difficulty int64 | source string | domain string | conversations list |
|---|---|---|---|
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Determine the sum of the squares of the roots of the equation\n\n$$x^3 + x^2 + 2x - 20 = 0$$\n\ngiven that one of the roots is a complex number of the form $x = a + bi$, where $a$ and $b$ are integers."
},
{
"from": "gpt",
"value": "<think> Okay, so I have to find th... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "In a triangle, the square of the length of the median to one side is equal to the product of the lengths of the segments into which it divides the opposite side. If the length of the median is 12 cm, and the lengths of the segments are in the ratio 2:3, find the area of the tria... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Find the product of the real roots of the polynomial $f(x) = x^4 - 6x^3 + 11x^2 - 6x - 1$."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to find the product of the real roots of the polynomial f(x) = x⁴ - 6x³ + 11x² - 6x - 1. Hmm, let me start by recalling s... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "In the figure, $ABCD$ is a rectangle with $AD = 4$ and $CD = 6$. The diagonals $\\overline{AC}$ and $\\overline{BD}$ intersect at $O$, and $\\triangle AOB$ is folded over $\\overline{AB}$ so that point $O$ coincides with point $P$ on $\\overline{AD}$. If $\\overline{OP} \\perp \... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "A sphere is inscribed in a right circular cylinder with radius 8 and height 20, so that it is tangent to the top and bottom bases of the cylinder. Find the radius of the sphere."
},
{
"from": "gpt",
"value": "<think> Okay, so there's this problem about a sphere inscr... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "A square and a regular triangle share a common side. Each side of the square is equal to the side length of the triangle. A circle is inscribed inside the square and another circle is inscribed inside the triangle. The area of the smaller circle is 0.71 times the area of the lar... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Let $\\textbf{A} = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}$ and $\\textbf{B} = \\begin{pmatrix} d & c \\\\ b & a \\end{pmatrix}$. If $\\textbf{AB} - \\textbf{BA} = \\begin{pmatrix} 0 & 2c \\\\ 2b & 0 \\end{pmatrix}$, what is $a + d$?"
},
{
"from": "gpt",
"va... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Suppose $f(x)$ is a linear function such that $f(f(f(x)))=x$ for all real numbers $x$ and $f(2)=7$. What is $f(7)$?"
},
{
"from": "gpt",
"value": "<think> Alright, so I have this problem here: \"Suppose \\( f(x) \\) is a linear function such that \\( f(f(f(x))) = x \... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "A positive number is 10 less than its cube root multiplied by 15. What is the largest integer with this property?"
},
{
"from": "gpt",
"value": "<think> Okay, let's try to tackle this problem step by step. The question says: \"A positive number is 10 less than its cu... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Three cylindrical cans have volumes of 50, 150, and 300 cubic centimeters. They are completely filled with paint. The same amount of paint is then poured out of each can. If the remaining paint in the first can fills half of its volume, and the remaining paint in the second can ... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Let $A=(0,0), B=(3,-4), C=(-2,5)$ be points in the plane. What is the minimum value of $|x+2|+|y-3|+|2x-3y-7|$ over all points $(x,y)$ in the triangle $ABC$?"
},
{
"from": "gpt",
"value": "<think> Alright, so I have this problem here: Given points A=(0,0), B=(3,-4), ... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Find the area of the shaded region in the figure, where three semicircles are constructed on the sides of a square with side length $a$.\n[asy]\ndefaultpen(linewidth(0.7pt));\nsize(150);\nreal a = 1; // define variable to be used in drawing\n\ndraw ((0,0)--(0,a)--(a,a)--(a,0)--c... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "The function $f(x) = \\sqrt{x} \\cdot \\sqrt[3]{x}$ has a domain of $x \\ge 0$. Find the largest value of $k$ such that $f(x) + kx + 1 > 0$ for all $x \\ge 0$."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to find the largest value of k such that the functio... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Consider the sequence $\\{b_k\\}$ of real numbers such that $b_1 = 1$ and $b_{m+n} = b_m \\cdot b_n \\cdot (m+n),$ for all positive integers $m$ and $n.$ Find $b_6.$"
},
{
"from": "gpt",
"value": "<think> Alright, so I need to find \\( b_6 \\) for this sequence \\( \... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "The volume of a sphere with radius 10 cm is V cm³. A cone is cut out of the sphere by a plane that is tangent to the sphere and perpendicular to a diameter of the sphere. The cone has a height of 8 cm. What is the volume of the cone in terms of π?"
},
{
"from": "gpt",
... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "The sum of the cubes of two consecutive positive integers is $1200$. Find the smaller integer."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to find two consecutive positive integers where the sum of their cubes is 1200. And I have to find the smaller one. L... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "In triangle $ABC$, $AB=\\sqrt{3}$, $AC=\\sqrt{107}$, and the altitude from $B$ to $\\overline{AC}$ meets $\\overline{AC}$ at $D$. If the medians $\\overline{BE}$ and $\\overline{CF}$ of triangle $ABC$ are perpendicular, find $BC^2$."
},
{
"from": "gpt",
"value": "<th... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Let f(n) be the integer closest to \\sqrt[4]{n}. Find \\sum_{k=1}^{1995}\\frac 1{f(k)}."
},
{
"from": "gpt",
"value": "<think> Alright, so I need to find the sum from k=1 to 1995 of 1 over f(k), where f(n) is the integer closest to the fourth root of n. Hmm, okay. Le... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Let $P$ be a transformation that maps a function $f(x)$ to another function $P(f(x))$ such that:\n(i) $P(f(x) + g(x)) = P(f(x)) + P(g(x))$ for all functions $f(x)$ and $g(x)$.\n(ii) $P(f(x) \\cdot g(x)) = P(f(x)) \\cdot P(g(x))$ for all functions $f(x)$ and $g(x)$.\n(iii) $P(x^2... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Let $a$ be a positive integer, and define the integers $b=a^3-3a^2+19$ and $c=3a^2-2a+1$. When dividing $b$ by $c$, the quotient is $a-1$, and the remainder is $18$. Find $a$."
},
{
"from": "gpt",
"value": "<think> Alright, so I have this problem here. Let me read it... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Let $f(x) = \\frac{p(x)}{q(x)}$ be a rational function, where $p(x)$ is linear and $q(x)$ is quadratic. The graph of $f(x)$ has a horizontal asymptote of $y=2$, and the point $(3,2)$ lies on the graph. Furthermore, $f(x)$ has a hole at $x=1$, and $f(1) = \\frac{5}{2}$. Find $p(x... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "In the coordinate plane, four points A(0,1), B(1,0), C(0,-1) and D(-1,0) are the vertices of a square. The lines AC and BD intersect at the origin, O. The other two diagonals, AB and CD, intersect at a point E. Let F be the midpoint of segment AB, and let G be the midpoint of se... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Let $a,$ $b,$ $c$ be complex numbers such that\n\\[a + b + c = ab + ac + bc = abc = 1.\\]Enter the values $a,$ $b,$ $c,$ separated by commas, in any order."
},
{
"from": "gpt",
"value": "<think> Alright, so I have this problem here where I need to find complex number... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Compute the product of the roots of the equation \\[x\\sqrt{x} + 4x - 5\\sqrt{x} - 2 = 0,\\]given that all of the roots are real and nonnegative."
},
{
"from": "gpt",
"value": "<think> Okay, so I have this equation to solve: \\(x\\sqrt{x} + 4x - 5\\sqrt{x} - 2 = 0\\)... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "The diagonals of a square intersect at a point $P$. Two lines, $y = x + 1$ and $y = kx + 2$, contain two pairs of opposite vertices of the square. Find the possible values of $k$."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to find the possible values of k... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Determine the number of complex numbers $z$ such that $z$, $|z|$ (the absolute value of $z$), and $\\arg z$ (the argument of $z$) form an arithmetic sequence. Consider only those complex numbers whose argument lies between $0$ and $\\pi$."
},
{
"from": "gpt",
"value"... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Let $x,$ $y,$ $z$ be real numbers, all greater than 4, so that\n\\[\\frac{(x + 3)^2}{y + z - 3} + \\frac{(y + 5)^2}{z + x - 5} + \\frac{(z + 7)^2}{x + y - 7} = 48.\\]Enter the ordered triple $(x,y,z).$"
},
{
"from": "gpt",
"value": "<think> Okay, so I have this probl... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Determine the largest positive integer that has exactly three distinct positive divisors, one of which is 8."
},
{
"from": "gpt",
"value": "<think> Alright, so I need to find the largest positive integer that has exactly three distinct positive divisors, and one of t... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Let g0(x) = x^2 + |x^2 - 1| and for n ≥ 1, let gn(x) = |gn-1(x)| - 2. For how many values of x is g200(x) = 0?"
},
{
"from": "gpt",
"value": "<think> Okay, so I need to figure out how many values of x satisfy the equation g₂₀₀(x) = 0. The functions are defined recurs... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Find the largest positive integer $x$ such that $5x^2 + 3x - 1$ is divisible by 11."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to find the largest positive integer x such that when you compute 5x² + 3x -1, the result is divisible by 11. Hmm, divisibility ... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Determine the value of parameter $t$ such that the parabolas given by $x^2-2ty+t=0$ and $y^2-2tx+t=0$ have only one intersection point."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to find the value of the parameter \\( t \\) such that the two parabolas giv... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "The function $f(x)$ is defined as follows: $f(0) = 1$, and for $x \\neq 0$, $f(x) = x + f(\\frac{1}{x})$. Determine the value of $f(-\\frac{3}{5})$."
},
{
"from": "gpt",
"value": "<think> Okay, so I have this function f(x) defined as f(0) = 1, and for x not equal to ... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Let $f(x)$ be a function such that $f(1)=2$, $f(2)=7$, and\n\n$$f(xf(y))=f(x)f(x+y)+y$$for all real numbers $x$ and $y$. Find $f(0)$."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to find f(0) given this functional equation. Let me start by writing down what... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Let $x$, $y$, and $z$ be real numbers such that\n\\[\\tan x + \\tan y + \\tan z = \\cot x + \\cot y + \\cot z = 0.\\]Find the product of all possible values of $\\sin 2x \\sin 2y \\sin 2z.$"
},
{
"from": "gpt",
"value": "<think> Okay, so I need to find the product of... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Given a triangle $ABC$ with sides of length 3, 4, and 5, a point $P$ is chosen inside the triangle such that the distance from $P$ to each side is 1. Find the length of the longest diagonal of the quadrilateral formed by connecting the feet of the perpendiculars from $P$ to the ... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Evaluate the series\n\\[\\frac{3}{2^2 \\cdot 5^2} + \\frac{7}{5^2 \\cdot 9^2} + \\frac{11}{9^2 \\cdot 13^2} + \\dotsb.\\]"
},
{
"from": "gpt",
"value": "<think> Alright, so I need to evaluate the series \\(\\frac{3}{2^2 \\cdot 5^2} + \\frac{7}{5^2 \\cdot 9^2} + \\fra... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "A bookstore has a collection of 8 novels written by different authors. Each novel has a unique genre (romance, sci-fi, mystery, etc.) and a unique color for its cover (red, blue, green, etc.). What is the greatest number of sets of 3 novels that can be chosen such that no two se... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Given that\n\n$$\\frac{\\sin^2 x}{\\cos^2 y} + \\frac{\\cos^2 x}{\\sin^2 y} = 2 \\quad \\text{and} \\quad \\frac{\\sin^2 y}{\\cos^2 x} + \\frac{\\cos^2 y}{\\sin^2 x} = 3,$$\n\nfind the value of $$\\frac{\\tan x \\tan y}{\\cot x \\cot y} + \\frac{\\cot x \\cot y}{\\tan x \\tan y}... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "I have a rectangular carpet with dimensions $x$ and $y$ (in feet), such that $x$ and $y$ are both integers greater than one. I would like to place this carpet in a room with dimensions $(x + 5)$ and $(2y + 1)$. If I measured the area of the room that is not covered by the carpe... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Problem:\nIn a large corporation, there are three departments: marketing, finance, and human resources. Each department has 5 employees, with 2 employees from each department being team leaders. A task force of 6 employees is to be formed, consisting of 2 team leaders and 4 non-... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Find the value of x for which the function\n$$g(x) = \\frac{2x^2 - 3x + 1}{x^2 - 2x - 1}$$\ntouches the line y=2."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to find the value of x where the function g(x) = (2x² - 3x + 1)/(x² - 2x - 1) touches the line y =... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Find the value of $a$ for which the equation $|x^2 - 4| = x^2 + ax + 3$ has exactly two solutions."
},
{
"from": "gpt",
"value": "<think> Alright, so I need to solve this equation: |x² - 4| = x² + ax + 3, and find the value of 'a' such that there are exactly two solu... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "A five-digit number $abcde$ is divisible by 4, where $a$, $b$, $c$, $d$ and $e$ represent distinct digits. The sum of the digits is equal to 25. The digits in the number are such that $e - d = 1$, $c - b = 2$ and $a - c = 3$. What is the value of $a$?"
},
{
"from": "gpt"... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Determine all polynomials P(x) such that\nP(0)=0,\nP(1)=1,\nand\nP(x)P(y)=P(x+y) for all real numbers x and y."
},
{
"from": "gpt",
"value": "<think> Alright, let's tackle this problem. I need to find all polynomials P(x) that satisfy three conditions: P(0) = 0, P(1)... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "A bookshelf has 5 shelves, and each shelf can hold exactly 8 books. A customer buys a number of boxes of books, with each box containing 12 books. After filling the bookshelf, the customer finds that she has 3 books left over. What is the SECOND smallest number of boxes of books... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Three types of tickets are available for a concert: economy, VIP, and deluxe. The economy tickets cost $20, VIP tickets cost $30, and deluxe tickets cost $40. A group of friends buys a total of 15 tickets, with at least one ticket of each type, for a total of $510. If the number... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Each of the nine dots in this figure is to be colored red, white, or blue. No two dots connected by a segment (with no other dots between) may be the same color. However, the dots are now connected by additional segments, as shown below. How many ways are there to color the dots... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "In a highly competitive music festival, three judges - Alice, Bob, and Charlie - are tasked with evaluating a large number of performances. Alice starts with the first performance and evaluates every $a$ th performance; Bob starts with the second performance and evaluates every ... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Let $\\varphi(x)$ be a continuous function such that for any integer $n$, $\\varphi(\\sin nx) = n\\varphi(\\sin x)$ for all $x \\in [0, \\pi/2]$.\nFind the smallest possible value of $\\varphi(1)$."
},
{
"from": "gpt",
"value": "<think> Okay, let me try to tackle thi... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Three numbers $a$, $b$, and $c$ form a geometric sequence. When written in base $b$, the numbers $a$ and $c$ have the same three digits in the same order. Find the smallest possible value of $b$."
},
{
"from": "gpt",
"value": "<think> Okay, let's tackle this problem ... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Problem:\n\nA professor is organizing a conference with 5 speakers and 3 sessions. Each speaker can present in any of the sessions, but no speaker can present more than once. The professor has received 12 abstracts, and after reviewing them, has decided to reject 4 of them. The ... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "A bookstore has 15 shelves, and the owner wants to display a different number of books on each shelf, with the number of books on each shelf being a multiple of 3. If the owner has 120 books to display, what is the maximum number of additional books that can be added to the shel... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "In triangle $PQR$, $PR$ is perpendicular to $PQ$, and $PS$ is an altitude to $QR$. Given that $\\angle QPS = \\angle SPR$, $PQ = 13$, and $PS = 2$, find $QS$."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to find the length of QS in triangle PQR where PR is ... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Let $a$, $b$, $c$ be nonzero real numbers such that $a^2+b^2+c^2=ab+bc+ca$ and $a^3+b^3+c^3=a^2b+a^2c+b^2a+b^2c+c^2a+c^2b$. Find the value of $a+b+c$."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to solve this problem where a, b, c are nonzero real numbers ... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "A quadrilateral has three sides of lengths 8, 10, and 12 units. How many different integer lengths can the fourth side be if it is guaranteed that at least one pair of opposite sides can form a triangle?"
},
{
"from": "gpt",
"value": "<think> Okay, so I have this pro... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "If $(2, -5)$ and $(8, 11)$ are the coordinates of two opposite vertices of a rectangle, what is the difference of the $x$-coordinates of the other two vertices?"
},
{
"from": "gpt",
"value": "<think> Okay, let me try to figure this problem out. So, we have a rectangl... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "In the Cartesian plane, the graph of the hyperbola $\\frac{x^2}{4}-y^2=1$ intersects the graph of the ellipse $x^2+4y^2=k$ in exactly one point. Find all possible values of $k.$"
},
{
"from": "gpt",
"value": "<think> Alright, so I have this problem here: the hyperbo... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "A circle with center $O$ is inscribed in triangle $PQR$, where $PQ = PR$. Point $S$ is the intersection of lines $OP$ and $QR$. Determine the measure of angle $\\angle PSR$ if $\\angle PQR = 45^\\circ$ and $\\angle OPR = 30^\\circ$."
},
{
"from": "gpt",
"value": "<th... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "A circle is inscribed in a square, and then a new square is circumscribed around the circle. If the area of the new square is 32 square units, what is the area of the original square?"
},
{
"from": "gpt",
"value": "<think> Okay, let's try to solve this problem step b... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "In the figure, $\\triangle ABC$ is an acute triangle with circumcenter $O$. The perpendicular bisectors of $AB$ and $AC$ intersect at $H$, and $HB=8$ while $HC=3$. Calculate $(AB)(AC)-(BC)^2$. [asy]\nsize(150); defaultpen(linewidth(0.8));\npair B = (0,0), C = (3,0), A = (2,2), H... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Six circles are arranged in a circle, with each circle externally tangent to its two neighbors. All six circles are internally tangent to a larger circle. The radius of each small circle is $r$. The area of the region inside the small circles and outside the large circle is $K$.... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "The median of the set $\\{-2, 4, x\\}$ is equal to the mode of the set $\\{-2, x, 4, 7, y\\}$. If $y$ is a whole number, what is the largest possible value of $y$?"
},
{
"from": "gpt",
"value": "<think> Okay, so I have this problem here: The median of the set {-2, 4,... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Emily had a bag of cookies, and none of the cookies could be broken into pieces. She ate $\\frac{1}{4}$ of them and then gave $\\frac{1}{5}$ of what remained to her friend Olivia. Emily and her mom then each ate 8 cookies from what Emily had left. Finally, Emily's sister took so... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "In triangle $ABC,$ $P$ is a point on side $\\overline{AB}$ such that $AP = 12$ and $BP = 4.$ If $\\sin \\angle CPD = 1/2,$ then find $BC$.\n\nNote: In the original problem, the angle is given in terms of tangent, whereas in this new problem, the angle is given in terms of sine,... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "The product of four different positive integers is equal to $5^4$. What is the sum of the four integers?"
},
{
"from": "gpt",
"value": "<think> Okay, so the problem is asking for four different positive integers whose product is 5^4, and then we need to find the sum ... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Determine the number of complex numbers $z$ such that $z$, $|z|$ (the absolute value of $z$), and $\\arg z$ (the argument of $z$) form an arithmetic sequence. Consider only those complex numbers whose argument lies between $0$ and $\\pi$."
},
{
"from": "gpt",
"value"... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "The mode of the dataset {3, 7, x, 11, 3} is equal to the median of the dataset {5, y, 12}. What is the value of x+y?"
},
{
"from": "gpt",
"value": "<think> Okay, so I have this problem here: The mode of the dataset {3, 7, x, 11, 3} is equal to the median of the datas... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "How many positive integers $m$ satisfy\n$\\dfrac{m+2000}{120} = \\lceil \\sqrt[3]{m} \\rceil$?\n(Recall that $\\lceil x\\rceil$ is the least integer greater than or equal to $x$.)"
},
{
"from": "gpt",
"value": "<think> Okay, so I have this problem here: I need to fin... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Determine the largest possible value of $m$ such that\n$$f(x) = (x^2-4x+4)(x^2-8x+4)(x^2+4x+4)(x^2+8x+4)$$\nis divisible by $x^m$, where $x$ is a multiple of $1280$."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to figure out the largest possible value of m ... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Melinda will roll three standard six-sided dice and make a three-digit number with the numbers she rolls. For example, if she rolls a 4, a 2, and a 5, she can form 425, 452, 524, 542, 245, or 254. What is the probability that she will be able to make an integer between 100 and 1... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "A point $P$ is located on the unit circle in the complex plane. If the angle between $P$ and the positive real axis is increased by $\\frac{\\pi}{6}$, the distance from $P$ to the origin doubles. What is the original distance from $P$ to the origin?"
},
{
"from": "gpt",
... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Let $a$, $b$, $c$ be positive real numbers such that\n\\[\\frac{a}{b} + \\frac{b}{c} + \\frac{c}{a} = 7.\\]Find the value of \\[\\frac{a + b + c}{ab + bc + ca}.\\]"
},
{
"from": "gpt",
"value": "<think> Okay, so I have this problem here where I need to find the value... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "A certain chocolate factory produces chocolate bars with a weight of 30 grams each. The bars are packed in boxes of 5 or 7 bars per box, with each box weighing exactly 1 kg. Due to a machine malfunction, some boxes were filled with the wrong number of bars. The total weight of t... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "There exist positive integers $a,$ $b,$ and $c$ such that\n\\[3 \\sqrt{\\sqrt[3]{5} - \\sqrt[3]{4}} = \\sqrt[3]{a} + \\sqrt[3]{b} - \\sqrt[3]{c}.\\]Find $a + b + c.$"
},
{
"from": "gpt",
"value": "<think> Alright, so I've got this problem here: \n\nThere exist positi... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "A cylindrical vessel with a height of 10 cm and a radius of 4 cm is filled with water. The water is then poured into a cone-shaped vessel with the same height and radius. What is the ratio of the height of the water in the cone to the height of the cone itself?"
},
{
"fr... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Find the value of the parameter a such that the polynomial\nx^2 + ax + a^2 - 3\ncan be divided without a remainder by the quadratic equation\nx^2 + 5x + 6\nbut with a remainder by its factor\nx + 2."
},
{
"from": "gpt",
"value": "<think> Okay, let me try to work thro... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "What is the product of all positive integers whose digits form a sequence of consecutive integers, and whose sum is divisible by the number of digits?"
},
{
"from": "gpt",
"value": "<think> Alright, so I came across this problem: \"What is the product of all positive... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Given the function $f(x)=x^{2}-2x$,\n\nCreate a new function $g(x)$ such that $g(g(x))=f(x)$. Determine the value of $g(1)$."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to create a function g(x) such that when I compose it with itself, I get f(x) = x² - 2x... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Let ABC be a triangle such that $\\angle A=\\angle B=60^\\circ$, $BC=4\\sqrt{3}$ and $\\overline{AC}$ is perpendicular to $\\overline{AD}$. If $D$ is a point on $\\overline{BC}$ such that the area of $\\triangle ACD$ is half of the area of $\\triangle ABC$, determine the area of... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "In the circle with center O, the measure of ∠RIP is 36∘ and OR=10 cm. The length of arc RP is x cm. If the circle is divided into two sectors, ROP and RIP, and the area of sector ROP is twice the area of sector RIP, find the value of x. Express your answer in terms of π."
},
... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "A necklace is a string of colored beads. How many different necklaces can be made using 6 beads of 3 different colors (2 beads of each color), considering two necklaces to be the same if one can be obtained from the other by rotation or reflection?"
},
{
"from": "gpt",
... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "A bookshelf has 5 shelves, and each shelf can hold a certain number of books. The number of books on each shelf is a consecutive integer from 1 to 5. If the bookshelf is currently empty, how many books need to be added to the bookshelf so that the total number of books on the bo... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "The average of all the possible values of $y$ that satisfy the equation $\\sqrt[3]{3y^3+2}=4$ is equal to $\\frac{\\sqrt[3]{55}-1}{3}$.\nFind the sum of all the possible values of $y$."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to solve the equation \\(\\... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "A cone is cut by a plane perpendicular to its axis, forming a circle. If the radius of the circle is 2/3 of the distance from the vertex of the cone to the plane, find the ratio of the volume of the cone below the plane to the volume of the cone above the plane."
},
{
"f... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "The diagonal of a rectangular prism is 25 units long, and the length of the longest edge is 15 units. If the surface area of the prism is 600 square units, find the volume of the prism."
},
{
"from": "gpt",
"value": "<think> Alright, so I need to find the volume of t... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Find the equation of the directrix of the parabola $y = \\frac{x^2 - 6x + 5}{12}.$\n\nAnother problem:\n\nFind the equation of the directrix of the parabola $y = x^2 + 2x + 1$, given that the parabola has been rotated $45^\\circ$ clockwise about its vertex."
},
{
"from":... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "A bookshelf has 5 shelves, and each shelf can hold a certain number of books. If the top shelf can hold 8 books, the second shelf can hold 12 books, the third shelf can hold 15 books, and so on, with each subsequent shelf holding 3 more books than the previous one, what is the g... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "A fair tetrahedron-shaped die has four congruent triangular faces, each of which is a different color: blue, green, red, and yellow. When the die is rolled, what is the probability that the colors on the two triangular faces that share an edge with the bottom face will be green ... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "In a quadrilateral $ABCD$, the midpoints of the sides $AB$ and $CD$ are $E$ and $F$, respectively. It is known that $\\triangle AED$ and $\\triangle BFC$ are congruent, and the length of $AB$ is twice the length of $CD$. Determine the length of $AD$ if $BC$ is $6$ cm."
},
{
... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "If $\\sqrt[3]{2} \\cdot \\sqrt[3]{3} \\cdot \\sqrt[3]{4} = p + \\sqrt[3]{q},$ where $p$ and $q$ are integers, what is $p + q$?"
},
{
"from": "gpt",
"value": "<think> Okay, so I have this problem here: If the cube root of 2 multiplied by the cube root of 3 multiplied ... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "In a circle of radius 4 cm, two non-overlapping sectors have areas 14.13 cm^2 and 28.26 cm^2 respectively. The endpoints of the two sectors divide the circle into 3 arcs, two of which are adjacent to one of the sectors. If these two arcs have equal lengths, find the length of th... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Let $g(x) = 2\\sin{x} - 3\\cos{x} + \\cot{x}$, using radian measure for the variable $x$. Let $s$ be the smallest positive value of $x$ for which $g(x) = 0$. Find $\\lceil s \\rceil.$"
},
{
"from": "gpt",
"value": "<think> Alright, so I need to solve the equation \\(... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Let $f(x)$ be a polynomial of degree greater than 3, such that the remainder when divided by $x(x-1)(x-2)$ is $x^2+2$. When $f(x)$ is divided by $(x+1)(x+2)(x+3)$, what is the remainder?"
},
{
"from": "gpt",
"value": "<think> Okay, let's see. I have this problem here... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Solve the equation\n$\\sqrt{y + 2 - 3 \\sqrt{y - 2}} + \\sqrt{y + 5 - 5 \\sqrt{y - 2}} = 2$."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to solve this equation: √(y + 2 - 3√(y - 2)) + √(y + 5 - 5√(y - 2)) = 2. Hmm, that looks a bit complicated with all the... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Problem:\nA right circular cone is inscribed in a right prism as shown. The volume of the cone is $\\frac{1}{3}$ of the volume of the prism. What is the ratio of the radius of the base of the cone to the height of the prism? Express your answer as a common fraction in terms of $... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "The graphs of $y=2-x^2+x^4$ and $y=1+x^2-x^4$ intersect in multiple points. Find the maximum difference between the $y$-coordinates of these intersection points."
},
{
"from": "gpt",
"value": "<think> Okay, let's see. I have to find the maximum difference between th... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "The hyperbola \\[\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1\\] and the circle $x^2+y^2=r^2$ have exactly one point of intersection. Find $r^2$ in terms of $a$ and $b$."
},
{
"from": "gpt",
"value": "<think> Okay, so I have this problem here: I need to find r squared in ... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Let $p(x) = x^7 + ax^6 + bx^5 + cx^4 + dx^3 + ex^2 + fx + g$ be a polynomial with integer coefficients such that $p(1) = p(2) = \\ldots = p(8) = 0$. Evaluate $p(0)$.\n\nNote: You can use the properties of polynomials and the factor theorem to approach this problem."
},
{
... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Find the ordered pair $(c,d)$ of rational numbers such that $$\\frac{\\tan 25^\\circ}{\\cos^2 25^\\circ} = c + d \\sec 25^\\circ.$$"
},
{
"from": "gpt",
"value": "<think> Okay, so I need to find rational numbers c and d such that (tan 25°)/(cos²25°) equals c + d sec2... |
null | ai2-adapt-dev/openmath-2-math | math | [
{
"from": "human",
"value": "Find the coefficient $b$ for which the graphs of $y=x^2+bx-1$ and $y=-x^2+bx+3$ intersect along a line segment."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to find the coefficient \\( b \\) such that the graphs of \\( y = x^2 + bx - 1 \\) and \\( y = -x^2 ... |
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