messages listlengths 3 3 |
|---|
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABCD\\) be a square with center \\(E\\). \\(M\\) is the midpoint of \\(AB\\). \\(G\\) is a point on side \\(BC\\) and \\(H\\) is a point on side \\(CD\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A permutation $a_1, a_2, \\cdots, a_6$ of the numbers $1, 2, \\cdots, 6$ can be transformed to $1, 2, \\cdots, 6$ by transposing two numbers exactly four time... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( a_{n} = \\sum_{i=1}^{n} \\exp \\left( \\frac{i}{n} \\right) \\). Study the existence and find the value, if it exists, of the limit \\( \\lim_{n \\to ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( f: \\mathbb{R} \\to \\mathbb{R} \\) be a function satisfying the inequality\n\\[ f(x) \\leq \\frac{f(x_1) \\sinh(x_2 - x) + f(x_2) \\sinh(x - x_1)}{\\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the least natural number $k$ such that there are only a finite number of pairs of consecutive primes whose sum is divisible by $k$."
},
{
"role":... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Describe the locus of points that are equidistant from two parallel lines AB and CD, which are 8 inches apart."
},
{
"role": "assistant",
"content... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( a_n = \\frac{n!}{(\\alpha+1)(\\alpha+2)\\cdots(\\alpha+n)} \\) for \\( \\alpha > 1 \\). Calculate the sum of the series \\( \\sum_{n=1}^{\\infty} a_n ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In $\\varDelta{ABC}$, the excircle opposite to $A$ is tangent to segment $BC$, line $AB$, and $AC$ at points $E$, $D$, and $F$ respectively. $EZ$ is the diame... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given that \\( f(0) = 0 \\) and \\( f'(x) \\in (0,1] \\), determine the least integer \\( k \\) such that \n\\[\n\\frac{\\int_{0}^{1} f'(x) \\, dx}{\\int_{0}^... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the following system of equations for real and complex values of \\(a\\), \\(b\\), and \\(c\\):\n\\[\na + b + c = 0\n\\]\n\\[\na^2 + b^2 + c^2 = 24\n\\]... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(AB\\) and \\(CD\\) be nonintersecting chords of a circle, and let \\(K\\) be a point on \\(CD\\). Construct a point \\(P\\) on the circle such that \\(... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( x, y, z \\) be positive real numbers such that \\( xyz = xy + xz + yz \\). Find the minimum value of \\( (x - 1)(y - 1)(z - 1) \\)."
},
{
"rol... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( p \\) be a prime number. Find the remainder when \\( \\binom{2p}{p} \\) is divided by \\( p \\)."
},
{
"role": "assistant",
"content": "<t... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Kelvin the Frog stands at one vertex of a square. Every minute, he rolls a standard die and moves that many vertices clockwise. After 4 minutes, what is the p... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In $\\triangle ABC$, where $AB=6$, $AC=8$, and $BC=10$, point $D$ lies on side $AC$. The incircles of $\\triangle ABD$ and $\\triangle ACD$ are drawn. What is... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Suppose $\\triangle ABC$ is an equilateral triangle with side length $3$. Points $D$, $E$, and $F$ are taken on sides $BC$, $CA$, and $AB$ respectively, such ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the integral \\( \\int_{0}^{1} \\frac{\\ln(x+1)}{\\ln(2+x-x^{2})}\\ dx \\)."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's t... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all functions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) such that:\n\\[ f(2yf(x) + x^2) + f(y^2) = f^2(x + y), \\quad \\forall x, y \\in \\mathbb{R... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the limit \\(\\lim_{x \\to \\infty} \\left( \\sum_{n=1}^{\\infty} \\left( \\frac{x}{n} \\right)^n \\right)^{1/x}\\)."
},
{
"role": "assistant... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given the equation \\((x-1)(x-2)(x-4)(x-5)(x-7)(x-8) = (x-3)(x-6)(x-9)\\) with 6 distinct real solutions \\(r_1, r_2, r_3, r_4, r_5, r_6\\), find the value of... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine if the set \\( H = \\left\\{ \\begin{pmatrix} 1 & 2 & 3 \\\\ 1 & 2 & 3 \\end{pmatrix}, \\begin{pmatrix} 1 & 2 & 3 \\\\ 2 & 3 & 1 \\end{pmatrix}, \\b... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "What is the expected magnitude of the sum of two unit vectors in the Cartesian plane if the directions of both vectors are randomly chosen from the interval $... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( n \\geq 2 \\) be an integer. Suppose there exist nonnegative integers \\( a \\) and \\( b \\) such that \n\\[ 2 + 3 + 4 + \\cdots + n = 2^a \\cdot 3^b... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "What is the smallest positive integer greater than 1 that leaves a remainder of 1 when divided by $2^{2^2}$ and $2^{3^3}$? (Note: $a^{b^c} = (a^b)^c$)"
},
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given that $\\int_{-\\infty}^{\\infty} e^{-\\frac{1}{2}x^2} dx = \\sqrt{2\\pi}$ and using the substitution $y = \\frac{1}{2} (\\sqrt{a}x - \\sqrt{b} /x)$, fin... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In triangle \\(ABC\\), let \\((O_9)\\) be the nine-point circle, \\(H\\) be the orthocenter, and \\(AD\\), \\(BE\\), \\(CF\\) be the altitudes. Let \\(M\\) an... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Calculate the integral \\( I = \\int \\frac{xe^x}{1+e^x} \\, dx \\)."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to solve the ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Monic quadratic polynomials \\( P(x) \\) and \\( Q(x) \\) have the property that \\( P(Q(x)) \\) has zeroes at \\( x = -23, -21, -17, \\text{and} -15 \\), and... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "If \\(a, b, c\\) are positive numbers such that \\(ab + bc + ca = 3\\) and \\(p \\ge 3 + \\sqrt{6}\\), calculate the minimum value of the expression\n\\[\n\\f... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For a prime number \\( p \\) and a natural number \\( n > 1 \\), let \\( k \\) be the largest integer such that \\( p^k \\) divides \\( \\binom{np}{p} - n \\)... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the points where the function $f$ is continuous."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to figure out where the func... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In an acute triangle \\(ABC\\) with \\(A > \\frac{\\pi}{3}\\), calculate the minimum value of the expression \\(2 \\sin A \\sin B \\sin C - \\cos A \\cos B \\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(a, b, c, d\\) be non-negative numbers such that \\(\\min(a, b, c, d) \\geq abcd\\). Define \\(f(a, b, c, d) = 4(1 + abcd) + 2\\sqrt{(1 + a^2)(1 + b^2)(... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For positive numbers \\(x, y, z\\), let \\(S_1 = \\sum_{cyc} \\frac{x}{\\sqrt{2(x^2 + y^2)}}\\) and \\(S_2 = \\sum_{cyc} \\frac{4x^2 + y^2}{x^2 + 4y^2}\\). Ca... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Describe the locus of points $K$ in $\\triangle ABC$ such that $\\frac{AB}{BK} = \\frac{AC}{CK}$."
},
{
"role": "assistant",
"content": "<think>\n... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given the equation \\( x^{2011} + x^{2010} + \\cdots + x + m = 0 \\) with \\( m \\in (0,1) \\) and the only real root being \\( a \\), compute the value of \\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider a square \\(ABCD\\) in 3D space where \\(A = (0, 0, 0)\\) and the side length of the square is an integer. Let \\(B\\) and \\(D\\) be primitive latti... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A number is chosen at random from the set of consecutive natural numbers $\\{1, 2, 3, \\ldots, 24\\}$. What is the probability that the number chosen is a fac... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In an equilateral triangle \\(ABC\\), point \\(P\\) is such that \\(PA = 4\\), \\(PB = 5\\), and \\(PC = 3\\). Find the perimeter \\(AB + BC + AC\\)."
},
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For which $ n \\in \\mathbb{N}$ do there exist rational numbers $ a, b$ which are not integers such that both $ a + b$ and $ a^n + b^n$ are integers?"
},
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For any positive integer \\( n \\), denote by \\( f(n) \\) the smallest positive integer \\( m \\) such that the sum \\( 1 + 2 + \\cdots + m \\) is divisible ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "If the sides of a triangle are 8, 15, and 17, what is the length of the shortest altitude to the nearest tenth?"
},
{
"role": "assistant",
"conten... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given a triangle \\(ABC\\) with incenter \\(I\\), let \\(XYZ\\) be the \\(I\\)-cevian triangle of \\(ABC\\). Points \\(A'\\), \\(B'\\), and \\(C'\\) are on se... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all functions $F \\in C(X)$ such that $F(\\mathbb{U}) \\subset \\mathbb{U}$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I have ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the value of $\\sum^{49}_{n=1}\\frac{1}{\\sqrt{n+\\sqrt{n^{2}-1}}}$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine the nature of the series $\\sum {u_n}^{a}$ where $a \\in \\mathbb{R}$ and $u_n=\\frac{1}{\\int_{n}^{n+1}\\ln(f(t))\\,dt}$ with $f(t)=\\int_{0}^{+\\i... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all pairs of 2018 rational numbers such that if any two numbers are deleted, the remaining numbers can be divided into 4 groups with the same product of ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given triangle \\(ABC\\) with \\(AB = AC\\) and \\(m(\\widehat{ABC}) = 54^\\circ\\). If \\(P\\) is a point inside the triangle such that \\(\\widehat{PBC} = 1... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In an isosceles triangle \\(ABC\\) with base angles at \\(B\\) and \\(C\\) measuring 40 degrees each, the bisector of angle \\(B\\) intersects \\(AC\\) at poi... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate $\\sum_{i=0}^{\\infty}\\sum_{j=0}^{\\infty}\\sum_{k=0}^{\\infty}\\frac{1}{5^{i}5^{j}5^{k}}$, where $i \\neq j \\neq k$."
},
{
"role": "assist... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( A, B, C, D, \\) and \\( E \\) be 5 points, in this order, on a semicircle with radius 1. Let \\( AB = a \\), \\( BC = b \\), \\( CD = c \\), and \\( D... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( x, y, z \\) be positive real numbers such that \\( x + y + z = 2 \\). Find the maximum value of:\n\\[ P = \\frac{x}{\\sqrt{x + \\frac{3}{4}yz}} + \\fr... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given \\( x = \\frac{a + b\\sqrt{c}}{d} \\), where \\( a, b, c, d \\) are integers, and the equation \\( \\frac{5x}{6} + 1 = \\frac{3}{x} \\), find the maximu... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Bob's password consists of a non-negative single-digit number followed by a letter and another non-negative single-digit number (which could be the same as th... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In \\(\\triangle ABC\\), \\(\\overline{AD} \\perp \\overline{BC}\\), \\(\\overline{AE}\\) bisects \\(\\angle BAC\\), and \\(\\overline{AF}\\) is the median of... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given that the sequence $a_0, a_1, a_2, \\ldots, a_{2015}$ satisfies $a_0 = a_{2015}$ and each term is a positive number. Suppose this sequence can also be de... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all sets $(p,q) \\in \\mathbb{N}^2$ such that for the integer sequence defined by $a_{n} = pa_{n-1} + qa_{n-2}$, the property $(a_{i}, a_{j}) = a_{(i,j)}... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider a triangle \\( A_1A_2A_3 \\) with circumcircle \\( \\Omega \\). For each \\( i \\in \\{1, 2, 3\\} \\), let \\( \\Gamma_i \\) be a circle passing thro... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find $\\Phi_{pq}(1)$ where $\\Phi_n(x)$ is the $n$-th cyclotomic polynomial and $p, q$ are distinct primes."
},
{
"role": "assistant",
"content": ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(a\\) and \\(b\\) be relatively prime positive integers such that \\[0.03 + 0.0033 + 0.000333 + 0.00003333 + \\ldots = \\frac{a}{b},\\] where each decim... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Rectangle \\( PQRS \\) lies in a plane with \\( PQ = RS = 2 \\) and \\( QR = SP = 6 \\). The rectangle is rotated \\( 90^\\circ \\) clockwise about \\( R \\),... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "What relationship do the convergents of $\\sqrt{d}$ have with the solutions to the equation $x^2 - d y^2 = 1$ or $x^2 - d y^2 = -1$?"
},
{
"role": "as... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine the largest integer \\(n\\) with \\(n \\leq 9,000,000\\) for which \\(\\sqrt{n+\\sqrt{n+\\sqrt{n+\\sqrt{n+\\cdots}}}}\\) is rational."
},
{
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Which is greater, $\\sqrt[2012]{2013!}$ or $\\sqrt[2013]{2012!}$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to figure out wh... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all integer solutions of the equation $x^4 + y^4 + z^4 = 2x^2y^2 + 2y^2z^2 + 2z^2x^2 + 24$."
},
{
"role": "assistant",
"content": "<think>\nO... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the function \\( f(\\theta) \\) such that\n\\[ f'(\\theta) = \\theta \\cos{2\\theta} + f(\\theta). \\]"
},
{
"role": "assistant",
"content": ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider the sequence of integers \\( n \\) such that \\( n^2 + 1 \\) is square-free. Let \\( S \\) be the set of the first 100 such integers \\( n \\). Calcu... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let a recursive sequence \\( x_0, x_1, x_2, \\ldots \\) satisfy \\( x_0^2 = 1 \\) and for all \\( n \\in \\mathbb{Z} \\),\n\n\\[ x_n = \\frac{\\cos\\left( n \... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Calculate the subtraction of the number of \"good\" numbers minus the number of \"bad\" numbers for integers less than or equal to 2023, where a \"good\" numb... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( S \\) be the point of intersection of the two lines \\( l_1 : 7x - 5y + 8 = 0 \\) and \\( l_2 : 3x + 4y - 13 = 0 \\). Let \\( P = (3, 7) \\) and \\( Q... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all the limit points of the sequence $n\\pi - \\lfloor n\\pi \\rfloor$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to fi... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In a work group with 25 members, meetings are held in three different rooms consisting of 3, 4, and 5 members each day. Each member must participate in a meet... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many natural numbers from 1 to 1000 have the property that every digit is greater than or equal to the previous digit? For example, 148, 112, etc."
},
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given positive real numbers \\(a, b, c\\) such that \\(a^2 + b^2 + c^2 = 1\\), calculate the maximum value of the expression\n\n\\[\n\\frac{a+b}{\\sqrt{ab+1}}... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given positive real numbers \\(a, b, c\\) such that \\(a + b + c = 1\\), find the minimum value of the expression \\(\\frac{1}{a^2 - 2a + 5} + \\frac{1}{b^2 -... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many possible values for $(a, b, c, d, e)$ satisfy the equation $a + b + c + d + e = 20$ and the inequality $a < b < c < d < e$, where $a, b, c, d, e$ are... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider the fields $\\mathbf{X_i}$ and $\\mathbf{Y_i}$ on a manifold $M$. Let $H \\in \\varepsilon(M)$ be a function such that $H = a_1\\mathbf{X_1} + a_2\\m... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many solutions in non-negative integers \\((a, b, c)\\) are there to the equation \\(2^a + 2^b = c!\\)?"
},
{
"role": "assistant",
"content": ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all functions \\( f: \\mathbb{R}^+ \\to \\mathbb{R}^+ \\) such that\n\\[ f(xf(y)) = yf(x) \\]\nand\n\\[ \\lim_{x \\to \\infty} f(x) = 0. \\]"
},
{
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "It takes 24 minutes for Jana to walk one mile. At that rate, how far in miles will she walk in 10 minutes? Express your answer as a decimal to the nearest ten... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For \\( a \\in \\mathbb{R} \\), find the minimum value of \\( \\int_0^{\\frac{\\pi}{2}} \\left| \\frac{\\sin 2x}{1 + \\sin^2 x} - a \\cos x \\right| \\, dx \\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine the smallest positive integer \\( n \\) such that for every integer \\( i \\) between 1 and 6, inclusive, there exists a positive integer \\( a_i \\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A mail carrier delivers mail to the nineteen houses on the east side of Elm Street. The carrier notices that no two adjacent houses ever get mail on the same ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many sequences $(u_n)$ satisfy $u_{2012} = \\frac{1}{2}$ and $u_{n+1} + 4u_n^2 - 4u_n = 0$ for all $n \\geq 1$?"
},
{
"role": "assistant",
"co... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( X = \\{1, 2, \\ldots, 12\\} \\). A subset \\( G \\) of \\( X \\) is called a \"good\" subset if for every element \\( a \\) in \\( G \\), the sets \\(... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given that \\(a_i \\in \\mathbb{R}\\) for \\(0 < i < 2009\\), and the conditions \\(a_1 + a_2 + \\cdots + a_{2008} = 0\\) and \\(a_1^2 + a_2^2 + \\cdots + a_{... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider a nontrivial group \\( G \\) such that \\( \\text{Aut}(G) \\) is transitive on \\( G \\setminus \\{e\\} \\). If \\( G \\) has a trivial center, calcu... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all integer numbers \\( A \\) such that \\( A \\) has 2009 digits, \\( 3 \\mid A \\), and all digits of \\( A \\) are in the set \\(\\{2, 3, 7, 9\\}\\)."... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In how many ways can 8 rooks be placed on a chessboard so that none threatens any other, if the rooks are distinguishable from each other?"
},
{
"role... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Deduce the value of:\n\\[ \\int^{\\frac{1}{2}}_{0} \\frac{\\sin(\\pi t)}{1-t} \\, dt \\]"
},
{
"role": "assistant",
"content": "<think>\nOkay, so ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For any finite set $X$, let $| X |$ denote the number of elements in $X$. Define $$S_n = \\sum | A \\cap B |,$$ where the sum is taken over all ordered pairs ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Alan and Charlie are located at vertices A and C of a rectangular track ABCD with dimensions AB=100 m and BC=50 m. Alan starts biking towards vertex B at a sp... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Calculate the value of \\(( [4x^2 + 3x + 1] - [x + 1]^{2005} ) ( [3x^{13} + 3] [2x + 4] )\\) in the field \\(F_{25} = Z_5[x] / \\langle x^2 + 2x + 3 \\rangle\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all functions \\( f : \\mathbb{R} \\rightarrow \\mathbb{R} \\) such that \\( f(x) + f(-x) = x f(x^2) + 2 \\) for all real numbers \\( x \\)."
},
{
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the solutions to the congruence $12x^2 + 25x \\equiv 10 \\pmod{11}$."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. I need ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let $x$ be a positive real number. What is the maximum value of $\\dfrac{2022x^2 \\log(x+2022)}{\\left(\\log(x+2022)\\right)^3 + 2x^3}$?"
},
{
"role":... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Bob thinks of a number between 1 and 32. Alice can ask Bob any yes-or-no questions she wants. How can Alice figure out Bob's number with at most five question... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider a finite group \\( G \\) of rank 2. Let \\( s(G) \\) denote the length of the commutator series of \\( G \\). Define \\( S \\) as the maximum value o... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given the sequence defined by \\( x_0 > 0 \\) and \\( x_{n+1} = x_n - e^{\\frac{-1}{x_n^2}} \\), find the asymptotic expansion of \\( x_n \\) with three terms... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given that $\\binom{17}{7}=19448$, $\\binom{17}{8}=24310$ and $\\binom{17}{9}=24310$, calculate $\\binom{19}{9}$."
},
{
"role": "assistant",
"cont... |
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