messages listlengths 3 3 |
|---|
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "The length of the base of an isosceles triangle is $20$. In the plane of the triangle, a point $4$ units from this base is $10$ units from each leg. Find the ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given the equations \\( y = 2x + \\frac{m-2}{m+1} \\) and \\( y + 3x = \\frac{9+3m}{m+1} \\), how do you eliminate \\( m \\) from the equations?"
},
{
... |
[
{
"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(i) \\( f(x + yf(x)) = f(x)f(y) \\) for all \\( x, y > 0 \\)\n(ii) there ar... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let $\\triangle ABC$ be a triangle with centroid $G$, incenter $I$, and side lengths $a$, $b$, and $c$. If $G$ is contained in the incircle $(I)$, calculate t... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A line segment is broken at two random points along its length. What is the probability that the three new segments form a triangle?"
},
{
"role": "as... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given \\(0 < x, y, z \\le 1\\) and \\(xy + yz + xz = 1\\), find the maximum value of the expression:\n\\[ P = \\left(\\dfrac{1-x^2}{1+x^2}\\right)^2 + \\sqrt{... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For \\( N \\in \\mathbb{N} \\), let \\( I(N) = \\int_{0}^{1} x^N (1 - x)^N \\, dx \\). Find the value of \\( \\frac{I(N)}{4^{-N}} \\)."
},
{
"role": "... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all possible values of $n$ such that $2^{2n+1} + 2^n + 1 = x^k$, where $k \\geq 2$ and $n$ are positive integers."
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that $\\sqrt{2 + \\sqrt{2}}$ can be written as $\\sqrt{a + \\sqrt{b}}$ where $a$ and $b$ are integers and $b$ is n... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the equation $\\sin(90-2x) = \\frac{5000^2-3750^2}{5000^2+3750^2}$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to solve... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A square pyramid with base $ABCD$ and vertex $E$ has eight edges of length 4. A plane passes through the midpoints of $\\overline{AE}$, $\\overline{BC}$, and ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Compute the number of integers \\( n \\) with \\( 1 < n \\leq 2010 \\) for which the decimal equivalent of \\( \\frac{1}{n} \\) begins its repetitive cycle im... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given a collection of 2018 rectangles with integer side lengths not exceeding 134, determine the maximum value of \\( k \\) such that there exist \\( k \\) re... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the integral\n$$\\int\\limits_{0}^{\\infty }e^{-t^2/2}\\cos \\frac{x^2}{2t^2}dt.$$"
},
{
"role": "assistant",
"content": "<think>\nOkay, ... |
[
{
"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 = 2007\\) and \\(a_{n+1} = \\frac{1}{a_n + \\frac{1}{a_n + \\frac{1}{a_n + \\cdots}}}\\), compute \\(\\lim_{n \\to \\inft... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Starting from an equilateral triangle with perimeter \\( P_0 \\), we carry out the following iterations: the first iteration consists of dividing each side of... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In an $n \\times n$ square where each cell can be colored or not with equal probability, what is the probability that a $3 \\times 3$ sub-square does not cont... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A fair coin is flipped $7$ times. What is the probability that at least $5$ consecutive flips come up heads?"
},
{
"role": "assistant",
"content":... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find a point \\( P \\) on the line \\( 3x + 2y + 10 = 0 \\) such that the difference \\( PA - PB \\) is maximized, where \\( A = (4, 2) \\) and \\( B = (2, 4)... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given a sequence of real numbers $(a_n)$ with $a_1=2$ and $a_n=\\left(\\frac{n+1}{n-1}\\right)(a_1+...+a_{n-1})$ for $n \\geq 2$, determine the term $a_{2013}... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Bruno Mars is at $1$ degree Fahrenheit right now (time $t = 0$), and his heat $h(t)$ is given by $h(t) = 2 \\cdot h(t - 1) + t$. If he will explode when he is... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the number of positive unequal integer solutions to the equation $x + y + z = 10$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many lines can be drawn from a given point in space such that the angles between any two of them are equal?"
},
{
"role": "assistant",
"conten... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( x^2 + y^2 + 2gx + 2fy + c = 0 \\) be the equation of a circle. Find the locus of the feet of the perpendiculars from the origin to chords that subtend... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider a sequence \\(\\{x_n\\}_{n\\geq 0}\\) of real numbers satisfying \\((x_{n+1} - x_n)(x_{n+1} + x_n + 1) \\leq 0\\) for all \\(n \\geq 0\\). If \\(x_0 ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the equation $x5^x = \\frac{7}{3}$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to solve the equation x times 5 to the p... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( a, b, \\) and \\( c \\) be positive real numbers with \\( abc = 1 \\). Calculate the minimum value of the expression:\n\\[ \\sqrt{9 + 16a} + \\sqrt{9 ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In trapezoid \\(ABCD\\) with \\(AD\\) and \\(BC\\) as the bases, \\(\\angle ABC = 90^\\circ\\). Point \\(E\\) is the intersection of \\(AB\\) and \\(CD\\), po... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A snowball is melting such that the radius is shrinking at a rate of 5 in/hr. At what rate will the volume be decreasing at the moment when the radius of the ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "What is the value of \\(-1\\) in \\(p\\)-adic numbers? What is the value of \\(\\frac{1}{2}\\) in \\(3\\)-adic numbers?"
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given \\(x, y, z \\geq 0\\) and \\(x^2 + y^2 + z^2 + 2xyz = 1\\), find the minimum value of the expression:\n\\[\n\\frac{(z + xy)^3}{(1 + x)^3} - 8y^2z^3x\n\\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the limit:\n$$\\lim _{n \\rightarrow \\infty} \\int_{\\log n}^{e^n} x^{2} e^{-x^9} \\, dx$$"
},
{
"role": "assistant",
"content": "<think... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "What is the inverse Laplace transform of \\( F(s) = s \\)?"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find the inverse Lap... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the integral $\\int_{0}^{1} \\lfloor -\\ln x \\rfloor \\ dx$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to evaluate... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In triangle \\(ABC\\) with \\(AB = BC\\) and \\(\\angle B = 20^\\circ\\), let \\(AB = x\\) and \\(AC = y\\). Calculate the value of \\(\\frac{x}{y}\\) and exp... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Define the sets:\n\\[\n\\Pi_1 = \\{6mn - m - n \\mid m, n \\in \\mathbb{N}\\}\n\\]\n\\[\n\\Pi_2 = \\{6mn + m + n \\mid m, n \\in \\mathbb{N}\\}\n\\]\n\\[\n\\P... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine the number of functions \\( f : \\{1, 2, \\ldots, 1999\\} \\to \\{0, 1, 2, 3\\} \\) such that the sum \\( f(1) + f(2) + \\cdots + f(1999) \\) is odd... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the greatest real number \\( r \\) such that\n\\[ ab \\ge r (1 - a - b) \\]\nfor all positive real numbers \\( a \\) and \\( b \\)."
},
{
"role":... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Triangle ABC has coordinates A(1,2), B(0,5), and C(5,4). \n\n Part A: State the coordinates of triangle A'B'C', the image of triangle ABC after the translati... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( x_1, x_2, \\ldots, x_{10} \\) be non-negative real numbers such that \\( \\frac{x_1}{1} + \\frac{x_2}{2} + \\cdots + \\frac{x_{10}}{10} \\leq 9 \\). F... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In $\\triangle{ABC}$, let $P$ and $Q$ denote the feet of the angle bisectors from $B$ and $C$, respectively. If $PQ$ is parallel to $BC$, $\\angle A = 30^\\ci... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find two linearly independent solutions of the differential equation \\( x^2y'' + 3x(1 + x)y' + (1 - 3x)y = 0 \\) in the vicinity of \\( x = 0 \\).\n\nGiven t... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given the polynomial \\( P(x) = a_{2020}x^{2020} + a_{2019}x^{2019} + \\cdots + a_1x + a_0 \\) where each coefficient \\( a_i = \\pm 1 \\) for \\( i = 0, 1, \... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( A \\) be the set of all permutations \\( a = (a_1, a_2, \\ldots, a_{2003}) \\) of the first 2003 positive integers such that there is no proper subset... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the maximum value of the expression \\( x^3 + y^3 + z^3 - 3xyz \\) where \\( x^2 + y^2 + z^2 = 1 \\) and \\( x, y, z \\) are real numbers."
},
{
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( x_n \\) and \\( y_n \\) be two sequences defined by the recurrence relations:\n\\[ x_{n+1} = 3x_n + 2y_n \\]\n\\[ y_{n+1} = 4x_n + 3y_n \\]\nwith init... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let $\\omega$ be the circumcircle of triangle $ABC$, and let $BB_1$ and $CC_1$ be the angle bisectors of $\\angle ABC$ and $\\angle ACB$, respectively. Let $I... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( p \\) be a fixed prime number. If \\( a(n) \\) denotes the exponent of \\( p \\) in the prime factorization of \\( n \\), determine the sum \\[ S(n) =... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given $\\sin\\left(\\frac{\\pi}{5}\\right) = k$ and $\\sin\\left(\\frac{\\pi}{40}\\right) = m$, find $\\sin\\left(\\frac{4\\pi}{10}\\right)\\sin\\left(\\frac{... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A function $f:\\mathbb{Z} \\to \\mathbb{Z}$ satisfies\n\\begin{align*}\nf(x+4)-f(x) &= 8x+20, \\\\\nf(x^2-1) &= (f(x)-x)^2+x^2-2\n\\end{align*}for all integer... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many prime numbers are there between $10^8$ and $10^9$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. The problem is asking... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the sum \\((\\tan 1^\\circ)^2 + (\\tan 2^\\circ)^2 + \\cdots + (\\tan 89^\\circ)^2\\)."
},
{
"role": "assistant",
"content": "<think>\nOk... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given a triangle \\(ABC\\) with midpoints \\(M\\), \\(N\\), and \\(P\\) of sides \\(BC\\), \\(CA\\), and \\(AB\\) respectively, and \\(F\\) as the Feuerbach p... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABCD\\) be a convex quadrilateral such that \\(AD = CD\\), \\(BD = BC\\), \\(\\angle ADC = 168^\\circ\\), and \\(\\angle ABC = 66^\\circ\\). Find \\(\\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( a, b, c, d > 0 \\) and \\( ab + bc + cd + da = 1 \\). Find the minimum value of \\( A = \\frac{a^3}{b + c + d} + \\frac{b^3}{c + d + a} + \\frac{c^3}{... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Construct the midpoint of a segment using an unmarked ruler and a trisector that marks the two points dividing the segment into three equal parts."
},
{
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( p \\) be a prime and \\( a, b, c \\) be integers such that \\( a^n + bp = b^n + cp = c^n + ap \\). Find the value of \\( a + b + c \\) given that \\( ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( P \\) be the product of the first 100 positive odd integers. Find the largest integer \\( k \\) such that \\( P \\) is divisible by \\( 3^k \\)."
},... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Express the derivative $\\frac{d}{dx}\\int_{0}^{x}tf(t)dt$ in terms of $x$, $f(x)$, $f'(x)$, and $\\int_{a}^{x}f(t)dt$."
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given that $\\sum a = d$ (a constant) and $a, b, c, k > 0$, find the maximum value of the function:\n\\[ F = 3 \\sum ab + \\sum \\frac{k(k+b)(k+c)}{k+a} \\]"
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( M = \\left\\{ x : \\sum_{k=1}^{n} x_k = 0 \\right\\} \\) in \\( l_2 \\). Find \\( M^{\\perp} \\)."
},
{
"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 nonnegative real numbers such that \\(a + b + c \\le 7\\). Find the maximum value of the expression \n\\[ \\frac{a}{1 + a + b + c} \\time... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "What is the probability that 3 randomly chosen points from 2010 evenly spaced points on a circle form the vertices of a right triangle?"
},
{
"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(69x) = f(x) \\text{ for all } x \\in \\mathbb{R}. \\]"
},
{
"role"... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the expression \\( \\frac{3}{2} + \\frac{5}{4} + \\frac{9}{8} + \\frac{17}{16} + \\frac{33}{32} + \\frac{65}{64} - 7 \\)."
},
{
"role": "assi... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In triangle \\(ABC\\), let \\(A'B'C'\\) be the orthic triangle, \\(DEF\\) the contact triangle, and \\(I\\) the incenter. Define \\(A''\\) as the intersection... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "When the price of a certain commodity is \\( p \\) dollars per unit, the manufacturer is willing to supply \\( x \\) hundred units, where \\( 110p^2 - 4x^3 = ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given a triangle with sides \\(a \\geq b \\geq c\\), determine \\(\\sup(\\min\\{a/b, b/c\\})\\). Here, \\(\\sup\\) denotes the least upper bound of the set."
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine all functions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) such that the set \n\\[\n\\left \\{ \\frac{f(x)}{x} : x \\neq 0 \\text{ and } x \\in \... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given that \\( p_1, p_2, p_3, p_4, p_5 \\) are prime numbers and \\( k \\) is a positive integer, find the smallest \\( k \\) such that the equation\n\\[ 23p_... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In an acute triangle \\( \\triangle ABC \\) with incenter \\( I \\), two common tangents of the incircle \\( (I) \\) and the circle \\( (AI) \\) intersect eac... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "An action with a probability of success of \\( p = 0.22 \\) is repeated 100 times. What is the probability of getting 19, 20, 21, or 22 successes?"
},
{
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all integers $a$, $b$, $c$, $d$, and $e$ such that\n\n\\begin{align*}\na^2&=a+b-2c+2d+e-8,\\\\\nb^2&=-a-2b-c+2d+2e-6,\\\\\nc^2&=3a+2b+c+2d+2e-31,\\\\\nd^... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "If $(x,y)$ is a solution to the system\n\\begin{align*}\nxy &= 6, \\\\\nx^2 y + xy^2 + x + y &= 63,\n\\end{align*}find $x^2 + y^2.$"
},
{
"role": "ass... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find \\(\\lim_{n \\to \\infty} \\left[ \\lim_{x \\to 0^+} \\left( (2\\cosh{\\sqrt{x}})^{(n)} \\right) \\right]\\), where \\( f^{(n)} \\) denotes the \\( n \\)... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABC\\) be a triangle with circumcircle \\(\\omega\\). The angle bisector of \\(\\angle CAB\\) intersects \\(BC\\) at \\(D\\). Let \\(M\\) be the midpoi... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given complex numbers \\(a, b, c\\) such that \\(a + b + c = \\omega\\), \\(a^2 + b^2 + c^2 = \\beta\\), and \\(a^3 + b^3 + c^3 = \\phi\\), find \\(a^m + b^m ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Compute \\(\\lim_{n \\to \\infty} \\left( \\sin\\frac{\\pi n}{6n+1} + \\cos\\frac{\\pi n}{3n+1} \\right)^n\\)."
},
{
"role": "assistant",
"content... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given the sequence defined by \\( x_1 = 1 \\) and \\( x_{n+1} = 20 + \\frac{10}{x_n} \\), find the limit of \\( x_n \\) as \\( n \\) approaches infinity."
}... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Alexis and Joshua are drawing symbols in the sand. Alexis draws stars in pairs, and Joshua draws squares in trios. They want to create rows of 15 adjacent sym... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Using the digits 0, 1, 2, 3, and 4, find the number of 13-digit sequences such that the difference between any two consecutive digits is 1. Examples of such s... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the integral $\\int \\sin(\\ln(x)) \\, dx$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to evaluate the integral of s... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In a city where each resident is assigned a unique three-digit number between 100 and 999, the first person to enter with a number that has already been recor... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine the period of the function \\( y = e^{-x} \\sin x \\)."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find the perio... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all integer solutions to the equation \\(x^{3}y + x + y = x^{2}y^{2} + 2xy^{3}\\)."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( P \\) and \\( Q \\) be isogonal conjugates with respect to the convex quadrilateral \\( ABCD \\). Given that \\( PA = 3 \\), \\( PB = 4 \\), \\( PC = ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For what base-6 digit $d$ is $2dd5_6$ divisible by the base 10 number 11? (Here $2dd5_6$ represents a base-6 number whose first digit is 2, whose last digit i... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "The average age of five children is 6 years old. Each child is a different age and there is a difference of two years in the ages of any two consecutive child... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "The average of the two bases of an isosceles trapezoid is $17$ cm. If the perimeter of the trapezoid is $97$ cm, what is the measure in cm of one of the non-p... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In the expansion of \\( A = (1+x)(1+x^2)(1+x^3) \\cdots (1+x^{63}) \\), what is the coefficient of \\( x^{2008} \\)?"
},
{
"role": "assistant",
"c... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many ways can 4 couples line up in a line such that no couple is standing adjacent to each other?"
},
{
"role": "assistant",
"content": "<thin... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the set of solutions to $\\lfloor x^2 \\rfloor = \\lfloor (x+3)^2 \\rfloor$."
},
{
"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": "Let \\(ABC\\) be a triangle, and let \\(D\\) be a point on the line \\(BC\\) such that \\(B\\) lies between \\(D\\) and \\(C\\) and \\(DB = AB\\). Let \\(M\\)... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "What is the smallest square frame that can cover a painting of dimensions 40 inches by 90 inches completely?"
},
{
"role": "assistant",
"content":... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A graph has $n$ points and 100 edges. A move consists of picking a point, removing all its edges, and joining it to any points which it was not joined to imme... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all natural numbers \\(a\\), \\(b\\), and \\(c\\) such that \\(a!b! = a! + b! + c!\\)."
},
{
"role": "assistant",
"content": "<think>\nOkay, ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Define a function $d: \\mathbb{N} \\rightarrow \\mathbb{Z}$ by $d(1)=0$, $d(p)=1$ for any prime $p$, and $d(mn)=nd(m)+md(n)$ for any integers $m,n$. Determine... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "There are 18 teams in a tournament. The tournament consists of $n$ rounds, where in each round, the 18 teams are split into 3 brackets, each containing 6 team... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find \\(a, b, c, d\\) such that the integral \\(\\int_{0}^{1}(e^t - at^3 - bt^2 - ct - d)^2 \\, dt\\) is minimized."
},
{
"role": "assistant",
"co... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "What is the least \\( x > 1 \\) that satisfies\n\\[\n\\frac{\\lfloor x \\rfloor \\cdot \\lfloor \\sqrt{x} \\rfloor}{\\lfloor \\sqrt[3]{x} \\rfloor \\cdot \\lf... |
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