messages listlengths 3 3 |
|---|
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For any number \\( x \\), let \\( \\lfloor x \\rfloor \\) denote the greatest integer less than or equal to \\( x \\). A sequence \\( a_1, a_2, \\cdots \\) is... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all functions \\( f : S_n \\to M_n(\\mathbb{R}) \\) such that \\( f(x \\circ y) = f(x) + f(y) \\) for all \\( x, y \\in S_n \\), where \\( S_n \\) is the... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find a closed-form for $\\sum_{k\\ge0}\\binom nk\\binom{n-k}{\\left\\lfloor\\frac{m-k}2\\right\\rfloor}(-2)^k$ as a function of $n$ and $m$."
},
{
"ro... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "$21^{105}$ is congruent to 27 modulo 666. Find $21^{970}$ modulo 666."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find 21^9... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A coin with a diameter of 8 units is tossed onto a 5 by 5 grid of squares, each having a side length of 10 units. The coin is in a winning position if no part... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In convex quadrilateral $ABCD$, the angles are given as $\\angle ABD = 12^\\circ$, $\\angle ACD = 24^\\circ$, $\\angle DBC = 36^\\circ$, and $\\angle BCA = 48... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "If \\(a, b, c\\) are three distinct complex numbers and \\(\\frac{a}{b-c} + \\frac{b}{c-a} + \\frac{c}{a-b} = 0\\), then find the value of \\(\\frac{a^2}{(b-c... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider a metric space \\( E \\). Let \\( A \\) be a subset of \\( E \\) that is both open and closed. Suppose \\( A \\) is the disjoint union of two nonempt... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given two positive integers \\( m, n \\geq 2 \\), find the maximum value of \\( \\min(ac, bd) \\) where \\( a, b, c, d \\) are positive integers such that \\(... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Test for convergence of the integral $\\int_2^\\infty \\frac{\\cos(x)}{(x+\\sin^2(x))(\\ln(x))^2} \\, dx$."
},
{
"role": "assistant",
"content": "... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In \\(\\triangle DES\\), \\(\\sin D = \\frac{1}{T}\\) and \\(\\sin E = \\frac{1}{20}\\). Given that \\(DS = 18\\), compute \\(ES\\)."
},
{
"role": "as... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine the number of good permutations for \\( n = 1999 \\), where a permutation \\( a_1, a_2, a_3, \\ldots, a_n \\) of \\( 1, 2, 3, \\ldots, n \\) is said... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In the sequence $1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, \\ldots$, where $n$ consecutive terms have the value $n$, find the $1025$th term."
},
{
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the minimum value of\n\\[4x + \\frac{1}{x^4}\\]for $x > 0.$"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find the minim... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(a, b, c \\in \\mathbb{C}\\) and consider the system of equations:\n\\[\n\\begin{cases}\n\\frac{a}{b+c} + \\frac{b}{c+a} + \\frac{c}{a+b} = 9 \\\\\n\\fr... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Point $A$ is on the semicircle $CB$, and points $X$ and $Y$ are on the line segment $BC$. The line $AX$ extended meets the line through $B$ perpendicular to $... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In triangle \\(ABC\\), \\(AB=13\\), \\(BC=14\\), and \\(CA=15\\). Distinct points \\(D\\), \\(E\\), and \\(F\\) lie on segments \\(\\overline{BC}\\), \\(\\ove... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In triangle \\(ABC\\), let point \\(D\\) be on \\(AB\\) such that \\(2\\angle{ADC} = \\angle{A}\\), \\(BD = 1\\), \\(CA = 6\\), and \\(BC = 11\\). Find the le... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many pairs of subsets \\(X\\) and \\(Y\\) of a set \\(S\\) with 6 elements are there such that \\(X\\) is a subset of \\(Y\\) and \\(X \\neq Y\\)?"
},
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all triplets of real numbers $(x,y,z)$ such that the following equations are satisfied simultaneously:\n\\[\n\\begin{align*}\nx^3 + y &= z^2, \\\\\ny^3 +... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given \\( x, y \\in (0,1) \\) such that \\( (x^3 + y^3)(x + y) = xy(1 - x)(1 - y) \\), find the maximum value of \n\\[ P = \\frac{1}{\\sqrt{1 + x^2}} + \\frac... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the sum \\(\\phi(p^0) + \\phi(p^1) + \\cdots + \\phi(p^k)\\) for a prime \\(p\\)."
},
{
"role": "assistant",
"content": "<think>\nOkay, s... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( k \\) and \\( n \\) be positive integers such that \\( k \\leq n - 1 \\). Let \\( S \\) be the set of the first \\( n \\) positive integers, and let \... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given \\(x, y\\) are positive and \\(x^2y^3 = 10\\), find the minimum value of \\(2x + y\\)."
},
{
"role": "assistant",
"content": "<think>\nOkay,... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the cardinality of the set \\(\\{(a, b, c, d, e) \\in \\mathbb{N}^5 : 0 \\leq a + b \\leq 2, 0 \\leq a + b + c + d + e \\leq 4\\}\\)."
},
{
"role... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( F_n \\) be a sequence defined recursively as \\( F_0 = 1 \\), \\( F_1 = 1 \\), \\( F_2 = 2 \\), and \\( F_n = F_{n-1} + F_{n-2} + F_{n-3} \\) for all ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In $\\triangle ABC$, $AB=13$, $AC=14$, and $BC=15$. Let $M$ denote the midpoint of $\\overline{AC}$. Point $P$ is placed on line segment $\\overline{BM}$ such... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given a sushi restaurant with $n$ types of rolls numbered from $1$ to $n$, where each roll has a price $P_i$ for $1 \\leq i \\leq n$, and the prices are $P_1 ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given the functions for the first few dimensions:\n\\[ f_1(x) = x, \\]\n\\[ f_2(x) = \\frac{x^2 - x}{2}, \\]\n\\[ f_3(x) = \\frac{x^3}{6} - \\frac{x^2}{2} + \... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the system of equations:\n\\[\n\\begin{cases}\n2y(1 + 13y^2) - x(1 + x^2) = 3xy(x + y) \\\\\nx^2(x^2 - 2) = 2(1 + 4y)\n\\end{cases}\n\\]"
},
{
"... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many ordered pairs \\((a, b)\\) of positive integers satisfy the equation \n\\[a \\cdot b + 63 = 20 \\cdot \\text{lcm}(a, b) + 12 \\cdot \\text{gcd}(a, b)... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( 0 < x_1 < 1 \\) be a real number and \\( x_{n+1} = x_n + \\left(\\frac{x_n}{n}\\right)^2 \\) for \\( n \\geq 1 \\). Define \\( a \\) as the limit of t... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In triangle $ABC$, point $M$ is the midpoint of side $BC$. The bisector of angle $AMB$ intersects side $AB$ at point $E$, and the bisector of angle $AMC$ inte... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Calculate the integral $\\int\\limits_{0}^{1} f(x) \\, dx$ where $f(x) = \\min_{y \\in [0,1]} \\max \\{4x - y, y\\}$."
},
{
"role": "assistant",
"... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the integral\n\n\\[ \\int \\frac{(\\sqrt{1-x^2}\\arccos{x})^{-1}}{\\ln \\left(1+\\frac{\\sin{(2x\\sqrt{1-x^2})}}{\\pi}\\right)}\\,dx \\]"
},
{
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all functions \\( f: \\mathbb{Z}_+ \\to \\mathbb{Z}_+ \\) such that for all positive integers \\( m > n \\),\n\\[ f(n!) = f(n)! \\]\n\\[ m - n \\mid f(m)... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Compute $\\displaystyle \\sum_{n=1} ^{99} \\lfloor 0.67n \\rfloor$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to compute the... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all positive integers $n$ such that $n^2 + 2^n$ is a perfect square."
},
{
"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": "In a braid starting with the order A, B, C, the outer left wire A is brought over wire B to the middle position, forming the order shown in step 1, then the o... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the positive numbers \\(a, b, c\\) if:\n\\[ a + b + c = 1 \\]\n\\[ \\frac{a}{2009b + c} + \\frac{b}{2009c + a} + \\frac{c}{2009a + b} = \\frac{1}{670} \\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For how many non-negative integers \\( n \\) from 0 to 300 (inclusive), is the combinatorial number \\( \\binom{300}{n} \\) not divisible by 5?"
},
{
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the integral $$\\int_0^1 \\frac{1-x+\\ln x}{(1-x)\\ln x} \\, dx.$$"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to eva... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), point \\( H \\) is on \\( \\overline{AC} \\) such that \\( \\overline{BH} \\perp \\overline{AC} \\), \\( \\measuredangle C... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the smallest multiple of $20$ that has exactly $20$ divisors."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find the sma... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "If \\(a\\), \\(b\\), and \\(c\\) are three real numbers such that \\(a + b + c = 1\\) and \\(a^2 + b^2 + c^2 = 1\\), find the smallest value of \\(a^3 + b^3 +... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( K > 12 \\) be a real number. Find the smallest possible value of \\( \\frac{(x+y)^5}{xy(x^3 + y^3)} \\) for positive real numbers \\( x \\) and \\( y ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the sum $$\\sum\\limits_{j=1}^{+\\infty }{\\left( 1-\\frac{1}{2}+\\frac{1}{3}-...+\\frac{\\left( -1 \\right)^{j-1}}{j}-\\ln 2 \\right)\\ln \\left( \\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally be... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find positive rational numbers \\(a, b, c,\\) and \\(r\\) given that \\(a, b, c\\) form an arithmetic sequence with common difference \\(r+1\\) and \\(a, b+2,... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many ways are there to remove an $11 \\times 11$ square from a $2011 \\times 2011$ square so that the remaining part can be tiled with dominoes ($1 \\time... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given that \\(46^3 - 1\\) is divisible by 103, find the integer \\(46 < n < 103\\) such that \\(n^3 - 1\\) is divisible by 103."
},
{
"role": "assista... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many functions $f:\\{1,2,3,4\\} \\rightarrow \\{1,2,3,4\\}$ satisfy $f(f(x)) = f(x)$ for all $x \\in \\{1,2,3,4\\}$?"
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all polynomials \\( P \\) with three variables such that \n\\[ P(a,b,c)P(d,e,f) = P(ad + bf + ce, ae + bd + cf, af + be + cd) \\]\nfor all \\( a, b, c, d... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In a city championship with 10 teams, each team plays one match per round over 9 rounds. How many teams can alternate between wins and defeats throughout the ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the equation \\(\\frac{x^{2009} - x}{x - 1} = 2^{2009} - 2\\)."
},
{
"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": "The parabola \\( y = x^2 + bx + c \\) has the vertex at \\( V(2, -3) \\). Find \\( b \\) and \\( c \\)."
},
{
"role": "assistant",
"content": "<th... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( f: \\mathbb{R} \\to (0, +\\infty) \\) be a function such that \\( f(x) \\log f(x) = e^x \\) for all \\( x \\in \\mathbb{R} \\). Evaluate the limit:\n\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Calculate the integral $$\\int_{0}^{+\\infty}\\frac{x^{-p}}{x^2 + 2x \\cos(r) + 1}dx,$$ where $-1 < p < 1$ and $-\\pi < r < \\pi$."
},
{
"role": "assi... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let the product $(12)(15)(16)$, each factor in base $b$, equal $3146$ in base $b$. Let $s = 12 + 15 + 16$, each term expressed in base $b$. Find the value of ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider the set $A=\\{0, 2+i, 1+3i, -1+2i, i, 2i, 1+i, 1+2i\\}$. For any complex number $a+bi$, where $a$ and $b$ are integers, find the sum of all distinct ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( m \\) and \\( n \\) be fixed natural numbers. Find the total number of possible orderings of \\( n+m-1 \\) points \\( P_1(x_1, y_1), P_2(x_2, y_2), \\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In a $5 \\times 5$ grid where each square represents either a kingdom or a disputed territory, the number of claims on a disputed territory is the number of k... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A large circle is just large enough to contain in its interior three non-overlapping, mutually tangent smaller circles, each of which has an area of one squar... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In $\\triangle ABC$, a circle $\\odot O$ passes through points $B$ and $C$ and intersects $AB$ and $AC$ again at points $D$ and $E$, respectively. Let $H$ be ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given a half circle with midpoint \\( O \\) and diameter \\( AB \\). Let \\( Z \\) be a random point inside the half circle, \\( X \\) be the intersection of ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many whole numbers from 0 to 999 do not have any digit that is a prime number?"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many positive integers \\( n \\), where \\( 10 \\leq n \\leq 100 \\), are such that \\( \\frac{n^2-9}{n^2-7} \\) is a fraction in lowest terms?"
},
{
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine all non-negative integral solutions \\((n_1, n_2, \\ldots, n_{14})\\) to the Diophantine equation\n\n\\[ n_1^4 + n_2^4 + \\cdots + n_{14}^4 = 1599. ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the sum of all real roots of the equation $$5^x + 5^{-x} - 3|x^3 - x| - 18x^4 + 171 = 0.$$"
},
{
"role": "assistant",
"content": "<think>\nOk... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Two points are chosen randomly inside a circle of radius 1. What is the expected value of the distance between these two points? (Or show that there is no suc... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the system of equations for \\(x, y, z, w \\in \\mathbb{N}\\):\n\\[ x^2 + y^2 = z^2 \\]\n\\[ x^2 - y^2 = w^2 \\]"
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( f:(0,\\infty) \\to \\mathbb{R} \\) be defined by \n\\[ f(x) = \\lim_{n \\to \\infty} \\cos^n \\left( \\frac{1}{n^x} \\right). \\]\nEvaluate \\( f \\) ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the values of $a$ and $b$ such that $\\lim_{x\\rightarrow -\\infty} (\\sqrt{x^2+2}-ax-b)=1.$"
},
{
"role": "assistant",
"content": "<think>\n... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider a convex $n$-gon (where $n \\ge 4$ is even) and an arbitrary point $X$ inside it that is inside a triangle two of whose sides are sides of the $n$-go... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all positive integers \\( n > 2 \\) such that for any positive real numbers \\( a_1, a_2, \\ldots, a_n \\), there exists a positive integer \\( M_n \\) s... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A $2007 \\times 2007$ board is filled with pluses. In the first move, we change all signs of the rows whose number is divisible by $1$. In the second move, we... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the integral \\( \\int \\left( \\cot 2x \\cot 3x - \\tan 2x \\tan 7x \\right) \\tan 5x \\, dx \\)."
},
{
"role": "assistant",
"content": ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the integral $\\int \\frac{\\sinh(x)}{e^x+e^{-x}}dx$ using Hyperbolic Trig."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I ne... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given $n$ types of weights, each having an infinite supply, where the $i$th type has weight $a_i$, find the number of distinct ways to combine these weights s... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For positive numbers \\(a\\) and \\(b\\), let \\(f(a, b) = \\sqrt[4]{\\frac{2a^2}{a^2+b^2}} + \\sqrt{\\frac{2b}{a+b}}\\). Find the maximum value of \\(f(a, b)... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "What is the largest number of regions into which the plane could be divided by a planar diagram composed of three circles and two lines?"
},
{
"role":... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given \\( n > 2 \\) and \\( t_1, t_2, \\ldots, t_n \\in \\mathbb{R} \\) which satisfy\n\\[ (t_1 + t_2 + \\cdots + t_n) \\left( \\frac{1}{t_1} + \\frac{1}{t_2}... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the integral \\(\\int_0^\\infty \\frac{(\\sin x)^2}{x^2+1} \\, dx\\)."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to ... |
[
{
"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 \\( f(a^4) + f(a^2) + 1 = (f(a^2) - f(a) + 1)(f(a^2) + f(a) + 1) \\)."
},
{
"role... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many different ways can six numbers \\( a, b, c, d, e, f \\) be ordered?"
},
{
"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": "Find all positive integers $m$ such that $m^2 - 1 \\mid 5^m + 3^m$."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. I need to fin... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For integer values of \\( n \\) with \\( 0 \\leq n \\leq 2012 \\), find the value of \\( n \\) that minimizes the expression \\( (2012 - n)!(n!)^2 \\)."
},
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For what value of $k$ is the series $\\sum_{n=1}^{\\infty} \\frac{1}{k^{\\ln n}}$ convergent?"
},
{
"role": "assistant",
"content": "<think>\nOkay... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the equation $[[ \\text{sgn}(x-2) - x ]] = 3$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I have this equation to solve: [[ sg... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In triangle \\(ABC\\), median \\(BM = 24\\). A perpendicular from \\(C\\) to ray \\(BM\\) meets \\(BM\\) at \\(T\\). If \\(CT = 6\\), determine the area of tr... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( M = \\{1, 2, 3, \\ldots, 40\\} \\). Find the smallest positive integer \\( n \\) for which it is possible to partition \\( M \\) into \\( n \\) disjoi... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given the sequence $(a_n)$ defined by $a_1=4$ and $a_{n+1}=\\frac{\\sqrt{a_n^2+a_n+2}}{3}+\\frac{2+\\frac{1}{n}}{a_n+5}$ for $n \\in \\mathbb{Z^+}$, find $\\l... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the formula for the sum $1^4 + 2^4 + 3^4 + \\cdots + n^4$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find the formul... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the system of equations:\n\\[\n\\begin{cases}\nx^2 - y \\sqrt{xy} = 36 \\\\\ny^2 - x \\sqrt{xy} = 72\n\\end{cases}\n\\]\nwhere \\(x, y \\in \\mathbb{R}\... |
[
{
"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(x+y) = f(x-y) + 2f(y) \\]\nfor all \\( x, y \\in \\mathbb{R} \\)."
},
{
"r... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the infinite product $\\frac{2^3-1}{2^3+1}\\cdot\\frac{3^3-1}{3^3+1}\\cdot\\frac{4^3-1}{4^3+1}\\ldots$"
},
{
"role": "assistant",
"conten... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABC\\) be a triangle, and let \\(D\\), \\(E\\), \\(F\\) be the points where the incircle of the triangle touches the sides \\(BC\\), \\(CA\\), and \\(A... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the minimum value of \\(25x^2 + 126xy + 162y^2\\) where \\(x, y \\in \\mathbb{Z}\\)."
},
{
"role": "assistant",
"content": "<think>\nAlright,... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(1 \\leq a, b, c \\leq 3\\) and satisfy the condition \\(a + b + c = 5\\) and \\(\\max\\{a, b, c\\} \\geq 2\\). Find the smallest possible value of \\(T... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In rectangle \\(PQRS\\) with \\(PQ = 8\\) and \\(QR = 6\\), points \\(A\\) and \\(B\\) lie on \\(\\overline{PQ}\\), points \\(C\\) and \\(D\\) lie on \\(\\ove... |
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