messages listlengths 3 3 |
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[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose you are chosen as a technology assistant by the director of the opening ceremony for the 2022 Winter Olympics, and your job is to evaluate the program proposals. One of t... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "(a) Does $\\sum_{i=1}^{p-1} \\frac{1}{i} \\equiv 0\\left(\\bmod p^{2}\\right)$ for all odd prime numbers $p$? (Note that $\\frac{1}{i}$ denotes the number such that $\\left.i \\c... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given vector $\\mathbf{u}=\\left(\\frac{1}{3}, \\frac{1}{3}, \\frac{1}{3} \\right)\\in\\mathbb{R}^3$ and recursively defined sequence of vectors $\\{\\mathbf{v}_n\\}_{n\\geq 0... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let \\[T=\\{(tq,1-t) \\in\\mathbb R^{2}| t \\in [0,1],q\\in\\mathbb Q\\}\\]Prove that each continuous function $f: T\\longrightarrow T$ has a fixed point.",
"role": "user"
... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathcal{R}$ be the region consisting of the set of points in the coordinate plane that satisfy both $|8 - x| + y \\le 10$ and $3y - x \\ge 15$ . When $\\mathcal{R}$ is rev... |
[{"content":"Please reason step by step, and put your final answer within \\boxed{}.","role":"system(...TRUNCATED) |
[{"content":"Please reason step by step, and put your final answer within \\boxed{}.","role":"system(...TRUNCATED) |
[{"content":"Please reason step by step, and put your final answer within \\boxed{}.","role":"system(...TRUNCATED) |
[{"content":"Please reason step by step, and put your final answer within \\boxed{}.","role":"system(...TRUNCATED) |
[{"content":"Please reason step by step, and put your final answer within \\boxed{}.","role":"system(...TRUNCATED) |
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