idx int64 0 1.25k | prompt stringlengths 235 7.69k | answers sequencelengths 16 16 | gt stringlengths 1 261 | rewards sequencelengths 16 16 |
|---|---|---|---|---|
393 | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $a, b, x, y$ be positive real numbers such that $a+b=1$. Prove that $\frac{1}{\frac{a}{x}+\frac{b}{y}}\leq ax+by$
and find when equality holds. Let's think step by step and output the final answer wi... | [
"I have this inequality to prove: $\\frac{1}{\\frac{a}{x} + \\frac{b}{y}} \\leq a x + b y$, given that $a + b = 1$ and all variables are positive real numbers. I need to find when equality holds.\n\nFirst, I'll try to simplify the left side of the inequality. The expression $\\frac{1}{\\frac{a}{x} + \\frac{b}{y}}$ ... | \frac{1}{\frac{a}{x} + \frac{b}{y}} \leq ax + by | [
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1,070 | "<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | ["So I've got this problem here: \"For what natural number \\( n \\) will the sum of the digits in t(...TRUNCATED) | 5 | [
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785 | "<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | ["So I've got this fraction: \\(\\frac{15}{25}\\), and I need to find which of these options is equi(...TRUNCATED) | \frac{3}{5} | [
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486 | "<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | ["Let's tackle this problem step by step. We have a table that defines an operation * for the number(...TRUNCATED) | 4 | [
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935 | "<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | ["To solve the equation\n\n$$\n\\frac{1}{4^{3x} - 13 \\cdot 4^{2x} + 51 \\cdot 4^{x} - 60} + \\frac{(...TRUNCATED) | x_1=0.5,\quadx_2\approx1.292 | [
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207 | "<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | ["So I've got this inequality to solve: 3|x - 1| + x² > 7. Alright, let's see. This one has an abso(...TRUNCATED) | (-\infty;-1)\cup(2;+\infty) | [
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943 | "<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | ["I'm trying to solve this problem: determine all real numbers \\( a \\) for which the equation\n\n\(...TRUNCATED) | 02023 | [
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669 | "<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | ["Let's tackle this interesting problem step by step. The problem states that \\( n \\) is a natural(...TRUNCATED) | 1,3,5,7 | [
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1,175 | "<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | ["So, Petya wants to create an unusual die that's shaped like a cube, with dots on each face, and th(...TRUNCATED) | 27 | [
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1,230 | "<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | ["So I've got this problem here: I need to find how many pairs of natural numbers (m, n), where both(...TRUNCATED) | 1706 | [
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