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# Problem 1
Let $a, b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove that
$$
\frac{a}{b}+\frac{b}{a}+\frac{b}{c}+\frac{c}{b}+\frac{c}{a}+\frac{a}{c}+6 \geq 2 \sqrt{2}\left(\sqrt{\frac{1-a}{a}}+\sqrt{\frac{1-b}{b}}+\sqrt{\frac{1-c}{c}}\right)
$$
When does equality hold?
| ## Solution
Replacing $1-a, 1-b, 1-c$ with $b+c, c+a, a+b$ respectively on the right hand side, the given inequality becomes
and equivalently
$$
\left(\frac{b+c}{a}-2 \sqrt{2} \sqrt{\frac{... | =b==\frac{1}{3} | Inequalities | proof | Yes | Yes | olympiads | false | [
"<think>\nAlright, let me try to tackle this problem. So, we have three positive real numbers a, b, c such that their sum is 1. We need to prove this inequality involving fractions and square roots. The equality condition is also required. Hmm, okay.\n\nFirst, let me write down the given condition and the inequalit... | 2 | [
false,
false
] | 0 | [
"<think>\nOkay, so I need to prove this inequality:\n\n$$\n\\frac{a}{b}+\\frac{b}{a}+\\frac{b}{c}+\\frac{c}{b}+\\frac{c}{a}+\\frac{a}{c}+6 \\geq 2 \\sqrt{2}\\left(\\sqrt{\\frac{1-a}{a}}+\\sqrt{\\frac{1-b}{b}}+\\sqrt{\\frac{1-c}{c}}\\right)\n$$\n\ngiven that \\( a, b, c \\) are positive real numbers with \\( a + b +... | [
true,
true
] | The provided solution methodically proves the inequality using appropriate mathematical principles including AM-GM and Cauchy-Schwarz. It correctly identifies that equality holds when a = b = c = 1/3, thus matching the ground truth solution. | 7 |
"## Problem 2\n\nLet the circles $k_{1}$ and $k_{2}$ intersect at two distinct points $A$ and $B$, a(...TRUNCATED) | "## Solution 1\n\nLet $P$ be the symmetric of $A$ with respect to $M$ (Figure 1). Then $A M=M P$ and(...TRUNCATED) | 45 | Geometry | math-word-problem | Yes | Yes | olympiads | false | ["<think>\nOkay, let me try to figure out this geometry problem. So, we have two circles, k₁ and k(...TRUNCATED) | 6 | [
true,
true,
true,
true,
true,
true
] | 6 | ["<think>\nOkay, let me try to solve this geometry problem step by step. So, we have two intersectin(...TRUNCATED) | [
true,
true,
true,
true,
true,
true
] | "The solution accurately derives the angle ∠NMB by analyzing geometric properties of the circles a(...TRUNCATED) | 13 |
"## Problem 1.\n\nFind all pairs $(a, b)$ of positive integers such that $a!+b$ and $b!+a$ are both (...TRUNCATED) | "\nSolution. The condition is symmetric so we can assume that $b \\leq a$.\n\nThe first case is when(...TRUNCATED) | (1,4),(4,1),(5,5) | Number Theory | math-word-problem | Yes | Yes | olympiads | false | ["<think>\nOkay, so I have this problem where I need to find all pairs of positive integers (a, b) s(...TRUNCATED) | 8 | [
false,
false,
false,
true,
false,
false,
true,
true
] | 3 | ["<think>\nOkay, so I have this problem where I need to find all pairs of positive integers (a, b) s(...TRUNCATED) | [
true,
true,
true,
true,
true,
true,
true,
true
] | "The solution correctly identifies all pairs (1,4), (4,1), and (5,5) where both expressions a! + b a(...TRUNCATED) | 47 |
"\nProblem 1. Find all distinct prime numbers $p, q$ and $r$ such that\n\n$$\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 | ["<think>\nOkay, so I have this problem where I need to find all distinct prime numbers p, q, and r (...TRUNCATED) | 8 | [
false,
true,
false,
true,
false,
false,
false,
false
] | 2 | ["<think>\nOkay, let's see. I have to find all distinct prime numbers p, q, and r such that 3p⁴ - (...TRUNCATED) | [
true,
true,
true,
true,
true,
true,
true,
true
] | "The model correctly identified distinct prime numbers p=5, q=3, and r=19 that satisfy the equation (...TRUNCATED) | 30 |
"\nProblem 1. Let $n(n \\geq 1)$ be an integer. Consider the equation\n\n$$\n2 \\cdot\\left\\lfloor\(...TRUNCATED) | "\nSolution. Let $k=\\left[\\frac{1}{2 x}\\right], k \\in \\mathbb{Z}$.\n\n(a) For $n=8$, the equati(...TRUNCATED) | Algebra | math-word-problem | Yes | Yes | olympiads | false | ["<think>\nOkay, let's tackle part (a) first. The problem is to solve the equation for n=8:\n\n2 · (...TRUNCATED) | 2 | [
false,
false
] | 0 | ["<think>\nOkay, so I need to solve part (a) where n=8. The equation is given as:\n\n2 · floor(1/(2(...TRUNCATED) | [
true,
true
] | "The provided solution correctly followed through the steps necessary to solve the given equations. (...TRUNCATED) | 21 | |
"\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 | ["<think>\nOkay, so I need to find the largest possible subset M of the integers from 1 to 2021 such(...TRUNCATED) | 5 | [
true,
true,
true,
true,
true
] | 5 | ["<think>\nOkay, so I need to find the largest subset M of {1,2,3,...,2021} such that for any three (...TRUNCATED) | [
true,
true,
true,
true,
true
] | "The model correctly identified that the maximum size of the set M which meets the condition |a + b (...TRUNCATED) | 24 |
"\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 | ["<think>\nOkay, so I need to find the area of quadrilateral AH₁BH₂ in terms of the area S of th(...TRUNCATED) | 4 | [
true,
true,
true,
true
] | 4 | ["<think>\nOkay, so I have this problem here about an acute triangle ABC with area S. There are seve(...TRUNCATED) | [
true,
true,
true,
true
] | "The provided solution correctly outlines the steps to find the area of quadrilateral AH₁BH₂ in (...TRUNCATED) | 37 |
"\nProblem 2. Let $a, b, c$ be positive real numbers such that $a+b+c=3$. Find the minimum value of (...TRUNCATED) | "## Solution:\n\nWe can rewrite $A$ as follows:\n\n$$\n\\begin{aligned}\n& A=\\frac{2-a^{3}}{a}+\\fr(...TRUNCATED) | 3 | Inequalities | math-word-problem | Yes | Yes | olympiads | false | ["<think>\nOkay, so I have this problem where I need to find the minimum value of the expression A, (...TRUNCATED) | 7 | [
true,
true,
true,
true,
true,
true,
true
] | 7 | ["<think>\nOkay, so I have this problem where I need to find the minimum value of the expression A, (...TRUNCATED) | [
true,
true,
true,
true,
true,
true,
true
] | "The solution correctly derives and evaluates the minimum value of the expression A given the constr(...TRUNCATED) | 71 |
"\nProblem 4. Find all pairs $(p, q)$ of prime numbers such that\n\n$$\n1+\\frac{p^{q}-q^{p}}{p+q}\n(...TRUNCATED) | "\nSolution. It is clear that $p \\neq q$. We set\n\n$$\n1+\\frac{p^{q}-q^{p}}{p+q}=r\n$$\n\nand we (...TRUNCATED) | (2,5) | Number Theory | math-word-problem | Yes | Yes | olympiads | false | ["<think>\nOkay, so I need to find all pairs of prime numbers (p, q) such that the expression 1 + (p(...TRUNCATED) | 6 | [
true,
true,
true,
true,
true,
true
] | 6 | ["<think>\nOkay, so I need to find all pairs of prime numbers (p, q) such that the expression 1 + (p(...TRUNCATED) | [
true,
true,
true,
true,
true,
true
] | "The solution accurately identifies the only valid pair of prime numbers (2, 5) for the given expres(...TRUNCATED) | 58 |
"\nProblem 4. A $9 \\times 7$ rectangle is tiled with tiles of the two types shown in the picture be(...TRUNCATED) | "\nSolution. Answer: 0 or 3 .\n\nDenote by $x$ the number of the pieces of the type ornerănd by $y$(...TRUNCATED) | 0or3 | Combinatorics | math-word-problem | Incomplete | Yes | olympiads | false | ["<think>\nOkay, so I need to figure out all possible numbers of 2x2 tiles (which are actually 2x2 s(...TRUNCATED) | 1 | [
false
] | 0 | ["<think>\nOkay, so I need to figure out all possible numbers of 2x2 tiles (which are actually 2x2 s(...TRUNCATED) | [
false
] | "The solution provided includes values of n that are not acceptable based on the established conditi(...TRUNCATED) | 19 |
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