messages listlengths 3 3 |
|---|
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate $\\int{\\frac{2x^3 - 1}{x + x^4}}{dx}$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to solve the integral of (2x³ - 1... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the natural number \\( n \\) such that there exist positive numbers \\( x, y, z \\) satisfying the properties:\n\\[ x^3 + y^3 + z^3 = x + y + z \\]\n\\[ ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given \\(a, b < 0\\) and \\(\\frac{1}{a} + \\frac{1}{b} = \\frac{1}{a-b}\\), find the value of \\(\\frac{a}{b}\\)."
},
{
"role": "assistant",
"con... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Use the reduction formula to find \\(\\int_{0}^{1} (1+x^{2})^{\\frac{3}{2}}\\ 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": "Evaluate the integral $\\int\\frac{(x-1)\\sqrt{x^4+2x^3-x^2+x+1}}{x^2(x+1)}dx$."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's try to... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABCD\\) be a cyclic quadrilateral with \\(BC = CD = 2\\). Let \\(I\\) be the incenter of triangle \\(ABD\\). If \\(AI = 2\\), find the minimum value of... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A line that passes through the origin intersects both the line \\(x=1\\) and the line \\(y=1+\\frac{\\sqrt{3}}{3}x\\). The three lines create an equilateral t... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(a, b, c\\) be complex numbers such that \\(ab + bc + ca = 61\\), \\(\\frac{1}{b+c} + \\frac{1}{c+a} + \\frac{1}{a+b} = 5\\), and \\(\\frac{a}{b+c} + \\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given three positive real numbers \\(a, b, c\\) such that \\(a^2 + b^2 + c^2 = 27\\), find the maximum value of \\(a^3 + b^3 + c^3\\)."
},
{
"role": "... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "What is the volume of tetrahedron $ABCD$ with edge lengths $AB=2, AC=3, AD=4, BC=\\sqrt{13}, BD=2\\sqrt{5},$ and $CD=5$?"
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let pyramid \\( S.ABCD \\) have a rectangular base \\( ABCD \\) with \\( AB = 2a \\). Given that \\( SA = SB \\) and the plane \\( (SAB) \\) is perpendicular ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( X \\) be a point lying in the interior of the acute triangle \\( ABC \\) such that \n\\[ \\angle BAX = 2\\angle XBA \\quad \\text{and} \\quad \\angle ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider a connected graph \\( G \\) with \\( n \\) vertices and \\( m \\) edges. Define \\( f(G) \\) as the number of edges that are not cut-edges. If \\( G ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many pairs of polynomials \\( P(x) \\) and \\( Q(x) \\) with real coefficients and \\( \\deg P > \\deg Q \\) satisfy the equation \\([P(x)]^2 + [Q(x)]^2 =... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the exact value of \\( \\frac{2\\tan 15^\\circ}{1 - \\tan^2 \\frac{\\pi}{12}} \\)."
},
{
"role": "assistant",
"content": "<think>\nOkay, let'... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given the linear transformation \\( T: P_3 \\to P_3 \\) defined by\n\\[ T(a + bx + cx^2 + dx^3) = a(1 + x^3) + b(x + x^3) + c(x^2 + x^3) + d(1 - x + x^2 + x^3... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine the area of the section enclosed by the functions $f(x) = |x-1| - |x-2|$ and $g(x) = |x-3|$."
},
{
"role": "assistant",
"content": "<thi... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( k \\) and \\( n \\) be two positive integers. Find the smallest positive integer \\( n \\) such that \\( 5^3 \\mid \\left( 2^{2n-1} + (-1)^{\\left\\lf... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Convert the repeating base 5 number \\(0.3404040\\ldots_5\\) to a base 6 decimal."
},
{
"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": "Find the dimension of the real linear space \\( V_t = \\{ \\mathbf{x} \\in \\mathbb{R}^3 \\mid A_t \\mathbf{x} = 0 \\} \\) where \\( A_t = \\begin{pmatrix} 2 ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Simplify $\\mathcal{F}(\\mathcal{F}(f)) = \\mathcal{F}^2(f)$, where $\\mathcal{F}(f)$ denotes the Fourier transform of $f$."
},
{
"role": "assistant",... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A school library was attended by 5 students on Monday, 6 on Tuesday, 4 on Wednesday, 8 on Thursday, and 7 on Friday. None of the students attended the library... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Study the nature of the series $\\sum \\frac{1}{x(n)^a}$ for $0 < a \\leq 1$, where $x(n) = \\sum_{k=1}^{n} |\\cos(2^k)|$."
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "On a table near the sea, there are $N$ glass boxes where $N<2021$, each containing exactly $2021$ balls. Sowdha and Rafi play a game by taking turns on the bo... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many different combinations of 3 blocks can be selected from a $4 \\times 4$ square grid so that no two blocks are in the same row or column?"
},
{
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In how many ways can John pay 2010 dollars using an unlimited supply of 2, 5, and 10 dollar notes?"
},
{
"role": "assistant",
"content": "<think>\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(a, b, c, k > 0\\) be such that the inequality\n\\[\n\\sum_{\\text{cyc}}{\\sqrt{\\frac{ab}{kb^2+(b-c+a)^2}}} \\ge \\frac{9}{\\sqrt{k + 1}} \\cdot \\frac... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( x \\) and \\( y \\) be positive real numbers satisfying \\(\\frac{1}{x+3y} + \\frac{1}{2x+y} = 1\\). Find the minimum value of \\( x + y \\)."
},
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "The equation \\( x^{10} + (13x-1)^{10} = 0 \\) has 10 complex roots \\( r_1, \\overline{r_1}, r_2, \\overline{r_2}, r_3, \\overline{r_3}, r_4, \\overline{r_4}... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In how many ways can a bug travel from the point (0,0) to the point (3,3) on a coordinate plane by traveling a path 9 units long?"
},
{
"role": "assis... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the last three digits of $2012^{2012}$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find the last three digits of 2012... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the integral $\\int{-8(\\sin(t))^2 - 2\\cos(t)\\sin(2\\sin(t))} \\, dt$."
},
{
"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": "Eleven students, including Ashley, Blake, and Chris, line up for movie tickets. If both Ashley and Blake are in front of Chris, how many possible ways are the... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the continuous function \\( f(x) \\) which satisfies the following equation for all real numbers \\( x \\):\n\n\\[\n\\int_0^x f(t) \\, dt + \\int_0^x t f... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given the equations \\(x + y + z = 24\\) and \\(x, y, z > 3\\), find the maximum value of \\(x^2 + y^2 + z^2\\)."
},
{
"role": "assistant",
"conte... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Professor Oak has 100 Pokémon, each with a container that has a total capacity of 100 kilograms combined. He wants to distribute 100 kilograms of food among t... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "If the vertices of an equilateral triangle lie on the curve \\(8x^3 + y^3 + 6xy = 1\\), find the area enclosed by the triangle."
},
{
"role": "assista... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABCD\\) be a trapezoid with \\(AB \\parallel CD\\), \\(AB = 11\\), \\(BC = 5\\), \\(CD = 19\\), and \\(DA = 7\\). The angle bisectors of \\(\\angle A\\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABC\\) be a triangle with \\(A'\\), \\(B'\\), and \\(C'\\) as the midpoints of \\(BC\\), \\(CA\\), and \\(AB\\) respectively. Define \\(I_1\\), \\(I_2\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABC\\) be an acute triangle with orthocenter \\(H\\). Let \\(\\omega_b\\) and \\(\\omega_c\\) be the incircles of triangles \\(HCA\\) and \\(HAB\\), re... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For how many positive integers \\( n < 2017 \\) is it possible to draw a right-angled isosceles triangle such that exactly \\( n \\) points on its perimeter, ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine if the function \\( f(x) = \\int_1^{\\infty} \\frac{\\cos t}{t^2 + x^2} \\, dt \\) is continuous and bounded."
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A machine for making pizza puts salami on every 18th pizza, pineapple on every 21st pizza, and peppers on every 20th pizza. How many pizzas have both pepper a... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine how many integers \\( x \\), \\( 0 \\leq x \\leq 2015 \\), satisfy \\( x^3 \\equiv -1 \\pmod{2016} \\)."
},
{
"role": "assistant",
"cont... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( f(x) \\) be a function such that \\( f(1) = 1 \\) and \\( f'(x) = \\frac{1}{x^2 + f(x)^2} \\) for \\( x \\geq 1 \\). Calculate the upper bound \\( M \... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the function \\( f : \\mathbb{Z} \\to \\mathbb{Z} \\) such that \\( f(x + y + f(y)) = f(x) + 2y \\)."
},
{
"role": "assistant",
"content": "<... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Write the complex number \\( z = \\frac{-\\sqrt{3} + i}{1 - i} \\cdot \\left( \\cos\\frac{\\pi}{12} - i \\sin\\frac{\\pi}{12} \\right) \\) in the exponential ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find a polynomial $p(x)$ with real coefficients such that $(p(x))^5 - x$ is divisible by $(x-1)(x+1)(x-32)$."
},
{
"role": "assistant",
"content":... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve in positive integers the equation $$(3a+b+1)^2=16ab.$$"
},
{
"role": "assistant",
"content": "<think>\nAlright, so I need to solve the equat... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How do you integrate $\\sin (x^3)$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to figure out how to integrate sin(x³). Hmm, i... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( X \\) and \\( Y \\) be independent random variables with \\( N(0,1) \\) distributions. Let \\( D_r \\) be the disk centered at the origin with radius ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider the equation \\(x^6 + y^6 = z^2\\). Let \\(f(x, y, z)\\) be the number of nonzero integral solutions to this equation for \\(|x|, |y|, |z| \\leq 10\\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In the same setup as Problem 1, let \\( DFe \\) intersect \\( PQ \\) at \\( W \\). \\( EW \\) and \\( FW \\) intersect \\( IF \\) and \\( IE \\) at \\( V \\) ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In the coordinate plane, a circle with radius 1 is centered at the origin. A point \\(P\\) on the circle has coordinates \\((\\cos \\theta, \\sin \\theta)\\).... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the series $$\\displaystyle{\\sum_{n\\geq1}\\left(\\sum_{k=1}^{n}\\frac{(-1)^{k-1}}{k}-\\ln2\\right)\\ln\\frac{n+1}{n}}.$$"
},
{
"role": "ass... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the arithmetic mean of all the three-digit palindromes. (Recall that a palindrome is a number that reads the same forward and backward, such as $777$ or ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the differential equation $x''(t) + x(t) = \\sin(t)$. Find the particular integral given that the complementary function is $A\\cos(t) + B\\sin(t)$."
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider the Fibonacci sequence \\( F_1, F_2, F_3, \\cdots \\) where \\( F_1 = 1 \\), \\( F_2 = 1 \\), and \\( F_n = F_{n-1} + F_{n-2} \\) for \\( n \\ge 3 \\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "\\( 6 \\) cats can eat \\( 6 \\) fish in \\( 1 \\) day. \\( c \\) cats can eat \\( 91 \\) fish in \\( d \\) days. Given that \\( c \\) and \\( d \\) are whole... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, making two ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In any triangle \\(ABC\\), let \\(p\\) be the semiperimeter, \\(R\\) be the circumradius, and \\(r\\) be the inradius. Given that \\(p = 10\\), calculate the ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Simplify the expression $\\frac{(n^{2})!}{(n!)^{2}}$."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. I need to simplify the expr... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(0 < a < b < \\infty\\) and \\(D = \\mathbb{R} \\setminus \\{1\\}\\). For \\(r \\in D\\), define \\(g_r: [a, b] \\to \\mathbb{R}\\) as \\(g_r(x) = x^{r-... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider the right isosceles triangle \\( \\vartriangle ABC \\) with the right angle at \\( A \\). Let \\( L \\) be the intersection of the angle bisector of ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given a quadrilateral $ABCD$ with tangents $EF$, $MN$, $HG$, and $OP$ to its incircle, where $O, E$ lie on $AB$, $F, M$ lie on $BC$, $N, H$ lie on $CD$, and $... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In rectangle \\(ABCD\\), diagonal \\(DB\\) is divided into 3 segments of length 1 by parallel lines \\(L\\) and \\(L'\\) that pass through \\(A\\) and \\(C\\)... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A swimming pool is 12 meters long, 6 meters wide, 1 meter deep at the shallow end, and 3 meters deep at the deep end. Water is being pumped into the pool at 1... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all graphs $G$ for which it is possible to assign a real number to each vertex (not all of them zero) such that for every vertex $v \\in G$, the sum of t... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In an acute triangle \\(ABC\\), point \\(D\\) is on \\(BC\\) such that \\(AD \\perp BC\\). Point \\(E\\) is on \\(AC\\) such that \\(\\angle ABE = \\angle EBC... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For how many pairs of positive integers $(a, b)$ is the sequence $10a$, $20b-a$, $2400$ a positive divisibility chain?"
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABC\\) be an acute triangle. Squares \\(AMNB\\) and \\(APQC\\) are constructed on sides \\(AB\\) and \\(AC\\) respectively, outside the triangle \\(ABC... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the area of the region \\( A \\) defined by \\( \\left\\{(x, y) \\mid x \\geq 0, y \\geq 0, \\left(\\frac{x}{a}\\right)^{\\frac{1}{2}} + \\left(\\frac{y}... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( a_0 = \\cos{\\frac{\\pi}{2010}} + i\\sin{\\frac{\\pi}{2010}} \\) and for all positive integers \\( n \\), \\( a_n = (a_{n-1})^{\\frac{13n + 2}{n + 1}}... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Ed has five identical green marbles and a large supply of identical red marbles. He arranges the green marbles and some of the red ones in a row such that the... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider the equation \\(\\left( x + 1 \\right) \\sqrt{y - 5} = y - 3x^2 + x - 2\\). Find the value of \\(y\\) in terms of \\(x\\) that satisfies this equatio... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Three numbers $x, y, z$ are selected at random, with replacement, from the set $\\{-3,-2,-1, 1, 2, 3\\}$. What is the probability that all three of the number... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find a formula for $\\binom{n}{0}+\\binom{n}{b}+\\cdots+\\binom{n}{kb}$ in terms of $n$, $b$, and $k$, where $n > kb$ but $n < (k+1)b$, and $k$ is a natural n... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABCD\\) be a parallelogram with \\(AB = a\\) and \\(AD = b\\), and let \\(M\\) and \\(N\\) be points such that \\(C\\) lies on \\(AM\\) and \\(D\\) lie... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( S(n) \\) denote the number of pairs of nonnegative integers \\((x, y)\\) such that \\( x^2 + y^2 \\le n^2 \\). Calculate the value of \\[ \\frac{S(100... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider an arc of a planar curve where the radius of curvature at any point is a differentiable function of the arc length, and its derivative is everywhere ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given a barn with side lengths 15 and 25, a goat is tethered to one corner of the barn with a rope of length 20. What is the area of grass the goat can feed o... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the local maximum of the function $2x(45-2x)(18-2x)$ without using derivatives."
},
{
"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": "What is the number of elements $r$ in the quotient ring $R=\\mathbb{Z}/2020\\mathbb{Z}$ such that $r^n=1$ for some integer $n \\ge 1$?"
},
{
"role": "... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Convert the fraction $\\frac{1}{16}$ from base 9 to base 4."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I have to convert the fractio... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Write the expression \\(\\tan(2 \\cos^{-1}(1/x))\\) as an algebraic expression in \\(x\\) for \\(x > 0\\)."
},
{
"role": "assistant",
"content": "... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given three linear functions \\( f(x) \\), \\( g(x) \\), and \\( h(x) \\) with integer coefficients, it is known that \\( f(3), g(3), h(3) \\) are \\( 3, 4, 5... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many solutions in positive integers (x, y) does the equation \\(2x + 3y = 563\\) have?"
},
{
"role": "assistant",
"content": "<think>\nOkay, l... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider a finite set of points in the plane, each with integer coordinates. Let \\( S \\) be such a set containing exactly 2023 points. Determine the minimum... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A cloth bag contains a solid pool ball. A second pool ball is chosen at random with probability $p$ of being a solid and $1-p$ of being a stripe. The second b... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In triangle $ABC$, angle bisectors $AD$ and $BE$ are drawn. It is given that $DE$ is the bisector of triangle $ADC$. Find the measure of angle $BAC$."
},
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABC\\) be a triangle with cevians \\(AD\\) and \\(CE\\) meeting at \\(G\\). Given that \\(BD:DC = 5:3\\) and \\(AE:EB = 1:2\\), determine the ratio \\(... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "The positive integers \\( x_1, x_2, \\ldots, x_7 \\) satisfy \\( x_6 = 144 \\) and \\( x_{n+3} = x_{n+2}(x_{n+1} + x_n) \\) for \\( n = 1, 2, 3, 4 \\). Find \... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all polynomials \\( P(x) \\) with real coefficients such that\n\\[ P(x^2 - 2x) = \\left( P(x - 2) \\right)^2. \\]"
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the ordered pair \\((A, B)\\) where \\(A\\) and \\(B\\) are radian measures of acute angles such that \\(\\frac{\\sin(A)}{\\sin(B)} = \\sqrt{3}\\) and \\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Numbers $1, 2, \\ldots, n$ are written on the vertices of a regular $n$-gon. In one step, we may choose two neighboring vertices and interchange the numbers o... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Two players take turns modifying an integer starting from $2011^{2011}$. Each move consists of either subtracting an integer between $1$ and $2010$ inclusive,... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(a, b, c > 0\\). Find the maximum value \\(k_{\\max}\\) such that the inequality\n\\[\n(a+b+c)\\left(\\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c}\\right) \\ge... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the remainder when $2^{1996}$ is divided by $37$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find the remainder when ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let $\\{A_n\\}_{n=1}^\\infty$ be a sequence of matrices where $A_n \\in M_n(\\mathbb{C})$ and $\\det(A_n) \\neq 0, 1$ for all $n \\in \\mathbb{N}$. Calculate ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine the number of odd perfect squares in the set $S = \\{8a + 1 : a \\in \\mathbb{N} \\text{ and } 1 \\leq a \\leq n-1\\}$. Use this to find the formula... |
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