messages listlengths 3 3 |
|---|
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "What is the minimum value of the function \n\n\\[ f(x) = \\frac{\\sin{x}}{\\sqrt{1-\\cos^2{x}}} + \\frac{\\cos{x}}{\\sqrt{1-\\sin^2{x}}} + \\frac{\\tan{x}}{\\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(n \\geq 3\\) and \\(x_1, x_2, \\ldots, x_n \\geq 0\\) such that \\(\\sum_{i=1}^n x_i = 1\\). Find the maximum value of the expression:\n\\[\nE = x_1^3 ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Define \\( (a_n) \\) recursively by \\( a_1 = 1 \\), \\( a_n = 20 \\cos (a_{n-1}^\\circ) \\). As \\( n \\) tends to infinity, \\( (a_n) \\) tends to \\( 18.91... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the limit \\(\\lim_{x \\to 1} \\frac{1 - x + \\ln x}{\\sin(\\pi \\cos(\\pi x))}\\)."
},
{
"role": "assistant",
"content": "<think>\nOkay,... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "If we divide the number of factors of \\((10!)^{10!}\\) by 210, the remainder will be \\(x\\). Find the value of \\(x\\)."
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( A \\) be a set with \\( n \\) elements and \\( B \\) be a set with \\( m \\) elements, where \\( m \\leq n \\). Given a surjective function \\( f: A \... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "If $z$ is a complex number such that $z^{23}=1$, find the value of $\\displaystyle \\sum^{22}_{k=1}\\frac{1}{1+z^{8k}+z^{16k}}$."
},
{
"role": "assist... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many positive numbers \\( n = a_1a_2a_3 \\ldots a_{2021}a_{2022} \\) are there such that:\n1. Each digit \\( a_i \\) is odd for all \\( i = 1, 2, \\ldots,... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given the sequence defined by \\(a_1 = 2\\), \\(a_2 = 8\\), and \\(a_{n+2} = 3a_{n+1} - a_n + 5(-1)^n\\) for \\(n \\geq 1\\), determine if \\(a_{2014}\\) is a... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Among all the words of length $n$ in $\\{H,T\\}^n$, which one takes the least amount of time to appear on average?"
},
{
"role": "assistant",
"con... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the area of the domain of the function \\( f(x, y) = \\sqrt{25 - x^2 - y^2} - \\sqrt{|x| - y} \\)."
},
{
"role": "assistant",
"content": "<th... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the function $f:\\mathbb{R}\\rightarrow \\mathbb{R}$ such that $2f(x) + (x^3 \\sin(f(x)) - x) \\frac{d}{dx}f(x) = 0$ for all $x \\in \\mathbb{R}$."
},
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the largest integer less than 74 that leaves a remainder of 3 when divided by 7."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I n... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider the chord PQ joining the points \\( \\theta_{1} \\) and \\( \\theta_{2} \\) on the hyperbola \\( \\frac{x^{2}}{a^{2}} - \\frac{y^{2}}{b^{2}} = 1 \\),... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A circle touches two adjacent sides of a square ABCD of unit area and passes through exactly one of its vertices. Find the radius \\( x \\) of the circle."
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many real values of \\( x \\) in the closed interval \\([0, 2024\\pi]\\) satisfy the equation \\(\\sin^2 x + \\sin x - 2 = 0\\)?"
},
{
"role": "as... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine all positive integers $n$ for which there exist positive integers $a_1, a_2, \\ldots, a_n$ such that $\\frac{A}{a_i} + a_i$ is a perfect square for ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Compute the value of the integral \\(\\int_0^{\\pi/4} \\frac{x}{\\cos x (\\cos x + \\sin x)} \\, dx.\\)"
},
{
"role": "assistant",
"content": "<th... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( x_0, x_1 \\in \\mathbb{R} \\) and \\( x_{n+2} = \\frac{1 + x_{n+1}}{x_n} \\). Find \\( x_{1998} \\)."
},
{
"role": "assistant",
"content":... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate $\\lfloor \\frac{2^0}{3} \\rfloor + \\lfloor \\frac{2^1}{3} \\rfloor + \\lfloor \\frac{2^2}{3} \\rfloor + \\ldots + \\lfloor \\frac{2^{100}}{3} \\rfl... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider the function \\( f(x) = \\begin{cases} 1 & \\text{if } x \\in \\mathbb{Q} \\\\ 0 & \\text{if } x \\in \\mathbb{R} \\setminus \\mathbb{Q} \\end{cases}... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given \\( k \\ge 2 \\), find the smallest \\( t < k^3 \\) such that \\( kt \\) has less than 5 distinct digits in its decimal representation. Calculate the va... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "What is the largest prime factor of $$(25!)^3 - (24!)^3$$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find the largest pri... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given that \\( x \\) and \\( y \\) are acute angles such that \\(\\sin y = 3 \\cos(x+y) \\sin x\\), find the maximum value of \\(\\tan y\\)."
},
{
"ro... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given a point \\((x_1, y_1)\\) on the unit circle such that \\(x_1^2 + y_1^2 = 1\\), what equation in the Desmos graphing calculator can be used to find the a... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given a cyclic convex quadrilateral \\(ABCD\\), let \\(F\\) be the intersection of \\(AC\\) and \\(BD\\), and let \\(E\\) be the intersection of \\(BC\\) and ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the best constant \\( k \\) such that the inequality\n\\[ \\sqrt{a^2 + kab + b^2} + \\sqrt{b^2 + kbc + c^2} + \\sqrt{c^2 + kca + a^2} \\ge k\\sqrt{ab + b... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Yizheng and Jennifer are playing a game of ping-pong. The winner of a match is given one point. If one person reaches 11 points before the other can reach 10 ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let\n$$f\\left(a\\right)=\\int_{0}^{1}\\frac{\\sin\\left(2023\\pi ax\\right)\\sin\\left(2023\\pi x\\right)\\cos\\left(2024\\pi x\\right)\\cos\\left(2024\\pi a... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Graph the function \\( y = -2(x + 3)^2 + 4 \\)."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. I need to graph the function y = ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine all functions \\( f : \\mathbb{R}^+ \\to \\mathbb{R}^+ \\) such that \n\\[ f(x + zf(yz)) = f(x) + yf(z^2) \\]\nfor all \\( x, y, z \\in \\mathbb{R}^... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the $1996$th term in the sequence of positive integers consisting of only $1$s, $6$s, and $9$s in ascending order."
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find every function \\( f: \\mathbb{R} \\to \\mathbb{R} \\) that satisfies the functional equation:\n\\[ f(f(x) + y) = 2x + f(f(f(y)) - x), \\quad \\forall x,... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine the remainder obtained when the expression \\( 2004^{2003^{2002^{2001}}} \\) is divided by 1000."
},
{
"role": "assistant",
"content": "... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(a, b, c\\) be positive real numbers such that \\(ab + bc + ca = 1\\). Find the minimum value of the function\n\\[ f(a, b, c) = \\frac{1}{a+b} + \\frac{... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Calculate the distance between line $FH$ and plane $\\alpha$ that contains $AC$ and is parallel to $FH$ in a cube $ABCD.EFGH$ with an edge length of 4 cm."
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all \\( x, y, z \\in \\mathbb{N}^* \\setminus \\{1\\} \\) such that \\( (x-y)(y-z)(z-x) \\mid xyz - 1 \\)."
},
{
"role": "assistant",
"conten... |
[
{
"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(x + f(y)) = f(x) f(y) f(xy) \\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": "Four points are chosen uniformly and independently at random in the interior of a given circle. Find the probability that they are the vertices of a convex qu... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the value of the limit $\\lim_{x \\to 0} \\frac{(1+x)^\\frac{1}{x} - e + \\frac{1}{2}ex}{x^2}$."
},
{
"role": "assistant",
"content": "<think... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( \\triangle ABC \\) be a triangle and \\( P \\) a point on the circumcircle \\( (ABC) \\). The tangent to \\( (ABC) \\) at \\( P \\) intersects the sid... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many ordered pairs of nonnegative integers $(n, k)$ are there such that $n \\le 100$ and $2^{n+1} - 1$ is a multiple of $k!$? (Note: $0! = 1$ by definitio... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given the function \\( F(X) = \\frac{X(X^4 + 10AX^2 + 5A^2)}{5X^4 + 10AX^2 + A^2} \\) and the sequence \\( (x_n)_{n \\ge 1} \\) defined by \\( x_{n+1} = F(x_n... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let there be four terms $\\alpha_1, \\alpha_2, \\alpha_3, \\alpha_4$ such that $\\alpha_1, \\alpha_2$ and $\\alpha_3$ form an arithmetic sequence and $\\alpha... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A long piece of paper 5 cm wide is wrapped 600 times around a cardboard tube of diameter 2 cm, forming a roll with a final diameter of 10 cm. Approximate the ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(a, b, c\\) be positive real numbers such that \\(a + b + c = 1\\). Find the maximum value of the expression\n\\[ \\frac{a}{a+bc} + \\frac{b}{b+ca} + \\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find positive integers \\(a\\), \\(b\\), and \\(c\\) such that \\(abc\\) divides \\((ab+1)(bc+1)(ca+1)\\)."
},
{
"role": "assistant",
"content": "... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( X_k = \\{1, 2, \\dots, k\\} \\) for every positive integer \\( k \\). Let \\( m \\) and \\( n \\) be positive integers such that \\( m \\geq n \\). Fi... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Point \\( P \\) moves on the line containing the side \\( BC \\) of a non-right triangle \\( ABC \\). The lines through \\( P \\) perpendicular to \\( AC \\) ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(Y\\) be a random variable defined on the sample space \\(W = \\{(x,y): 0 \\leq x \\leq 1 \\text{ and } 0 \\leq y \\leq 1\\}\\) with the probability giv... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A natural number \\( n \\) has exactly 81 natural divisors and can be arranged in a 9 by 9 table such that for any two numbers in the same row or column, one ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In triangle \\(ABC\\), the incircle touches sides \\(BC\\) and \\(CA\\) at points \\(D\\) and \\(E\\), respectively. The segment \\(AD\\) intersects the incir... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine all complex numbers $z$ which satisfy the inequality \\[|z^{n+1}-z^n|\\geqslant|z^{n+1}-1|+|z^{n+1}-z|\\] for an integer $n \\geqslant 2$."
},
{... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all pairs $(a, b)$ of positive integers such that $ab + 1$ divides $b^3 + 1$."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's try... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given positive integers \\(a, b, c\\) and a positive integer \\(k\\), let \\(q\\) be the maximum prime factor of \\(\\Phi_p(k)\\). Find the number of pairs \\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Point P lies inside the square ABCD such that \\( AP + CP = 27 \\), \\( BP - DP = 17 \\), and \\( \\angle DAP = \\angle DCP \\). Find the area of the square."... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Shrek is on a game show trying to win an autobiography DVD. Initially, there are $5$ doors labeled $1$, $2$, $3$, $4$, and $5$, and the DVD has an $x/15$ chan... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In an acute triangle \\( \\triangle ABC \\), let \\( I \\) be the incenter, \\( X \\) be the midpoint of the arc \\( BC \\) of the circumcircle of \\( \\trian... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the value of the sum $$\\sum_{r=0}^{2017}\\frac{1}{3^r+\\sqrt{3^{2017}}}.$$"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given a line segment BC with length 6, point A is chosen such that angle BAC is a right angle. A point D is chosen on BC such that AD is perpendicular to BC. ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( P(x) \\) be the unique monic polynomial of minimal degree that satisfies the following conditions:\n- \\( 1 \\) is a root of \\( P(x) - 1 \\),\n- \\( ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Suppose that $Z_{10}$ is a homomorphic image of a finite group $G$. What can we say about the order $|G|$?"
},
{
"role": "assistant",
"content": "... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the maximum value of \\( \\sin(\\cos x) + \\cos(\\sin x) \\) for \\( x \\in \\mathbb{R} \\)."
},
{
"role": "assistant",
"content": "<think>\n... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In a triangle where the side lengths form an arithmetic progression and the measure of the angle opposite the longest side is three times the measure of the a... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Use a Maclaurin polynomial for the function \\( f(x) = \\arccos(0.5 + x) \\) to estimate \\( \\arccos(0.52) \\) correct to 6 decimal places."
},
{
"ro... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all real numbers \\( x \\) such that \\( [x^2] = ([x])^2 \\), where \\( [x] \\) denotes the integral part of \\( x \\)."
},
{
"role": "assistant"... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given that \\(88^4 + 82^4 + 63^4 + 28^4 + 6^4 = k^4\\) with \\(k\\) an integer, find \\(k\\)."
},
{
"role": "assistant",
"content": "<think>\nOkay... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many possible values of \\( x \\), where \\( 1 \\le x \\le 100 \\), satisfy the equation \\( x^2 - \\lfloor x^2 \\rfloor = (x - \\lfloor x \\rfloor)^2 \\)... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A microbe either divides into two identical copies of itself or dies. If the probability of division is \\( p = \\frac{1}{3} \\), what is the probability that... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "What is the largest integer \\( n \\) such that the equation \\( 13a + 18b = n \\) has no solutions for positive integers \\( a \\) and \\( b \\)?"
},
{
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABCD\\) be a convex quadrilateral with \\(AB = BC\\), \\(AC = BD\\), \\(\\measuredangle ABD = 80^\\circ\\), and \\(\\measuredangle CBD = 20^\\circ\\). ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In triangle \\(ABC\\) with altitude \\(AH\\), point \\(K\\) is chosen on segment \\(AH\\) such that \\(AH = 3KH\\). Let \\(O\\) be the circumcenter of triangl... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate $\\left\\lceil\\sqrt{2}\\,\\right\\rceil+\\left\\lceil\\sqrt{22}\\,\\right\\rceil+\\left\\lceil\\sqrt{222}\\,\\right\\rceil$."
},
{
"role": "... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the remainder when $x^3$ is divided by $x^2 + 5x + 1.$"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find the remainder ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "On a rectangular billiard table of dimensions $2 \\times 1$ with pockets at its corners and at the midpoints of its two largest sides, find the minimal number... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the equation: $[\\sin{x}] \\cdot \\{\\sin{x}\\} = \\sin{x}$\n\nHere, $[ \\cdot ]$ denotes the greatest integer function (floor function) and $\\{ \\cdot... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the integral\n$$\\int e^{\\sin(x)}(e^x\\cos(e^x)+\\sin(e^x)\\cos(x))\\,dx$$"
},
{
"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 a continuous function \\( f : [0,2] \\rightarrow \\mathbb{R} \\), simplify the limit\n\n\\[\n\\lim_{n \\to \\infty} \\int_0^2 \\frac{nf(x)}{4 + n^2x^2} ... |
[
{
"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 exact value of $a_{p,2}$, where $a_{p,2}$ is the maximum number of permutations of a set $X$ with $p$ elements such that e... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given the quadratic expression \\((x^2) + 5x + 16 + \\left(\\frac{10}{x-1}\\right)\\), how do you solve for \\(x\\) using the quadratic formula \\(x = \\frac{... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine all points $P$ in the plane for which the distance sum $|AP| + |BP| + |CP| + |MP|$ is minimal, where $ABC$ is an equilateral triangle with center $M... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( \\mu \\) be a finitely additive set function on the algebra \\( \\mathcal{A} \\). Write a corresponding formula for \\( \\mu(E_1 \\cup E_2 \\cup \\cdo... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given that \\(a, b, c > 0\\), find the minimum value of the expression:\n\\[\n\\sum_{\\mathrm{cyc}} \\frac{a^4}{1 + a^2 b} - \\frac{abc(a + b + c)}{1 + abc}\n... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all polynomials $p \\in \\mathbb{R}[x]$ such that\n\\[p(\\sin x + \\cos x) = p(\\sin x) + p(\\cos x)\\]\nfor all real numbers $x$."
},
{
"role": ... |
[
{
"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(2x + f(y)) = x + y + f(f(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": "A curve \\( C \\) which is not a straight line lies in the first quadrant. The tangent at any point on \\( C \\) meets the positive directions of the coordina... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all pairs of natural numbers \\((m, n)\\) such that it is possible to color each unit square of an \\(m \\times n\\) rectangle either white or black so t... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given a non-degenerate triangle \\(ABC\\), find the locus of points \\(D\\) in the plane of \\(ABC\\) such that \\(ABCD\\) forms a non-degenerate convex quadr... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given the equations $w+4x+9y+16z=6$, $4w+9x+16y+25z=7$, and $9w+16x+25y+36z=12$, find the value of $w+x+y+z$."
},
{
"role": "assistant",
"content"... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In a $2n \\times k$ rectangular grid where each unit square is colored with one of $n$ colors, find the maximum value of $k$ such that no four unit squares wh... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine the smallest integer $n \\ge 1$ such that an $n \\times n$ square can be cut into square pieces of size $1 \\times 1$ and $2 \\times 2$ with both ty... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the sum \\( \\binom{n}{0} + 2\\binom{n}{1} + 3\\binom{n}{2} + \\cdots + (n+1)\\binom{n}{n} \\)."
},
{
"role": "assistant",
"content": "<think... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many ways can a solid pyramid TABCD, with a quadrilateral base ABCD, be colored using five available colors such that no two faces with a common edge have... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many equiangular octagons can be created with side lengths exactly the first 8 positive integers, where each side has a different length?"
},
{
"r... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider the ellipse $(E):\\frac{x^2}{8}+\\frac{y^2}{4}=1$. A line with slope $m$ passing through $(1,0)$ intersects $(E)$ at two points $A$ and $B$ such that... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the equation \\(\\sum_{k=1}^{2n} \\sqrt{x^2 - 2kx + k^2} = |2nx - n - 2n^2|\\) for \\(n \\in \\mathbb{N}\\)."
},
{
"role": "assistant",
"con... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the triplets of integers $(x, y, z)$ such that:\n\\[\n\\frac{x^3-1}{27} + y^3 + 27z^3 = 3xyz.\n\\]"
},
{
"role": "assistant",
"content": "<th... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many ordered triples $(x,y,z)$ of real numbers satisfy the system of equations $$x^2+y^2+z^2=9,$$ $$x^4+y^4+z^4=33,$$ $$xyz=-4?$$"
},
{
"role": "a... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( A \\) be a set of 8 elements. Find the maximum number of 3-element subsets of \\( A \\) such that the intersection of any two of them is not a 2-eleme... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In triangle $\\triangle ABC$ with $AB \\ne AC$, let $O_1$ and $O_2$ be the centers of circles $\\omega_1$ and $\\omega_2$ with diameters $AB$ and $BC$, respec... |
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