chosen listlengths 3 3 | rejected listlengths 3 3 |
|---|---|
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $G$ be a finite group and let $x_1,…,x_n$ be an enumeration of its elements. We consider the matrix $(a_{ij})_{1 \\le i,j \\le n},$ where $a_{ij}=0$ if $x_ix_j^{-1}=... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $G$ be a finite group and let $x_1,…,x_n$ be an enumeration of its elements. We consider the matrix $(a_{ij})_{1 \\le i,j \\le n},$ where $a_{ij}=0$ if $x_ix_j^{-1}=... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a $1 \\mathrm{~cm}$ long slit with negligible height. First, we divide the slit into thirds and cover the middle third. Then, we perform the same steps on the two shorte... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a $1 \\mathrm{~cm}$ long slit with negligible height. First, we divide the slit into thirds and cover the middle third. Then, we perform the same steps on the two shorte... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The following image is 1024 pixels by 1024 pixels, and each pixel is either black or white. The border defines the boundaries of the image, but is not part of the image. Let $a$ ... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The following image is 1024 pixels by 1024 pixels, and each pixel is either black or white. The border defines the boundaries of the image, but is not part of the image. Let $a$ ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider the sequence of numbers defined recursively by $t_1=1$ and for $n>1$ by $t_n=1+t_{(n/2)}$ when $n$ is even and by $t_n=\\frac{1}{t_{(n-1)}}$ when $n$ is odd.... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider the sequence of numbers defined recursively by $t_1=1$ and for $n>1$ by $t_n=1+t_{(n/2)}$ when $n$ is even and by $t_n=\\frac{1}{t_{(n-1)}}$ when $n$ is odd.... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $a, b, c$, and $d$ be positive real numbers such that\n\\[\\begin{array}{c@{\\hspace{3pt}}c@{\\hspace{3pt}}c@{\\hspace{3pt}}c@{\\hspace{3pt}}c}a^2+b^2&=&c^2+d^2&=&2008,\\\\ a... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $a, b, c$, and $d$ be positive real numbers such that\n\\[\\begin{array}{c@{\\hspace{3pt}}c@{\\hspace{3pt}}c@{\\hspace{3pt}}c@{\\hspace{3pt}}c}a^2+b^2&=&c^2+d^2&=&2008,\\\\ a... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all possible values of $a_1$ for which there exists a sequence $a_1, a_2, \\dots$ of rational numbers satisfying $$ a_{n+1}^2-a_{n+1}=a_n $$ for all positive int... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all possible values of $a_1$ for which there exists a sequence $a_1, a_2, \\dots$ of rational numbers satisfying $$ a_{n+1}^2-a_{n+1}=a_n $$ for all positive int... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Call a three-term strictly increasing arithmetic sequence of integers special if the sum of the squares of the three terms equals the product of the middle term and the square of... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Call a three-term strictly increasing arithmetic sequence of integers special if the sum of the squares of the three terms equals the product of the middle term and the square of... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Is the Fourier transform of the signal $x_1(t)=\\left\\{\\begin{array}{cc}\\sin \\omega_0 t, & -\\frac{2 \\pi}{\\omega_0} \\leq t \\leq \\frac{2 \\pi}{\\omega_0} \\\\ 0, & \\text... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Is the Fourier transform of the signal $x_1(t)=\\left\\{\\begin{array}{cc}\\sin \\omega_0 t, & -\\frac{2 \\pi}{\\omega_0} \\leq t \\leq \\frac{2 \\pi}{\\omega_0} \\\\ 0, & \\text... |
[{"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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