conversations listlengths 2 2 | thinking stringclasses 1
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|---|---|---|
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nEvaluate the definite integral $\\int_0^1(1-\\sqrt[2024]x)^{2024}\\,dx$."
},
{
"role": "assistant",
"content": "We need to evaluate the definite integral I = ∫_0^1 (... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nDetermine the quadrant in which the point $(\\cos{B}-\\sin{A}, \\sin{B}-\\cos{A})$ lies, given that $\\triangle{ABC}$ is a triangle with three acute angles A, B, and C."
},
... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nZoey read 15 books, one at a time. The first book took her 1 day to read, the second book took her 2 days to read, the third book took her 3 days to read, and so on, with each b... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \\( A \\) be the set of the 16 first positive integers. Find the least positive integer \\( k \\) such that in every \\( k \\)-subset of \\( A \\), there exist two distinct ... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nHow many three-digit natural numbers have digits that sum to 13 and leave a remainder of 1 when divided by 4?"
},
{
"role": "assistant",
"content": "We need to count... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind the nth derivative of \\( f(x) = \\frac{x^n}{1 - x} \\)."
},
{
"role": "assistant",
"content": "We need to find the nth derivative of f(x) = x^n/(1-x). Actually... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all the positive divisors of the integers:\ni) \\( A = p^k \\), where \\( p \\) is a prime number and \\( k \\) is a positive integer.\nii) \\( B = p^k \\cdot q^l \\), wher... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nEvaluate the integral $\\displaystyle \\int^{\\infty}_{0}\\sin(x^2)dx$."
},
{
"role": "assistant",
"content": "We need to evaluate ∫_0^∞ sin(x^2) dx. This is a known... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven that \\(\\sin A + \\sin B = 1\\) and \\(\\cos A + \\cos B = 0\\), find the value of \\(12 \\cos 2A + 4 \\cos 2B\\)."
},
{
"role": "assistant",
"content": "We n... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nThe regular octagon \\( ABCDEFGH \\) has an area of \\( n \\). Let \\( m \\) be the area of quadrilateral \\( ACEG \\). Calculate \\( \\frac{m}{n} \\)."
},
{
"role": "as... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the equation $x(x^{2}+11) + 3(x^{2}-3) + (x-11) \\sqrt{3-x} = 0$."
},
{
"role": "assistant",
"content": "We are given the equation:\n\nx(x^2 + 11) + 3(x^2 - 3)... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all values of \\( n \\) such that the equation \\( x^3 + y^3 + z^3 = n(x^2 y^2 z^2) \\) has non-trivial solutions in positive integers."
},
{
"role": "assistant",
... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nEvaluate the integral \\( \\int_0^{\\frac{\\pi}{2}} (x \\cos x + \\sin^2 x) \\sin x \\, dx \\)."
},
{
"role": "assistant",
"content": "We are asked to evaluate the i... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSuppose \\( A = [a_{ij}] \\in \\mathbb{R}^{n \\times n} \\) has the property that there exist constant real numbers \\( c_1, c_2, \\cdots, c_n \\) such that\n\\[ \\sum_{i=1}^k a... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all primes \\( p < q < r \\) such that \\( 25pq + r = 2004 \\) and \\( pqr + 1 \\) is a perfect square."
},
{
"role": "assistant",
"content": "We need to find a... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven two sinusoids with frequencies of $8 \\text{ kHz}$ and $3.5 \\text{ kHz}$, is the function obtained by summing these two sinusoids periodic? If so, what is the period of t... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the system of equations $x + y + z = 1$ and $xyz = 1$ where $x, y, z$ are complex numbers with absolute value equal to $1$."
},
{
"role": "assistant",
"content... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nDetermine all positive integers that are equal to 300 times the sum of their digits."
},
{
"role": "assistant",
"content": "We need to find all positive integers N s... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nA number is called flippy if its digits alternate between two distinct digits. For example, 2020 and 37373 are flippy, but 3883 and 123123 are not. How many five-digit flippy nu... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nCalculate $\\lim_{n\\to \\infty} \\sqrt[n]{{nk}\\choose{n}}$, where $k$ is an arbitrarily fixed integer greater than $1$."
},
{
"role": "assistant",
"content": "We n... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nWithout using a calculator, determine which is greater: $\\sin 28^\\circ$ or $\\tan 21^\\circ$."
},
{
"role": "assistant",
"content": "We are asked: \"Without using ... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFor certain pairs $(m, n)$ of positive integers with $m \\ge n$, there are exactly 50 distinct positive integers $k$ such that $|\\log m - \\log k| < \\log n$. Find the sum of a... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet $n$ be a positive integer. Ana arranges $2n$ cups in a row on a table, each facing upside-down. She places a ball under one cup and makes a hole in the table under another c... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIn the complex plane, let points \\( A(x) \\), \\( B(y) \\), and \\( C(z) \\) be such that \\( |x| = |y| = |z| = m \\) where \\( m \\in \\mathbb{R}^+ \\), and \\( m^2 (x + y + z... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all pairs \\((m, n) \\in \\mathbb{N}^2\\) such that \\(n^{n^n} = m^m\\)."
},
{
"role": "assistant",
"content": "We need to find all pairs of positive integers (... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nConsider a sequence of integers \\(a_1, a_2, a_3, \\ldots\\) where \\(a_1 > 2\\). The sequence is defined by \\(a_{n+1} = \\frac{a_n(a_n - 1)}{2}\\). For what values of \\(a_1\\... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven 8 piles of bricks with different amounts of bricks, each pile can be removed and its bricks can be scattered to the other piles so that the remaining piles have the same n... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven that \\( n + m = 2014 \\), find the maximum and minimum values of \\( n! \\cdot m! \\)."
},
{
"role": "assistant",
"content": "We are given n + m = 2014. n and... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nA word is a combination of 8 letters, each either A or B. Let x and y be two words that differ in exactly 3 places. How many words differ from each of x and y in at least 5 plac... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nA particular right prism has a rectangular octagonal base with each base edge measuring 4 cm and a height of 3 cm. Its volume can be expressed in the form \\(a(b + \\sqrt{c})\\)... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the equation \\(\\dfrac{1}{x+2009} + \\dfrac{2010}{y+2010} = 1\\) for natural numbers \\(x\\) and \\(y\\)."
},
{
"role": "assistant",
"content": "We need to so... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven a triangle with vertices \\((x_1, y_1)\\), \\((x_2, y_2)\\), and \\((x_3, y_3)\\), find the coordinates of the incenter of the triangle."
},
{
"role": "assistant",... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the equation in real numbers:\n$$[x] + [2x] + [3x] + \\cdots + [2017x] = 1 + 2 + 3 + \\cdots + 2017$$"
},
{
"role": "assistant",
"content": "We need to solve f... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nA pipe is being carried down a hallway that is 8 ft wide. At the end of the hallway, there is a right-angled turn into a narrower hallway 7 ft wide. Find the length of the longe... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIn parallelogram $ABCD$, angles $B$ and $D$ are acute while angles $A$ and $C$ are obtuse. The perpendicular from $C$ to $AB$ and the perpendicular from $A$ to $BC$ intersect at... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind the coefficient of \\( a^8 b^4 c^9 d^9 \\) in the expansion of \\( (ab(c+d) + cd(a+b))^{10} \\)."
},
{
"role": "assistant",
"content": "We are asked: Find the c... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all triples of positive integers \\((x, y, z)\\) such that \\(2015x^3 + y^3 = 2014z^3\\)."
},
{
"role": "assistant",
"content": "We need to find all positive in... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nDetermine the value of the expression $ \\sqrt {5 + \\sqrt {5^2 + \\sqrt {5^4 + \\sqrt {5^8 + \\ldots}}}}$."
},
{
"role": "assistant",
"content": "We need to evaluat... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind \\( \\lim_{x \\to 1} \\frac{\\ln(x-1) + \\tan\\left(\\frac{\\pi x}{2}\\right)}{\\cot(\\pi x)} \\)"
},
{
"role": "assistant",
"content": "We need to compute:\n\n... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nWhat is the largest $n$ for which there exists a convex $n$-gon whose diagonal lengths are at most two different values?"
},
{
"role": "assistant",
"content": "We ne... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind the number of solutions in non-negative reals to the following system of equations:\n\\[\n\\begin{cases}\nx_1 + x_n^2 = 4x_n \\\\\nx_2 + x_1^2 = 4x_1 \\\\\n\\vdots \\\\\nx_... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nA right-angled triangle is inscribed in the parabola $y = x^2$, with its hypotenuse parallel to the $x$-axis. What can be the altitude of the triangle drawn to the hypotenuse?"
... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nConsider a sequence \\(a_1, a_2, \\cdots, a_{100}\\) of integers where \\(a_1 = 0\\) and \\(a_{100} = 0\\). For each term \\(a_i\\) (where \\(2 \\leq i \\leq 99\\)), \\(a_i\\) i... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIn a cyclic quadrilateral, each side has a length of either $6$ or $8$, and one of the diagonals has a length of $10$. Find the radius of its circumscribed circle."
},
{
... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven a triangle \\( \\triangle ABC \\) with \\( \\angle ACB = 60^\\circ \\) and \\( AC = BC \\). Point \\( M \\) is such that \\( AM = \\sqrt{2} \\), \\( BM = 2 \\), and \\( \\... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the system of equations for \\(a, b, c \\in \\mathbb{R}\\):\n\n\\[\n\\begin{aligned}\na^2 + b^2 + c^2 &= \\frac{5}{3} \\\\\nabc &= 1\n\\end{aligned}\n\\]"
},
{
"ro... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nWhat is the greatest integer of the product \\((1+1)(1+\\frac{1}{2})(1+\\frac{1}{4})(1+\\frac{1}{8})(1+\\frac{1}{16}) \\cdots (1+\\frac{1}{1024})(1+\\frac{1}{2048})\\)?"
},
... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nDetermine the remainder when \\( a_{83} = 6^{83} + 8^{83} \\) is divided by 49."
},
{
"role": "assistant",
"content": "We are asked: Determine the remainder when a_{... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven that \\( x^2 + y^2 = \\sqrt{3} \\), find the minimum value of \\( \\frac{3}{xy} + \\frac{x}{y} - \\frac{y}{x} \\)."
},
{
"role": "assistant",
"content": "We ar... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind the largest natural number \\( n \\) such that \\( 4^n + 4^{1004} + 4^{88} \\) is a perfect square."
},
{
"role": "assistant",
"content": "We need to find the l... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\n$A$ and $B$ travel around a circular track at uniform speeds in opposite directions, starting from diametrically opposite points. If they start at the same time, meet first afte... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \\( P(x) \\) be a polynomial of degree 2010. What is the maximum degree of the polynomial \\( P(x-1) - 3P(x) + 3P(x+1) - P(x+2) \\)?"
},
{
"role": "assistant",
"... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nWhat is the difference between the largest and smallest value for \\( \\text{lcm}(a, b, c) \\), where \\( a, b \\), and \\( c \\) are distinct positive integers between 1 and 10... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the system:\n\\[ x^3 + x(y - z)^2 = 2 \\]\n\\[ y^3 + y(z - x)^2 = 30 \\]\n\\[ z^3 + z(x - y)^2 = 16 \\]"
},
{
"role": "assistant",
"content": "We are given a s... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all functions \\( f\\colon \\mathbb{R}\\to\\mathbb{R} \\) such that \\( f(f(x)) - 4f^3(x) + 4f(x) = -12x^3 + 21x - 10 \\)."
},
{
"role": "assistant",
"content":... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet $a, b, c$, and $m$ be positive real numbers. Find for which $m$, independently of $a, b$, and $c$, at least one of the following three equations has real roots:\n\\[ ax^2 + ... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nEvaluate \\(\\lim_{n\\to\\infty} nI_n^2\\) where \\(I_n = \\int_0^{\\frac{\\pi}{2}} \\sin^n \\theta \\, d\\theta\\) for \\(n = 0, 1, 2, \\ldots\\)."
},
{
"role": "assist... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve for \\( m \\) such that \\( K(m,3) = 64m^3 + 702046 \\), where \\( K(m,n) \\) is the smallest natural number exceeding \\( (\\sqrt{m+3} + \\sqrt{m+2})^{2n} \\)."
},
{
... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \\(a, b, c, d\\) be the roots of the equation \\(t^4 - t^3 - 4t^2 + t + 1 = 0\\). Solve the system of equations:\n\n\\[\n\\begin{cases}\n\\frac{x}{a} + \\frac{y}{a-1} + \\fr... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all solutions $(r,s,t) \\in \\mathbb{N}^3$ of the equation $1 + 3^r + 3^s = 7^t$."
},
{
"role": "assistant",
"content": "We need to find all natural number solu... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven that \\( f(xy) = f(x) + f(y) + 1 \\) for \\( x, y > 0 \\) and \\( f(2) = 0 \\), find \\( f(2018) \\)."
},
{
"role": "assistant",
"content": "We are given a fun... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the system of equations for \\( x, y, z \\in \\mathbb{R} \\):\n\\[\n\\begin{cases} \n\\sqrt{x+y} + \\sqrt{z} = 7 \\\\\n\\sqrt{x+z} + \\sqrt{y} = 7 \\\\\n\\sqrt{y+z} + \\sq... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nWhat is the value of $$\\sum_{b=5}^\\infty \\frac{15b^2+68b+120}{2^b(b^4+64)}$$?"
},
{
"role": "assistant",
"content": "We need to evaluate the infinite sum:\n\nS = ... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) such that for all \\( x, y \\in \\mathbb{R} \\), the equation \\( f(xf(y) + f(x)) = 2f(x) + xy \\) holds."
},
{
... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven a triangle \\(ABC\\), locate its orthocenter using only a compass and an unmarked ruler. No other tools are allowed (besides a pencil)."
},
{
"role": "assistant",
... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIn $\\triangle ABC$, points $M$, $I$, and $H$ are the feet of the median, bisector, and height, respectively, drawn from $A$. Given that $BC = 2$, $MI = 2 - \\sqrt{3}$, and $AB ... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all functions \\( f: \\mathbb{Z} \\longrightarrow \\mathbb{R} \\) such that for every \\( k \\in \\mathbb{Z} \\setminus \\{0\\} \\),\n\\[ \\sum_{s \\in \\mathbb{Z}} f(s)f(s... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven the matrix \\( A = \\begin{pmatrix} 2 & 1 & 0 \\\\ 0 & 2 & 0 \\\\ 2 & 3 & 1 \\end{pmatrix} \\) and the minimal polynomial \\( m(x) = (x-2)^2 (x-1) \\), find the matrix \\(... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nCompare \\( A = \\frac{2007^{2006} + 1}{2007^{2007} + 1} \\) and \\( B = \\frac{2007^{2007} + 1}{2007^{2008} + 1} \\)."
},
{
"role": "assistant",
"content": "We need... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nUse the Taylor Series found in part (a) to find the limit as \\( x \\) approaches 0 of \\( \\frac{1 - x^2 - f(x)}{x^4} \\)."
},
{
"role": "assistant",
"content": "We... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIf $z$ 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": "assistant",
"content"... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven various trapezoids whose two nonparallel sides and the smaller base are equal to $a$. Find the value of the greater base of the trapezoid which has the greatest area."
}... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nEvaluate the integral $\\int_{0}^{\\frac{\\pi}{4}} \\cos^{\\frac{3}{2}}(2x) \\cos(x) \\, dx$."
},
{
"role": "assistant",
"content": "We need to evaluate the definite... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet A be the point (7,4) and D be (-5,-3). What is the length of the shortest path ABCD, where B is a point (x, 2) and C is a point (x, 0)? This path consists of three connected... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFor \\( n \\geq 2 \\), let \\( a_n x + b_n \\) be the remainder when \\( x^n \\) is divided by \\( x^2 - 6x - 12 \\). Find, for each \\( n \\), all prime numbers that are common... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven complex numbers \\( z_1, z_2, z_3 \\) satisfying \\( |z_1| \\leq 1 \\), \\( |z_2| \\leq 1 \\), \\( |z_3| \\leq 1 \\), and \\( z_1 + z_2 + z_3 = 0 \\), find the maximum val... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the following equations and inequalities: $\\sqrt {\\frac{x^2-2x-2}{x^2+4x+2}}+\\sqrt {\\frac{x^2+4x+2}{x^2-2x-2}}=2$."
},
{
"role": "assistant",
"content": "W... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind the sum of the squares of the digits of the least positive integer having 72 positive divisors."
},
{
"role": "assistant",
"content": "We need to find the least... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven three non-negative real numbers \\(a, b, c\\) such that \\(a + b + c = 6\\), find the maximal value of \\(\\sqrt[3]{a^2 + bc} + \\sqrt[3]{b^2 + ca} + \\sqrt[3]{c^2 + ab}\\... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all positive integers $n$ such that the sum of the digits of $2^n$ is equal to the sum of the digits of $5^n$."
},
{
"role": "assistant",
"content": "We need to... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \\(a, b, x, y\\) be real numbers such that \\(ax + by = 1\\), \\(ax^2 + by^2 = 2\\), \\(ax^3 + by^3 = 8\\), and \\(ax^5 + by^5 = 100\\). Find \\(ax^4 + by^4\\)."
},
{
... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nCompute the limit\n\\[\n\\lim_{x \\to 1} \\left( \\frac{H_x}{x(x-1)} - \\frac{1}{(x-1)(2x-1)} \\right),\n\\]\nwhere \\( H_x \\) represents the fractional harmonic number."
},
... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the following system of equations for the components of $\\vec{E}$:\n\n\\[\nE_x=\\frac{D_x}{1+4\\pi\\left(\\chi_1+\\chi_k\\left(E_x^2+E_y^2+E_z^2\\right)\\right)}\n\\]\n\n... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nConsider the 13 cards in a standard playing deck of cards that belong to the hearts suit. Let U be the set of all four card arrangements chosen from the hearts suit with replace... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nDetermine the quadrant in which the orthocenter of the triangle formed by the lines \\( x + y = 1 \\), \\( 2x + 3y = 6 \\), and \\( 4x - y + 4 = 0 \\) lies."
},
{
"role"... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nEvaluate the integral $\\int \\frac{e^{x} ( x^{2} - x +2)}{(x+1)^{3}} \\ dx$."
},
{
"role": "assistant",
"content": "We need to evaluate the indefinite integral: ∫ e... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \\(ABC\\) be a triangle with \\(AB=13\\), \\(BC=14\\), and \\(CA=15\\). The incircle of \\(ABC\\) meets \\(BC\\) at \\(D\\). Line \\(AD\\) meets the circle through \\(B\\), ... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSolve the system of matrix equations in the unknowns \\(X, Y \\in \\mathcal{M}_2(\\mathbb{C})\\):\n\\[ X^2 = Y^2 \\]\n\\[ XY \\neq YX \\]"
},
{
"role": "assistant",
... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven the sequence defined by $a_{1}=0$ and $a_{n}=2a_{n-1} + (-1)^n$, find a closed-form expression for $a_n$ as a function of $n$."
},
{
"role": "assistant",
"cont... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIn triangle \\(ABC\\), \\(\\angle B = 30^\\circ\\) and \\(\\angle C = 105^\\circ\\). Point \\(D\\) is the midpoint of side \\(BC\\). Find \\(\\angle DAC\\)."
},
{
"role"... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nEach of 999 numbers placed in a circular way is either 1 or -1. Both values appear. Consider the total sum of the products of every 10 consecutive numbers. Find the maximal poss... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nWhat is the number of ordered pairs $(m,n)$ of positive integers such that $3m^2+8mn+3n^2=383$?"
},
{
"role": "assistant",
"content": "We need to count ordered pairs... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \\(X, Y, Z\\) be independent random variables uniformly distributed on \\((0,1)\\). Define \\(Z_1 = \\max(X, Y)\\) and \\(Z_2 = \\min(Y, Z)\\). Find \\(E(Z_1, I(Z_1 = Z_2))\... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nEvaluate:\n\\[ \\sum_{n=1}^{\\infty} \\arctan \\left( \\frac{\\sqrt{3}}{1 + 3n + 3n^{2}} \\right) \\]"
},
{
"role": "assistant",
"content": "We need to evaluate the ... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all positive integers \\( x, y, z \\) such that \\( 5^x + 3^y = 1 + 7^z \\)."
},
{
"role": "assistant",
"content": "We are asked: Find all positive integers x, ... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nAn isosceles triangle has integer side lengths. A cevian separates the base into two segments of lengths 41 and 49. Find the smallest possible integer length of the cevian."
}... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nEvaluate the integral $$\\int_0^1 \\frac{\\ln^2 x}{x^2+x+1}dx.$$"
},
{
"role": "assistant",
"content": "We need to evaluate I = ∫_0^1 (ln^2 x)/(x^2 + x + 1) dx.\n\nT... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFor which \\( x \\in \\mathbb{R} \\) will the infinite series \\( \\sum_{n=0}^{\\infty} \\sin(nx) \\) be convergent?"
},
{
"role": "assistant",
"content": "We are as... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \\( a_{t+1} = \\frac{1}{3 - a_t} \\). Given \\( a_1 = 2009 \\), \\( a_{\\infty} \\) can be expressed in the form \\( \\frac{x + \\sqrt{y}}{z} \\), where \\( x \\), \\( y \\)... | on | success |
[
{
"role": "user",
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind the twice continuously differentiable solutions \\( u: \\mathbb{R}^{2} \\rightarrow \\mathbb{R} \\) of the partial differential equation:\n\\[\n\\partial_{1} u(x,y) \\parti... | on | success |
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