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"# Problem 1 \n\nLet $a, b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove that\n\n$$\n(...TRUNCATED) | "## Solution\n\nReplacing $1-a, 1-b, 1-c$ with $b+c, c+a, a+b$ respectively on the right hand side, (...TRUNCATED) | =b==\frac{1}{3} | Inequalities | proof | Yes | Yes | olympiads | false | 2 | [
false,
false
] | 0 | ["<think>\nAlright, let me try to tackle this problem. So, we have three positive real numbers a, b,(...TRUNCATED) | ["<think>\nAlright, let me try to tackle this problem. So, we have three positive real numbers a, b,(...TRUNCATED) | "<think>\nAlright, let me try to tackle this problem. So, we have three positive real numbers a, b, (...TRUNCATED) | [{"from":"human","value":"# Problem 1 \n\nLet $a, b$ and $c$ be positive real numbers such that $a+b(...TRUNCATED) |
"# Problem 1 \n\nLet $a, b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove that\n\n$$\n(...TRUNCATED) | "## Solution\n\nReplacing $1-a, 1-b, 1-c$ with $b+c, c+a, a+b$ respectively on the right hand side, (...TRUNCATED) | =b==\frac{1}{3} | Inequalities | proof | Yes | Yes | olympiads | false | 2 | [
false,
false
] | 0 | ["<think>\nAlright, let me try to tackle this problem. So, we have three positive real numbers a, b,(...TRUNCATED) | ["<think>\nAlright, let me try to tackle this problem. So, we have three positive real numbers a, b,(...TRUNCATED) | "<think>\nOkay, so I need to prove this inequality:\n\n$$\n\\frac{a}{b}+\\frac{b}{a}+\\frac{b}{c}+\\(...TRUNCATED) | [{"from":"human","value":"# Problem 1 \n\nLet $a, b$ and $c$ be positive real numbers such that $a+b(...TRUNCATED) |
"\nProblem 4. Let $M$ be a subset of the set of 2021 integers $\\{1,2,3, \\ldots, 2021\\}$ such that(...TRUNCATED) | "\nSolution. The set $M=\\{1016,1017, \\ldots, 2021\\}$ has 1006 elements and satisfies the required(...TRUNCATED) | 1006 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 5 | [
true,
true,
true,
true,
true
] | 5 | ["<think>\nOkay, so I need to find the largest possible subset M of the integers from 1 to 2021 such(...TRUNCATED) | ["<think>\nOkay, so I need to find the largest possible subset M of the integers from 1 to 2021 such(...TRUNCATED) | "<think>\nOkay, so I need to find the largest possible subset M of the integers from 1 to 2021 such (...TRUNCATED) | [{"from":"human","value":"\nProblem 4. Let $M$ be a subset of the set of 2021 integers $\\{1,2,3, \\(...TRUNCATED) |
"\nProblem 4. Let $M$ be a subset of the set of 2021 integers $\\{1,2,3, \\ldots, 2021\\}$ such that(...TRUNCATED) | "\nSolution. The set $M=\\{1016,1017, \\ldots, 2021\\}$ has 1006 elements and satisfies the required(...TRUNCATED) | 1006 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 5 | [
true,
true,
true,
true,
true
] | 5 | ["<think>\nOkay, so I need to find the largest possible subset M of the integers from 1 to 2021 such(...TRUNCATED) | ["<think>\nOkay, so I need to find the largest possible subset M of the integers from 1 to 2021 such(...TRUNCATED) | "<think>\nOkay, let me try to figure out this problem. So, we need to find the largest subset M of t(...TRUNCATED) | [{"from":"human","value":"\nProblem 4. Let $M$ be a subset of the set of 2021 integers $\\{1,2,3, \\(...TRUNCATED) |
"\nProblem 4. Let $M$ be a subset of the set of 2021 integers $\\{1,2,3, \\ldots, 2021\\}$ such that(...TRUNCATED) | "\nSolution. The set $M=\\{1016,1017, \\ldots, 2021\\}$ has 1006 elements and satisfies the required(...TRUNCATED) | 1006 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 5 | [
true,
true,
true,
true,
true
] | 5 | ["<think>\nOkay, so I need to find the largest possible subset M of the integers from 1 to 2021 such(...TRUNCATED) | ["<think>\nOkay, so I need to find the largest possible subset M of the integers from 1 to 2021 such(...TRUNCATED) | "<think>\nOkay, so I need to find the largest subset M of {1,2,3,...,2021} such that for any three e(...TRUNCATED) | [{"from":"human","value":"\nProblem 4. Let $M$ be a subset of the set of 2021 integers $\\{1,2,3, \\(...TRUNCATED) |
"\nProblem 4. Let $M$ be a subset of the set of 2021 integers $\\{1,2,3, \\ldots, 2021\\}$ such that(...TRUNCATED) | "\nSolution. The set $M=\\{1016,1017, \\ldots, 2021\\}$ has 1006 elements and satisfies the required(...TRUNCATED) | 1006 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 5 | [
true,
true,
true,
true,
true
] | 5 | ["<think>\nOkay, so I need to find the largest possible subset M of the integers from 1 to 2021 such(...TRUNCATED) | ["<think>\nOkay, so I need to find the largest possible subset M of the integers from 1 to 2021 such(...TRUNCATED) | "<think>\nOkay, so I have this problem here where I need to find the largest possible subset M of th(...TRUNCATED) | [{"from":"human","value":"\nProblem 4. Let $M$ be a subset of the set of 2021 integers $\\{1,2,3, \\(...TRUNCATED) |
"\nProblem 4. Let $M$ be a subset of the set of 2021 integers $\\{1,2,3, \\ldots, 2021\\}$ such that(...TRUNCATED) | "\nSolution. The set $M=\\{1016,1017, \\ldots, 2021\\}$ has 1006 elements and satisfies the required(...TRUNCATED) | 1006 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 5 | [
true,
true,
true,
true,
true
] | 5 | ["<think>\nOkay, so I need to find the largest possible subset M of the integers from 1 to 2021 such(...TRUNCATED) | ["<think>\nOkay, so I need to find the largest possible subset M of the integers from 1 to 2021 such(...TRUNCATED) | "<think>\nOkay, let me try to work through this problem step by step. The problem is about finding t(...TRUNCATED) | [{"from":"human","value":"\nProblem 4. Let $M$ be a subset of the set of 2021 integers $\\{1,2,3, \\(...TRUNCATED) |
"\nProblem 1. Find all distinct prime numbers $p, q$ and $r$ such that\n\n$$\n3 p^{4}-5 q^{4}-4 r^{2(...TRUNCATED) | "\nSolution. First notice that if both primes $q$ and $r$ differ from 3 , then $q^{2} \\equiv r^{2} (...TRUNCATED) | p=5,q=3,r=19 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 8 | [
false,
true,
false,
true,
false,
false,
false,
false
] | 2 | ["<think>\nOkay, so I have this problem where I need to find all distinct prime numbers p, q, and r (...TRUNCATED) | ["<think>\nOkay, let's see. I need to find distinct prime numbers p, q, and r such that 3p⁴ - 5q(...TRUNCATED) | "<think>\nOkay, let's see. I need to find distinct prime numbers p, q, and r such that 3p⁴ - 5q⁴(...TRUNCATED) | [{"from":"human","value":"\nProblem 1. Find all distinct prime numbers $p, q$ and $r$ such that\n\n$(...TRUNCATED) |
"\nProblem 1. Find all distinct prime numbers $p, q$ and $r$ such that\n\n$$\n3 p^{4}-5 q^{4}-4 r^{2(...TRUNCATED) | "\nSolution. First notice that if both primes $q$ and $r$ differ from 3 , then $q^{2} \\equiv r^{2} (...TRUNCATED) | p=5,q=3,r=19 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 8 | [
false,
true,
false,
true,
false,
false,
false,
false
] | 2 | ["<think>\nOkay, so I have this problem where I need to find all distinct prime numbers p, q, and r (...TRUNCATED) | ["<think>\nOkay, let's see. I need to find distinct prime numbers p, q, and r such that 3p⁴ - 5q(...TRUNCATED) | "<think>\nOkay, let's see. I have to find all distinct prime numbers p, q, and r such that 3p⁴ - 5(...TRUNCATED) | [{"from":"human","value":"\nProblem 1. Find all distinct prime numbers $p, q$ and $r$ such that\n\n$(...TRUNCATED) |
"\nProblem 2. Consider an acute triangle $A B C$ with area S. Let $C D \\perp A B \\quad(D \\in A B)(...TRUNCATED) | "\nSolution 1. Let $O, P, K, R$ and $T$ be the mid-points of the segments $C D, M N$, $C N, C H_{1}$(...TRUNCATED) | S | Geometry | math-word-problem | Yes | Yes | olympiads | false | 4 | [
true,
true,
true,
true
] | 4 | ["<think>\nOkay, so I need to find the area of quadrilateral AH₁BH₂ in terms of the area S of th(...TRUNCATED) | ["<think>\nOkay, so I need to find the area of quadrilateral AH₁BH₂ in terms of the area S of th(...TRUNCATED) | "<think>\nOkay, so I need to find the area of quadrilateral AH₁BH₂ in terms of the area S of the(...TRUNCATED) | [{"from":"human","value":"\nProblem 2. Consider an acute triangle $A B C$ with area S. Let $C D \\pe(...TRUNCATED) |
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