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{"problem_id": "imo-bench_number_theory-013-part9", "problem": " For a positive integer $n \\geq 2$, let the set $C_n$ be the set of integers $2^n - 2^i$ for integers $i$ such that $0 \\leq i < n$. Find the largest positive integer that cannot be expressed as a sum of numbers in $C_n$ (where the same number can be used... |
{"problem_id": "imo-bench_number_theory-013-part10", "problem": " For a positive integer $n \\geq 2$, let the set $C_n$ be the set of integers $2^n - 2^i$ for integers $i$ such that $0 \\leq i < n$. Find the largest positive integer that cannot be expressed as a sum of numbers in $C_n$ (where the same number can be use... |
{"problem_id": "imo-bench_number_theory-013-part11", "problem": " For a positive integer $n \\geq 2$, let the set $C_n$ be the set of integers $2^n - 2^i$ for integers $i$ such that $0 \\leq i < n$. Find the largest positive integer that cannot be expressed as a sum of numbers in $C_n$ (where the same number can be use... |
{"problem_id": "imo-bench_number_theory-013-part12", "problem": " For a positive integer $n \\geq 2$, let the set $C_n$ be the set of integers $2^n - 2^i$ for integers $i$ such that $0 \\leq i < n$. Find the largest positive integer that cannot be expressed as a sum of numbers in $C_n$ (where the same number can be use... |
{"problem_id": "imo-bench_number_theory-013-part13", "problem": " For a positive integer $n \\geq 2$, let the set $C_n$ be the set of integers $2^n - 2^i$ for integers $i$ such that $0 \\leq i < n$. Find the largest positive integer that cannot be expressed as a sum of numbers in $C_n$ (where the same number can be use... |
{"problem_id": "imo-bench_number_theory-013-part14", "problem": " For a positive integer $n \\geq 2$, let the set $C_n$ be the set of integers $2^n - 2^i$ for integers $i$ such that $0 \\leq i < n$. Find the largest positive integer that cannot be expressed as a sum of numbers in $C_n$ (where the same number can be use... |
{"problem_id": "imo-bench_number_theory-013-part15", "problem": " For a positive integer $n \\geq 2$, let the set $C_n$ be the set of integers $2^n - 2^i$ for integers $i$ such that $0 \\leq i < n$. Find the largest positive integer that cannot be expressed as a sum of numbers in $C_n$ (where the same number can be use... |
{"problem_id": "imo-bench_number_theory-013-part16", "problem": " For a positive integer $n \\geq 2$, let the set $C_n$ be the set of integers $2^n - 2^i$ for integers $i$ such that $0 \\leq i < n$. Find the largest positive integer that cannot be expressed as a sum of numbers in $C_n$ (where the same number can be use... |
{"problem_id": "imo-bench_number_theory-013-part17", "problem": " For a positive integer $n \\geq 2$, let the set $C_n$ be the set of integers $2^n - 2^i$ for integers $i$ such that $0 \\leq i < n$. Find the largest positive integer that cannot be expressed as a sum of numbers in $C_n$ (where the same number can be use... |
{"problem_id": "imo-bench_number_theory-013-part18", "problem": " For a positive integer $n \\geq 2$, let the set $C_n$ be the set of integers $2^n - 2^i$ for integers $i$ such that $0 \\leq i < n$. Find the largest positive integer that cannot be expressed as a sum of numbers in $C_n$ (where the same number can be use... |
{"problem_id": "imo-bench_number_theory-013-part19", "problem": " For a positive integer $n \\geq 2$, let the set $C_n$ be the set of integers $2^n - 2^i$ for integers $i$ such that $0 \\leq i < n$. Find the largest positive integer that cannot be expressed as a sum of numbers in $C_n$ (where the same number can be use... |
{"problem_id": "imo-bench_geometry-069", "problem": "Circles $\\Omega_1$ and $\\Omega_2$ intersect at two points $M$ and $N$, and their common tangent line closer to $M$ intersects $\\Omega_1$ and $\\Omega_2$ at points $C$ and $D$, respectively. The line parallel to line $CD$ that passes through $M$ intersects $\\Omega... |
{"problem_id": "imo-bench_geometry-069-part1", "problem": "Circles $\\Omega_1$ and $\\Omega_2$ intersect at two points $M$ and $N$, and their common tangent line closer to $M$ intersects $\\Omega_1$ and $\\Omega_2$ at points $C$ and $D$, respectively. The line parallel to line $CD$ that passes through $M$ intersects $\... |
{"problem_id": "imo-bench_geometry-069-part2", "problem": "Circles $\\Omega_1$ and $\\Omega_2$ intersect at two points $M$ and $N$, and their common tangent line closer to $M$ intersects $\\Omega_1$ and $\\Omega_2$ at points $C$ and $D$, respectively. The line parallel to line $CD$ that passes through $M$ intersects $\... |
{"problem_id": "imo-bench_geometry-069-part3", "problem": "Circles $\\Omega_1$ and $\\Omega_2$ intersect at two points $M$ and $N$, and their common tangent line closer to $M$ intersects $\\Omega_1$ and $\\Omega_2$ at points $C$ and $D$, respectively. The line parallel to line $CD$ that passes through $M$ intersects $\... |
{"problem_id": "imo-bench_geometry-069-part4", "problem": "Circles $\\Omega_1$ and $\\Omega_2$ intersect at two points $M$ and $N$, and their common tangent line closer to $M$ intersects $\\Omega_1$ and $\\Omega_2$ at points $C$ and $D$, respectively. The line parallel to line $CD$ that passes through $M$ intersects $\... |
{"problem_id": "imo-bench_geometry-069-part5", "problem": "Circles $\\Omega_1$ and $\\Omega_2$ intersect at two points $M$ and $N$, and their common tangent line closer to $M$ intersects $\\Omega_1$ and $\\Omega_2$ at points $C$ and $D$, respectively. The line parallel to line $CD$ that passes through $M$ intersects $\... |
{"problem_id": "imo-bench_geometry-069-part6", "problem": "Circles $\\Omega_1$ and $\\Omega_2$ intersect at two points $M$ and $N$, and their common tangent line closer to $M$ intersects $\\Omega_1$ and $\\Omega_2$ at points $C$ and $D$, respectively. The line parallel to line $CD$ that passes through $M$ intersects $\... |
{"problem_id": "imo-bench_geometry-069-part7", "problem": "Circles $\\Omega_1$ and $\\Omega_2$ intersect at two points $M$ and $N$, and their common tangent line closer to $M$ intersects $\\Omega_1$ and $\\Omega_2$ at points $C$ and $D$, respectively. The line parallel to line $CD$ that passes through $M$ intersects $\... |
{"problem_id": "imo-bench_geometry-069-part8", "problem": "Circles $\\Omega_1$ and $\\Omega_2$ intersect at two points $M$ and $N$, and their common tangent line closer to $M$ intersects $\\Omega_1$ and $\\Omega_2$ at points $C$ and $D$, respectively. The line parallel to line $CD$ that passes through $M$ intersects $\... |
{"problem_id": "imo-bench_geometry-069-part9", "problem": "Circles $\\Omega_1$ and $\\Omega_2$ intersect at two points $M$ and $N$, and their common tangent line closer to $M$ intersects $\\Omega_1$ and $\\Omega_2$ at points $C$ and $D$, respectively. The line parallel to line $CD$ that passes through $M$ intersects $\... |
{"problem_id": "imo-bench_geometry-069-part10", "problem": "Circles $\\Omega_1$ and $\\Omega_2$ intersect at two points $M$ and $N$, and their common tangent line closer to $M$ intersects $\\Omega_1$ and $\\Omega_2$ at points $C$ and $D$, respectively. The line parallel to line $CD$ that passes through $M$ intersects $... |
{"problem_id": "imo-bench_combinatorics-085", "problem": "Evan fills the fields of an $78 \\times 78$ board with numbers from 1 to $6084$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such that:\n\... |
{"problem_id": "imo-bench_combinatorics-085-part1", "problem": "Evan fills the fields of an $78 \\times 78$ board with numbers from 1 to $6084$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such th... |
{"problem_id": "imo-bench_combinatorics-085-part2", "problem": "Evan fills the fields of an $78 \\times 78$ board with numbers from 1 to $6084$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such th... |
{"problem_id": "imo-bench_combinatorics-085-part3", "problem": "Evan fills the fields of an $78 \\times 78$ board with numbers from 1 to $6084$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such th... |
{"problem_id": "imo-bench_combinatorics-085-part4", "problem": "Evan fills the fields of an $78 \\times 78$ board with numbers from 1 to $6084$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such th... |
{"problem_id": "imo-bench_combinatorics-085-part5", "problem": "Evan fills the fields of an $78 \\times 78$ board with numbers from 1 to $6084$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such th... |
{"problem_id": "imo-bench_combinatorics-085-part6", "problem": "Evan fills the fields of an $78 \\times 78$ board with numbers from 1 to $6084$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such th... |
{"problem_id": "imo-bench_combinatorics-085-part7", "problem": "Evan fills the fields of an $78 \\times 78$ board with numbers from 1 to $6084$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such th... |
{"problem_id": "imo-bench_combinatorics-085-part8", "problem": "Evan fills the fields of an $78 \\times 78$ board with numbers from 1 to $6084$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such th... |
{"problem_id": "imo-bench_combinatorics-085-part9", "problem": "Evan fills the fields of an $78 \\times 78$ board with numbers from 1 to $6084$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such th... |
{"problem_id": "imo-bench_combinatorics-085-part10", "problem": "Evan fills the fields of an $78 \\times 78$ board with numbers from 1 to $6084$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such t... |
{"problem_id": "imo-bench_combinatorics-085-part11", "problem": "Evan fills the fields of an $78 \\times 78$ board with numbers from 1 to $6084$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such t... |
{"problem_id": "imo-bench_combinatorics-085-part12", "problem": "Evan fills the fields of an $78 \\times 78$ board with numbers from 1 to $6084$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such t... |
{"problem_id": "imo-bench_combinatorics-085-part13", "problem": "Evan fills the fields of an $78 \\times 78$ board with numbers from 1 to $6084$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such t... |
{"problem_id": "imo-bench_combinatorics-085-part14", "problem": "Evan fills the fields of an $78 \\times 78$ board with numbers from 1 to $6084$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such t... |
{"problem_id": "imo-bench_combinatorics-085-part15", "problem": "Evan fills the fields of an $78 \\times 78$ board with numbers from 1 to $6084$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such t... |
{"problem_id": "imo-bench_combinatorics-085-part16", "problem": "Evan fills the fields of an $78 \\times 78$ board with numbers from 1 to $6084$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such t... |
{"problem_id": "imo-bench_combinatorics-085-part17", "problem": "Evan fills the fields of an $78 \\times 78$ board with numbers from 1 to $6084$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such t... |
{"problem_id": "imo-bench_combinatorics-085-part18", "problem": "Evan fills the fields of an $78 \\times 78$ board with numbers from 1 to $6084$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such t... |
{"problem_id": "imo-bench_combinatorics-085-part19", "problem": "Evan fills the fields of an $78 \\times 78$ board with numbers from 1 to $6084$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such t... |
{"problem_id": "imo-bench_combinatorics-085-part20", "problem": "Evan fills the fields of an $78 \\times 78$ board with numbers from 1 to $6084$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such t... |
{"problem_id": "imo-bench_number_theory-076", "problem": "(a) Show that there exists a degree 3 monic polynomial $P(x)$ with integer coefficients such that for an integer $n$, $P(n)$ is a square of an integer if and only if $n = 2024$ or $2025$.\n\n(b) For such a polynomial $P(x)$ in (a), find all possible values for $... |
{"problem_id": "imo-bench_number_theory-076-part1", "problem": "(a) Show that there exists a degree 3 monic polynomial $P(x)$ with integer coefficients such that for an integer $n$, $P(n)$ is a square of an integer if and only if $n = 2024$ or $2025$.\n\n(b) For such a polynomial $P(x)$ in (a), find all possible values... |
{"problem_id": "imo-bench_number_theory-076-part2", "problem": "(a) Show that there exists a degree 3 monic polynomial $P(x)$ with integer coefficients such that for an integer $n$, $P(n)$ is a square of an integer if and only if $n = 2024$ or $2025$.\n\n(b) For such a polynomial $P(x)$ in (a), find all possible values... |
{"problem_id": "imo-bench_number_theory-076-part3", "problem": "(a) Show that there exists a degree 3 monic polynomial $P(x)$ with integer coefficients such that for an integer $n$, $P(n)$ is a square of an integer if and only if $n = 2024$ or $2025$.\n\n(b) For such a polynomial $P(x)$ in (a), find all possible values... |
{"problem_id": "imo-bench_number_theory-076-part4", "problem": "(a) Show that there exists a degree 3 monic polynomial $P(x)$ with integer coefficients such that for an integer $n$, $P(n)$ is a square of an integer if and only if $n = 2024$ or $2025$.\n\n(b) For such a polynomial $P(x)$ in (a), find all possible values... |
{"problem_id": "imo-bench_number_theory-076-part5", "problem": "(a) Show that there exists a degree 3 monic polynomial $P(x)$ with integer coefficients such that for an integer $n$, $P(n)$ is a square of an integer if and only if $n = 2024$ or $2025$.\n\n(b) For such a polynomial $P(x)$ in (a), find all possible values... |
{"problem_id": "imo-bench_number_theory-076-part6", "problem": "(a) Show that there exists a degree 3 monic polynomial $P(x)$ with integer coefficients such that for an integer $n$, $P(n)$ is a square of an integer if and only if $n = 2024$ or $2025$.\n\n(b) For such a polynomial $P(x)$ in (a), find all possible values... |
{"problem_id": "imo-bench_number_theory-076-part7", "problem": "(a) Show that there exists a degree 3 monic polynomial $P(x)$ with integer coefficients such that for an integer $n$, $P(n)$ is a square of an integer if and only if $n = 2024$ or $2025$.\n\n(b) For such a polynomial $P(x)$ in (a), find all possible values... |
{"problem_id": "imo-bench_number_theory-076-part8", "problem": "(a) Show that there exists a degree 3 monic polynomial $P(x)$ with integer coefficients such that for an integer $n$, $P(n)$ is a square of an integer if and only if $n = 2024$ or $2025$.\n\n(b) For such a polynomial $P(x)$ in (a), find all possible values... |
{"problem_id": "imo-bench_number_theory-076-part9", "problem": "(a) Show that there exists a degree 3 monic polynomial $P(x)$ with integer coefficients such that for an integer $n$, $P(n)$ is a square of an integer if and only if $n = 2024$ or $2025$.\n\n(b) For such a polynomial $P(x)$ in (a), find all possible values... |
{"problem_id": "imo-bench_number_theory-076-part10", "problem": "(a) Show that there exists a degree 3 monic polynomial $P(x)$ with integer coefficients such that for an integer $n$, $P(n)$ is a square of an integer if and only if $n = 2024$ or $2025$.\n\n(b) For such a polynomial $P(x)$ in (a), find all possible value... |
{"problem_id": "imo-bench_number_theory-028", "problem": "Given a positive integer $n$, there exists an integer $a$ such that the sequence $\\{a_k\\}$ defined by $a_0 = a$ and $a_k = \\frac{a_{k-1}}{k} + k^{n-1}$ consists only of integers. Find the possible values of the remainder when $n$ is divided by 3.\n", "origina... |
{"problem_id": "imo-bench_number_theory-028-part1", "problem": "Given a positive integer $n$, there exists an integer $a$ such that the sequence $\\{a_k\\}$ defined by $a_0 = a$ and $a_k = \\frac{a_{k-1}}{k} + k^{n-1}$ consists only of integers. Find the possible values of the remainder when $n$ is divided by 3.\n", "o... |
{"problem_id": "imo-bench_number_theory-028-part2", "problem": "Given a positive integer $n$, there exists an integer $a$ such that the sequence $\\{a_k\\}$ defined by $a_0 = a$ and $a_k = \\frac{a_{k-1}}{k} + k^{n-1}$ consists only of integers. Find the possible values of the remainder when $n$ is divided by 3.\n", "o... |
{"problem_id": "imo-bench_number_theory-028-part3", "problem": "Given a positive integer $n$, there exists an integer $a$ such that the sequence $\\{a_k\\}$ defined by $a_0 = a$ and $a_k = \\frac{a_{k-1}}{k} + k^{n-1}$ consists only of integers. Find the possible values of the remainder when $n$ is divided by 3.\n", "o... |
{"problem_id": "imo-bench_number_theory-028-part4", "problem": "Given a positive integer $n$, there exists an integer $a$ such that the sequence $\\{a_k\\}$ defined by $a_0 = a$ and $a_k = \\frac{a_{k-1}}{k} + k^{n-1}$ consists only of integers. Find the possible values of the remainder when $n$ is divided by 3.\n", "o... |
{"problem_id": "imo-bench_number_theory-028-part5", "problem": "Given a positive integer $n$, there exists an integer $a$ such that the sequence $\\{a_k\\}$ defined by $a_0 = a$ and $a_k = \\frac{a_{k-1}}{k} + k^{n-1}$ consists only of integers. Find the possible values of the remainder when $n$ is divided by 3.\n", "o... |
{"problem_id": "imo-bench_number_theory-028-part6", "problem": "Given a positive integer $n$, there exists an integer $a$ such that the sequence $\\{a_k\\}$ defined by $a_0 = a$ and $a_k = \\frac{a_{k-1}}{k} + k^{n-1}$ consists only of integers. Find the possible values of the remainder when $n$ is divided by 3.\n", "o... |
{"problem_id": "imo-bench_number_theory-028-part7", "problem": "Given a positive integer $n$, there exists an integer $a$ such that the sequence $\\{a_k\\}$ defined by $a_0 = a$ and $a_k = \\frac{a_{k-1}}{k} + k^{n-1}$ consists only of integers. Find the possible values of the remainder when $n$ is divided by 3.\n", "o... |
{"problem_id": "imo-bench_number_theory-028-part8", "problem": "Given a positive integer $n$, there exists an integer $a$ such that the sequence $\\{a_k\\}$ defined by $a_0 = a$ and $a_k = \\frac{a_{k-1}}{k} + k^{n-1}$ consists only of integers. Find the possible values of the remainder when $n$ is divided by 3.\n", "o... |
{"problem_id": "imo-bench_number_theory-028-part9", "problem": "Given a positive integer $n$, there exists an integer $a$ such that the sequence $\\{a_k\\}$ defined by $a_0 = a$ and $a_k = \\frac{a_{k-1}}{k} + k^{n-1}$ consists only of integers. Find the possible values of the remainder when $n$ is divided by 3.\n", "o... |
{"problem_id": "imo-bench_number_theory-028-part10", "problem": "Given a positive integer $n$, there exists an integer $a$ such that the sequence $\\{a_k\\}$ defined by $a_0 = a$ and $a_k = \\frac{a_{k-1}}{k} + k^{n-1}$ consists only of integers. Find the possible values of the remainder when $n$ is divided by 3.\n", "... |
{"problem_id": "imo-bench_number_theory-028-part11", "problem": "Given a positive integer $n$, there exists an integer $a$ such that the sequence $\\{a_k\\}$ defined by $a_0 = a$ and $a_k = \\frac{a_{k-1}}{k} + k^{n-1}$ consists only of integers. Find the possible values of the remainder when $n$ is divided by 3.\n", "... |
{"problem_id": "imo-bench_number_theory-028-part12", "problem": "Given a positive integer $n$, there exists an integer $a$ such that the sequence $\\{a_k\\}$ defined by $a_0 = a$ and $a_k = \\frac{a_{k-1}}{k} + k^{n-1}$ consists only of integers. Find the possible values of the remainder when $n$ is divided by 3.\n", "... |
{"problem_id": "imo-bench_number_theory-028-part13", "problem": "Given a positive integer $n$, there exists an integer $a$ such that the sequence $\\{a_k\\}$ defined by $a_0 = a$ and $a_k = \\frac{a_{k-1}}{k} + k^{n-1}$ consists only of integers. Find the possible values of the remainder when $n$ is divided by 3.\n", "... |
{"problem_id": "imo-bench_number_theory-028-part14", "problem": "Given a positive integer $n$, there exists an integer $a$ such that the sequence $\\{a_k\\}$ defined by $a_0 = a$ and $a_k = \\frac{a_{k-1}}{k} + k^{n-1}$ consists only of integers. Find the possible values of the remainder when $n$ is divided by 3.\n", "... |
{"problem_id": "imo-bench_number_theory-028-part15", "problem": "Given a positive integer $n$, there exists an integer $a$ such that the sequence $\\{a_k\\}$ defined by $a_0 = a$ and $a_k = \\frac{a_{k-1}}{k} + k^{n-1}$ consists only of integers. Find the possible values of the remainder when $n$ is divided by 3.\n", "... |
{"problem_id": "imo-bench_number_theory-028-part16", "problem": "Given a positive integer $n$, there exists an integer $a$ such that the sequence $\\{a_k\\}$ defined by $a_0 = a$ and $a_k = \\frac{a_{k-1}}{k} + k^{n-1}$ consists only of integers. Find the possible values of the remainder when $n$ is divided by 3.\n", "... |
{"problem_id": "imo-bench_number_theory-028-part17", "problem": "Given a positive integer $n$, there exists an integer $a$ such that the sequence $\\{a_k\\}$ defined by $a_0 = a$ and $a_k = \\frac{a_{k-1}}{k} + k^{n-1}$ consists only of integers. Find the possible values of the remainder when $n$ is divided by 3.\n", "... |
{"problem_id": "imo-bench_number_theory-028-part18", "problem": "Given a positive integer $n$, there exists an integer $a$ such that the sequence $\\{a_k\\}$ defined by $a_0 = a$ and $a_k = \\frac{a_{k-1}}{k} + k^{n-1}$ consists only of integers. Find the possible values of the remainder when $n$ is divided by 3.\n", "... |
{"problem_id": "imo-bench_geometry-078", "problem": "Let $XYZ$ be a triangle. How many points $Q$ on segment $YZ$ are there satisfying the following property: If $A$ and $B$ are the intersections of line $XQ$ with the common external tangent lines of the circumcircles of triangles $XQY$ and $XQZ$, then\\[\\left(\\frac{... |
{"problem_id": "imo-bench_geometry-078-part1", "problem": "Let $XYZ$ be a triangle. How many points $Q$ on segment $YZ$ are there satisfying the following property: If $A$ and $B$ are the intersections of line $XQ$ with the common external tangent lines of the circumcircles of triangles $XQY$ and $XQZ$, then\\[\\left(\... |
{"problem_id": "imo-bench_geometry-078-part2", "problem": "Let $XYZ$ be a triangle. How many points $Q$ on segment $YZ$ are there satisfying the following property: If $A$ and $B$ are the intersections of line $XQ$ with the common external tangent lines of the circumcircles of triangles $XQY$ and $XQZ$, then\\[\\left(\... |
{"problem_id": "imo-bench_geometry-078-part3", "problem": "Let $XYZ$ be a triangle. How many points $Q$ on segment $YZ$ are there satisfying the following property: If $A$ and $B$ are the intersections of line $XQ$ with the common external tangent lines of the circumcircles of triangles $XQY$ and $XQZ$, then\\[\\left(\... |
{"problem_id": "imo-bench_geometry-078-part4", "problem": "Let $XYZ$ be a triangle. How many points $Q$ on segment $YZ$ are there satisfying the following property: If $A$ and $B$ are the intersections of line $XQ$ with the common external tangent lines of the circumcircles of triangles $XQY$ and $XQZ$, then\\[\\left(\... |
{"problem_id": "imo-bench_geometry-078-part5", "problem": "Let $XYZ$ be a triangle. How many points $Q$ on segment $YZ$ are there satisfying the following property: If $A$ and $B$ are the intersections of line $XQ$ with the common external tangent lines of the circumcircles of triangles $XQY$ and $XQZ$, then\\[\\left(\... |
{"problem_id": "imo-bench_geometry-078-part6", "problem": "Let $XYZ$ be a triangle. How many points $Q$ on segment $YZ$ are there satisfying the following property: If $A$ and $B$ are the intersections of line $XQ$ with the common external tangent lines of the circumcircles of triangles $XQY$ and $XQZ$, then\\[\\left(\... |
{"problem_id": "imo-bench_geometry-078-part7", "problem": "Let $XYZ$ be a triangle. How many points $Q$ on segment $YZ$ are there satisfying the following property: If $A$ and $B$ are the intersections of line $XQ$ with the common external tangent lines of the circumcircles of triangles $XQY$ and $XQZ$, then\\[\\left(\... |
{"problem_id": "imo-bench_geometry-078-part8", "problem": "Let $XYZ$ be a triangle. How many points $Q$ on segment $YZ$ are there satisfying the following property: If $A$ and $B$ are the intersections of line $XQ$ with the common external tangent lines of the circumcircles of triangles $XQY$ and $XQZ$, then\\[\\left(\... |
{"problem_id": "imo-bench_geometry-078-part9", "problem": "Let $XYZ$ be a triangle. How many points $Q$ on segment $YZ$ are there satisfying the following property: If $A$ and $B$ are the intersections of line $XQ$ with the common external tangent lines of the circumcircles of triangles $XQY$ and $XQZ$, then\\[\\left(\... |
{"problem_id": "imo-bench_geometry-078-part10", "problem": "Let $XYZ$ be a triangle. How many points $Q$ on segment $YZ$ are there satisfying the following property: If $A$ and $B$ are the intersections of line $XQ$ with the common external tangent lines of the circumcircles of triangles $XQY$ and $XQZ$, then\\[\\left(... |
{"problem_id": "imo-bench_geometry-078-part11", "problem": "Let $XYZ$ be a triangle. How many points $Q$ on segment $YZ$ are there satisfying the following property: If $A$ and $B$ are the intersections of line $XQ$ with the common external tangent lines of the circumcircles of triangles $XQY$ and $XQZ$, then\\[\\left(... |
{"problem_id": "imo-bench_geometry-078-part12", "problem": "Let $XYZ$ be a triangle. How many points $Q$ on segment $YZ$ are there satisfying the following property: If $A$ and $B$ are the intersections of line $XQ$ with the common external tangent lines of the circumcircles of triangles $XQY$ and $XQZ$, then\\[\\left(... |
{"problem_id": "imo-bench_geometry-078-part13", "problem": "Let $XYZ$ be a triangle. How many points $Q$ on segment $YZ$ are there satisfying the following property: If $A$ and $B$ are the intersections of line $XQ$ with the common external tangent lines of the circumcircles of triangles $XQY$ and $XQZ$, then\\[\\left(... |
{"problem_id": "imo-bench_geometry-078-part14", "problem": "Let $XYZ$ be a triangle. How many points $Q$ on segment $YZ$ are there satisfying the following property: If $A$ and $B$ are the intersections of line $XQ$ with the common external tangent lines of the circumcircles of triangles $XQY$ and $XQZ$, then\\[\\left(... |
{"problem_id": "imo-bench_geometry-078-part15", "problem": "Let $XYZ$ be a triangle. How many points $Q$ on segment $YZ$ are there satisfying the following property: If $A$ and $B$ are the intersections of line $XQ$ with the common external tangent lines of the circumcircles of triangles $XQY$ and $XQZ$, then\\[\\left(... |
{"problem_id": "imo-bench_geometry-078-part16", "problem": "Let $XYZ$ be a triangle. How many points $Q$ on segment $YZ$ are there satisfying the following property: If $A$ and $B$ are the intersections of line $XQ$ with the common external tangent lines of the circumcircles of triangles $XQY$ and $XQZ$, then\\[\\left(... |
{"problem_id": "imo-bench_geometry-078-part17", "problem": "Let $XYZ$ be a triangle. How many points $Q$ on segment $YZ$ are there satisfying the following property: If $A$ and $B$ are the intersections of line $XQ$ with the common external tangent lines of the circumcircles of triangles $XQY$ and $XQZ$, then\\[\\left(... |
{"problem_id": "imo-bench_geometry-078-part18", "problem": "Let $XYZ$ be a triangle. How many points $Q$ on segment $YZ$ are there satisfying the following property: If $A$ and $B$ are the intersections of line $XQ$ with the common external tangent lines of the circumcircles of triangles $XQY$ and $XQZ$, then\\[\\left(... |
{"problem_id": "imo-bench_geometry-078-part19", "problem": "Let $XYZ$ be a triangle. How many points $Q$ on segment $YZ$ are there satisfying the following property: If $A$ and $B$ are the intersections of line $XQ$ with the common external tangent lines of the circumcircles of triangles $XQY$ and $XQZ$, then\\[\\left(... |
{"problem_id": "imo-bench_geometry-078-part20", "problem": "Let $XYZ$ be a triangle. How many points $Q$ on segment $YZ$ are there satisfying the following property: If $A$ and $B$ are the intersections of line $XQ$ with the common external tangent lines of the circumcircles of triangles $XQY$ and $XQZ$, then\\[\\left(... |
{"problem_id": "imo-bench_geometry-078-part21", "problem": "Let $XYZ$ be a triangle. How many points $Q$ on segment $YZ$ are there satisfying the following property: If $A$ and $B$ are the intersections of line $XQ$ with the common external tangent lines of the circumcircles of triangles $XQY$ and $XQZ$, then\\[\\left(... |
{"problem_id": "imo-bench_geometry-078-part22", "problem": "Let $XYZ$ be a triangle. How many points $Q$ on segment $YZ$ are there satisfying the following property: If $A$ and $B$ are the intersections of line $XQ$ with the common external tangent lines of the circumcircles of triangles $XQY$ and $XQZ$, then\\[\\left(... |
{"problem_id": "imo-bench_geometry-078-part23", "problem": "Let $XYZ$ be a triangle. How many points $Q$ on segment $YZ$ are there satisfying the following property: If $A$ and $B$ are the intersections of line $XQ$ with the common external tangent lines of the circumcircles of triangles $XQY$ and $XQZ$, then\\[\\left(... |
{"problem_id": "imo-bench_algebra-020", "problem": "Find all $P:\\mathbb{R}\\rightarrow \\mathbb{R}$ such that $P$ is not identically zero and there exists $Q:\\mathbb{R}\\rightarrow \\mathbb{R}$ satisfying\n\n\\[\nQ(P(a))-P(b)=(b+a)Q(2a-2b)\n\\]\n\nfor all real numbers $a,b$.\n", "original_problem": "Find all $P:\\mat... |
{"problem_id": "imo-bench_algebra-020-part1", "problem": "Find all $P:\\mathbb{R}\\rightarrow \\mathbb{R}$ such that $P$ is not identically zero and there exists $Q:\\mathbb{R}\\rightarrow \\mathbb{R}$ satisfying\n\n\\[\nQ(P(a))-P(b)=(b+a)Q(2a-2b)\n\\]\n\nfor all real numbers $a,b$.\n", "original_problem": "Find all $P... |
{"problem_id": "imo-bench_algebra-020-part2", "problem": "Find all $P:\\mathbb{R}\\rightarrow \\mathbb{R}$ such that $P$ is not identically zero and there exists $Q:\\mathbb{R}\\rightarrow \\mathbb{R}$ satisfying\n\n\\[\nQ(P(a))-P(b)=(b+a)Q(2a-2b)\n\\]\n\nfor all real numbers $a,b$.\n", "original_problem": "Find all $P... |
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