{"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"}
{"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"}
{"problem": "Calculate the value of $\\sqrt{9+8\\cos 20^{\\circ }}-\\sec 20^{\\circ }$.", "answer": "3", "subject": "Algebra", "unique_id": "OlymMATH-EASY-2-EN"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}