messages listlengths 3 3 |
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[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For any number \\( x \\), let \\( \\lfloor x \\rfloor \\) denote the greatest integer less than or equal to \\( x \\). A sequence \\( a_1, a_2, \\cdots \\) is... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all functions \\( f : S_n \\to M_n(\\mathbb{R}) \\) such that \\( f(x \\circ y) = f(x) + f(y) \\) for all \\( x, y \\in S_n \\), where \\( S_n \\) is the... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find a closed-form for $\\sum_{k\\ge0}\\binom nk\\binom{n-k}{\\left\\lfloor\\frac{m-k}2\\right\\rfloor}(-2)^k$ as a function of $n$ and $m$."
},
{
"ro... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "$21^{105}$ is congruent to 27 modulo 666. Find $21^{970}$ modulo 666."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find 21^9... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A coin with a diameter of 8 units is tossed onto a 5 by 5 grid of squares, each having a side length of 10 units. The coin is in a winning position if no part... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In convex quadrilateral $ABCD$, the angles are given as $\\angle ABD = 12^\\circ$, $\\angle ACD = 24^\\circ$, $\\angle DBC = 36^\\circ$, and $\\angle BCA = 48... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "If \\(a, b, c\\) are three distinct complex numbers and \\(\\frac{a}{b-c} + \\frac{b}{c-a} + \\frac{c}{a-b} = 0\\), then find the value of \\(\\frac{a^2}{(b-c... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider a metric space \\( E \\). Let \\( A \\) be a subset of \\( E \\) that is both open and closed. Suppose \\( A \\) is the disjoint union of two nonempt... |
[{"role":"system","content":"Please reason step by step, and put your final answer within \\boxed{}.(...TRUNCATED) |
[{"role":"system","content":"Please reason step by step, and put your final answer within \\boxed{}.(...TRUNCATED) |
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