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"
},
{
"content": "Two identical moving coil galvanometers have $10 \\Omega$ resistance and full scale deflection at $2 \\mu \\mathrm{A}$ current. One of them is converted into a voltmeter of $100 ... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Two identical moving coil galvanometers have $10 \\Omega$ resistance and full scale deflection at $2 \\mu \\mathrm{A}$ current. One of them is converted into a voltmeter of $100 ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Does there exist a finite group $ G$ with a normal subgroup $ H$ such that $ |\\text{Aut } H| > |\\text{Aut } G|$ ? Disprove or provide an example. Here the notation $ |\\t... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Does there exist a finite group $ G$ with a normal subgroup $ H$ such that $ |\\text{Aut } H| > |\\text{Aut } G|$ ? Disprove or provide an example. Here the notation $ |\\t... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Compute the number of triples $(f, g, h)$ of permutations on $\\{1,2,3,4,5\\}$ such that $$ \\begin{aligned} & f(g(h(x)))=h(g(f(x)))=g(x), \\\\ & g(h(f(x)))=f(h(g(x)))=h(x), \\te... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Compute the number of triples $(f, g, h)$ of permutations on $\\{1,2,3,4,5\\}$ such that $$ \\begin{aligned} & f(g(h(x)))=h(g(f(x)))=g(x), \\\\ & g(h(f(x)))=f(h(g(x)))=h(x), \\te... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $s(n)$ denote the sum of the digits (in base ten) of a positive integer $n$. Compute the number of positive integers $n$ at most $10^{4}$ that satisfy $$s(11 n)=2 s(n)$$",
... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $s(n)$ denote the sum of the digits (in base ten) of a positive integer $n$. Compute the number of positive integers $n$ at most $10^{4}$ that satisfy $$s(11 n)=2 s(n)$$",
... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Triangle $ABC$ has side lengths $AB = 9$ , $BC =$ $5\\sqrt{3}$ , and $AC = 12$ . Points $A = P_{0}, P_{1}, P_{2}, ... , P_{2450} = B$ are on segment $\\overline{AB}$ with $P_{k}$... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Triangle $ABC$ has side lengths $AB = 9$ , $BC =$ $5\\sqrt{3}$ , and $AC = 12$ . Points $A = P_{0}, P_{1}, P_{2}, ... , P_{2450} = B$ are on segment $\\overline{AB}$ with $P_{k}$... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A function $f$ is defined for all real numbers and satisfies $f(2+x)=f(2-x)$ and $f(7+x)=f(7-x)$ for all $x$ . If $x=0$ is a root for $f(x)=0$ , what is the least number of roots... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A function $f$ is defined for all real numbers and satisfies $f(2+x)=f(2-x)$ and $f(7+x)=f(7-x)$ for all $x$ . If $x=0$ is a root for $f(x)=0$ , what is the least number of roots... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "c. Assuming the radius is the value you found in part (b), how much energy is stored in the electric field of the electron? Express your answer in terms of $E_{0}=m c^{2}$ and th... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "c. Assuming the radius is the value you found in part (b), how much energy is stored in the electric field of the electron? Express your answer in terms of $E_{0}=m c^{2}$ and th... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that the electron number density n(r) in a region is proportional to the cube of a characteristic momentum, pF(r)^3. The differential number density of electrons F(p) aro... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that the electron number density n(r) in a region is proportional to the cube of a characteristic momentum, pF(r)^3. The differential number density of electrons F(p) aro... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Define the function $f(x) = \\lfloor x \\rfloor + \\lfloor \\sqrt{x} \\rfloor + \\lfloor \\sqrt{\\sqrt{x}} \\rfloor$ for all positive real numbers $x$ . How many integers from... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Define the function $f(x) = \\lfloor x \\rfloor + \\lfloor \\sqrt{x} \\rfloor + \\lfloor \\sqrt{\\sqrt{x}} \\rfloor$ for all positive real numbers $x$ . How many integers from... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In trapezoid $ABCD$ , $AD$ is parallel to $BC$ . Knowing that $AB=AD+BC$ , prove that the bisector of $\\angle A$ also bisects $CD$ .",
"role": "user"
},
{
"c... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In trapezoid $ABCD$ , $AD$ is parallel to $BC$ . Knowing that $AB=AD+BC$ , prove that the bisector of $\\angle A$ also bisects $CD$ .",
"role": "user"
},
{
"c... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that $X$ and $Y$ are metric spaces and $f:X \\longrightarrow Y$ is a continious function. Also $f_1: X\\times \\mathbb R \\longrightarrow Y\\times \\mathbb R$ wit... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that $X$ and $Y$ are metric spaces and $f:X \\longrightarrow Y$ is a continious function. Also $f_1: X\\times \\mathbb R \\longrightarrow Y\\times \\mathbb R$ wit... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "(a) Compute the electric potential inside and outside the sphere.",
"role": "user"
},
{
"content": "Here's a breakdown of the thought process to solve this problem:\n... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "(a) Compute the electric potential inside and outside the sphere.",
"role": "user"
},
{
"content": "Here's a breakdown of the thought process to solve this problem:\n... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $|x|$ is less than 0.7, then if we use fifth Maclaurin polynomial approximate $sin(x)$ the error is less than 0.0001. Is this correct? Answer True or False.",
"role": "use... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $|x|$ is less than 0.7, then if we use fifth Maclaurin polynomial approximate $sin(x)$ the error is less than 0.0001. Is this correct? Answer True or False.",
"role": "use... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $v$ and $w$ be distinct, randomly chosen roots of the equation $z^{1997}-1=0$ . Let $m/n$ be the probability that $\\sqrt{2+\\sqrt{3}}\\le |v+w|$ , where $m$ and $n$ are rela... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $v$ and $w$ be distinct, randomly chosen roots of the equation $z^{1997}-1=0$ . Let $m/n$ be the probability that $\\sqrt{2+\\sqrt{3}}\\le |v+w|$ , where $m$ and $n$ are rela... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Prove that a nontrivial finite ring is not a skew field if and only if the equation $ x^n+y^n=z^n $ has nontrivial solutions in this ring for any natural number $ n. $ ",
... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Prove that a nontrivial finite ring is not a skew field if and only if the equation $ x^n+y^n=z^n $ has nontrivial solutions in this ring for any natural number $ n. $ ",
... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The function f is defined on the set of integers and satisfies $f(n)= \\begin{cases} n-3 & \\mbox{if }n\\ge 1000 \\\\ f(f(n+5)) & \\mbox{if }n<1000 \\end{cases}$ Find $f(84)$ .... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The function f is defined on the set of integers and satisfies $f(n)= \\begin{cases} n-3 & \\mbox{if }n\\ge 1000 \\\\ f(f(n+5)) & \\mbox{if }n<1000 \\end{cases}$ Find $f(84)$ .... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The points $A$ , $B$ and $C$ lie on the surface of a sphere with center $O$ and radius $20$ . It is given that $AB=13$ , $BC=14$ , $CA=15$ , and that the distance from $O$ to tri... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The points $A$ , $B$ and $C$ lie on the surface of a sphere with center $O$ and radius $20$ . It is given that $AB=13$ , $BC=14$ , $CA=15$ , and that the distance from $O$ to tri... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $w$ and $z$ be complex numbers such that $|w| = 1$ and $|z| = 10$ . Let $\\theta = \\arg \\left(\\tfrac{w-z}{z}\\right)$ . The maximum possible value of $\\tan^2 \\theta$ can... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $w$ and $z$ be complex numbers such that $|w| = 1$ and $|z| = 10$ . Let $\\theta = \\arg \\left(\\tfrac{w-z}{z}\\right)$ . The maximum possible value of $\\tan^2 \\theta$ can... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A spherical shell with a radius R has an electric charge distribution on its surface. The surface charge density is described by σ(θ)=4 cos(θ), where θ is the angle with respect ... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A spherical shell with a radius R has an electric charge distribution on its surface. The surface charge density is described by σ(θ)=4 cos(θ), where θ is the angle with respect ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The numbers $x_1,...x_{100}$ are written on a board so that $ x_1=\\frac{1}{2}$ and for every $n$ from $1$ to $99$ , $x_{n+1}=1-x_1x_2x_3*...*x_{100}$ . Prove that $x_... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The numbers $x_1,...x_{100}$ are written on a board so that $ x_1=\\frac{1}{2}$ and for every $n$ from $1$ to $99$ , $x_{n+1}=1-x_1x_2x_3*...*x_{100}$ . Prove that $x_... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Square $ABCD$ is inscribed in a circle. Square $EFGH$ has vertices $E$ and $F$ on $\\overline{CD}$ and vertices $G$ and $H$ on the circle. The ratio of the area of square $EFGH$ ... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Square $ABCD$ is inscribed in a circle. Square $EFGH$ has vertices $E$ and $F$ on $\\overline{CD}$ and vertices $G$ and $H$ on the circle. The ratio of the area of square $EFGH$ ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A bee starts flying from point $P_0$. She flies $1$ inch due east to point $P_1$. For $j \\ge 1$, once the bee reaches point $P_j$, she turns $30^{\\circ}$ counterclockwise and t... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A bee starts flying from point $P_0$. She flies $1$ inch due east to point $P_1$. For $j \\ge 1$, once the bee reaches point $P_j$, she turns $30^{\\circ}$ counterclockwise and t... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Perform 2 iterations with the Müller method for the following equation: log_{10}(x) - x + 3 = 0, x_0 = 1/4, x_1 = 1/2, x_2 = 1. What's the decimal value of x_3?",
"role": "us... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Perform 2 iterations with the Müller method for the following equation: log_{10}(x) - x + 3 = 0, x_0 = 1/4, x_1 = 1/2, x_2 = 1. What's the decimal value of x_3?",
"role": "us... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider all rectangles lying in the region\n\n\\[\n\\left\\{(x, y) \\in \\mathbb{R} \\times \\mathbb{R}: 0 \\leq x \\leq \\frac{\\pi}{2} \\text { and } 0 \\leq y \\leq 2 \\sin (... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider all rectangles lying in the region\n\n\\[\n\\left\\{(x, y) \\in \\mathbb{R} \\times \\mathbb{R}: 0 \\leq x \\leq \\frac{\\pi}{2} \\text { and } 0 \\leq y \\leq 2 \\sin (... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $z_1,z_2,z_3,\\dots,z_{12}$ be the 12 zeroes of the polynomial $z^{12}-2^{36}$ . For each $j$ , let $w_j$ be one of $z_j$ or $i z_j$ . Then the maximum possible value of the ... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $z_1,z_2,z_3,\\dots,z_{12}$ be the 12 zeroes of the polynomial $z^{12}-2^{36}$ . For each $j$ , let $w_j$ be one of $z_j$ or $i z_j$ . Then the maximum possible value of the ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given complex number $z$, define sequence $z_{0}, z_{1}, z_{2}, \\ldots$ as $z_{0}=z$ and $z_{n+1}=2 z_{n}^{2}+2 z_{n}$ for $n \\geq 0$. Given that $z_{10}=2017$, find the minimu... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given complex number $z$, define sequence $z_{0}, z_{1}, z_{2}, \\ldots$ as $z_{0}=z$ and $z_{n+1}=2 z_{n}^{2}+2 z_{n}$ for $n \\geq 0$. Given that $z_{10}=2017$, find the minimu... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the sum of all positive integers $n$ such that $\\sqrt{n^2+85n+2017}$ is an integer.",
"role": "user"
},
{
"content": "The problem asks for the sum of all positi... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the sum of all positive integers $n$ such that $\\sqrt{n^2+85n+2017}$ is an integer.",
"role": "user"
},
{
"content": "The problem asks for the sum of all positi... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are ... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Two three-letter strings, $aaa^{}_{}$ and $bbb^{}_{}$ , are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Two three-letter strings, $aaa^{}_{}$ and $bbb^{}_{}$ , are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Triangle $ABC_0$ has a right angle at $C_0$ . Its side lengths are pairwise relatively prime positive integers, and its perimeter is $p$ . Let $C_1$ be the foot of the altitude t... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Triangle $ABC_0$ has a right angle at $C_0$ . Its side lengths are pairwise relatively prime positive integers, and its perimeter is $p$ . Let $C_1$ be the foot of the altitude t... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $a$ and $b$ be positive integers satisfying $\\frac{ab+1}{a+b} < \\frac{3}{2}$ . The maximum possible value of $\\frac{a^3b^3+1}{a^3+b^3}$ is $\\frac{p}{q}$ , where $p$ and $... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $a$ and $b$ be positive integers satisfying $\\frac{ab+1}{a+b} < \\frac{3}{2}$ . The maximum possible value of $\\frac{a^3b^3+1}{a^3+b^3}$ is $\\frac{p}{q}$ , where $p$ and $... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $x,$ $y,$ $z$ be real numbers such that $x + y + z = 1,$ and $x \\ge -\\frac{1}{3},$ $y \\ge -1,$ and $z \\ge -\\frac{5}{3}.$ Find the maximum value of\n\\[\\sqrt{3x + 1} + ... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $x,$ $y,$ $z$ be real numbers such that $x + y + z = 1,$ and $x \\ge -\\frac{1}{3},$ $y \\ge -1,$ and $z \\ge -\\frac{5}{3}.$ Find the maximum value of\n\\[\\sqrt{3x + 1} + ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A hexagon that is inscribed in a circle has side lengths $22$ , $22$ , $20$ , $22$ , $22$ , and $20$ in that order. The radius of the circle can be written as $p+\\sqrt{q}$ , whe... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A hexagon that is inscribed in a circle has side lengths $22$ , $22$ , $20$ , $22$ , $22$ , and $20$ in that order. The radius of the circle can be written as $p+\\sqrt{q}$ , whe... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider two systems of binary black holes at distance DA and DB. Binary system A is composed of two black holes with masses m1=10 and m2= 25 in units of solar masses. Masses of ... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider two systems of binary black holes at distance DA and DB. Binary system A is composed of two black holes with masses m1=10 and m2= 25 in units of solar masses. Masses of ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A right rectangular prism $P_{}$ (i.e., a rectangular parallelepiped) has sides of integral length $a, b, c,$ with $a\\le b\\le c.$ A plane parallel to one of the faces of $P_{}$... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A right rectangular prism $P_{}$ (i.e., a rectangular parallelepiped) has sides of integral length $a, b, c,$ with $a\\le b\\le c.$ A plane parallel to one of the faces of $P_{}$... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Prove that $$ \\int^{\\frac\\pi3}_{\\frac\\pi6}\\frac u{\\sin u}du=\\frac83\\sum_{k=0}^\\infty\\frac{(-1)^k}{3^k(2k+1)^2}+\\frac{\\pi\\ln3}{3\\sqrt3}-\\frac{4C}3+\\frac\\pi6\\ln... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Prove that $$ \\int^{\\frac\\pi3}_{\\frac\\pi6}\\frac u{\\sin u}du=\\frac83\\sum_{k=0}^\\infty\\frac{(-1)^k}{3^k(2k+1)^2}+\\frac{\\pi\\ln3}{3\\sqrt3}-\\frac{4C}3+\\frac\\pi6\\ln... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $u_0, u_1, u_2, \\ldots$ be integers such that $u_0 = 100$ ; $u_{k+2} \\geqslant 2 + u_k$ for all $k \\geqslant 0$ ; and $u_{\\ell+5} \\leqslant 5 + u_\\ell$ for all ... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $u_0, u_1, u_2, \\ldots$ be integers such that $u_0 = 100$ ; $u_{k+2} \\geqslant 2 + u_k$ for all $k \\geqslant 0$ ; and $u_{\\ell+5} \\leqslant 5 + u_\\ell$ for all ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Two electromagnetic (em) fields are applied on the 1s->2p transition of a Hydrogen atomic vapor sample. The field#1 is polarized in z-direction, propagating in x-direction, with ... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Two electromagnetic (em) fields are applied on the 1s->2p transition of a Hydrogen atomic vapor sample. The field#1 is polarized in z-direction, propagating in x-direction, with ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $a_{1}, a_{2}, \\ldots$ be a sequence of positive integers such that for integers $n>2, a_{n}=$ $3 a_{n-1}-2 a_{n-2}$. How many such sequences $\\left\\{a_{n}\\right\\}$ are ... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $a_{1}, a_{2}, \\ldots$ be a sequence of positive integers such that for integers $n>2, a_{n}=$ $3 a_{n-1}-2 a_{n-2}$. How many such sequences $\\left\\{a_{n}\\right\\}$ are ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Triangles $\\triangle ABC$ and $\\triangle A'B'C'$ lie in the coordinate plane with vertices $A(0,0)$ , $B(0,12)$ , $C(16,0)$ , $A'(24,18)$ , $B'(36,18)$ , $C'(24,2)$ . A rotatio... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Triangles $\\triangle ABC$ and $\\triangle A'B'C'$ lie in the coordinate plane with vertices $A(0,0)$ , $B(0,12)$ , $C(16,0)$ , $A'(24,18)$ , $B'(36,18)$ , $C'(24,2)$ . A rotatio... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "There are $200$ numbers on a blackboard: $ 1! , 2! , 3! , 4! , ... ... , 199! , 200!$ .\nJulia erases one of the numbers. When Julia multiplies the remaining $199$ numbe... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "There are $200$ numbers on a blackboard: $ 1! , 2! , 3! , 4! , ... ... , 199! , 200!$ .\nJulia erases one of the numbers. When Julia multiplies the remaining $199$ numbe... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Alice and Bob live on the same road. At time $t$ , they both decide to walk to each other's houses at constant speed. However, they were busy thinking about math so that they di... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Alice and Bob live on the same road. At time $t$ , they both decide to walk to each other's houses at constant speed. However, they were busy thinking about math so that they di... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "There are real numbers $a, b, c,$ and $d$ such that $-20$ is a root of $x^3 + ax + b$ and $-21$ is a root of $x^3 + cx^2 + d.$ These two polynomials share a complex root $m + \\s... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "There are real numbers $a, b, c,$ and $d$ such that $-20$ is a root of $x^3 + ax + b$ and $-21$ is a root of $x^3 + cx^2 + d.$ These two polynomials share a complex root $m + \\s... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Points $A(0,0), B(9,6)$ and $C(6,12)$ are vertices of triangle $ABC$. Point $D$ is on segment $AB$ such that $2(AD) = DB$, point $E$ is on segment $BC$ such that $2(BE) = EC$ and... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Points $A(0,0), B(9,6)$ and $C(6,12)$ are vertices of triangle $ABC$. Point $D$ is on segment $AB$ such that $2(AD) = DB$, point $E$ is on segment $BC$ such that $2(BE) = EC$ and... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $P$ denote the set of all ordered pairs $ \\left(p,q\\right)$ of nonnegative integers. Find all functions $f: P \\rightarrow \\mathbb{R}$ satisfying\n\\[ f(p,q) \\equal... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $P$ denote the set of all ordered pairs $ \\left(p,q\\right)$ of nonnegative integers. Find all functions $f: P \\rightarrow \\mathbb{R}$ satisfying\n\\[ f(p,q) \\equal... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $G_1$ and $G_2$ be two finite groups such that for any finite group $H$ , the number of group homomorphisms from $G_1$ to $H$ is equal to the number of group homomor... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $G_1$ and $G_2$ be two finite groups such that for any finite group $H$ , the number of group homomorphisms from $G_1$ to $H$ is equal to the number of group homomor... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the least positive integer $m$ such that $m^2 - m + 11$ is a product of at least four not necessarily distinct primes.",
"role": "user"
},
{
"content": "The prob... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the least positive integer $m$ such that $m^2 - m + 11$ is a product of at least four not necessarily distinct primes.",
"role": "user"
},
{
"content": "The prob... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In the far future, the Earth received an enormous amount of charge as a result of Mad Scientist ecilA's nefarious experiments. Specifically, the total charge on Earth is $Q=1.0 \... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In the far future, the Earth received an enormous amount of charge as a result of Mad Scientist ecilA's nefarious experiments. Specifically, the total charge on Earth is $Q=1.0 \... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $S$ be the set of integers between 1 and $2^{40}$ whose binary expansions have exactly two 1's. If a number is chosen at random from $S,$ the probability that it is divisible... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $S$ be the set of integers between 1 and $2^{40}$ whose binary expansions have exactly two 1's. If a number is chosen at random from $S,$ the probability that it is divisible... |
[
{
"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": "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) Let $n$ is a positive integer. Calculate $\\displaystyle \\int_0^1 x^{n-1}\\ln x\\,dx$ .\n\n(b) Calculate $\\displaystyle \\sum_{n=0}^{\\infty}(-1)^n\\left(\\frac{1}{(n+1... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "(a) Let $n$ is a positive integer. Calculate $\\displaystyle \\int_0^1 x^{n-1}\\ln x\\,dx$ .\n\n(b) Calculate $\\displaystyle \\sum_{n=0}^{\\infty}(-1)^n\\left(\\frac{1}{(n+1... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Mr. Jackson bought his house in 1995, and financed the loan for 30 years at an interest rate of 7.8%. His monthly payment was $1260. In 2015, Mr. Jackson decides to pay off the l... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Mr. Jackson bought his house in 1995, and financed the loan for 30 years at an interest rate of 7.8%. His monthly payment was $1260. In 2015, Mr. Jackson decides to pay off the l... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a forward contract on a 4-year bond with maturity 1 year. The current value of the bond is $1018.86, it has a face value of $1000 and a coupon rate of 10% per annum. A c... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a forward contract on a 4-year bond with maturity 1 year. The current value of the bond is $1018.86, it has a face value of $1000 and a coupon rate of 10% per annum. A c... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine which set of states mentioned below are only entangled states:\n(a) (1/√30)* (|00>+ 2i|01>− 3|10>− 4i|11>)\n(b) (1/5)* (|00>+ 2i|01>− 2|10>− 4i|11>)\n(c) (1/2)(|00>+ |0... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine which set of states mentioned below are only entangled states:\n(a) (1/√30)* (|00>+ 2i|01>− 3|10>− 4i|11>)\n(b) (1/5)* (|00>+ 2i|01>− 2|10>− 4i|11>)\n(c) (1/2)(|00>+ |0... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "An angle $x$ is chosen at random from the interval $0^\\circ < x < 90^\\circ.$ Let $p$ be the probability that the numbers $\\sin^2 x, \\cos^2 x,$ and $\\sin x \\cos x$ are not t... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "An angle $x$ is chosen at random from the interval $0^\\circ < x < 90^\\circ.$ Let $p$ be the probability that the numbers $\\sin^2 x, \\cos^2 x,$ and $\\sin x \\cos x$ are not t... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A stationary source emits sound of frequency $f_{0}=492 \\mathrm{~Hz}$. The sound is reflected by a large car approaching the source with a speed of $2 \\mathrm{~ms}^{-1}$. The r... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A stationary source emits sound of frequency $f_{0}=492 \\mathrm{~Hz}$. The sound is reflected by a large car approaching the source with a speed of $2 \\mathrm{~ms}^{-1}$. The r... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In triangle $ABC$ , $AB = 125$ , $AC = 117$ , and $BC = 120$ . The angle bisector of angle $A$ intersects $\\overline{BC}$ at point $L$ , and the angle bisector of angle $B$ int... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In triangle $ABC$ , $AB = 125$ , $AC = 117$ , and $BC = 120$ . The angle bisector of angle $A$ intersects $\\overline{BC}$ at point $L$ , and the angle bisector of angle $B$ int... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathcal{S}$ be the set of all perfect squares whose rightmost three digits in base $10$ are $256$ . Let $\\mathcal{T}$ be the set of all numbers of the form $\\frac{x-256}... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathcal{S}$ be the set of all perfect squares whose rightmost three digits in base $10$ are $256$ . Let $\\mathcal{T}$ be the set of all numbers of the form $\\frac{x-256}... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine the smallest positive value of $x,$ in degrees, for which\n\\[\\tan (x + 100^{\\circ}) = \\tan (x + 50^{\\circ}) \\tan x \\tan (x - 50^{\\circ}).\\]",
"role": "user... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine the smallest positive value of $x,$ in degrees, for which\n\\[\\tan (x + 100^{\\circ}) = \\tan (x + 50^{\\circ}) \\tan x \\tan (x - 50^{\\circ}).\\]",
"role": "user... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a three-person game involving the following three types of fair six-sided dice. - Dice of type $A$ have faces labelled $2,2,4,4,9,9$. - Dice of type $B$ have faces label... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a three-person game involving the following three types of fair six-sided dice. - Dice of type $A$ have faces labelled $2,2,4,4,9,9$. - Dice of type $B$ have faces label... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "FemtoPravis is walking on an $8 \\times 8$ chessboard that wraps around at its edges (so squares on the left edge of the chessboard are adjacent to squares on the right edge, and... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "FemtoPravis is walking on an $8 \\times 8$ chessboard that wraps around at its edges (so squares on the left edge of the chessboard are adjacent to squares on the right edge, and... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "f. What is the increase in the entropy of the universe as a result of the free expansion? Give your answer to first order in $\\frac{a^{\\prime} N}{V k_{B} T}$.",
"role": "us... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "f. What is the increase in the entropy of the universe as a result of the free expansion? Give your answer to first order in $\\frac{a^{\\prime} N}{V k_{B} T}$.",
"role": "us... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Solve the crossword puzzle. You are presented with a clue as input and the number of letters in brackets.\n\nDevour Dickensian work and get into debt (4,2,4,5)",
"role": "use... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Solve the crossword puzzle. You are presented with a clue as input and the number of letters in brackets.\n\nDevour Dickensian work and get into debt (4,2,4,5)",
"role": "use... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "How many positive integers less than or equal to 240 can be expressed as a sum of distinct factorials? Consider 0 ! and 1 ! to be distinct.",
"role": "user"
},
{
"con... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "How many positive integers less than or equal to 240 can be expressed as a sum of distinct factorials? Consider 0 ! and 1 ! to be distinct.",
"role": "user"
},
{
"con... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Among the following stars, which one will appear redder than if it were located next to the Sun? Please assume that all the stars have the same parallax (1 milliarcsecond) and th... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Among the following stars, which one will appear redder than if it were located next to the Sun? Please assume that all the stars have the same parallax (1 milliarcsecond) and th... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "An infinite sequence of real numbers $a_1, a_2, \\dots$ satisfies the recurrence\n\\[ a_{n+3} = a_{n+2} - 2a_{n+1} + a_n \\]for every positive integer $n$. Given that $a_1 = a_3 ... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "An infinite sequence of real numbers $a_1, a_2, \\dots$ satisfies the recurrence\n\\[ a_{n+3} = a_{n+2} - 2a_{n+1} + a_n \\]for every positive integer $n$. Given that $a_1 = a_3 ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\{ z_n \\}_{n \\ge 1}$ be a sequence of complex numbers, whose odd terms are real, even terms are purely imaginary, and for every positive integer $k$ , $|z_k z_{k+1}|=... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\{ z_n \\}_{n \\ge 1}$ be a sequence of complex numbers, whose odd terms are real, even terms are purely imaginary, and for every positive integer $k$ , $|z_k z_{k+1}|=... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A sphere is inscribed in the tetrahedron whose vertices are $A = (6,0,0), B = (0,4,0), C = (0,0,2),$ and $D = (0,0,0).$ The radius of the sphere is $m/n,$ where $m$ and $n$ are r... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A sphere is inscribed in the tetrahedron whose vertices are $A = (6,0,0), B = (0,4,0), C = (0,0,2),$ and $D = (0,0,0).$ The radius of the sphere is $m/n,$ where $m$ and $n$ are r... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "There is a unique angle $\\theta$ between $0^\\circ$ and $90^\\circ$ such that for nonnegative integers $n,$ the value of $\\tan(2^n\\theta)$ is positive when $n$ is a multiple o... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "There is a unique angle $\\theta$ between $0^\\circ$ and $90^\\circ$ such that for nonnegative integers $n,$ the value of $\\tan(2^n\\theta)$ is positive when $n$ is a multiple o... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of distinct numbers in the list\n\\[\\left\\lfloor \\frac{1^2}{1000} \\right\\rfloor, \\ \\left\\lfloor \\frac{2^2}{1000} \\right\\rfloor, \\ \\left\\lfloor \\fra... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of distinct numbers in the list\n\\[\\left\\lfloor \\frac{1^2}{1000} \\right\\rfloor, \\ \\left\\lfloor \\frac{2^2}{1000} \\right\\rfloor, \\ \\left\\lfloor \\fra... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $P$ be a point chosen uniformly at random in the interior of the unit square with vertices at $(0,0), (1,0), (1,1)$ , and $(0,1)$ . The probability that the slope of the line... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $P$ be a point chosen uniformly at random in the interior of the unit square with vertices at $(0,0), (1,0), (1,1)$ , and $(0,1)$ . The probability that the slope of the line... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Jackie and Phil have two fair coins and a third coin that comes up heads with probability $\\frac47$ . Jackie flips the three coins, and then Phil flips the three coins. Let $\\f... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Jackie and Phil have two fair coins and a third coin that comes up heads with probability $\\frac47$ . Jackie flips the three coins, and then Phil flips the three coins. Let $\\f... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "How many functions $f$ from \\{-1005, \\ldots, 1005\\} to \\{-2010, \\ldots, 2010\\} are there such that the following two conditions are satisfied? - If $a<b$ then $f(a)<f(b)$. ... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "How many functions $f$ from \\{-1005, \\ldots, 1005\\} to \\{-2010, \\ldots, 2010\\} are there such that the following two conditions are satisfied? - If $a<b$ then $f(a)<f(b)$. ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A number is called a visible factor number if it is divisible by each of its non-zero digits. For example, 102 is divisible by 1 and 2, so it is a visible factor number. How many... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A number is called a visible factor number if it is divisible by each of its non-zero digits. For example, 102 is divisible by 1 and 2, so it is a visible factor number. How many... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In quantum mechanics, when calculating the interaction between the electron with the proton in a hydrogen atom, it is necessary to compute the following volume integral (over all... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In quantum mechanics, when calculating the interaction between the electron with the proton in a hydrogen atom, it is necessary to compute the following volume integral (over all... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For her zeroth project at Magic School, Emilia needs to grow six perfectly-shaped apple trees. First she plants six tree saplings at the end of Day $0$ . On each day afterwards,... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For her zeroth project at Magic School, Emilia needs to grow six perfectly-shaped apple trees. First she plants six tree saplings at the end of Day $0$ . On each day afterwards,... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "<span style=\"color:darkred\"> Let $(R,+,\\cdot)$ be a ring and let $f$ be a surjective endomorphism of $R$ such that $[x,f(x)]=0$ for any $x\\in R$ , where $[a,b]=ab-... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "<span style=\"color:darkred\"> Let $(R,+,\\cdot)$ be a ring and let $f$ be a surjective endomorphism of $R$ such that $[x,f(x)]=0$ for any $x\\in R$ , where $[a,b]=ab-... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "There is a polynomial $P$ such that for every real number $x$,\n\\[\n x^{512} + x^{256} + 1 = (x^2 + x + 1) P(x).\n\\]When $P$ is written in standard polynomial form, how many o... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "There is a polynomial $P$ such that for every real number $x$,\n\\[\n x^{512} + x^{256} + 1 = (x^2 + x + 1) P(x).\n\\]When $P$ is written in standard polynomial form, how many o... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the surface generated by the solutions of \\[ \\frac {dx}{yz} = \\frac {dy}{zx} = \\frac{dz}{xy}, \\] which intersects the circle $y^2+ z^2 = 1, x = 0.$ ",
"role": "use... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the surface generated by the solutions of \\[ \\frac {dx}{yz} = \\frac {dy}{zx} = \\frac{dz}{xy}, \\] which intersects the circle $y^2+ z^2 = 1, x = 0.$ ",
"role": "use... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $n\\geq1$ be a positive integer. $n$ lamps are placed in a line. At minute 0, some lamps are on (maybe all of them). Every minute the state of the lamps changes: A lamp i... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $n\\geq1$ be a positive integer. $n$ lamps are placed in a line. At minute 0, some lamps are on (maybe all of them). Every minute the state of the lamps changes: A lamp i... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider an ultra high energy proton coming from deep space. This proton collide with a photon from the cosmic microwave background. This collision is head-on. During this proces... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider an ultra high energy proton coming from deep space. This proton collide with a photon from the cosmic microwave background. This collision is head-on. During this proces... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Solve the crossword puzzle. You are presented with a clue as input and the number of letters in brackets.\n\nThey patronise beginning of Easter, crossing part of threshold in chu... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Solve the crossword puzzle. You are presented with a clue as input and the number of letters in brackets.\n\nThey patronise beginning of Easter, crossing part of threshold in chu... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Submit an integer between $0$ and $100$ inclusive as your answer to this problem. Suppose that $Q_1$ and $Q_3$ are the medians of the smallest $50\\%$ and largest $50\... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Submit an integer between $0$ and $100$ inclusive as your answer to this problem. Suppose that $Q_1$ and $Q_3$ are the medians of the smallest $50\\%$ and largest $50\... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that $x$ and $y$ are both integers between $100$ and $999$ , inclusive; $y$ is the number formed by reversing the digits of $x$ ; and $z=|x-y|$ . How many distinct values o... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that $x$ and $y$ are both integers between $100$ and $999$ , inclusive; $y$ is the number formed by reversing the digits of $x$ ; and $z=|x-y|$ . How many distinct values o... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Call a positive number $n$ *fine*, if there exists a prime number $p$ such that $p|n$ and $p^2\\nmid n$ . Prove that at least 99% of numbers $1, 2, 3, \\ldots, 10^{12}$ ... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Call a positive number $n$ *fine*, if there exists a prime number $p$ such that $p|n$ and $p^2\\nmid n$ . Prove that at least 99% of numbers $1, 2, 3, \\ldots, 10^{12}$ ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "On a long straight stretch of one-way single-lane highway, cars all travel at the same speed and all obey the safety rule: the distance from the back of the car ahead to the fron... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "On a long straight stretch of one-way single-lane highway, cars all travel at the same speed and all obey the safety rule: the distance from the back of the car ahead to the fron... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A function $f:[0,\\infty)\\to[0,\\infty)$ is integrable and $$ \\int_0^\\infty f(x)^2 dx<\\infty,\\quad \\int_0^\\infty xf(x) dx <\\infty $$ Prove the following inequality. ... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A function $f:[0,\\infty)\\to[0,\\infty)$ is integrable and $$ \\int_0^\\infty f(x)^2 dx<\\infty,\\quad \\int_0^\\infty xf(x) dx <\\infty $$ Prove the following inequality. ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the ratio of forward-bias to reverse-bias currents when the same voltage 1.5 V is applied in both forward and reverse. Assume room temperature 293 K.",
"role": "user"
... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the ratio of forward-bias to reverse-bias currents when the same voltage 1.5 V is applied in both forward and reverse. Assume room temperature 293 K.",
"role": "user"
... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Prove that for any interval $[a,b]$ of real numbers and any positive integer $n$ there exists a positive integer $k$ and a partition of the given interval\n\\[a = x (0) < x... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Prove that for any interval $[a,b]$ of real numbers and any positive integer $n$ there exists a positive integer $k$ and a partition of the given interval\n\\[a = x (0) < x... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Joanie takes a $\\$6,\\!000$ loan to pay for her car. The annual interest rate on the loan is $12\\%$. She makes no payments for 4 years, but has to pay back all the money she ... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Joanie takes a $\\$6,\\!000$ loan to pay for her car. The annual interest rate on the loan is $12\\%$. She makes no payments for 4 years, but has to pay back all the money she ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathcal{P}$ be the unique parabola in the $xy$ -plane which is tangent to the $x$ -axis at $(5,0)$ and to the $y$ -axis at $(0,12)$ . We say a line $\\ell$ is ... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathcal{P}$ be the unique parabola in the $xy$ -plane which is tangent to the $x$ -axis at $(5,0)$ and to the $y$ -axis at $(0,12)$ . We say a line $\\ell$ is ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider the following molecules: $\\mathrm{Br}_{3} \\mathrm{O}_{8}, \\mathrm{~F}_{2} \\mathrm{O}, \\mathrm{H}_{2} \\mathrm{~S}_{4} \\mathrm{O}_{6}, \\mathrm{H}_{2} \\mathrm{~S}_... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider the following molecules: $\\mathrm{Br}_{3} \\mathrm{O}_{8}, \\mathrm{~F}_{2} \\mathrm{O}, \\mathrm{H}_{2} \\mathrm{~S}_{4} \\mathrm{O}_{6}, \\mathrm{H}_{2} \\mathrm{~S}_... |
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