chosen listlengths 3 3 | rejected listlengths 3 3 |
|---|---|
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "$H_{\\alpha}$ is a prominent absorption line in the sun's spectrum with $\\lambda=656.281$. For an observer on $\\alpha$-Centauri, assuming the Jupiter-Sun orbital plane is in th... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "$H_{\\alpha}$ is a prominent absorption line in the sun's spectrum with $\\lambda=656.281$. For an observer on $\\alpha$-Centauri, assuming the Jupiter-Sun orbital plane is in th... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider any rectangular table having finitely many rows and columns, with a real \n\n number $a(r, c)$ in the cell in row $r$ and column $c$. A pair $(R, C)$, where $R$ is a set... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider any rectangular table having finitely many rows and columns, with a real \n\n number $a(r, c)$ in the cell in row $r$ and column $c$. A pair $(R, C)$, where $R$ is a set... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The numbers $1, 2, \\ldots, 2012$ are written on a blackboard. Each minute, a student goes up to the board, chooses two numbers $x$ and $y$ , erases them, and writes the num... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The numbers $1, 2, \\ldots, 2012$ are written on a blackboard. Each minute, a student goes up to the board, chooses two numbers $x$ and $y$ , erases them, and writes the num... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\lambda$ be a constant, $0 \\le \\lambda \\le 4,$ and let $f : [0,1] \\to [0,1]$ be defined by\n\\[f(x) = \\lambda x(1 - x).\\]Find the values of $\\lambda,$ $0 \\le \\lamb... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\lambda$ be a constant, $0 \\le \\lambda \\le 4,$ and let $f : [0,1] \\to [0,1]$ be defined by\n\\[f(x) = \\lambda x(1 - x).\\]Find the values of $\\lambda,$ $0 \\le \\lamb... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $A$ be a finite ring. Show that there exists two natural numbers $m,p$ where $m> p\\ge 1$ , such that $a^m=a^p$ for all $a\\in A$ .",
"role": "user"
},
{
... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $A$ be a finite ring. Show that there exists two natural numbers $m,p$ where $m> p\\ge 1$ , such that $a^m=a^p$ for all $a\\in A$ .",
"role": "user"
},
{
... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Every group of order $5\\cdot7\\cdot47=1645 is abelian, and cyclic. Is this true? Answer true or false.",
"role": "user"
},
{
"content": "The problem asks whether eve... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Every group of order $5\\cdot7\\cdot47=1645 is abelian, and cyclic. Is this true? Answer true or false.",
"role": "user"
},
{
"content": "The problem asks whether eve... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Imagine a radioactive nuclei X(Z,A) can decay into Y(Z-2, A-4) by emitting an alpha particle with partial half life 3.0 minutes. X(Z,A) can also decay into Q(Z+1,A) by decaying a... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Imagine a radioactive nuclei X(Z,A) can decay into Y(Z-2, A-4) by emitting an alpha particle with partial half life 3.0 minutes. X(Z,A) can also decay into Q(Z+1,A) by decaying a... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $a_{1}=3$, and for $n>1$, let $a_{n}$ be the largest real number such that $$4\\left(a_{n-1}^{2}+a_{n}^{2}\\right)=10 a_{n-1} a_{n}-9$$ What is the largest positive integer l... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $a_{1}=3$, and for $n>1$, let $a_{n}$ be the largest real number such that $$4\\left(a_{n-1}^{2}+a_{n}^{2}\\right)=10 a_{n-1} a_{n}-9$$ What is the largest positive integer l... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "We define a sequence of natural numbers by the initial values $a_0 = a_1 = a_2 = 1$ and the recursion $$ a_n = \\bigg \\lfloor \\frac{n}{a_{n-1}a_{n-2}a_{n-3}} \\bigg \\rflo... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "We define a sequence of natural numbers by the initial values $a_0 = a_1 = a_2 = 1$ and the recursion $$ a_n = \\bigg \\lfloor \\frac{n}{a_{n-1}a_{n-2}a_{n-3}} \\bigg \\rflo... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider the ellipse\n\n\\[\n\\frac{x^{2}}{4}+\\frac{y^{2}}{3}=1\n\\]\n\nLet $H(\\alpha, 0), 0<\\alpha<2$, be a point. A straight line drawn through $H$ parallel to the $y$-axis ... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider the ellipse\n\n\\[\n\\frac{x^{2}}{4}+\\frac{y^{2}}{3}=1\n\\]\n\nLet $H(\\alpha, 0), 0<\\alpha<2$, be a point. A straight line drawn through $H$ parallel to the $y$-axis ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A simple calculator is given to you. (It contains 8 digits and only does the operations +,-,*,/, $ \\sqrt{\\mbox{}}$ ) How can you find $ 3^{\\sqrt{2}}$ with accuracy of 6 digi... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A simple calculator is given to you. (It contains 8 digits and only does the operations +,-,*,/, $ \\sqrt{\\mbox{}}$ ) How can you find $ 3^{\\sqrt{2}}$ with accuracy of 6 digi... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "2. Determine the angle $\\alpha$ that $\\boldsymbol{\\mu}$ makes with $\\boldsymbol{B}_{\\text {eff }}$. Also, prove that the magnetization varies with time as\n\n$$\nM(t)=N \\m... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "2. Determine the angle $\\alpha$ that $\\boldsymbol{\\mu}$ makes with $\\boldsymbol{B}_{\\text {eff }}$. Also, prove that the magnetization varies with time as\n\n$$\nM(t)=N \\m... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The function $f(x)$ is of the form $a x^{2}+b x+c$ for some integers $a, b$, and $c$. Given that $\\{f(177883), f(348710), f(796921), f(858522)\\} = \\{1324754875645,178222546669... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The function $f(x)$ is of the form $a x^{2}+b x+c$ for some integers $a, b$, and $c$. Given that $\\{f(177883), f(348710), f(796921), f(858522)\\} = \\{1324754875645,178222546669... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Compute the mean molecular speed v in the light gas hydrogen (H2) in m/s",
"role": "user"
},
{
"content": "Here's a thought process to arrive at the solution for the ... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Compute the mean molecular speed v in the light gas hydrogen (H2) in m/s",
"role": "user"
},
{
"content": "Here's a thought process to arrive at the solution for the ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "How many positive integers have exactly three proper divisors (positive integral divisors excluding itself), each of which is less than 50?",
"role": "user"
},
{
"con... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "How many positive integers have exactly three proper divisors (positive integral divisors excluding itself), each of which is less than 50?",
"role": "user"
},
{
"con... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $f:[0,1] \\rightarrow[0,1]$ be the function defined by $f(x)=\\frac{x^{3}}{3}-x^{2}+\\frac{5}{9} x+\\frac{17}{36}$. Consider the square region $S=[0,1] \\times[0,1]$. Let $G=... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $f:[0,1] \\rightarrow[0,1]$ be the function defined by $f(x)=\\frac{x^{3}}{3}-x^{2}+\\frac{5}{9} x+\\frac{17}{36}$. Consider the square region $S=[0,1] \\times[0,1]$. Let $G=... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $S_{0}$ be a unit square in the Cartesian plane with horizontal and vertical sides. For any $n>0$, the shape $S_{n}$ is formed by adjoining 9 copies of $S_{n-1}$ in a $3 \\ti... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $S_{0}$ be a unit square in the Cartesian plane with horizontal and vertical sides. For any $n>0$, the shape $S_{n}$ is formed by adjoining 9 copies of $S_{n-1}$ in a $3 \\ti... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $f: \\mathbb{N} \\rightarrow \\mathbb{N}$ be a function satisfying the following conditions: (a) $f(1)=1$ (b) $f(a) \\leq f(b)$ whenever $a$ and $b$ are positive integers wit... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $f: \\mathbb{N} \\rightarrow \\mathbb{N}$ be a function satisfying the following conditions: (a) $f(1)=1$ (b) $f(a) \\leq f(b)$ whenever $a$ and $b$ are positive integers wit... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the least positive integer $n$ such that when $3^n$ is written in base $143$ , its two right-most digits in base $143$ are $01$ .",
"role": "user"
},
{
"content"... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the least positive integer $n$ such that when $3^n$ is written in base $143$ , its two right-most digits in base $143$ are $01$ .",
"role": "user"
},
{
"content"... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In the last few decades, reverberation mapping, interferometry and various other observations across the electromagnetic spectrum, have been used to probe the inner structure of ... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In the last few decades, reverberation mapping, interferometry and various other observations across the electromagnetic spectrum, have been used to probe the inner structure of ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Carl chooses a *functional expression** $E$ which is a finite nonempty string formed from a set $x_1, x_2, \\dots$ of variables and applications of a function $f$ , together... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Carl chooses a *functional expression** $E$ which is a finite nonempty string formed from a set $x_1, x_2, \\dots$ of variables and applications of a function $f$ , together... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a hexagon with vertices labeled $M$ , $M$ , $A$ , $T$ , $H$ , $S$ in that order. Clayton starts at the $M$ adjacent to $M$ and $A$ , and writes the letter do... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a hexagon with vertices labeled $M$ , $M$ , $A$ , $T$ , $H$ , $S$ in that order. Clayton starts at the $M$ adjacent to $M$ and $A$ , and writes the letter do... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For positive integers $i = 2, 3, \\ldots, 2020$ , let \\[\na_i = \\frac{\\sqrt{3i^2+2i-1}}{i^3-i}.\n\\]Let $x_2$ , $\\ldots$ , $x_{2020}$ be positive reals such that $x_2^4... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For positive integers $i = 2, 3, \\ldots, 2020$ , let \\[\na_i = \\frac{\\sqrt{3i^2+2i-1}}{i^3-i}.\n\\]Let $x_2$ , $\\ldots$ , $x_{2020}$ be positive reals such that $x_2^4... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $z_1 = 18 + 83i$ , $z_2 = 18 + 39i,$ and $z_3 = 78 + 99i,$ where $i = \\sqrt{-1}$ . Let $z$ be the unique complex number with the properties that $\\frac{z_3 - z_1}{z_2 - z_1... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $z_1 = 18 + 83i$ , $z_2 = 18 + 39i,$ and $z_3 = 78 + 99i,$ where $i = \\sqrt{-1}$ . Let $z$ be the unique complex number with the properties that $\\frac{z_3 - z_1}{z_2 - z_1... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The vector $\\begin{pmatrix} 1 \\\\ 2 \\\\ 2 \\end{pmatrix}$ is rotated $90^\\circ$ about the origin. During the rotation, it passes through the $x$-axis. Find the resulting ve... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The vector $\\begin{pmatrix} 1 \\\\ 2 \\\\ 2 \\end{pmatrix}$ is rotated $90^\\circ$ about the origin. During the rotation, it passes through the $x$-axis. Find the resulting ve... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Today is Barbara's birthday, and Alberto wants to give her a gift playing the following game. The numbers 0,1,2,...,1024 are written on a blackboard. First Barbara erases $2^{9}... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Today is Barbara's birthday, and Alberto wants to give her a gift playing the following game. The numbers 0,1,2,...,1024 are written on a blackboard. First Barbara erases $2^{9}... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Astronomers are studying a star with a 1.4 solar radius and 1.2 solar masses. When the star's surface is not covered by dark spots, its Teff is 6000K. However, when 40% of its su... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Astronomers are studying a star with a 1.4 solar radius and 1.2 solar masses. When the star's surface is not covered by dark spots, its Teff is 6000K. However, when 40% of its su... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The escalator of the department store, which at any given time can be seen at $75$ steps section, moves up one step in $2$ seconds. At time $0$ , Juku is standing on an esca... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The escalator of the department store, which at any given time can be seen at $75$ steps section, moves up one step in $2$ seconds. At time $0$ , Juku is standing on an esca... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For any positive integer $a,$ $\\sigma(a)$ denotes the sum of the positive integer divisors of $a$ . Let $n$ be the least positive integer such that $\\sigma(a^n)-1$ is divisible... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For any positive integer $a,$ $\\sigma(a)$ denotes the sum of the positive integer divisors of $a$ . Let $n$ be the least positive integer such that $\\sigma(a^n)-1$ is divisible... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The filament of a light bulb has surface area $64 \\mathrm{~mm}^{2}$. The filament can be considered as a black body at temperature $2500 \\mathrm{~K}$ emitting radiation like a ... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The filament of a light bulb has surface area $64 \\mathrm{~mm}^{2}$. The filament can be considered as a black body at temperature $2500 \\mathrm{~K}$ emitting radiation like a ... |
[
{
"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 when $1^3+2^3+3^3+\\cdots +n^3$ is divided by $n+5$ , the remainder is $17$ .",
"role": "user"
},
{
"content": "Th... | [
{
"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 when $1^3+2^3+3^3+\\cdots +n^3$ is divided by $n+5$ , the remainder is $17$ .",
"role": "user"
},
{
"content": "Th... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathcal{P}_1$ and $\\mathcal{P}_2$ be two parabolas with distinct directrices $\\ell_1$ and $\\ell_2$ and distinct foci $F_1$ and $F_2$ respectively. It is kno... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathcal{P}_1$ and $\\mathcal{P}_2$ be two parabolas with distinct directrices $\\ell_1$ and $\\ell_2$ and distinct foci $F_1$ and $F_2$ respectively. It is kno... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Denote $g(k)$ as the greatest odd divisor of $k$ . Put $f(k) = \\dfrac{k}{2} + \\dfrac{k}{g(k)}$ for $k$ even, and $2^{(k+1)/2}$ for $k$ odd. Define the sequence $x_1... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Denote $g(k)$ as the greatest odd divisor of $k$ . Put $f(k) = \\dfrac{k}{2} + \\dfrac{k}{g(k)}$ for $k$ even, and $2^{(k+1)/2}$ for $k$ odd. Define the sequence $x_1... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A scientist is doing an experiment with a setup consisting of 2 ideal solenoids that share the same axis. The lengths of the solenoids are both $\\ell$, the radii of the solenoid... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A scientist is doing an experiment with a setup consisting of 2 ideal solenoids that share the same axis. The lengths of the solenoids are both $\\ell$, the radii of the solenoid... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $A = A(x,y)$ and $B = B(x,y)$ be two-variable polynomials with real coefficients. Suppose that $A(x,y)/B(x,y)$ is a polynomial in $x$ for infinitely many values of $... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $A = A(x,y)$ and $B = B(x,y)$ be two-variable polynomials with real coefficients. Suppose that $A(x,y)/B(x,y)$ is a polynomial in $x$ for infinitely many values of $... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A2 Find the electron number density $n_{e}$ at equilibrium when both external ionizers are switched on simultaneously.",
"role": "user"
},
{
"content": "Let's break d... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A2 Find the electron number density $n_{e}$ at equilibrium when both external ionizers are switched on simultaneously.",
"role": "user"
},
{
"content": "Let's break d... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "(a) A function $f:\\mathbb{Z} \\rightarrow \\mathbb{Z}$ is called $\\mathbb{Z}$ -good if $f(a^2+b)=f(b^2+a)$ for all $a, b \\in \\mathbb{Z}$ . What is the largest possible ... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "(a) A function $f:\\mathbb{Z} \\rightarrow \\mathbb{Z}$ is called $\\mathbb{Z}$ -good if $f(a^2+b)=f(b^2+a)$ for all $a, b \\in \\mathbb{Z}$ . What is the largest possible ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the ei... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the ei... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider the additive group $\\mathbb{Z}^{2}$ . Let $H$ be the smallest subgroup containing $(3,8), (4,-1)$ and $(5,4)$ .\nLet $H_{xy}$ be the smallest subgroup containin... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider the additive group $\\mathbb{Z}^{2}$ . Let $H$ be the smallest subgroup containing $(3,8), (4,-1)$ and $(5,4)$ .\nLet $H_{xy}$ be the smallest subgroup containin... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. When at the beginning of the turn there are $n\\geq 1$ marbles ... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. When at the beginning of the turn there are $n\\geq 1$ marbles ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at ... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $m \\circ n=(m+n) /(m n+4)$. Compute $((\\cdots((2005 \\circ 2004) \\circ 2003) \\circ \\cdots \\circ 1) \\circ 0)$.",
"role": "user"
},
{
"content": "The problem... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $m \\circ n=(m+n) /(m n+4)$. Compute $((\\cdots((2005 \\circ 2004) \\circ 2003) \\circ \\cdots \\circ 1) \\circ 0)$.",
"role": "user"
},
{
"content": "The problem... |
[
{
"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 ... | [
{
"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 ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $S$ be the set of positive integer divisors of $20^9.$ Three numbers are chosen independently and at random with replacement from the set $S$ and labeled $a_1,a_2,$ and $a_3$... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $S$ be the set of positive integer divisors of $20^9.$ Three numbers are chosen independently and at random with replacement from the set $S$ and labeled $a_1,a_2,$ and $a_3$... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The positive numbers $a_1, a_2,...$ satisfy $a_1 = 1$ and $(m+n)a_{m+n }\\le a_m +a_n$ for all positive integers $m$ and $n$ . Show that $\\frac{1}{a_{200}} > 4 \\cdo... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The positive numbers $a_1, a_2,...$ satisfy $a_1 = 1$ and $(m+n)a_{m+n }\\le a_m +a_n$ for all positive integers $m$ and $n$ . Show that $\\frac{1}{a_{200}} > 4 \\cdo... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the arc length of the curve, where x=t, y=t^2 and z=2*t^3/3.",
"role": "user"
},
{
"content": "The curve is given by the parametric equations x=t, y=t^2, and z=2... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the arc length of the curve, where x=t, y=t^2 and z=2*t^3/3.",
"role": "user"
},
{
"content": "The curve is given by the parametric equations x=t, y=t^2, and z=2... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $|M|$ denote the determinant of a square matrix $M$. Let $g:\\left[0, \\frac{\\pi}{2}\\right] \\rightarrow \\mathbb{R}$ be the function defined by\n\nwhere\n\n\\[\ng(\\theta)... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $|M|$ denote the determinant of a square matrix $M$. Let $g:\\left[0, \\frac{\\pi}{2}\\right] \\rightarrow \\mathbb{R}$ be the function defined by\n\nwhere\n\n\\[\ng(\\theta)... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider the multiplicative group $A=\\{z\\in\\mathbb{C}|z^{2006^k}=1, 0<k\\in\\mathbb{Z}\\}$ of all the roots of unity of degree $2006^k$ for all positive integers $k$ .\n\... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider the multiplicative group $A=\\{z\\in\\mathbb{C}|z^{2006^k}=1, 0<k\\in\\mathbb{Z}\\}$ of all the roots of unity of degree $2006^k$ for all positive integers $k$ .\n\... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given an integer $n\\geq 2$ . There are $N$ distinct circle on the plane such that any two circles have two distinct intersections and no three circles have a common intersect... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given an integer $n\\geq 2$ . There are $N$ distinct circle on the plane such that any two circles have two distinct intersections and no three circles have a common intersect... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Welcome to the USAYNO, where each question has a yes/no answer. Choose any subset of the following six problems to answer. If you answer $n$ problems and get them all correct, yo... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Welcome to the USAYNO, where each question has a yes/no answer. Choose any subset of the following six problems to answer. If you answer $n$ problems and get them all correct, yo... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Dissociating gas cycle\n\nDinitrogen tetroxide forms an equilibrium mixture with nitrogen dioxide:\n\n$$\n\\mathrm{N}_{2} \\mathrm{O}_{4}(g) \\rightleftharpoons 2 \\mathrm{NO}_{2... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Dissociating gas cycle\n\nDinitrogen tetroxide forms an equilibrium mixture with nitrogen dioxide:\n\n$$\n\\mathrm{N}_{2} \\mathrm{O}_{4}(g) \\rightleftharpoons 2 \\mathrm{NO}_{2... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\sigma, \\tau$ be two permutations of the quantity $\\{1, 2,. . . , n\\}$ . \nProve that there is a function $f: \\{1, 2,. . . , n\\} \\to \\{-1, 1\\}$ such that for... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\sigma, \\tau$ be two permutations of the quantity $\\{1, 2,. . . , n\\}$ . \nProve that there is a function $f: \\{1, 2,. . . , n\\} \\to \\{-1, 1\\}$ such that for... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A sequence is defined as follows $a_1=a_2=a_3=1,$ and, for all positive integers $n, a_{n+3}=a_{n+2}+a_{n+1}+a_n.$ Given that $a_{28}=6090307, a_{29}=11201821,$ and $a_{30}=20603... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A sequence is defined as follows $a_1=a_2=a_3=1,$ and, for all positive integers $n, a_{n+3}=a_{n+2}+a_{n+1}+a_n.$ Given that $a_{28}=6090307, a_{29}=11201821,$ and $a_{30}=20603... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Eight spheres of radius 100 are placed on a flat surface so that each sphere is tangent to two others and their centers are the vertices of a regular octagon . A ninth sphere is... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Eight spheres of radius 100 are placed on a flat surface so that each sphere is tangent to two others and their centers are the vertices of a regular octagon . A ninth sphere is... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A trinitro compound, 1,3,5-tris-(4-nitrophenyl)benzene, on complete reaction with an excess of $\\mathrm{Sn} / \\mathrm{HCl}$ gives a major product, which on treatment with an ex... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A trinitro compound, 1,3,5-tris-(4-nitrophenyl)benzene, on complete reaction with an excess of $\\mathrm{Sn} / \\mathrm{HCl}$ gives a major product, which on treatment with an ex... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "2.2.2. Calculate the numerical value of $\\Gamma$.",
"role": "user"
},
{
"content": "Let's first understand what $\\Gamma$ refers to in the context of the textbook ex... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "2.2.2. Calculate the numerical value of $\\Gamma$.",
"role": "user"
},
{
"content": "Let's first understand what $\\Gamma$ refers to in the context of the textbook ex... |
[
{
"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 \\[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": "2.3. Assume that the atmospheric temperature depends on altitude according to the relation $T(z)=T(0)-\\Lambda z$, where $\\Lambda$ is a constant. Find the dependence of the parc... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "2.3. Assume that the atmospheric temperature depends on altitude according to the relation $T(z)=T(0)-\\Lambda z$, where $\\Lambda$ is a constant. Find the dependence of the parc... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose $E$ , $I$ , $L$ , $V$ are (not necessarily distinct) nonzero digits in base ten for which\n\n\n- the four-digit number $\\underline{E}\\ \\underline{V}\\ \\underlin... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose $E$ , $I$ , $L$ , $V$ are (not necessarily distinct) nonzero digits in base ten for which\n\n\n- the four-digit number $\\underline{E}\\ \\underline{V}\\ \\underlin... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $x,$ $y,$ $z$ be positive real numbers such that\n\\[\\left( \\frac{x}{y} + \\frac{y}{z} + \\frac{z}{x} \\right) + \\left( \\frac{y}{x} + \\frac{z}{y} + \\frac{x}{z} \\right)... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $x,$ $y,$ $z$ be positive real numbers such that\n\\[\\left( \\frac{x}{y} + \\frac{y}{z} + \\frac{z}{x} \\right) + \\left( \\frac{y}{x} + \\frac{z}{y} + \\frac{x}{z} \\right)... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the vector $\\mathbf{v}$ such that\n\\[\\mathbf{i} \\times [(\\mathbf{v} - \\mathbf{j}) \\times \\mathbf{i}] + \\mathbf{j} \\times [(\\mathbf{v} - \\mathbf{k}) \\times \\mat... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the vector $\\mathbf{v}$ such that\n\\[\\mathbf{i} \\times [(\\mathbf{v} - \\mathbf{j}) \\times \\mathbf{i}] + \\mathbf{j} \\times [(\\mathbf{v} - \\mathbf{k}) \\times \\mat... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "You have a 10 uL aliquot of a 10 uM DNA template of a protein library. The template contains 12 NNK codons in the coding region. What is the order of magnitude of the maximum pos... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "You have a 10 uL aliquot of a 10 uM DNA template of a protein library. The template contains 12 NNK codons in the coding region. What is the order of magnitude of the maximum pos... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by $\\overline{AB}$ , has length 31. Find the sum of the lengths of the three diagonal... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by $\\overline{AB}$ , has length 31. Find the sum of the lengths of the three diagonal... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For a positive real number $x > 1,$ the Riemann zeta function $\\zeta(x)$ is defined by\n\\[\\zeta(x) = \\sum_{n = 1}^\\infty \\frac{1}{n^x}.\\]Compute\n\\[\\sum_{k = 2}^\\infty ... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For a positive real number $x > 1,$ the Riemann zeta function $\\zeta(x)$ is defined by\n\\[\\zeta(x) = \\sum_{n = 1}^\\infty \\frac{1}{n^x}.\\]Compute\n\\[\\sum_{k = 2}^\\infty ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $R$ be a square region and $n \\ge 4$ an integer. A point $X$ in the interior of $R$ is called *n-ray partitional* if there are $n$ rays emanating from $X$ that d... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $R$ be a square region and $n \\ge 4$ an integer. A point $X$ in the interior of $R$ is called *n-ray partitional* if there are $n$ rays emanating from $X$ that d... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For integers $k\\ (0\\leq k\\leq 5)$ , positive numbers $m,\\ n$ and real numbers $a,\\ b$ , let $f(k)=\\int_{-\\pi}^{\\pi}(\\sin kx-a\\sin mx-b\\sin nx)^{2}\\ dx$ , \r $p(k... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For integers $k\\ (0\\leq k\\leq 5)$ , positive numbers $m,\\ n$ and real numbers $a,\\ b$ , let $f(k)=\\int_{-\\pi}^{\\pi}(\\sin kx-a\\sin mx-b\\sin nx)^{2}\\ dx$ , \r $p(k... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $f : \\mathbb{C} \\to \\mathbb{C} $ be defined by $ f(z) = z^2 + iz + 1 $. How many complex numbers $z $ are there such that $ \\text{Im}(z) > 0 $ and both the real and the i... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $f : \\mathbb{C} \\to \\mathbb{C} $ be defined by $ f(z) = z^2 + iz + 1 $. How many complex numbers $z $ are there such that $ \\text{Im}(z) > 0 $ and both the real and the i... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $ f(t)$ be a continuous function on the interval $ 0 \\leq t \\leq 1$ , and define the two sets of points \\[ A_t\\equal{}\\{(t,0): t\\in[0,1]\\} , B_t\\equal{}\\{(f(t),1)... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $ f(t)$ be a continuous function on the interval $ 0 \\leq t \\leq 1$ , and define the two sets of points \\[ A_t\\equal{}\\{(t,0): t\\in[0,1]\\} , B_t\\equal{}\\{(f(t),1)... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $D$ be the complex unit disk $D=\\{z \\in \\mathbb{C}: |z|<1\\}$ , and $0<a<1$ a real number. Suppose that $f:D \\to \\mathbb{C}\\setminus \\{0\\}$ is a holomorphic fu... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $D$ be the complex unit disk $D=\\{z \\in \\mathbb{C}: |z|<1\\}$ , and $0<a<1$ a real number. Suppose that $f:D \\to \\mathbb{C}\\setminus \\{0\\}$ is a holomorphic fu... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Paul and Sara are playing a game with integers on a whiteboard, with Paul going first. When it is Paul's turn, he can pick any two integers on the board and replace them with the... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Paul and Sara are playing a game with integers on a whiteboard, with Paul going first. When it is Paul's turn, he can pick any two integers on the board and replace them with the... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The cards in a stack of $2n$ cards are numbered consecutively from 1 through $2n$ from top to bottom. The top $n$ cards are removed, kept in order, and form pile $A.$ The remaini... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The cards in a stack of $2n$ cards are numbered consecutively from 1 through $2n$ from top to bottom. The top $n$ cards are removed, kept in order, and form pile $A.$ The remaini... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "During a quantum communication task of a quantum dialogue for a two-mode entangled squeezed continuous variable state under noiseless quantum channel such that the two modes gene... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "During a quantum communication task of a quantum dialogue for a two-mode entangled squeezed continuous variable state under noiseless quantum channel such that the two modes gene... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a 900 Kbytes file stored in a web server. Client A sends a request to the server to retrieve the file from a remote location. There are 3 links (2 intermediate nodes) be... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a 900 Kbytes file stored in a web server. Client A sends a request to the server to retrieve the file from a remote location. There are 3 links (2 intermediate nodes) be... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\omega = e^{2 \\pi i/1729}.$ Compute\n\\[\\sum_{k = 1}^{1728} \\frac{1}{1 + \\omega^k + \\omega^{2k} + \\omega^{3k}}.\\]",
"role": "user"
},
{
"content": "The ... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\omega = e^{2 \\pi i/1729}.$ Compute\n\\[\\sum_{k = 1}^{1728} \\frac{1}{1 + \\omega^k + \\omega^{2k} + \\omega^{3k}}.\\]",
"role": "user"
},
{
"content": "The ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In decimal representation $$ \\text {34!=295232799039a041408476186096435b0000000}. $$ Find the numbers $a$ and $b$ .",
"role": "user"
},
{
"content": "The prob... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In decimal representation $$ \\text {34!=295232799039a041408476186096435b0000000}. $$ Find the numbers $a$ and $b$ .",
"role": "user"
},
{
"content": "The prob... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let \\[P(x) = (2x^4 - 26x^3 + ax^2 + bx + c)(5x^4 - 80x^3 + dx^2 + ex + f),\\]where $a, b, c, d, e, f$ are real numbers. Suppose that the set of all complex roots of $P(x)$ is $\... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let \\[P(x) = (2x^4 - 26x^3 + ax^2 + bx + c)(5x^4 - 80x^3 + dx^2 + ex + f),\\]where $a, b, c, d, e, f$ are real numbers. Suppose that the set of all complex roots of $P(x)$ is $\... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In trapezoid $ABCD$ with $\\overline{BC}\\parallel\\overline{AD}$, let $BC = 1000$ and $AD = 2008$. Let $\\angle A = 37^\\circ$, $\\angle D = 53^\\circ$, and $M$ and $N$ be the m... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In trapezoid $ABCD$ with $\\overline{BC}\\parallel\\overline{AD}$, let $BC = 1000$ and $AD = 2008$. Let $\\angle A = 37^\\circ$, $\\angle D = 53^\\circ$, and $M$ and $N$ be the m... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $S$ be the sum of all numbers of the form $\\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive divisors of $1000.$ What is the greatest integer that does not exce... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $S$ be the sum of all numbers of the form $\\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive divisors of $1000.$ What is the greatest integer that does not exce... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $z$ be a non-real complex number with $z^{23}=1$. Compute $$ \\sum_{k=0}^{22} \\frac{1}{1+z^{k}+z^{2 k}} $$",
"role": "user"
},
{
"content": "The problem asks for... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $z$ be a non-real complex number with $z^{23}=1$. Compute $$ \\sum_{k=0}^{22} \\frac{1}{1+z^{k}+z^{2 k}} $$",
"role": "user"
},
{
"content": "The problem asks for... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $n$ be a positive integer. Given a sequence $\\varepsilon_1$ , $\\dots$ , $\\varepsilon_{n - 1}$ with $\\varepsilon_i = 0$ or $\\varepsilon_i = 1$ for each $i = 1... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $n$ be a positive integer. Given a sequence $\\varepsilon_1$ , $\\dots$ , $\\varepsilon_{n - 1}$ with $\\varepsilon_i = 0$ or $\\varepsilon_i = 1$ for each $i = 1... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $f(t)$ be the cubic polynomial for $t$ such that $\\cos 3x=f(\\cos x)$ holds for all real number $x$ .\r\nEvaluate\r\n\r\n\\[\\int_0^1 \\{f(t)\\}^2 \\sqrt{1-t^2}dt\\]"... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $f(t)$ be the cubic polynomial for $t$ such that $\\cos 3x=f(\\cos x)$ holds for all real number $x$ .\r\nEvaluate\r\n\r\n\\[\\int_0^1 \\{f(t)\\}^2 \\sqrt{1-t^2}dt\\]"... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $ A$ be a unitary finite ring with $ n$ elements, such that the equation $ x^n\\equal{}1$ has a unique solution in $ A$ , $ x\\equal{}1$ . Prove that \r\n\r\na) $ 0$... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $ A$ be a unitary finite ring with $ n$ elements, such that the equation $ x^n\\equal{}1$ has a unique solution in $ A$ , $ x\\equal{}1$ . Prove that \r\n\r\na) $ 0$... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "2.1 Find the ratio $\\frac{\\rho_{i} T_{i}}{\\rho_{a} T_{a}}$ in terms of $\\gamma, P_{a}$ and $R_{0}$.",
"role": "user"
},
{
"content": "The problem is about finding... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "2.1 Find the ratio $\\frac{\\rho_{i} T_{i}}{\\rho_{a} T_{a}}$ in terms of $\\gamma, P_{a}$ and $R_{0}$.",
"role": "user"
},
{
"content": "The problem is about finding... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Show that there exists a continuos function $f: [0,1]\\rightarrow [0,1]$ such that it has no periodic orbit of order $3$ but it has a periodic orbit of order $5$ .",
"ro... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Show that there exists a continuos function $f: [0,1]\\rightarrow [0,1]$ such that it has no periodic orbit of order $3$ but it has a periodic orbit of order $5$ .",
"ro... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Compute the nearest integer to $$100 \\sum_{n=1}^{\\infty} 3^{n} \\sin ^{3}\\left(\\frac{\\pi}{3^{n}}\\right)$$",
"role": "user"
},
{
"content": "The problem asks for... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Compute the nearest integer to $$100 \\sum_{n=1}^{\\infty} 3^{n} \\sin ^{3}\\left(\\frac{\\pi}{3^{n}}\\right)$$",
"role": "user"
},
{
"content": "The problem asks for... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The oxidation state of a carbon atom in an organic molecule varies depending on the number and nature of directly attached substituents. Depending upon the oxidation state of the... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The oxidation state of a carbon atom in an organic molecule varies depending on the number and nature of directly attached substituents. Depending upon the oxidation state of the... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a file with a size of 350 Kbytes storing in a web server. Client A sends a request to the server to retrieve the file from a remote location. It is known that the link c... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a file with a size of 350 Kbytes storing in a web server. Client A sends a request to the server to retrieve the file from a remote location. It is known that the link c... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all sets of real numbers $S$ such that:\n\n\n\n- $1$ is the smallest element of $S$ ,\n- for all $x,y\\in S$ such that $x>y$ , $\\sqrt{x^2-y^2}\\in S$\n\n\n*Adi... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all sets of real numbers $S$ such that:\n\n\n\n- $1$ is the smallest element of $S$ ,\n- for all $x,y\\in S$ such that $x>y$ , $\\sqrt{x^2-y^2}\\in S$\n\n\n*Adi... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The root of the equation x = (1 / 2) + sin x by using the iteration method: x_{k+1} = 1/2 + sin(x_k), x_0 = 1 correct to o six decimals is x = 1.497300. Determine the number of i... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The root of the equation x = (1 / 2) + sin x by using the iteration method: x_{k+1} = 1/2 + sin(x_k), x_0 = 1 correct to o six decimals is x = 1.497300. Determine the number of i... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For nonnegative integers $a$ and $b$ with $a + b \\leq 6$ , let $T(a, b) = \\binom{6}{a} \\binom{6}{b} \\binom{6}{a + b}$ . Let $S$ denote the sum of all $T(a, b)$ , where $a$ an... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For nonnegative integers $a$ and $b$ with $a + b \\leq 6$ , let $T(a, b) = \\binom{6}{a} \\binom{6}{b} \\binom{6}{a + b}$ . Let $S$ denote the sum of all $T(a, b)$ , where $a$ an... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A decrease in activity in the electrophilic mononitration reaction (given that in all cases the mechanism of the reaction is the same) is observed in the sequence:\n1.\t 3-methyl... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A decrease in activity in the electrophilic mononitration reaction (given that in all cases the mechanism of the reaction is the same) is observed in the sequence:\n1.\t 3-methyl... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let G, H be two finite groups, and let $\\varphi, \\psi: G \\to H$ be two surjective but non-injective homomorphisms. Prove that there exists an element of G that is not the id... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let G, H be two finite groups, and let $\\varphi, \\psi: G \\to H$ be two surjective but non-injective homomorphisms. Prove that there exists an element of G that is not the id... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For $1 \\leq i \\leq 215$ let $a_i = \\dfrac{1}{2^{i}}$ and $a_{216} = \\dfrac{1}{2^{215}}$. Let $x_1, x_2, \\dots, x_{216}$ be positive real numbers such that $\\sum_{i=1}^{216}... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For $1 \\leq i \\leq 215$ let $a_i = \\dfrac{1}{2^{i}}$ and $a_{216} = \\dfrac{1}{2^{215}}$. Let $x_1, x_2, \\dots, x_{216}$ be positive real numbers such that $\\sum_{i=1}^{216}... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In trapezoid $ABCD$ with $\\overline{BC}\\parallel\\overline{AD}$ , let $BC = 1000$ and $AD = 2008$ . Let $\\angle A = 37^\\circ$ , $\\angle D = 53^\\circ$ , and $M$ and $N$ be t... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In trapezoid $ABCD$ with $\\overline{BC}\\parallel\\overline{AD}$ , let $BC = 1000$ and $AD = 2008$ . Let $\\angle A = 37^\\circ$ , $\\angle D = 53^\\circ$ , and $M$ and $N$ be t... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "You are given integrable random variables \\( X, Y_0 \\) and \\( Z_0 \\) on the same probability space \\( (\\Omega, \\mathcal{F}, \\mathbb{P}) \\), and two \\( \\sigma \\)-algeb... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "You are given integrable random variables \\( X, Y_0 \\) and \\( Z_0 \\) on the same probability space \\( (\\Omega, \\mathcal{F}, \\mathbb{P}) \\), and two \\( \\sigma \\)-algeb... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Alice Czarina is bored and is playing a game with a pile of rocks. The pile initially contains 2015 rocks. At each round, if the pile has $N$ rocks, she removes $k$ of them, wher... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Alice Czarina is bored and is playing a game with a pile of rocks. The pile initially contains 2015 rocks. At each round, if the pile has $N$ rocks, she removes $k$ of them, wher... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $N$ be the number of ordered triples $(A,B,C)$ of integers satisfying the conditions: (a) $0\\le A<B<C\\le99$ , (b) there exist integers $a$ , $b$ , and $c$ , and prime $p$ w... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $N$ be the number of ordered triples $(A,B,C)$ of integers satisfying the conditions: (a) $0\\le A<B<C\\le99$ , (b) there exist integers $a$ , $b$ , and $c$ , and prime $p$ w... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "(1) Find the range of $a$ for which there exist two common tangent lines of the curve $y=\\frac{8}{27}x^3$ and the parabola $y=(x+a)^2$ other than the $x$ axis.\n\n(2) Fo... | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "(1) Find the range of $a$ for which there exist two common tangent lines of the curve $y=\\frac{8}{27}x^3$ and the parabola $y=(x+a)^2$ other than the $x$ axis.\n\n(2) Fo... |
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