messages listlengths 2 2 | domain stringclasses 1
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|---|---|---|---|
[
{
"content": "Five of the eight numbers 2317, 1993, 2019, 2025, 2109, 2121, 2234, and 2125 have a median (middle value) of 2010. What is the median (middle value) of the other three numbers?",
"role": "user"
},
{
"content": "<think> Okay, so the problem says that five out of these eight numbers have... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A convex polygon is inscribed in a circle such that each vertex of the polygon is a point on the circle. The polygon has 15 sides, and its diagonals intersect the circle at 30 distinct points. How many of these points are inside the polygon?",
"role": "user"
},
{
"content": "<think> Al... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Find the largest base $b$ such that the base-$b$ digits of $111^3$ do not add up to $3^3$.",
"role": "user"
},
{
"content": "<think> Okay, so I need to find the largest base \\( b \\) such that when I cube the number 111 (which is in base \\( b \\)), the sum of its digits in that base ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A sequence of integers is called \"tasty\" if the sum of the absolute values of its elements is equal to 1000. What is the smallest possible value of the largest element in a tasty sequence?",
"role": "user"
},
{
"content": "<think> Okay, let me try to figure out this problem. So, we n... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Six points are marked on a circle. How many ways are there to connect the points with line segments so that each point is connected to exactly two other points, and no two line segments intersect?",
"role": "user"
},
{
"content": "<think> Okay, so I need to figure out how many ways the... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let $P(x)$ be a polynomial such that when it's multiplied by $x+2$ the product is\n$$P(x)\\cdot (x+2)=25x^2+33x+10$$.\nDetermine $P(x)$.",
"role": "user"
},
{
"content": "<think> Okay, so I need to find the polynomial P(x) such that when it's multiplied by (x + 2), the result is 25x² +... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "You start at the origin and take turns flipping a fair coin and moving to the left if it's heads or to the right if it's tails. If you ever reach a position more than 4 units to the left or right, you immediately go back to the origin and start over. What is the expected number of turns it will ta... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let $x$ and $y$ be positive integers such that $y<x\\leq 100$ and $\\frac{x}{y}$ is an integer. How many ordered pairs $(x,y)$ are there such that $x+y$ is a perfect square?",
"role": "user"
},
{
"content": "<think> Okay, so I have this problem here: I need to find the number of ordere... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Luna listed all the single-digit base-$c$ numbers and multiplied them in base $c$, getting $102_c$.\n\nWhat is $c$?",
"role": "user"
},
{
"content": "<think> Okay, let's try to solve this problem. So, Luna listed all the single-digit base-c numbers and multiplied them in base c, gettin... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A runner ran at a constant speed for the first 45 minutes of a race, covering a certain distance. For the remaining 30 minutes, she ran at a speed that was 2 miles per hour faster than her initial speed, covering an additional 6 miles. If her total distance covered was 16 miles, what was her initi... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A box contains 10 light bulbs, with each bulb having a different wattage from 1 to 10 watts. If 3 bulbs are randomly chosen and connected in series, what is the probability that the total resistance of the circuit is greater than 15 ohms?",
"role": "user"
},
{
"content": "<think> Okay,... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Find the sum of all the integer values of $x$ that satisfy the inequality $x^4 - 25x^2 + 144 > 0$.",
"role": "user"
},
{
"content": "<think> Okay, so I need to find all integer values of x that satisfy the inequality x⁴ - 25x² + 144 > 0, and then sum them up. Hmm, quartic inequality. T... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In the diagram, $\\triangle{ABC}$ is a right triangle with a right angle at $B$, and $\\angle{CAD}=90^{\\circ}$. Given that $AB=15$ cm, $BC=20$ cm, and $CD=8$ cm, find the length of $AD$, to the nearest tenth of a centimeter.",
"role": "user"
},
{
"content": "<think> Okay, let's see he... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In a geometric sequence, the fifth term is twice the second term, and the sum of the first three terms is equal to the sum of the next three terms. What is the common ratio?",
"role": "user"
},
{
"content": "<think> Okay, let's tackle this problem about the geometric sequence. Alright,... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In a given rectangular coordinate system, the graph of an ellipse is tangent to the line $y = x + 2$ at the point $(0, 2)$ and tangent to the line $y = -x + 4$ at the point $(2, 2)$. Find the equation of the ellipse and the area enclosed by it.",
"role": "user"
},
{
"content": "<think>... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let $\\mathbf{C}$ and $\\mathbf{D}$ be matrices such that\n\\[\\mathbf{C} - \\mathbf{D} = \\mathbf{C} \\mathbf{D}.\\]If $\\mathbf{C} \\mathbf{D} = \\begin{pmatrix} 1 & -2 \\\\ 3 & -4 \\end{pmatrix},$ find $\\mathbf{D} \\mathbf{C}.$",
"role": "user"
},
{
"content": "<think> Okay, let me... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "What is the largest negative integer that is a perfect cube and 5 more than a prime number?",
"role": "user"
},
{
"content": "<think> Okay, so the problem is asking for the largest negative integer that is a perfect cube and is 5 more than a prime number. Hmm, let's see. Let me break t... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "An artist has 15 identical squares of colored paper and wants to arrange them into a rectangular mosaic with a specific pattern. The mosaic must consist of a combination of 2x2 squares and 1x1 squares, with the total number of 1x1 squares being exactly 4 more than the total number of 2x2 squares. ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "How many 5-digit numbers are there in which the sum of the digits is divisible by 6 and there is exactly one pair of equal digits among the five?",
"role": "user"
},
{
"content": "<think> Alright, so I have this problem here: \"How many 5-digit numbers are there in which the sum of the... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Determine the number of perfect squares that can be expressed as the sum of two consecutive perfect cubes.",
"role": "user"
},
{
"content": "<think> Okay, so I need to figure out how many perfect squares are the sum of two consecutive perfect cubes. Hmm, let me start by understanding t... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let $\\triangle ABC$ be a right triangle such that $B$ is a right angle. A circle with diameter of $BC$ meets side $AC$ at $D.$ If $AD = 1$ and $BD = 4,$ then what is $CD$?",
"role": "user"
},
{
"content": "<think> Okay, so here's this geometry problem: we have a right triangle ABC wit... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A school has 20 classes, each with 25 students. The average height of the students in each class is 160 cm, with a standard deviation of 5 cm. If the average height of all 500 students in the school is 161 cm, what is the standard deviation of the average heights of the classes?",
"role": "use... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Find the greatest common divisor of two numbers, knowing that their product is 5832 and their sum is 252.",
"role": "user"
},
{
"content": "<think> Okay, so I have this problem here: Find the greatest common divisor (GCD) of two numbers. The product of these numbers is 5832, and their ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "The quadratic function $f(x) = ax^2 + bx + c$ has the property that the sum of its values at any two points $x$ and $y$ is equal to the product of the $x$-coordinates of those points. Find the value of $b$.\n\n(Note: I've created a new problem that requires a different approach to solve, involving... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A right triangle and an equilateral triangle of equal height are placed side by side, as shown, with both bases on the x-axis. The lower right vertex of the right triangle and the lower left vertex of the equilateral triangle are at (16, 0). The hypotenuse of the right triangle and the base of the... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Triangle $MNO$ has vertices $M(0, 4)$, $N(3, 1)$, $O(2, 6)$. A circle with center $P$ on the side $NO$ passes through $M$ and cuts the area of $\\triangle MNO$ in half; find the distance from $P$ to the midpoint of $NO$.",
"role": "user"
},
{
"content": "<think> Okay, so I have this p... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Given a line $L$ in the $xy$-plane defined by the equation $y = 2x + 1$, find the matrix $\\mathbf{T}$ such that for any point $\\mathbf{p}$ in the plane, $\\mathbf{T} \\mathbf{p}$ is the point obtained by rotating $\\mathbf{p}$ about $L$ by $90^\\circ$ counterclockwise.",
"role": "user"
},
... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let $a$ and $b$ be positive integers such that $ab=432$.\nHow many ordered pairs $(a,b)$ are there such that $a$ and $b$ have the same number of positive divisors?",
"role": "user"
},
{
"content": "<think> Okay, so I have this problem here: Let a and b be positive integers such that ab... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A city has a set of streets that form a grid of squares, with each square representing a city block. There are $n$ different types of restaurants, and the city wants to place one restaurant of each type on a city block, such that no two restaurants of the same type are on adjacent blocks (horizont... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "The polynomial $P(x)$ with real coefficients satisfies the condition:\n\n\\[P(\\sin x) \\ge P(\\cos x)\\]\n\nfor all real numbers $x.$ Find the sum of the roots of $P(x)$.",
"role": "user"
},
{
"content": "<think> Okay, so I have this problem here: A polynomial P(x) with real coeffici... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let $\\overline{AB}$ be a diameter of circle $\\omega$. Extend $\\overline{AB}$ through $A$ to $C$. Point $T$ lies on $\\omega$ so that line $CT$ is tangent to $\\omega$. Point $P$ is the foot of the perpendicular from $A$ to line $CT$. Suppose $\\overline{AB} = 18$, and let $m$ denote the maximum... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "An $800$-page book is divided into four sections. If the number of pages in each section is proportional to the number of coins in each of Adam's four coin jars, and the first section has $160$ more pages than the second section, what is the maximum number of coins that the third jar could contai... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A bakery sells a total of 1500 muffins and cakes on a certain day. The muffins are packed in boxes of 6, while the cakes are packed in boxes of 8. At the end of the day, the bakery has 5 boxes of muffins and 3 boxes of cakes left over, but they still have a few unpacked muffins and cakes. If the t... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "The equation $x^3 - 2x^2 - 5x + 1 = 0$ has three solutions. One of the solutions has the form $\\frac{a}{b}$ for positive natural numbers $a$ and $b$ where $a > b$. What is $a - b$?",
"role": "user"
},
{
"content": "<think> Okay, so I need to solve this cubic equation: \\( x^3 - 2x^2 -... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Find all possible integer values of $n$ such that the quadratic equation $x^2 + nx + 16 = 0$ has two distinct integer roots, one of which is positive and the other negative, and the sum of their absolute values is a perfect square.",
"role": "user"
},
{
"content": "<think> Okay, let me... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Two natural numbers, $a$ and $b$, are chosen randomly and independently. What is the probability that the greatest common divisor of $a$ and $b$ is equal to the least common multiple of the remainders when $a$ is divided by $b$ and $b$ is divided by $a$?",
"role": "user"
},
{
"content"... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Determine the value of $b$ that satisfies the equation $417_{b}+529_{b}=94C_{b}$, where $C_{b}=12_{10}$.",
"role": "user"
},
{
"content": "<think> Okay, so I have this equation here: 417 base b plus 529 base b equals 94C base b, and they tell me that C in base b is 12 in decimal. I nee... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Determine all possible values of $x$ for which the equation\n$$2^{\\sin x}+2^{\\cos x}=2^{\\sin x+\\cos x}$$ holds true.",
"role": "user"
},
{
"content": "<think> Okay, so I have this equation to solve: \\(2^{\\sin x} + 2^{\\cos x} = 2^{\\sin x + \\cos x}\\). Hmm, I need to find all x ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A triangle is formed by three vertices, each representing a point in the Cartesian plane. The vertices are $A (p, q), B (m, n), and $C (r, s). The equation of the line containing the median from vertex $A$ to side $BC$ is $x - y = 1. The coordinates of $B$ and $C$ are $(3, -5)$ and $(4, -2)$, resp... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let $N$ be a three-digit number that is divisible by 5 and has the property that when its digits are reversed, the resulting number is $\\frac{3}{5}N$. How many such numbers $N$ exist?",
"role": "user"
},
{
"content": "<think> Okay, let's try to figure out this problem. We need to find... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In a small pond, two types of water lilies are growing: Type X and Type Y. The number of Type X lilies doubles every week, while the number of Type Y lilies triples every week. On the first week, the total number of lilies in the pond is 20. After 4 weeks, the total number of lilies is 244. How ma... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A convex hexagon has interior angles with measures $x$, $2x$, $3x$, $4x$, $5x$, and $6x$ degrees. The sum of the measures of any three consecutive angles is equal to 540 degrees. What is the measure of the smallest angle?",
"role": "user"
},
{
"content": "<think> Okay, so I need to s... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A company is planning to distribute a total of 480 bonus shares to its employees. The number of shares each employee receives is directly proportional to the number of years they have worked at the company. If the employee who has worked the longest receives 5 times as many shares as the employee ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "The numbers $a$ and $b$ are such that $3\\le \\ds\\f{2a+3b}{a+b}<6$. Given that $\\ds\\f{2a}{3b}$ is a prime number, find its value.",
"role": "user"
},
{
"content": "<think> Okay, so I need to solve this problem where we have two numbers, a and b, and they satisfy the inequality 3 ≤ (... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Consider a sequence of integers $\\{a_n\\}$ such that $a_1 = 12$ and for each $n > 1$, $a_n$ is the smallest integer greater than $a_{n-1}$ that can be expressed as the sum of two distinct primes. Determine the smallest possible value of $n$ such that $a_n > 100$.",
"role": "user"
},
{
... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "What is the area of the portion of the circle defined by $x^2-12x+y^2=28$ that lies above the $x$-axis and to the right of the line $y=6-x$?",
"role": "user"
},
{
"content": "<think> Okay, so I need to find the area of the portion of the circle defined by the equation \\( x^2 - 12x + y... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Pentagon $ABCDE$ has right angles at $B$, $C$, and $D$, $\\triangle ABC \\sim \\triangle BCD \\sim \\triangle CDE$, and $AB > BC > CD$. There is a point $F$ in the interior of $ABCDE$ such that $\\triangle ABC \\sim \\triangle CFB$ and the area of $\\triangle AEF$ is $22$ times the area of $\\tria... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "The 20 members of a school's Drama Club are lining up for a group picture. When they line up in rows of 4, there are 4 members in the last row. When they line up in rows of 5, there are 2 members in the last row. When they line up in rows of 7, there are 3 members in the last row. How many differe... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A fruit salad consists of apples, bananas, and grapes in the ratio of 3:4:5. If there are 6 more apples than grapes, and there are a total of 96 fruits in the salad, how many bananas are there?",
"role": "user"
},
{
"content": "<think> Okay, let's see. The problem is about a fruit sala... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "There are $n$ bags numbered from 1 to $n$. Each bag contains exactly $n-1$ different colored balls. Prove that it is possible to choose one ball from each bag such that the $n$ balls chosen all have distinct colors.",
"role": "user"
},
{
"content": "<think> Okay, so there's this proble... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A wealthy pig has $\\$15.00$ in coins consisting of dimes and nickels only. Each time it runs around a muddy pit, it loses one dime. What is the maximum number of laps it can run without having an amount of money that cannot be made with dimes and nickels?",
"role": "user"
},
{
"conten... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Problem:\n\nA bookshelf has 5 shelves, and each shelf can hold a certain number of books. The number of books on each shelf is a multiple of the number of books on the previous shelf. If the total number of books on the bookshelf is 2011, what is the largest number of books that can be placed on a... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "The sum of the squares of two consecutive integers is 241. What is the larger integer?",
"role": "user"
},
{
"content": "<think> Okay, so I need to find two consecutive integers where the sum of their squares is 241. And I have to figure out which one is the larger integer. Let me star... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A right pyramid has a square base with side length $s$ cm. Its peak is $h$ cm above the center of its base. If the total surface area of the pyramid is twice the area of the base, find the ratio $h/s$.",
"role": "user"
},
{
"content": "<think> Okay, so I have this problem about a right... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A quadrilateral has sides of lengths $2x+3$, $3x+1$, $4x-2$, and $x+5$. The sum of the lengths of the two shorter sides is equal to the sum of the lengths of the two longer sides. What is the length of the longest side of the quadrilateral?",
"role": "user"
},
{
"content": "<think> Oka... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Five congruent circles are arranged in a cross shape such that their centers form a square of side length $2$ and each circle is tangent to its neighbors. A square is inscribed in each circle. Find the perimeter of the square that can be formed by connecting the centers of the squares.",
"role... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A polynomial $p(x)$ is called \"self-centered\" if it has integer coefficients and $p(100) = 100.$ If $p(x)$ is a quadratic self-centered polynomial, what is the maximum number of integer solutions $k$ to the equation $p(k) = k^3$?",
"role": "user"
},
{
"content": "<think> Okay, let m... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "What is the value of the sum\n\\[\n \\sum_z \\frac{1}{{\\left|1 - z\\right|}^2} \\, ,\n\\]where $z$ ranges over all 7 solutions (real and nonreal) of the equation $z^7 = -1$?",
"role": "user"
},
{
"content": "<think> Okay, so I need to find the sum of 1 over |1 - z|² where z are all t... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "There are three points that are $7$ units from the line $y=10$ and $10$ units from the point $(5,10)$. What is the sum of the $x$- and $y$-coordinates of all three of these points?",
"role": "user"
},
{
"content": "<think> Okay, so I need to find three points that are 7 units away fro... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Given that $\\log_{a} (7x + 10) = \\log_{b} (3 - 2x)$ and $a, b > 0, a \\neq 1, b \\neq 1$, find $x$ such that $a^2 \\cdot b^2 = 4$.",
"role": "user"
},
{
"content": "<think> Okay, let's see. I need to find x such that log base a of (7x + 10) equals log base b of (3 - 2x), and also a s... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Find the coordinates of the vertex of the parabola described by the equation $y = x^2 + ax + b$, given that the solutions to the inequality $y \\leq 0$ are the intervals $[-2, 1]$ and $[3, 4]$.",
"role": "user"
},
{
"content": "<think> Okay, so I need to find the coordinates of the ver... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Find the maximum value of $\\sin(\\theta_1-\\theta_2)+\\sin(\\theta_2-\\theta_3)+\\sin(\\theta_3-\\theta_4)+\\sin(\\theta_4-\\theta_5)+\\sin(\\theta_5-\\theta_1)$, where $\\theta_1$, $\\theta_2$, $\\theta_3$, $\\theta_4$, and $\\theta_5$ are real numbers such that $\\sum_{i=1}^{5} \\theta_i = 0$."... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Find the range of real numbers $a$ for which there exist two distinct real numbers $x$ and $y$ satisfying $\\cos (\\cos x) = \\cos (\\cos y)$ and $\\cos (\\cos (x+a)) = \\cos (\\cos (y+a))$.",
"role": "user"
},
{
"content": "<think> Okay, so I need to find the range of real numbers \\(... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In a certain tennis tournament, 16 players are competing in a single-elimination bracket. Each player has a unique skill level, and the winner of each match is the player with the higher skill level. If the players are randomly seeded into the bracket, how many different orderings of the players b... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let a planar polygonal line consisting of segments $\\overline{OA}, \\overline{AB}, \\ldots, \\overline{PQ}, \\overline{QO}$ have the following properties: \n\n1. The segments $OA$ and $QO$ lie on the coordinate axes; \n2. The projection of the polygonal line on both axes is equal to $2$; \n3. For... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "At the annual meeting of the League of Brilliant Mathematicians, each of the 40 attendees will participate in a special ceremony where they will shake hands with each other exactly once. Each participant will wear a badge with their name and a unique ID number from 1 to 40. What is the smallest po... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Regions A, B, and C are bounded by rectangles. The perimeter of region A is 20 units and the perimeter of region B is 40 units. What is the ratio of the area of region A to the area of region C? Express your answer as a common fraction.\n\n[asy]\ndraw((0,0)--(12,0)--(12,8)--(0,8)--(0,0)--cycle,lin... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In a right-angled triangle, the hypotenuse $AB$ is twice as long as one of its legs, say $AC$. Point $P$ is the midpoint of $BC$ and $AP$ is perpendicular to $BC$. Given that $AC$ is $10$ units, determine the length of $AP$.",
"role": "user"
},
{
"content": "<think> Okay, so I need to ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "The equation of a circle is given by\n\\[x^2 + y^2 + Ax + By + C = 0.\\]\nSuppose that the circle intersects the $x$-axis in two points whose distance from the origin is equal to 10. Find the value of $C$ that maximizes the area of the circle.",
"role": "user"
},
{
"content": "<think> ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Count the number of different necklaces that can be created with 3 red beads, 2 green beads, and 1 blue bead. (Note: Two necklaces are considered the same if one can be obtained from the other by rotation or reflection.)",
"role": "user"
},
{
"content": "<think> Okay, so I need to figu... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let $ABCD$ be a trapezoid such that $AD = 2x$ and $BC = 3x$. The midsegment $MN$ of $ABCD$ is divided into five equal parts by points $E$, $F$, $G$, and $H$. If $NH = 2$, what is the number of units in the length of $\\overline{AD}$?",
"role": "user"
},
{
"content": "<think> Okay, so I... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A semicircle of diameter $a$ is inscribed inside a circle of radius $r$ so that its diameter lies on the circumference of the larger circle. If the area between the two circles is equal to the area of a circle with radius 4, find the value of $r$.",
"role": "user"
},
{
"content": "<thi... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A particular fortune cookie lists four three-digit positive integers as your lucky numbers. The first three are 123, 456, and 789, but you got sauce on the last one and can't read it. If the sum of the digits of all four numbers equals the number of digits of the sum of all four numbers, what is t... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In a parabola $y^2=4x$ a chord $\\overline{AB}$ is perpendicular to its axis of symmetry and its midpoint lies at the focus $F$. Given that $AF=x+1$ and $BF=x-1$, find the equation of the directrix.",
"role": "user"
},
{
"content": "<think> Okay, let's see. I need to find the equation ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Find the number of positive integers less than 1000 that are divisible by 7 and have exactly two digits that are the same.",
"role": "user"
},
{
"content": "<think> Okay, so the problem is asking for the number of positive integers less than 1000 that are divisible by 7 and have exactl... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Factor the polynomial\n\\[x^6 + 2x^4 + 4x^2 + 8\\]\ninto the product of two monic non-constant polynomials with integer coefficients.\nDetermine $p(-1) \\cdot q(-1),$ where $p(x)$ and $q(x)$ are the resulting polynomials.",
"role": "user"
},
{
"content": "<think> Okay, so I need to fac... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Find the greatest possible difference between the volumes of two rectangular prisms, each with an integer length, width, and height, and a surface area of 600 square cm.",
"role": "user"
},
{
"content": "<think> Okay, so I need to find the greatest possible difference between the volum... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "To express 20 as a sum of different powers of 2, we would write $20 = 2^4 + 2^2$. The sum of the exponents of these powers is $4 + 2 = 6$. If 400 were expressed as a sum of at least two distinct powers of 2, what would be the least possible sum of the exponents of these powers?",
"role": "user... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A bookstore has 5 shelves, each with 3 sections. The store owner wants to display 15 books of different genres (5 fiction, 5 non-fiction, and 5 biographies) in such a way that each shelf contains exactly 3 books, and each section contains at most one book from each genre. How many ways can the own... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let $g(x)$ be a polynomial function whose graph passes through the points $(-1,1),$ $(0,0),$ $(1,-1),$ and $(2,0)$. Define the function $h(x)=ag(x)+b$, where $a$ and $b$ are constants. If $h(x)$ passes through the points $(0,-2)$ and $(2,4)$, find the value of $h(-1)$.",
"role": "user"
},
... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In the diagram below, $ABCD$ is a trapezoid such that $\\overline{AB}\\parallel \\overline{DC}$ and $\\overline{AD}\\perp\\overline{DC}$. If $DC = 20$, $\\tan C = 3$, and $\\tan A = 4$, then what is $AD$?\n\n\n\n[asy]\n\npair A,B,C,D;\n\nC = (0,0);\n\nD = (20,0);\n\nA = (20,60);\n\nB= (60,60);\n\... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A small wooden cube is cut into 8 smaller congruent cubes with side length $\\frac{1}{2}$ of the original cube. Two of the smaller cubes are then removed. What is the total surface area of the remaining pieces?",
"role": "user"
},
{
"content": "<think> Okay, so I need to figure out the... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Find the number of negative solutions to\n\\[\\cos (\\arcsin( \\cot (\\arccos x) ) ) ) = x.\\]",
"role": "user"
},
{
"content": "<think> Okay, so I need to solve the equation cos(arcsin(cot(arccos x))) = x and find the number of negative solutions. Hmm, that looks a bit complicated wit... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Rachel has 4 sons and no daughters. Some of her sons have 4 sons, and the rest have none. Rachel has a total of 28 sons and grandsons, and no great-grandsons. How many of Rachel's sons and grandsons have no sons?",
"role": "user"
},
{
"content": "<think> Alright, let me try to work thr... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Three congruent circles each with a radius of 2 units are placed in such a way that they touch each other externally. A triangle is formed by connecting the centers of these circles. The area of the triangle is 18 square units. What is the number of square units in the area of the sector formed by... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Two circles have the same area and their circumferences differ by 8 units. The ratio of the radii of the two circles is 5:4. Find the maximum possible value of their common area.",
"role": "user"
},
{
"content": "<think> Okay, so I have this problem here: Two circles have the same area... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In the Cartesian plane, there is a parabola defined by the equation $y=x^2-2x+1$. Also, there is a circle with center at the vertex of the parabola and radius equal to $\\sqrt 2$. The circle intersects the parabola at two points. Find the distance between the tangent lines to the parabola at these... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A binary number is multiplied by $111_2$ and the result is $11011_2$. Determine the original binary number.",
"role": "user"
},
{
"content": "<think> Okay, so I have this problem here: A binary number is multiplied by 111₂, and the result is 11011₂. I need to find the original binary n... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "For how many values of $c$ in the interval $[0, 1000]$ does the equation \\[7 \\lfloor x \\rfloor + 2 \\lceil x \\rceil = c\\] have a solution for $x$?\n\nThis problem requires finding the number of values of $c$ that satisfy the given equation, which involves the floor and ceiling functions.\n\nL... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A large box contains 90 ping-pong balls, half of which are painted white and the other half black. The balls are packed in 10 layers of 9 balls each. If the top and bottom layers consist only of white balls, and in every other layer there are exactly 3 black balls, how many black balls are not tou... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "The sum of the squares of two positive integers is 245, and the positive difference of their squares is 81. What is the sum of the two integers?",
"role": "user"
},
{
"content": "<think> Okay, so I have this problem here: The sum of the squares of two positive integers is 245, and the ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A set of $n$ points in a three-dimensional space has the following properties: the distance between each pair of points is at least $5$ units, and the sum of the squares of the distances between all pairs of points is equal to $n^2(n-1)$. Find the smallest possible value of $n$.",
"role": "use... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "The sides of a hexagon are all parallel to the sides of a rectangle with sides of length 6 and 5. The hexagon shares a side with the rectangle. Prove that the perimeter of the hexagon is 22 units.",
"role": "user"
},
{
"content": "<think> Okay, so I need to prove that the perimeter of ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In a large corporation, there are 500 employees, each having a unique job title and a unique level of seniority (entry-level, mid-level, senior-level, or executive-level). Each employee can be promoted only one level at a time, and only if their current supervisor has left the company. Initially, ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A subset of size $n$ of the set $\\{1,2,3,\\ldots,2n\\}$ is selected, such that for any two elements $a$ and $b$ in the subset, if $a\\leq n$ and $b>n$, then $a<b$. Find the number of such subsets.",
"role": "user"
},
{
"content": "<think> Okay, so I have this problem here: I need to f... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Three circles, each of radius 1, are drawn with centers $A = (14,92)$, $B = (92,14)$, and $C = (14,14)$. A line passing through $A$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side o... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In the figure below, triangle $AOF$ is right and the angle measure of $\\angle FOG$ is $16^\\circ$ more than that of $\\angle GOF$. Find the measure of $\\angle FOG$.\n\n[asy]\nimport olympiad;\ndraw((0,0)--(0,10),linewidth(1));\ndraw((0,0)--(20,0),linewidth(1));\ndraw((0,0)--(16.3,8.2),linewidth(... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "The product of 7 consecutive integers is $6^4$. What is the largest integer among them?",
"role": "user"
},
{
"content": "<think> Okay, so I have this problem here: The product of 7 consecutive integers is 6^4. What is the largest integer among them? Hmm, let's try to figure this out s... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Three different integers between $10$ and $25$ are chosen. When the product of the three integers is divided by their sum, which of the following integers could be obtained?\n\n$$\\text A. \\ \\ 28 \\qquad \\text B. \\ \\ 47 \\qquad \\text C. \\ \\ 61 \\qquad \\text D. \\ \\ 84 \\qquad \\text E. \... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "The sum of the first 30 positive odd integers is also the sum of five consecutive odd integers. What is the largest of these five integers?",
"role": "user"
},
{
"content": "<think> Alright, let me try to work through this problem step by step. So the question is: The sum of the first ... | math | ai2-adapt-dev/openmath-2-math | null |