+
{"problem": "一个班有 25 名学生. 老师想要准备 $N$ 块糖果举办竞赛, 并按照成绩分配糖果 (分数相同得到相同数目的糖果, 分数越少得到的糖果越少, 可以是 0 块). 求 $N$ 的最小值, 使得无论竞赛有多少题目, 以及学生的答题情况如何, 都可以这样分配糖果.", "answer": "600", "subject": "组合", "unique_id": "OlymMATH-HARD-14-ZH"}
|
| 16 |
+
{"problem": "设 $n$ 为正整数, 集合 $T_{n}$ 为数集 $A_{n}=\\{k \\mid k \\in \\mathbf{Z}_{+}, 且 k \\leqslant n\\}$ 的一个子集, 且 $T_{n}$ 中任意两个数之差不等于 4 或 7. 若 $T_{n}$ 的元素个数的最大值记为 $f_{n}$ (如 $f_{1}=1$, $f_{2}=2$), 求 $\\sum_{n=1}^{2023}f_{n}$ 的值.", "answer": "932604", "subject": "组合", "unique_id": "OlymMATH-HARD-15-ZH"}
|
| 17 |
+
{"problem": "试求最大的正整数 $n \\le 2025$ 使得存在正整数数列 $a_1 < a_2 < \\cdots < a_n$, 使得 $a_i + a_j (1 \\le i < j \\le n)$ 互不相同, 且模 $4$ 意义下各余数出现的次数相同.", "answer": "1296", "subject": "数论", "unique_id": "OlymMATH-HARD-16-ZH"}
|
| 18 |
+
{"problem": "记 $[x]$ 表示不超过实数 $x$ 的最大整数. 数列 $\\{x_n\\}$ 满足: $x_1 = 1$, $x_{n+1} = 4x_n + [\\sqrt{11}x_n]$. 求 $x_{2021}$ 的个位数字.", "answer": "9", "subject": "数论", "unique_id": "OlymMATH-HARD-17-ZH"}
|
| 19 |
+
{"problem": "已知正三棱锥 $P-ABC$ 与正三棱锥 $Q-ABC$ 内接于同一个单位球 $O$, 两顶点 $P$, $Q$ 在底面 $ABC$ 的异侧. 设二面角 $P-AB-C$, 二面角 $Q-AB-C$ 的平面角分别为 $\\alpha$, $\\beta$. 求 $AB \\tan(\\alpha + \\beta)$ 的值.", "answer": "-\\frac{4\\sqrt{3}}{3}", "subject": "几何", "unique_id": "OlymMATH-HARD-18-ZH"}
|
| 20 |
+
{"problem": "在 $\\triangle ABC$ 中, $AB = AC$, $\\angle BAC = 30^\\circ$. 在边 $AB$ 上取五等分点 $T_1$, $T_2$, $T_3$, $T_4$, 点 $A$, $T_1$, $T_2$, $T_3$, $T_4$, $B$ 顺次排列. 记 $\\theta_k = \\angle BT_k C$ ($k = 1, 2, 3, 4$). 求 $\\tan A \\cdot \\tan \\theta_1 + \\sum_{k=1}^3 \\tan \\theta_k \\cdot \\tan \\theta_{k+1} - \\tan \\theta_4 \\cdot \\tan B$ 的值.", "answer": "-5 - \\frac{10 \\sqrt{3}}{3}", "subject": "代数", "unique_id": "OlymMATH-HARD-19-ZH"}
|
| 21 |
+
{"problem": "给定一个 $2022 \\times 2022$ 的方格表.在方格表的每格中填入 $A$, $B$, $C$, $D$ 四种颜色之一. 若方格表中每一个 $2 \\times 2$ 的正方形都有四种颜色, 求有多少种不同的完美表格.", "answer": "12 \\times 2^{2022} - 24", "subject": "组合", "unique_id": "OlymMATH-HARD-20-ZH"}
|
| 22 |
+
{"problem": "设正整数 $a$, $b$, $c$, $d$ 满足 $a < b < c < d$, 且其中任意三个两两不同的数都可作为一个钝角三角形的三边长. 求 $d$ 的最小值.", "answer": "14", "subject": "数论", "unique_id": "OlymMATH-HARD-21-ZH"}
|
| 23 |
+
{"problem": "设函数 $f(x)=\\sin^4 \\omega x - \\sin \\omega x \\cdot \\cos \\omega x + \\cos^4 \\omega x (\\omega > 0)$. 若存在 $a, b \\in [0, \\pi]$, 使得 $f(a) + f(b) = \\frac{9}{4}$, 求 $\\omega$ 的最小值.", "answer": "\\frac{7}{12}", "subject": "代数", "unique_id": "OlymMATH-HARD-22-ZH"}
|
| 24 |
+
{"problem": "给定一个3行2025列的方格表, 一只蚂蚁从左下角的格出发, 每次可移动到有公共边的格. 若蚂蚁不重复地走遍方格表的每个格并且最终到达方格表的右上角, 求不同的走法有多少种.", "answer": "2^2023", "subject": "组合", "unique_id": "OlymMATH-HARD-23-ZH"}
|
| 25 |
+
{"problem": "已知双曲线 $x^2 - \\frac{y^2}{3} = 1$ 的左, 右焦点为 $F_1$, $F_2$, 过 $F_2$ 的直线与双曲线右支交于 $A$, $B$ 两点. 求 $\\triangle AF_1F_2$, $\\triangle BF_1F_2$ 的内切圆半径之和的取值范围.", "answer": "\\left[2, \\frac{4}{3}\\sqrt{3}\\right)", "subject": "几何", "unique_id": "OlymMATH-HARD-24-ZH"}
|
| 26 |
+
{"problem": "已知对任意的正整数 $n$, $\\tau(n)$ 表示 $n$ 的正约数的个数, $\\varphi(n)$ 表示与 $n$ 互质但是小于 $n$ 的正整数的个数. 若正整数 $n$ 满足, $n, \\tau(n), \\varphi(n)$ 中其中之一为其他两数的算术平均, 则称为好数. 求共有多少个好数.", "answer": "4", "subject": "数论", "unique_id": "OlymMATH-HARD-25-ZH"}
|
| 27 |
+
{"problem": "设 $a, b, c$ 是正有理数, 且 $a+1/b, b+1/c, c+1/a$ 均为整数. $a + b + c$ 的所有可能的值构成了集合 $S$, 求 $S$ 中所有元素之积.", "answer": "\\frac{21}{2}", "subject": "数论", "unique_id": "OlymMATH-HARD-26-ZH"}
|
| 28 |
+
{"problem": "已知椭圆 $C: x^{2} / a^{2}+y^{2} / b^{2}=1$ $(a>b>0)$ 的离心率 $e=4 / 5$, $P$ 为椭圆上异于长轴左右顶点 $A$, $B$ 的任意一点, $F_{1}$, $F_{2}$ 分别为椭圆的左右焦点, 且 $\\angle APB=2 \\alpha$, $\\angle F_{1} P F_{2}=2 \\beta$, 求 $\\tan \\beta \\cdot \\tan 2 \\alpha$ 的最小值.", "answer": "-\\frac{5}{2}", "subject": "几何", "unique_id": "OlymMATH-HARD-27-ZH"}
|
| 29 |
+
{"problem": "已知 $\\frac{by}{z}+\\frac{cz}{y}=a$, $\\frac{cz}{x}+\\frac{ax}{z}=b$, $\\frac{ax}{y}+\\frac{by}{x}=c$, 且 $abc=1$, 求 $a^{3}+b^{3}+c^{3}$ 的值.", "answer": "5", "subject": "代数", "unique_id": "OlymMATH-HARD-28-ZH"}
|
| 30 |
+
{"problem": "在矩形 $ABCD$ 中, $AB=2$, $AD=4$, 点 $E$ 在线段 $AD$ 上, 且 $AE=3$. 现分别沿 $BE$、$CE$ 将 $\\triangle ABE$、$\\triangle DCE$ 翻折, 使得点 $D$ 落在线段 $AE$ 上. 求二面角 $D-EC-B$ 的余弦值.", "answer": "\\frac{7}{8}", "subject": "几何", "unique_id": "OlymMATH-HARD-29-ZH"}
|
| 31 |
+
{"problem": "设 $f(x) = || \\cdots || x^{10} - 2^{2007}| - 2^{2006}| - \\cdots - 2^2| - 2| $. 求 $f(2007)$ 的值.", "answer": "1", "subject": "代数", "unique_id": "OlymMATH-HARD-30-ZH"}
|
| 32 |
+
{"problem": "用六种不同的颜色给正四面体 $ABCD$ 各棱染色, 每条棱只能染一种颜色且共顶点的棱不同色. 求所有棱均不同色的概率.", "answer": "\\frac{3}{17}", "subject": "组合", "unique_id": "OlymMATH-HARD-31-ZH"}
|
| 33 |
+
{"problem": "对��正方体 $ABCD-A_{1}B_{1}C_{1}D_{1}$, 将 $1, 2, \\cdots, 8$ 分别放在正方体的八个顶点上, 要求每一个面上的任意三个数之和均不小于 $10$. 求不同放法的个数.", "answer": "480", "subject": "组合", "unique_id": "OlymMATH-HARD-32-ZH"}
|
| 34 |
+
{"problem": "给定直线上 $2024$ 个点. 现在随机将所有点配成 $1012$ 对, 连成 $1012$ 条线段. 求找到一条线段, 与其余 $1011$ 条线段都相交的概率.", "answer": "\\frac{2}{3}", "subject": "组合", "unique_id": "OlymMATH-HARD-33-ZH"}
|
| 35 |
+
{"problem": "五名网球选手进行单循环赛（任意两人之间恰好比赛一场）, 且没有平局. 在这十场比赛中的每一场, 两名选手的获胜概率都是 $50\\%$, 且各场比赛的结果相互独立. 求对于整个比赛过程, 存在四位两两不同的选手 $P_1$、$P_2$、$P_3$、$P_4$, 满足 $P_1$ 胜 $P_2$、$P_2$ 胜 $P_3$、$P_3$ 胜 $P_4$、$P_4$ 胜 $P_1$ 的概率.", "answer": "\\frac{49}{64}", "subject": "组合", "unique_id": "OlymMATH-HARD-34-ZH"}
|
| 36 |
+
{"problem": "在椭圆 $\\Gamma: \\frac{x^{2}}{2019} + \\frac{y^{2}}{2018} = 1$ 中, $F$ 为左焦点. 过右焦点的直线 $l$ 与椭圆 $\\Gamma$ 的左准线及椭圆 $\\Gamma$ 依次交于点 $C, A, B$. 若 $\\angle FAB = 40^{\\circ}$, $\\angle FBA = 10^{\\circ}$, 求 $\\angle FCA$ 的值.", "answer": "15^{\\circ}", "subject": "几何", "unique_id": "OlymMATH-HARD-35-ZH"}
|
| 37 |
+
{"problem": "初始时, 饲养员在一个 $20 \\times 20$ 的方格左上角的格内放置一根质量为 $a$ 的胡萝卜和一只兔子. 接下来, 若兔子与胡萝卜在同一格中, 它将吃掉 $\\frac{1}{20}a$ 质量的胡萝卜, 同时, 饲养员将剩下的胡萝卜等可能地放置到某个格 (有可能是当前的格) 中; 否则, 兔子会移动到一个相邻的格 (两个格相邻当且仅当它们有一条公共边) 中, 且该移动会缩短它与胡萝卜之间的距离. 求兔子吃完整根胡萝卜时, 其移动次数的期望.", "answer": "\\frac{2318}{5}", "subject": "组合", "unique_id": "OlymMATH-HARD-36-ZH"}
|
| 38 |
+
{"problem": "有一个 $n \\times n$（$n \\geqslant 2$, $n \\in \\mathbb{Z}_{+}$）的方格表, 称其中每个 $1 \\times 1$ 的格为单位格. 在每个单位格里要么放上一枚棋子要么不放任何东西. 若放完棋子之后, 发现对于任意一个单位格, 必有与其相邻（即异于该单位格且与该单位格至少有一个公共顶点的单位格）的某个单位格中有一枚棋子, 则称放入棋子的总数目为 \"$n$-好数\". 对于每个 $n \\geqslant 2(n \\in \\mathbb{Z}_{+})$, 记 $f(n)$ 是所有的 $n$-好数的最小值. 若常数 $c$ 满足: 对一切 $n \\geqslant 2(n \\in \\mathbb{Z}_{+})$, 均有 $f(n) \\geqslant cn^{2}$ 成立, 求 $c$ 的最大值.", "answer": "\\frac{1}{7}", "subject": "组合", "unique_id": "OlymMATH-HARD-37-ZH"}
|
| 39 |
+
{"problem": "已知线段 $x+y=1$ ($x\\geqslant 0$, $y\\geqslant 0$) 上有 $2020$ 个点. 求最小的正整数 $k$, 使得对任意这样 $2020$ 个点, 总能存在一种方法将这 $2020$ 个点分为两组, 其中一组点的纵坐标之和不大于 $k$, 另一组的横坐标之和不大于 $k$ (这 $2020$ 个点可以重合).", "answer": "506", "subject": "组合", "unique_id": "OlymMATH-HARD-38-ZH"}
|
| 40 |
+
{"problem": "设集合 $A = \\{1, 2, \\cdots, 5\\}$, 由集合 $A$ 的全体子集组成的集合称为 $A$ 的幂集, 记作 $2^A$. 称映射 $f: 2^A \\rightarrow A$ 是 ``完美映射'', 若对任意的 $X, Y \\in 2^A$, 均有 $f(X \\cap Y) = \\min\\{f(X), f(Y)\\}$. 求完美映射的个数.", "answer": "4425", "subject": "组合", "unique_id": "OlymMATH-HARD-39-ZH"}
|
| 41 |
+
{"problem": "已知一个正多边形的每条边和对角线恰各染成 $2018$ 种颜色之一, 且所有边及对角线不全同色. 若正多边形中不存在两色三角形 (即三角形的三边恰被染成两种颜色), 则称该多边形的染色是 ``和谐的''. 求最大的正整数 $N$, 使得存在一个和谐的染色正 $N$ 边形.", "answer": "2017^2", "subject": "组合", "unique_id": "OlymMATH-HARD-40-ZH"}
|
| 42 |
+
{"problem": "定义函数 $f: \\mathbb{Z} \\rightarrow \\mathbb{Z}$, 对任意的 $x, y \\in \\mathbb{Z}$, 均有 $f(x^2 - 3y^2) + f(x^2 + y^2) = 2(x+y)f(x-y)$. 若 $n > 0$, 则 $f(n) > 0$, 且 $f(2015)f(2016)$ 为完全平方数. 求 $f(1) + f(2)$ 的最小值.", "answer": "246", "subject": "数论", "unique_id": "OlymMATH-HARD-41-ZH"}
|
| 43 |
+
{"problem": "设集合 $X=\\{1,2,\\cdots ,2022\\}$. 集族 $\\mathcal{F}$ 由 $X$ 的若干互不相同的子集组成, 满足: 对任意的 $F\\in \\mathcal{F}$, 均有 $|F| \\geqslant 800$; 且对任意的 $x\\in X$, 使得 $x\\in F$ 的集合 $F\\in \\mathcal{F}$ 至少有 $800$ 个. 求最小的正整数 $m$, 使得一定存在 $\\mathcal{F}$ 中的 $m$ 个集合的并集为 $X$.", "answer": "1222", "subject": "组合", "unique_id": "OlymMATH-HARD-42-ZH"}
|
| 44 |
+
{"problem": "已知在平面直角坐标系内, 点 $P(x, y)$ 的轨迹满足方程组\n$\\begin{cases}\na^{2}x-axy-y=0, \\\\\na^{2}y+axy+x=0,\n\\end{cases}$.\n点 $A(1,t)$ 与 $B(s,2)$ 关于原点中心对称. 求 $\\overrightarrow{AP} \\cdot \\overrightarrow{BP}$ 的最小值.", "answer": "6\\sqrt{3}-5", "subject": "几何", "unique_id": "OlymMATH-HARD-43-ZH"}
|
| 45 |
+
{"problem": "已知与圆柱 $OO'$ 底面成 $60^\\circ$ 角的截面 $\\alpha$ 截圆柱侧面所得的平面图形为椭圆, 球 $C_1$、$C_2$ 位于截面 $\\alpha$ 的两侧, 分别与圆柱的侧面、一个底面及截面 $\\alpha$ 相切. 设球 $C_1$、$C_2$、圆柱 $OO'$ 的体积分别为 $V_1$、$V_2$、$V$. 求 $\\frac{V_1+V_2}{V}$ 的值.", "answer": "\\frac{4}{9}", "subject": "几何", "unique_id": "OlymMATH-HARD-44-ZH"}
|
| 46 |
+
{"problem": "将 $8 \\times 8$ 方格表的 $64$ 个格编号为 $1, 2, \\cdots, 64$, 使得对一切 $1 \\le i \\le 63$, 编号为 $i$ 与 $i+1$ 的两个格均有一条公共边. 求主对角线上的八个格的编号之和的最大值.", "answer": "432", "subject": "组合", "unique_id": "OlymMATH-HARD-45-ZH"}
|
| 47 |
+
{"problem": "在 $101 \\times 101$ 的方格表中, 每个格填入集合 $\\{1, 2, \\cdots, 101^2\\}$ 中的一个数, 且集合中的每个数恰用一次. 方格表的左右边界视为同一条线, 上下边界也视为同一条线 (即为一个圆环面). 若无论怎么填, 均存在两个相邻格 (有公共边的) 其所填两数之差不小于 $M$, 求 $M$ 的最大值.", "answer": "201", "subject": "组合", "unique_id": "OlymMATH-HARD-46-ZH"}
|
| 48 |
+
{"problem": "有红色、绿色、白色、蓝色 (除颜色外均相同) 的棋子各两枚. 现从中选取七枚镶嵌在一个正六棱锥的顶点处, 每个顶点镶嵌一枚棋子. 求不同的镶嵌方法种数.", "answer": "424", "subject": "组合", "unique_id": "OlymMATH-HARD-47-ZH"}
|
| 49 |
+
{"problem": "抛物线 $y^2=2px$ 的内接 $\\mathrm{Rt}\\triangle ABC$, 斜边 $BC \\perp x$ 轴于点 $M$, 延长 $MA$ 到点 $D$, 使得以 $AD$ 为直径的 $\\odot N$ 与 $x$ 轴切于点 $E$, 联结 $BE$, 与抛物线交于点 $F$. 若四边形 $AFBC$ 的面积为 $8p^2$, 且 $A, F$ 点不重合, $p^2=\\sqrt{2}$, 求 $\\triangle ACD$ 的面积.", "answer": "\\frac{15\\sqrt{2}}{2}", "subject": "几何", "unique_id": "OlymMATH-HARD-48-ZH"}
|
| 50 |
+
{"problem": "给定整数集合 $A = \\{1, 2, \\cdots, 100\\}$. 设函数 $f: A \\rightarrow A$ 满足: （1）对任意的 $1 \\leqslant i \\leqslant 99$, 均有 $|f(i) - f(i+1)| \\leqslant 1$；（2）对任意的 $1 \\leqslant i \\leqslant 100$, 均有 $f(f(i)) = 100$. 求 $\\sum_{i=1}^{100} f(i)$ 的最小可能值.", "answer": "8350", "subject": "组合", "unique_id": "OlymMATH-HARD-49-ZH"}
|
| 51 |
+
{"problem": "设集合 $A = \\{0, 1, \\cdots, 2018\\}$. 若 $x, y, z \\in A$, 且 $x^2 + y^2 - z^2 = 2019^2$, 求 $x + y + z$ 的最大值与最小值之和.", "answer": "7962", "subject": "数论", "unique_id": "OlymMATH-HARD-50-ZH"}
|
| 52 |
+
{"problem": "若不等式 $2\\sin^2 C + \\sin A \\cdot \\sin B > k \\sin B \\cdot \\sin C$ 对任意的 $\\triangle ABC$ 均成立, 求实数 $k$ 的最大值.", "answer": "2\\sqrt{2}-1", "subject": "几何", "unique_id": "OlymMATH-HARD-51-ZH"}
|
| 53 |
+
{"problem": "求最小的实数 $a$, 使得对所有正整数 $n$ 和实数 $0 = x_0 < x_1 < \\cdots < x_n$ 满足\n$$a \\sum_{k=1}^{n} \\frac{\\sqrt{(k+1)^3}}{\\sqrt{x_k^2 - x_{k-1}^2}} \\geq \\sum_{k=1}^{n} \\frac{k^2 + 3k + 3}{x_k}.$$", "answer": "\\frac{16\\sqrt{2}}{9}", "subject": "代数", "unique_id": "OlymMATH-HARD-52-ZH"}
|
| 54 |
+
{"problem": "已知 $\\overrightarrow{OA}$、$\\overrightarrow{OB}$ 为非零的不共线的向量. 设 $\\overrightarrow{OC} = \\frac{1}{1+r} \\overrightarrow{OA} + \\frac{r}{1+r} \\overrightarrow{OB}$. 定义点集 $M = \\{K \\mid \\frac{\\overrightarrow{KA} \\cdot \\overrightarrow{KC}}{|\\overrightarrow{KA}|} = \\frac{\\overrightarrow{KB} \\cdot \\overrightarrow{KC}}{|\\overrightarrow{KB}|} \\}$. 当 $K_1$、$K_2 \\in M$ 时, 若对任意的 $r \\geq 2$, 不等式 $|\\overrightarrow{K_1 K_2}| \\leq c |\\overrightarrow{AB}|$ 恒成立, 求实数 $c$ 的最小值.", "answer": "\\frac{4}{3}", "subject": "几何", "unique_id": "OlymMATH-HARD-53-ZH"}
|
| 55 |
+
{"problem": "在平面区域 $M = \\{(x, y) | 0 \\le y \\le 2 - x, 0 \\le x \\le 2 \\}$ 内任取 $k$ 个点, 均能将这 $k$ 个点分成 $A$、 $B$ 两组, 使得 $A$ 组所有点的横坐标之和不大于 $6$, 而 $B$ 组所有点的纵坐标之和不大于 $6$. 求正整数 $k$ 的最大值.", "answer": "11", "subject": "组合", "unique_id": "OlymMATH-HARD-54-ZH"}
|
| 56 |
+
{"problem": "设方程 $4^{1-2x} + \\log_2 x = 0$ 的三个根为 $x_1, x_2, x_3$ ($x_1 < x_2 < x_3$). 求 $\\frac{\\log_2 x_2}{x_1 x_2 x_3}$ 的值.", "answer": "-32", "subject": "代数", "unique_id": "OlymMATH-HARD-55-ZH"}
|
| 57 |
+
{"problem": "求最大的 $C \\in \\mathbf{R}_{+}$, 使得从任何实数列 $a_{1}, a_{2}, \\ldots, a_{2022}$ 中均可以选取一部分项, 同时满足以下条件: (1) 任何连续三项不同时被取出；(2) 任何连续三项至少有一项被取出；(3) 被取出的各项之和的绝对值不小于 $C(|a_{1}| + |a_{2}| + \\cdots + |a_{2022}|)$.", "answer": "\\frac{1}{6}", "subject": "组合", "unique_id": "OlymMATH-HARD-56-ZH"}
|
| 58 |
+
{"problem": "已知对于任意的实数 $x$, 均有 $f(x) = 1 - a \\cos x - b \\sin x - A \\cos 2x - B \\sin 2x \\ge 0$. 求 $(A^2 + B^2)(a^2 + b^2)$ 的最大值.", "answer": "2", "subject": "代数", "unique_id": "OlymMATH-HARD-57-ZH"}
|
| 59 |
+
{"problem": "已知区间 $[0, 1]$ 内有若干个数（可以相同）, 其和不超过 $S$. 求 $S$ 的最大值, 使得总可以将这些数分成两组, 每组中的各数之和均不超过 $11$.", "answer": "\\frac{253}{12}", "subject": "组合", "unique_id": "OlymMATH-HARD-58-ZH"}
|
| 60 |
+
{"problem": "设 $\\begin{cases} \\sin \\alpha = \\sin(\\alpha + \\beta + \\gamma) + 1, \\\\ \\sin \\beta = 3\\sin(\\alpha + \\beta + \\gamma) + 2, \\\\ \\sin \\gamma = 5\\sin(\\alpha + \\beta + \\gamma) + 3. \\end{cases}$ 求 $\\sin \\alpha \\cdot \\sin \\beta \\cdot \\sin \\gamma$ 的所有可能值的乘积.", "answer": "\\frac{3}{512}", "subject": "代数", "unique_id": "OlymMATH-HARD-59-ZH"}
|
| 61 |
+
{"problem": "设 $n \\in \\mathbf{Z}_{+}$, $n \\geqslant 2$, $a_{1}, a_{2}, \\cdots, a_{n} \\in \\mathbf{R}$, 且 $a_{1} + a_{2} + \\cdots + a_{n} = 1$. 令 $b_{k} = \\sqrt{1 - \\frac{1}{16^{k}}} \\sqrt{a_{1}^{2} + a_{2}^{2} + \\cdots + a_{k}^{2}}$ $(1 \\leqslant k \\leqslant n)$. 求 $b_{1} + b_{2} + \\cdots + b_{n-1} + \\frac{4}{3} b_{n}$ 的最小值.", "answer": "\\frac{\\sqrt{15}}{3}", "subject": "代数", "unique_id": "OlymMATH-HARD-60-ZH"}
|
| 62 |
+
{"problem": "定义异面棱长相等的四面体为等腰四面体. 设等腰四面体 $DBMN$ 的外接球半径为 $R$, $\\triangle BMN$ 的外接圆半径为 $r$. 已知 $DB=MN=a$, $DM=BN=b$, $DN=BM=c$. 求 $\\frac{r}{R}$ 的取值范围.", "answer": "\\left[\\frac{2\\sqrt{2}}{3},1\\right)", "subject": "几何", "unique_id": "OlymMATH-HARD-61-ZH"}
|
| 63 |
+
{"problem": "已知 $n = \\overline{d_1 d_2 \\cdots d_{2017}}$, 其中 $d_i \\in \\{1, 3, 5, 7, 9\\}$ $(i = 1, 2, \\cdots, 2017)$, 且 $\\sum_{i=1}^{1009} d_i d_{i+1} \\equiv 1 \\pmod{4}$, $\\sum_{i=1010}^{2016} d_i d_{i+1} \\equiv 1 \\pmod{4}$. 求满足条件的 $n$ 的个数.", "answer": "6 \\times 5^{2015}", "subject": "数论", "unique_id": "OlymMATH-HARD-62-ZH"}
|
| 64 |
+
{"problem": "设 $x\\in (0,1)$, $\\frac{1}{x}\\notin \\mathbf{Z}$, $a_{n}=\\frac{nx}{(1-x)(1-2x)\\cdots (1-nx)}$, 其中, $n=1, 2, {\\ldots}$. 称 $x$ 为 ``好数'' 当且仅当 $x$ 使上述所定义的 $\\{a_{n}\\}$ 满足 $a_{1}+a_{2}+\\cdots +a_{10}> -1$ 且 $a_{1}a_{2}\\cdots a_{10}> 0$. 求全体好数在数轴上对应的所有区间的长度之和.", "answer": "\\frac{61}{210}", "subject": "代数", "unique_id": "OlymMATH-HARD-63-ZH"}
|
| 65 |
+
{"problem": "已知 $a>0$, $b\\in \\mathbf{R}$. 若 $|ax^3-bx^2+ax|\\leqslant bx^4+(a+2b)x^2+b$ 对任意 $x\\in [\\frac{1}{2},2]$ 都成立, 求 $\\frac{b}{a}$ 的取值范围.", "answer": "\\left[\\frac{\\sqrt{2}-1}{2},+\\infty \\right)", "subject": "代数", "unique_id": "OlymMATH-HARD-64-ZH"}
|
| 66 |
+
{"problem": "已知 $P$ 为正方体 $ABCD-A_1B_1C_1D_1$ 棱 $AB$ 上的一点, 满足直线 $A_1B$ 与平面 $B_1CP$ 所成角为 $60^\\circ$. 求二面角 $A_1-B_1P-C$ 的正切值.", "answer": "-\\sqrt{5}", "subject": "几何", "unique_id": "OlymMATH-HARD-65-ZH"}
|
| 67 |
+
{"problem": "设 $x \\in [0, 2\\pi]$, 求函数 $f(x) = \\sqrt{4\\cos^2x + 4\\sqrt{6}\\cos x + 6} + \\sqrt{4\\cos^2x - 8\\sqrt{6}\\cos x + 4\\sqrt{2}\\sin x + 22}$ 的最大值.", "answer": "2(\\sqrt{6}+\\sqrt{2})", "subject": "代数", "unique_id": "OlymMATH-HARD-66-ZH"}
|
| 68 |
+
{"problem": "求所有的素数 $p$, 使得 $p^2 - 87p + 729$ 为完全立方数.", "answer": "2011", "subject": "数论", "unique_id": "OlymMATH-HARD-67-ZH"}
|
| 69 |
+
{"problem": "对任意正实数 $a_1, a_2, \\cdots, a_5$, 若 $\\sum_{i=1}^{5}\\frac{a_i}{\\sqrt{a_i^2+2^{i-1}a_{i+1}a_{i+2}}}\\geqslant \\lambda$, 求 $\\lambda$ 的最大值.", "answer": "1", "subject": "代数", "unique_id": "OlymMATH-HARD-68-ZH"}
|
| 70 |
+
{"problem": "若实数 $x$, $y$ 满足条件 $x^2 - y^2 = 4$, 求 $\\frac{1}{x^2} - \\frac{y}{x}$ 的取值范围.", "answer": "\\left[-1, \\frac{5}{4}\\right]", "subject": "代数", "unique_id": "OlymMATH-HARD-69-ZH"}
|
| 71 |
+
{"problem": "求不定方程 $\\arctan \\frac{1}{m} + \\arctan \\frac{1}{n} + \\arctan \\frac{1}{p} = \\frac{\\pi}{4}$ 的正整数解的组数.", "answer": "15", "subject": "数论", "unique_id": "OlymMATH-HARD-70-ZH"}
|
| 72 |
+
{"problem": "在 $\\triangle ABC$ 中, 内切圆分别与边 $AB$, $AC$ 切于点 $E$, $F$, $AD$ 为 $\\triangle ABC$ 的边 $BC$ 上的高, 且 $AE+AF=AD$. 求 $\\sin \\frac{A}{2}$ 的取值范围.", "answer": "\\left[\\frac{3}{5},\\frac{\\sqrt{2}}{2}\\right)", "subject": "几何", "unique_id": "OlymMATH-HARD-71-ZH"}
|
| 73 |
+
{"problem": "已知函数 $f(x) = a(|\\sin x| + |\\cos x|) - 3\\sin 2x - 7$, 其中, $a$ 为实参数. 设数对 $(a, n)$ ($n \\in \\mathbf{Z}_{+}$), 使得函数 $y = f(x)$ 在区间 $(0, n\\pi)$ 内恰有 $2019$ 个零点, 所有这样的数对构成了集合 $S$, 求 $\\sum_{(a_0, n_0)\\in S} (a_0^2+n_0)$.", "answer": "4650", "subject": "代数", "unique_id": "OlymMATH-HARD-72-ZH"}
|
| 74 |
+
{"problem": "对于正四面体 $ABCD$, $M$、$N$ 分别为棱 $AB$、$AC$ 的中点, $P$、$Q$ 分别为面 $ACD$、面 $ABD$ 的中心. 求 $MP$ 与 $NQ$ 所成的角.", "answer": "\\arccos \\frac{7}{18}", "subject": "几何", "unique_id": "OlymMATH-HARD-73-ZH"}
|
| 75 |
+
{"problem": "空间有四个点 $A$、$B$、$C$、$D$, 满足 $AB = BC = CD$. 若 $\\angle ABC = \\angle BCD = \\angle CDA = 36^{\\circ}$, 求直线 $AC$ 与 $BD$ 所成角的大小的所有可能值之和.", "answer": "126^{\\circ}", "subject": "几何", "unique_id": "OlymMATH-HARD-74-ZH"}
|
| 76 |
+
{"problem": "对任意 $2016$ 个复数 $z_{1}, z_{2}, \\cdots, z_{2016}$, 均有 $\\sum_{k=1}^{2016} | z_{k} |^{2} \\geq \\lambda \\min_{1 \\leq k \\leq 2016} \\{ | z_{k+1} - z_{k} |^{2} \\}$, 其中, $z_{2017} = z_{1}$. 求 $\\lambda$ 的最大值.", "answer": "504", "subject": "代数", "unique_id": "OlymMATH-HARD-75-ZH"}
|
| 77 |
+
{"problem": "设 $\\triangle ABC$ 为椭圆 $\\Gamma: \\frac{x^2}{4} + y^2 = 1$ 的内接三角形, 其中 $A$ 为椭圆 $\\Gamma$ 与 $x$ 轴正半轴的交点, 直线 $AB$, $AC$ 斜率的乘积为 $-\\frac{1}{4}$, $G$ 为 $\\triangle ABC$ 的重心. 求 $|GA| + |GB| + |GC|$ 的取值范围.", "answer": "\\left[\\frac{2\\sqrt{13}+4}{3}, \\frac{16}{3}\\right)", "subject": "几何", "unique_id": "OlymMATH-HARD-76-ZH"}
|
| 78 |
+
{"problem": "已知 $O$ 为坐标原点, $F$ 为椭圆 $C: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$ 的右焦点, 过点 $F$ 的直线 $l$ 与椭圆 $C$ 交于 $A$, $B$ 两点, 椭圆上两点 $P$, $Q$ 满足 $\\overrightarrow{OP} + \\overrightarrow{OA} + \\overrightarrow{OB} = \\overrightarrow{OP} + \\overrightarrow{OQ} = \\mathbf{0}$ 且 $P$, $A$, $Q$, $B$ 四点共圆. 求椭圆 $C$ 的离心率.", "answer": "\\frac{\\sqrt{2}}{2}", "subject": "几何", "unique_id": "OlymMATH-HARD-77-ZH"}
|
| 79 |
+
{"problem": "已知 $P(x) = x^8 + 3x^7 + 6x^6 + 10x^5 + 15x^4 + 21x^3 + 28x^2 + 36x + 45$, $z = \\cos \\frac{2\\pi}{11} + i\\sin \\frac{2\\pi}{11}$. 求 $P(z)P(z^2)\\cdots P(z^{10})$ 的值.", "answer": "11^8 (5^{11} - 4^{11})", "subject": "代数", "unique_id": "OlymMATH-HARD-78-ZH"}
|
| 80 |
+
{"problem": "已知抛物线 $C_{1}: x^{2}=y$, 圆 $C_{2}: x^{2}+(y-4)^{2}=1$, $P$, $A$, $B$ 是抛物线 $C_{1}$ 上互异的三点, 且点 $P$ 异于原点. 已知直线 $PA$, $PB$ 都与圆 $C_{2}$ 相切, $|PA|=|PB|$. 求点 $P$ 的纵坐标.", "answer": "\\frac{23}{5}", "subject": "几何", "unique_id": "OlymMATH-HARD-79-ZH"}
|
| 81 |
+
{"problem": "设 $n = 108$, $n$ 项正数列 $x_1, x_2, \\dots, x_n$ 满足 $0 < x_1 \\leqslant x_2 \\leqslant \\cdots \\leqslant x_n$ 及 $x_1 + x_2 \\leqslant x_n$. 求 $\\left(x_1 + x_2 + \\cdots + x_n\\right)\\left(\\frac{1}{x_1} + \\frac{1}{x_2} + \\cdots + \\frac{1}{x_n}\\right)$ 的最小值.", "answer": "210\\sqrt{10}+11035", "subject": "代数", "unique_id": "OlymMATH-HARD-80-ZH"}
|
| 82 |
+
{"problem": "过正四面体 $ABCD$ 的顶点 $A$ 作一个形状为等腰三角形的截面, 且该截面与面 $BCD$ 所成角为 $75 ^{\\circ}$, 求这样的截面有多少个.", "answer": "18", "subject": "几何", "unique_id": "OlymMATH-HARD-81-ZH"}
|
| 83 |
+
{"problem": "设数列 $\\{a_n\\}$ 满足 $a_0=0$, $a_{n+1}=\\frac{8}{5}a_n+\\frac{6}{5}\\sqrt{4^n-a_n^2}\\left(n\\in\\mathbb{N}\\right)$. 求 $\\sum_{k=0}^{2005} a_k$ 的小数部分(用小数表示).", "answer": "0.84", "subject": "代数", "unique_id": "OlymMATH-HARD-82-ZH"}
|
| 84 |
+
{"problem": "平面直角坐标系中, $A(-1, 0), B(1, 0), C(0, 1)$. 假如存在参数 $a$, 使得直线 $l:y=ax+b$ 将 $\\triangle ABC$ 分割成面积相等的两部分, 求 $b$ 的取值范围.", "answer": "\\left[1-\\frac{1}{\\sqrt{2}}, \\frac{1}{2}\\right)", "subject": "几何", "unique_id": "OlymMATH-HARD-83-ZH"}
|
| 85 |
+
{"problem": "$a_1, a_2, \\cdots, a_{2016}$ 为 $1, 2, \\cdots, 2016$ 的排列, 且满足 $2017 | (a_1 a_2 + a_2 a_3 + \\cdots + a_{2015} a_{2016})$. 这样的排列有 $K$ 个, 求 $K$ 模 $4066272$ 的余数.", "answer": "2016", "subject": "数论", "unique_id": "OlymMATH-HARD-84-ZH"}
|
| 86 |
+
{"problem": "已知 $n$ 为不大于 2021 的正整数, 且满足 $\\left( \\left[ \\sqrt{n} \\right]^2 + 1 \\right) | \\left( n^2 + 1 \\right)$, 求 $n$ 的个数.", "answer": "47", "subject": "数论", "unique_id": "OlymMATH-HARD-85-ZH"}
|
| 87 |
+
{"problem": "求所有满足下述条件的有序正整数对 $(m,k)$ 的个数, 其中 $3 \\leqslant k \\leqslant 12$, $2 \\leqslant m \\leqslant 20$. 同时, 用 $m$ 进制循环小数表示 $\\frac{1}{k}$ 时, 循环节内各数字互异, 且删除小数部分前几位可以得到 $\\frac{2}{k}, \\cdots, \\frac{k-1}{k}$ 的 $m$ 进制循环小数表示.", "answer": "21", "subject": "数论", "unique_id": "OlymMATH-HARD-86-ZH"}
|
| 88 |
+
{"problem": "已知正整数 $n$ 满足: 在任意连续 $n$ 个正整数中, 一定可以挑出两个数 $a$、$b(a\\neq b)$, 且存在正整数 $k$, 使得 $210|(a^k-b^k)$. 求满足条件的 $n$ 的最小值.", "answer": "9", "subject": "数论", "unique_id": "OlymMATH-HARD-87-ZH"}
|
| 89 |
+
{"problem": "四面体 $ABCD$ 的顶点为 $A, B, C, D$. $M_1, \\cdots, M_6$ 为六条棱的中点. 在这 $10$ 个点中任取 $4$ 点, 求它们不共面的概率.", "answer": "\\frac{37}{70}", "subject": "组合", "unique_id": "OlymMATH-HARD-88-ZH"}
|
| 90 |
+
{"problem": "设 $a_1, a_2, a_3, a_4, a_5 \\in [0, 1]$, 求 $\\prod_{1 \\le i < j \\le 5} |a_i - a_j|$ 的最大值.", "answer": "\\frac{3\\sqrt{21}}{38416}", "subject": "代数", "unique_id": "OlymMATH-HARD-89-ZH"}
|
| 91 |
+
{"problem": "求 $\\sum_{k=0}^{1234}\\binom{2016\\times 1234}{2016k}$ 在模 $2017^2$ 下的余数（回答在 $[0, 2017^2)$ 上的那个值）.", "answer": "1581330", "subject": "数论", "unique_id": "OlymMATH-HARD-90-ZH"}
|
| 92 |
+
{"problem": "从左到右依次写出 $1$ 到 $10000$ 的全部正整数, 然后去掉那些能被 $5$ 或 $7$ 整除的数, 将剩下的数连成一排组成一个新数, 求新数被 $11$ 除的余数（回答在 $[0, 11)$ 上的那个值）.", "answer": "8", "subject": "数论", "unique_id": "OlymMATH-HARD-91-ZH"}
|
| 93 |
+
{"problem": "将形如 $0.a_1 a_2^{(k)} \\cdots a_n^{(k)} \\cdots$ 的十进制小数记为 $A(a_1, k)$, 其数字 $a_1$ 为 $1$ 至 $9$ 的任何一个自然数. 而当 $a_1$ 给定后, $a_2^{(k)}$ 等于乘积 $ka_1$ 的个位数字, $a_n^{(k)}$ 等于乘积 $ka_{n-1}^{(k)}$ 的个位数字, 其中 $n=3, 4, \\cdots$. 求 $\\sum_{k=1}^9 \\sum_{a_1=1}^9 A(a_1, k)$ 的值.", "answer": "\\frac{401}{9}", "subject": "数论", "unique_id": "OlymMATH-HARD-92-ZH"}
|
| 94 |
+
{"problem": "设集合 $S\\subset \\{1, 2, \\cdots, 100\\}$. 已知对 $S$ 中任意两个不同元素 $a, b$, 都存在正整数 $k$ 和 $S$ 中的两个不同元素 $c, d$（允许等于 $a$ 或 $b$）, 使 $c < d$, 且 $a + b = c^k d$. 求集合 $S$ 元素个数的最大值.", "answer": "48", "subject": "数论", "unique_id": "OlymMATH-HARD-93-ZH"}
|
| 95 |
+
{"problem": "设数列 $\\{a_n\\}$ 满足: (1) $a_1$ 是完全平方数 (2) 对任意正整数 $n$, $a_{n + 1}$ 是使 $2^na_1+2^{n-1}a_2+\\cdots+2a_n+a_{n+1}$ 为完全平方数的最小的正整数. 若存在正整数 $s$, 使得 $a_s = a_{s + 1} = t$, 求 $t$ 的最小可能值.", "answer": "31", "subject": "数论", "unique_id": "OlymMATH-HARD-94-ZH"}
|
| 96 |
+
{"problem": "设正整数 $x_1, x_2, \\cdots, x_{2005}$ 满足 $\\sum_{i = 1} ^ {2005} x_i = 432972$, 求 $\\sum_{i = 1} ^ {2005} \\gcd(x_i, x_{i+1}, x_{i+2})$ 的最大值, 其中下标按模 $2005$ 理解.", "answer": "432756", "subject": "数论", "unique_id": "OlymMATH-HARD-95-ZH"}
|
| 97 |
+
{"problem": "求最小的整数 $m\\ge 2017$, 使得对任意整数 $a_1, a_2, \\cdots, a_{m}$, 存在 $1 < i_1 < i_2 < \\cdots < i_{2017} \\le m$ 及 $\\varepsilon_1, \\varepsilon_2, \\cdots, \\varepsilon_{2017} \\in \\{-1, 1\\}$, 使得 $\\sum_{j=1}^{2017}\\varepsilon_j a_{i_j}$ 能被 $2017$ 整除.", "answer": "2027", "subject": "数论", "unique_id": "OlymMATH-HARD-96-ZH"}
|
| 98 |
+
{"problem": "称一个正整数是``好数'', 如果它可以表示为 $1893$ 个整数的两两之差的平方和. 求最小的非完全平方数的正整数 $a$, 使得任一好数的 $a$ 倍仍然是好数.", "answer": "43", "subject": "数论", "unique_id": "OlymMATH-HARD-97-ZH"}
|
| 99 |
+
{"problem": "设 $a_1, a_2, \\cdots, a_{20}$ 是 $20$ 个两两不同的正整数, 且集合 $\\{a_i + a_j | 1 \\le i, j \\le 20\\}$ 中有 $201$ 个不同的元素. 求集合 $\\{|a_i - a_j| | 1 \\le i, j \\le 20\\}$ 中不同元素个数的最小可能值.", "answer": "100", "subject": "组合", "unique_id": "OlymMATH-HARD-98-ZH"}
|
| 100 |
+
{"problem": "求不超过 $2009$ 的正整数 $t$ 的个数, 使得对所有自然数 $n$, 有 $\\sum_{k = 0}^n \\binom{2n+1}{2k+1} t^k$ 与 $2009$ 互质.", "answer": "980", "subject": "数论", "unique_id": "OlymMATH-HARD-99-ZH"}
|