{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}
{"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"}