messages listlengths 2 2 | domain stringclasses 1 value | source stringclasses 1 value | difficulty null |
|---|---|---|---|
[
{
"content": "A bookshelf has 5 shelves, and each shelf can hold up to 3 books. In how many ways can 6 distinct books be placed on the bookshelf such that no shelf contains more than 2 books by the same author, and no two books by the same author are on adjacent shelves?",
"role": "user"
},
{
"conte... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "If $x^4 + 3x^3 - 5x^2 + rx + s$ is divisible by $(x^2 + 2x + 1)(x - 1),$ find the ordered pair $(r,s).$",
"role": "user"
},
{
"content": "Given the polynomial \\( P(x) = x^4 + 3x^3 - 5x^2 + rx + s \\), we need to find the ordered pair \\((r, s)\\) such that \\( P(x) \\) is divisible by... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A linear function $f(x) = ax + b$ is defined on the interval $[-10, 10]$ such that the sum of the absolute values of $f(x)$ at $100$ evenly spaced points in the interval is equal to $1000$. Let $M$ be the maximum possible value of $|f(0)|$ and let $m$ be the minimum possible value of $|f(0)|$. Wha... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In a mathematical competition, six teams are participating, and each team has four members. The teams are arranged in two rows of three teams each, with each team facing another team in the opposite row. The sum of the ages of the members of each team is written on a card, and the six cards are pl... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "The number $x$ can be expressed as a three-place decimal $0.abc,$ where $a, b,$ and $c$ represent digits, any of which could be zero. It is desired to approximate $x$ by a fraction whose numerator is 1 or 3 and whose denominator is an integer. The closest such fraction to $x$ is $\\frac 13.$ What ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A kite is flying at an altitude of 25 meters, and its shadow on the ground is 15 meters long. Suddenly, the wind changes, and the kite begins to move in the opposite direction at a constant speed of 5 meters per second. At what rate is the length of the shadow changing when the kite is 30 meters a... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let $f(x) = x^3 + ax^2 + bx + c$ be a cubic polynomial with rational coefficients, and let $\\alpha$, $\\beta$, and $\\gamma$ be its roots. If $f(x)$ has a local maximum at $\\alpha$, and $\\beta$ and $\\gamma$ are positive integers, find the minimum possible value of $\\beta + \\gamma$.",
"ro... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Suppose that the equation \\[\\frac{2x}{x^2+2x+1}=\\frac{A}{x+1}+\\frac{Bx+C}{x^2+1}\\] holds for all real values of $x$. What is the value of $A+B+C$?",
"role": "user"
},
{
"content": "Given the equation:\n\n\\[\n\\frac{2x}{x^2 + 2x + 1} = \\frac{A}{x+1} + \\frac{Bx + C}{x^2 + 1}\n\\]... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "The vertices of a regular tetrahedron have coordinates $A=(x_1, y_1, z_1), B=(x_2, y_2, z_2), C=(x_3, y_3, z_3),$ and $D=(x_4, y_4, z_4).$ The midpoints of the edges $\\overline{BC}$ and $\\overline{AD}$ are $M$ and $N,$ respectively. The midpoints of the segments $\\overline{AM}$ and $\\overline{... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In a round-robin chess tournament, each player earns a certain number of points for each match: 3 points for a win, 1 point for a draw, and 0 points for a loss. If there are 10 players in the tournament, and the total number of points earned by all players is 140, how many matches ended in a draw?... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A solid hemisphere has a curved surface area of $18\\pi$ square centimeters and a height of 9 centimeters. The hemisphere is then stretched while maintaining its curved surface area, resulting in a new height. If the volume of the stretched hemisphere is $135\\pi$ cubic centimeters, what is the ra... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Find all values of $x$ such that $x + \\lfloor x \\rfloor^2 = 22.5$.\n\n(Note: $\\lfloor x \\rfloor$ represents the greatest integer less than or equal to $x$)",
"role": "user"
},
{
"content": "Given the equation \\( x + \\lfloor x \\rfloor^2 = 22.5 \\), we start by letting \\( n = \\l... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let $m$ and $n$ be positive integers such that the product of their sum and difference is 320. If $m$ and $n$ are relatively prime, what is the absolute difference between $m$ and $n$?",
"role": "user"
},
{
"content": "Given positive integers \\( m \\) and \\( n \\) such that the produ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In the triangle $MNO$ with $\\angle M = 45^\\circ$, the side opposite to vertex $M$ has length of $5$ units. The altitude to this side has length of $\\sqrt{2}+1$ units. What is the length of the segment connecting $M$ to the foot of this altitude?",
"role": "user"
},
{
"content": "Giv... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In pentagon ABCDE, AB = BC = CD = DE = 2 and ∠B = ∠C = ∠D = 90°. Find AE.",
"role": "user"
},
{
"content": "To find the length of AE in pentagon ABCDE with AB = BC = CD = DE = 2 and angles at B, C, D being 90°, we can use coordinate geometry and vector analysis.\n\n1. **Assigning Coord... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A bookshelf contains 20 books, each written by a different author. If you want to arrange the books on the shelf such that no two books written by authors whose names start with the same letter are next to each other, what is the maximum number of different letters that the authors' names can star... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Find the largest positive integer that is not the sum of a positive integral multiple of 28 and a positive composite integer.",
"role": "user"
},
{
"content": "To solve the problem of finding the largest positive integer that cannot be expressed as the sum of a positive integral multip... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "The function\n\\[f(z)=\\frac{z^3}{1+z^4}\\]\ntakes a real value for every complex number $z$ satisfying $|z|=1$.\nFind the maximum value of $f(z)$.",
"role": "user"
},
{
"content": "Given the function \\( f(z) = \\frac{z^3}{1 + z^4} \\), we need to determine if it takes a real value fo... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "There are three boxes containing $n$ balls each. Each ball is one of three different colors, and there is an equal number of each color in each box. One ball is selected at random from each box, and the three balls are placed in a bag. If three balls of the same color are selected, what is the pro... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let $p(x)$ be a polynomial with integer coefficients, such that for every integer $n$, the remainder of the division of $p(n)$ by $n$ is $n-1$. Find the smallest possible degree of $p(x)$.",
"role": "user"
},
{
"content": "Given a polynomial \\( p(x) \\) with integer coefficients such ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let $m$ and $n$ be positive integers such that $n^2-m^2=105$. If $n>5$ and $\\text{gcd}(m, n)=3$, what is the largest possible value of $n$?",
"role": "user"
},
{
"content": "Given the problem where \\( n \\) and \\( m \\) are positive integers such that \\( n^2 - m^2 = 105 \\), \\( n ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Jane is 25 years old. Dick is older than Jane. In $n$ years, where $n$ is a positive integer, Dick's age and Jane's age will both be two-digit number and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let $d$ be Dick's present age. How many ordered pa... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A nonzero polynomial with rational coefficients has all of the numbers \n\\[1 + \\sqrt{3}, \\; 2 + \\sqrt{5}, \\; 3 + \\sqrt{7}, \\; \\dots, \\; 500 + \\sqrt{1001}\\]\nas roots. What is the smallest possible degree of such a polynomial?",
"role": "user"
},
{
"content": "To determine th... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "For what values of the constant $k$ does the graph of $f(x) = \\frac{x^3+x^2+kx-1}{x^4-2x^2+1}$ have exactly two horizontal asymptotes?",
"role": "user"
},
{
"content": "To determine the values of \\( k \\) for which the graph of \\( f(x) = \\frac{x^3 + x^2 + kx - 1}{x^4 - 2x^2 + 1} \\... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let $X$ be a set of integers, and let $f:X \\to X$ be a function with the following properties:\n\n(1) If $n \\in X$ is odd, then $f(n)$ is even.\n\n(2) If $n \\in X$ is even, then $f(n)$ is odd.\n\n(3) $f(n) > n$ for all $n \\in X.$\n\nProve that $f$ is not surjective.",
"role": "user"
},
... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In $\\triangle ABC$ from the diagram, $CX$ bisects $\\angle ACB$, $AB = AC$ and $AY$ bisects $\\angle CAB$.\n\n[asy]\nimport markers;\n\nreal t=.56;\npair A=(0,0);\npair B=(3,2);\npair C=(.5,1.5);\npair X=t*A+(1-t)*B;\npair Y=(0,0);\n\ndraw(C--A--B--C--X);\n\nlabel(\"$A$\",A,SW);\nlabel(\"$B$\",B,... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A circle is inscribed in the region bounded by the curve $y = x^2 - 4x + 4$ and the x-axis. What is the radius of the circle?",
"role": "user"
},
{
"content": "The given curve is \\( y = x^2 - 4x + 4 \\), which can be rewritten in vertex form as \\( y = (x - 2)^2 \\). This parabola has... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let $f(x)$ be a function defined on the set of positive integers, such that\n$$f(1) = 2, f(2) = 4, f(3) = 6,$$\nand $f(x+3) = f(x) + f(x+1) + f(x+2)$ for all $x>0$.\nProve that the limit\n$$\\lim_{n\\to\\infty}\\frac{f(n)}{n!}$$\nexists and calculate its value.",
"role": "user"
},
{
"c... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In a large library, there are 15 bookshelves, each with a different height. The librarian wants to place a ladder at each pair of adjacent bookshelves to facilitate book retrieval. However, the ladders come in different lengths, and the librarian wants to minimize the total number of ladders neede... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In triangle $ABC$, the altitude $\\overline{AH}$ and the median $\\overline{AM}$ intersect at $Q$. If $AB = 4$, $AC = 6$, and $BC = 8$, find $\\frac{AQ}{QM}$.",
"role": "user"
},
{
"content": "To solve the problem, we start by setting up the coordinates for triangle \\(ABC\\) with \\(B... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In a small village, there are five shops selling different types of traditional crafts. Each shop has exactly three neighboring shops with which they exchange goods regularly. Let M be the number of ways these exchanges can occur, considering two exchange arrangements different if and only if at l... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In triangle $DEF,$ $G$ lies on $\\overline{DE}$ such that $DG:GE = 3:2,$ and $H$ lies on $\\overline{DF}$ such that $DH:HF = 2:3.$ Let $Q$ be the intersection of $\\overline{EG}$ and $\\overline{FH}.$ Then\n\\[\\overrightarrow{Q} = a \\overrightarrow{D} + b \\overrightarrow{E} + c \\overrightarr... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "What is the smallest positive integer $k$ such that $k\\sqrt{2}$ can be expressed as a sum of two perfect squares?",
"role": "user"
},
{
"content": "To find the smallest positive integer \\( k \\) such that \\( k\\sqrt{2} \\) can be expressed as a sum of two perfect squares, we conside... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Find the smallest positive integer $m$ such that there exist exactly two distinct positive integers $a$ and $b$ satisfying $\\frac{1}{3} < \\frac{a}{m} < \\frac{b}{m} < \\frac{2}{5}$.",
"role": "user"
},
{
"content": "To find the smallest positive integer \\( m \\) such that there exis... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "The graphs of $y = \\frac{x^2 + 3x + 4}{x^2 - x - 6}$ and $y = \\frac{x^2 + ax + b}{x^2 + cx + d}$ intersect at two distinct points. If the two lines through these points are perpendicular, find $a - c$.",
"role": "user"
},
{
"content": "To solve the problem, we start by equating the ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Points $A$ and $B$ are on the hyperbola $y^2 - x^2 = 1$, and the midpoint of $\\overline{AB}$ is the point $(1,1)$. Find the product of the slopes of the lines through $A$ and $B$ that are perpendicular to $\\overline{AB}$.",
"role": "user"
},
{
"content": "Given points \\( A \\) and \... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Find the range of the function\n\n$$f(x) = \\frac{x+2}{x^2+2} + \\frac{x^2+2}{x^3+2}$$\n\nExpress your answer as a union of intervals.",
"role": "user"
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{
"content": "To find the range of the function \\( f(x) = \\frac{x+2}{x^2+2} + \\frac{x^2+2}{x^3+2} \\), we analyze the behavior o... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let $S$ denote the set of all the integers that can be represented as $6x^2+19x+13$ or $6x^2+45x+41$, where $x$ is an integer.\nFind the smallest positive integer that is not an element of $S$.",
"role": "user"
},
{
"content": "To find the smallest positive integer that cannot be repre... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "The positive integers $M$ and $M^3$ both end in the same sequence of four digits $pqrs$ when written in base $10$, where digit p is not zero. Find the three-digit number $pqr$.",
"role": "user"
},
{
"content": "To solve the problem, we need to find a positive integer \\( M \\) such tha... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A group of 6 people are planning to attend a concert, but they realize that their tickets are in different rows. The venue has a strict policy that allows only 2 people from the same row to sit together. How many ways can the 6 people be seated if they want to sit in pairs from the same row, and e... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "The expression $\\frac{\\sqrt{2500}}{10} + \\sqrt{90} - 3\\sqrt{2}$ can be simplified to $\\sqrt{M}$, where $M$ is an integer. Determine the value of $M$.",
"role": "user"
},
{
"content": "First, we simplify each term in the expression \\(\\frac{\\sqrt{2500}}{10} + \\sqrt{90} - 3\\sqrt... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "The complex roots of the equation $z^4 - 2z^3 + 2z^2 - 2z + 1 = 0$ form a regular polygon when plotted in the complex plane. Find the area of this polygon.",
"role": "user"
},
{
"content": "To solve the problem, we start by factoring the given polynomial \\( z^4 - 2z^3 + 2z^2 - 2z + 1... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A bookstore has a special display of 20 books, with 5 shelves containing 4 books each. Each book has a unique ranking in terms of sales, from 1st to 20th. If the best-selling book is placed on the top shelf, and the worst-selling book is placed on the bottom shelf, what is the minimum number of bo... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A five-digit palindrome is a number that reads the same forwards and backwards, such as 32123. How many five-digit palindromes are divisible by 3 and have exactly 3 odd digits?",
"role": "user"
},
{
"content": "To solve the problem of finding the number of five-digit palindromes divisi... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A museum has a collection of 8 paintings, and wants to display them in a circular exhibition hall with 8 slots. However, due to the curvature of the hall, some of the paintings will be partially obscured by the pillars supporting the roof. In particular, if a painting is placed in slot 1, it will ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In a regular hexagon, six equilateral triangles are drawn from each vertex to the center of the hexagon. Each triangle is then divided into four smaller equilateral triangles, and the process continues indefinitely. What fractional part of the figure is shaded if the largest triangles are shaded i... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In a regular octahedron, consider the set of all line segments connecting two vertices, where no vertex connects to itself. Two of these segments are chosen randomly without replacement. What is the probability that the two selected segments lie in the same plane?",
"role": "user"
},
{
... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In triangle $ABC$, angle $A$ measures 45 degrees and angle $B$ measures 75 degrees. The angle bisectors of angles $A$ and $B$ intersect at point $D$. What is the measure of obtuse angle $ADB$?",
"role": "user"
},
{
"content": "In triangle \\(ABC\\), angle \\(A\\) measures \\(45^\\circ\... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Eight teams participate in a competition. Each team plays a match against every other team, and each match results in a win or a loss. If no team wins all its matches, and no team loses all its matches, what is the minimum number of matches that must be played?",
"role": "user"
},
{
"c... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let $f(x)=\\lceil x\\rceil - \\lfloor x \\rfloor$. Find the sum of all values of $x$ such that\n\\[\\lfloor x \\rfloor \\cdot f(x) = 3 \\quad \\mbox{and} \\quad f(x^2)=5.\\]",
"role": "user"
},
{
"content": "Given the function \\( f(x) = \\lceil x \\rceil - \\lfloor x \\rfloor \\), we ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A bookshelf consists of five shelves, and each shelf can hold the same number of books. If Tom has 18 books and Alex has 30 books, and they want to put all their books on the bookshelf such that each shelf has the same number of books, what is the smallest possible number of books that each shelf ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In a parallelogram $ABCD$, the diagonals $AC$ and $BD$ intersect at point $O$. If $\\angle AOB = 100^{\\circ}$ and $\\angle BOC = 70^{\\circ}$, what is the degree measure of $\\angle BAD$?",
"role": "user"
},
{
"content": "Given a parallelogram \\(ABCD\\) with diagonals \\(AC\\) and \\... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "The sum of the digits of a positive integer is equal to 24. What is the smallest number that can be added to this integer to make it a palindrome?",
"role": "user"
},
{
"content": "Given a positive integer with a digit sum of 24, we need to find the smallest number that can be added to... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Given the rational function:\n\\[g(x)=\\frac{x^2-4}{x^3+px^2+qx+r}\\]\nwith vertical asymptotes at $x=-1, 2, 3$ and a hole at $x=5$, find the sum of $p$, $q$, and $r$.",
"role": "user"
},
{
"content": "Given the rational function:\n\\[ g(x) = \\frac{x^2 - 4}{x^3 + px^2 + qx + r} \\]\nw... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let $a$ and $b$ be the roots of the equation $x^2 - 5x - 6 = 0$. Let $c$ and $d$ be the roots of the equation $2x^2 + 3x - 4 = 0$. If $c$ and $d$ are each three less than $a$ and $b$, respectively, what is the value of $ab + cd$?",
"role": "user"
},
{
"content": "Given the problem, we ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In triangle $ABC$, $D$ is the midpoint of $\\overline{AB}$, and $E$ is the midpoint of $\\overline{BC}$. Lines $DE$ and $AC$ intersect at $F$. If $G$ is the midpoint of $\\overline{DE}$, find the ratio of the area of triangle $DGF$ to the area of triangle $EGF$.",
"role": "user"
},
{
"... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "The harmonic mean of three positive numbers is 4. Two of the numbers are 5 and 12, and the third is unknown. If the geometric mean of the three numbers is 6, what is the value of the unknown number?",
"role": "user"
},
{
"content": "Given the problem, we need to find the unknown number... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "The polynomial $Q(x) = x^4 + dx^3 + ex^2 + fx + 3$ has the property that the median of its zeros, the sum of its zeros, and the product of its coefficients are all equal. If the $y$-intercept of the graph of $y=Q(x)$ is 3, what is $e$?",
"role": "user"
},
{
"content": "Given the polyno... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Determine the smallest positive integer $m$ such that the remainder when $12m$ is divided by 25 is the same as the remainder when $9m$ is divided by 23.",
"role": "user"
},
{
"content": "To determine the smallest positive integer \\( m \\) such that the remainder when \\( 12m \\) is di... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Find the maximum value of\n\\[\\tan \\theta_1 \\cot \\theta_2 + \\tan \\theta_2 \\cot \\theta_3 + \\tan \\theta_3 \\cot \\theta_4 + \\tan \\theta_4 \\cot \\theta_5 + \\tan \\theta_5 \\cot \\theta_1,\\]over all real numbers $\\theta_1,$ $\\theta_2,$ $\\theta_3,$ $\\theta_4,$ and $\\theta_5.$",
... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "For how many values of $n$ is $2^{12}$ the greatest common divisor of the positive integers $n^2$, $n^3$, and $3n+1$?",
"role": "user"
},
{
"content": "To determine how many values of \\( n \\) make the greatest common divisor (gcd) of \\( n^2 \\), \\( n^3 \\), and \\( 3n + 1 \\) equal... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let $p(x)$ be a polynomial of degree 3 such that $p(0) = p(1) = p(2) = 0$ and $p(3) = 1$. Find all real numbers $x$ that satisfy the inequality\n\\[|p(x)| + |p(x + 1)| + |p(x + 2)| \\ge 1.\\]",
"role": "user"
},
{
"content": "Given the polynomial \\( p(x) \\) of degree 3 with roots at ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let $a$ and $b$ be positive integers. Consider the product of two numbers, $3^a$ and $2^b$, that ends with the digits...16. Find the smallest possible value of $a+b$.",
"role": "user"
},
{
"content": "To solve the problem of finding the smallest possible value of \\(a + b\\) such that ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A company has 100 employees, and each employee has a unique ID number from 1 to 100. The company wants to assign a 5-digit PIN number to each employee, such that no two employees have the same PIN number, and no PIN number can be obtained from another by changing only one digit. What is the minimu... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In triangle $ABC$, $AX = XY = YB = BC$ and the measure of angle $ABC$ is 120 degrees. What is the number of degrees in the measure of angle $BAC$?\n\n[asy]\npair A,X,Y,B,C;\nX = A + dir(30); Y = X + dir(0); B = Y + dir(60); C = B + dir(-30);\ndraw(B--Y--X--B--C--A--X);\nlabel(\"$A$\",A,W); label(\... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Find the number of integer values of $b$ such that the equation\n$$x^3 + bx^2 + (b-1)x + 1 = 0$$\nhas at least one integer solution for $x$.",
"role": "user"
},
{
"content": "To find the number of integer values of \\( b \\) such that the equation \\( x^3 + bx^2 + (b-1)x + 1 = 0 \\) ha... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "What is the positive difference between the two largest prime factors of 198721?",
"role": "user"
},
{
"content": "To find the positive difference between the two largest prime factors of 198,721, we start by factorizing the number.\n\nFirst, we check divisibility by smaller primes:\n-... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Given the plane $x - y + 2z = 3$, find the equation of the line passing through the point $(1,1,0)$ that makes an angle of 60 degrees with the plane. \n\n[asy]\nimport three;\n\nsize(150);\ncurrentprojection = perspective(6,3,2);\n\ntriple I = (1,0,0), J = (0,1,0), K = (0,0,1), O = (0,0,0);\n\ndra... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "The equation\n\n\\[x^4 - 2x^3 + 7x^2 - 11x - 9 = 0\\]\n\nhas two pure imaginary solutions. Find these solutions, separated by commas.",
"role": "user"
},
{
"content": "To find the pure imaginary solutions of the equation \n\n\\[x^4 - 2x^3 + 7x^2 - 11x - 9 = 0,\\]\n\nwe substitute \\(x ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In $\\triangle DEF$, $DE= 300$, $EF=280$, and $DF=350$. An interior point $Q$ is then drawn, and segments are drawn through $Q$ parallel to the sides of the triangle. If these three segments are of an equal length $e$, find $e$.",
"role": "user"
},
{
"content": "To solve the problem, w... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In a unit cube, three circular holes of diameter $1/2$ are drilled along the three diagonals that meet at a vertex. The cube is then placed on a table with the holes facing upwards. What is the maximum volume of a sphere that can fit inside the cube without falling into any of the holes?",
"ro... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A function $f(x) = ax^3 + bx^2 + cx + d$ has three real roots at $x = -3, 0, 4$ and a local maximum at $x = 1$. If $f(1) = 2$, find the value of $a$.",
"role": "user"
},
{
"content": "Given a cubic function \\( f(x) = ax^3 + bx^2 + cx + d \\) with three real roots at \\( x = -3, 0, 4 \... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "The students at a school are lining up for a group photo. Each student has a different colored shirt, either blue, green, or yellow, and each student has a different favorite sport, either basketball, football, or soccer. The students are arranged in a line such that no two students wearing the sa... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "The least positive integer $n$ such that $n$ and $4n$ have the same number of digits in base 5 is $n = 12240$. Determine the smallest possible number of digits of $n$ in base 5.",
"role": "user"
},
{
"content": "To determine the smallest possible number of digits of \\( n \\) in base 5... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In the figure, $ABCD$ is a rectangle with $AB = 12$ cm and $BC = 6$ cm. The points $E$ and $F$ are on the side $AB$ such that $AE = 4$ cm and $BF = 3$ cm. The area of the quadrilateral $ECFD$ is equal to the area of the triangle $AEB$. What is the area of the quadrilateral $ECFD$?",
"role": "u... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A rectangle with sides of length 12 and 16 is inscribed in a circle. Two bugs start simultaneously from one of the vertices of the rectangle and crawl along the circumference of the circle in opposite directions at the same speed. They meet at a point on the circle. What is the length of the arc b... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "On the graph of the function $f(x) = x^2$, two points $A(x_1, y_1)$ and $B(x_2, y_2)$ are chosen, where $0 < x_1 < x_2$. The points $C$ and $D$ bisect $\\overline{AB}$, with $AC < CB$. Through $C$ a vertical line is drawn to cut the curve at $E(x_3, y_3)$. Determine $x_3$ if $x_1 = 1$ and $x_2 = 3... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Consider a circle centered at the origin with radius $R,$ and let $A$ and $B$ be two points on the circle such that the line segment $\\overline{AB}$ passes through a fixed point $C = (0, c),$ where $c$ is a constant. Suppose that the quantity $t = \\frac{AC}{AB} + \\frac{BC}{AB}$ is constant for ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In a triangle $ABC$, the altitude from vertex $A$ intersects side $BC$ at point $H$, forming right triangle $AHC$. Given that $\\tan A = \\frac{2}{3}$ and $\\cos B = \\frac{1}{2}$, find $\\sin C$.",
"role": "user"
},
{
"content": "Given a triangle \\(ABC\\) with an altitude from vertex... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In quadrilateral $ABCD$, $\\angle A = 60^\\circ$, $\\angle B = 90^\\circ$, $AB = 4$, and $BC = 6$. Diagonals $AC$ and $BD$ intersect at $E$. Find $AE:EC$.",
"role": "user"
},
{
"content": "In quadrilateral \\(ABCD\\), we are given \\(\\angle A = 60^\\circ\\), \\(\\angle B = 90^\\circ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Determine all values of the positive integer $n$ such that $n^2+5n+16$ and $n+8$ share exactly three common positive divisors.",
"role": "user"
},
{
"content": "To determine all positive integers \\( n \\) such that \\( n^2 + 5n + 16 \\) and \\( n + 8 \\) share exactly three common pos... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In a trapezoid $ABCD$ with $\\overline{AB}$ parallel to $\\overline{CD}$, the area of triangle $ABC$ is 24 square units, and the area of triangle $BCD$ is 30 square units. What is the length of $\\overline{AB}$?",
"role": "user"
},
{
"content": "Given a trapezoid \\(ABCD\\) with \\( \\... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Freddy wants to rewrite the cubic polynomial $2x^3 + 12x^2 - 7x + 1$ in the form of $(x + d)(ex^2 + fx + g)$, where $d, e, f,$ and $g$ are all integers. What is the value of $d + e + f + g$?",
"role": "user"
},
{
"content": "Freddy wants to rewrite the cubic polynomial \\(2x^3 + 12x^2 ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Given a circle with center $O$ and radius $r=6$, points $A$, $B$, and $C$ are on the circle such that $\\angle AOB = \\angle BOC = \\angle COA$. Point $P$ is inside the circle so that $\\triangle PAB$, $\\triangle PBC$, and $\\triangle PCA$ are similar isosceles triangles. Find $\\cos \\angle AOP$... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let $f$ be a function that maps real numbers to real numbers and has the following properties:\n\n(1) $f(ax + by) = af(x) + bf(y)$ for all real numbers $x$ and $y,$ and for all rational numbers $a$ and $b.$\n\n(2) $f(x^2) = f(x)^2$ for all rational numbers $x.$\n\n(3) $f(1) = 2$ and $f(2) = 3.$\n\... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Three positive integers have a product of 27000 and are in an arithmetic progression. What is the middle integer?",
"role": "user"
},
{
"content": "Given three positive integers in an arithmetic progression (AP) whose product is 27000, we need to find the middle integer.\n\n1. **Expres... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "A music teacher is planning to divide her students into groups for a performance. She wants to divide them into groups of 3, 4, or 5, but each group must have at least 2 students playing the same instrument. If she has 15 students playing the violin, 10 playing the piano, and 8 playing the guitar,... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let $a,$ $b,$ $c$ be distinct, nonzero real numbers such that\n\\[a^2 + \\frac{1}{b^2} = b^2 + \\frac{1}{c^2} = c^2 + \\frac{1}{a^2}.\\]\nFind $|abc|.$",
"role": "user"
},
{
"content": "Given distinct, nonzero real numbers \\(a\\), \\(b\\), and \\(c\\) such that:\n\n\\[\na^2 + \\frac{1... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let $a_1,$ $a_2,$ $\\dots,$ $a_n$ be positive integers such that\n$a_1 a_2 \\dots a_n = 2^{100}$\nand\n$\\frac{1}{a_1} + \\frac{1}{a_2} + \\dots + \\frac{1}{a_n} \\ge 10.$\nFind the largest possible value of $n$.",
"role": "user"
},
{
"content": "Given the problem of finding the larges... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Anna wants to tile a rectangular room with square tiles of two different sizes: 1 foot by 1 foot and 2 feet by 2 feet. If she has an unlimited number of each size, what is the smallest possible length of a side of the room that can be tiled so that the number of small tiles is equal to the number ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Four positive integers $M$, $N$, $P$, and $Q$ have a sum of 96. If $M+4 = N-2 = P \\times 2 = Q \\div 2$, what is the value of the product $M \\times N \\times P \\times Q$?",
"role": "user"
},
{
"content": "Given four positive integers \\( M \\), \\( N \\), \\( P \\), and \\( Q \\) wi... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "The positive integers $a$, $b$, and $c$ are pairwise relatively prime and share a common divisor $d>1$. If the least common multiple of $a$, $b$, and $c$ is $2^4 \\cdot 3^2 \\cdot 5$, and $d$ is the largest divisor of $2^4 \\cdot 3^2 \\cdot 5$ that is less than $2^4 \\cdot 3^2 \\cdot 5$, determine... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "What is the smallest three-digit number in Pascal's triangle that is also a perfect square?",
"role": "user"
},
{
"content": "To find the smallest three-digit number in Pascal's triangle that is also a perfect square, we need to consider the binomial coefficients, which form Pascal's t... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "The line $y = x-2$ with $0 < 2 < 6$ intersects the $x$-axis at $P$ and the line $y=6$ at $S$. If the ratio of the area of triangle $QRS$ to the area of triangle $QOP$ is 4:9, what is the value of the $x$-coordinate of point $Q$? Express the answer as a decimal to the nearest tenth.\n\n[asy]\ndraw(... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that $P(1) = P(4) = P(7) = a$ and $P(2) = P(3) = P(5) = P(6) = -a$. What is the smallest possible value of $a$?",
"role": "user"
},
{
"content": "Given \\(a > 0\\) and a polynomial \\(P(x)\\) with integer coeffi... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Two students, Alice and Bob, are working on a project together. They need to share a packet of colored pencils, which contains 12 pencils of different colors. Alice wants to use $\\frac{1}{3}$ of the pencils for her part of the project, while Bob wants to use $\\frac{2}{7}$ of the pencils for his ... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let $f(x)$ be a real-valued function such that $f(x) = f(1-x)$ for all $x \\in [0,1].$ Find the maximum value of\n\\[\\frac{f(0) + f(1/3) + f(2/3) + f(1)}{\\sqrt{\\int_0^1 f(x) dx}}.\\]",
"role": "user"
},
{
"content": "Given a real-valued function \\( f(x) \\) such that \\( f(x) = f(1... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Find the smallest natural number that can be added to the decimal representation of $\\sqrt{99}$ so that the resulting number is a palindrome.",
"role": "user"
},
{
"content": "To solve the problem of finding the smallest natural number \\( N \\) that can be added to the decimal repres... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "Let $\\overline{CE}$ be an altitude of $\\triangle DEF$. Let $T$ and $U$ be the points where the circles inscribed in the triangles $DEC$ and $FEC$ are tangent to $\\overline{CE}$. If $DE = 2001$, $DF = 2002$, and $EF = 2003$, then $TU$ can be expressed as $p/q$, where $p$ and $q$ are relatively p... | math | ai2-adapt-dev/openmath-2-math | null |
[
{
"content": "In $\\triangle ABC$, medians $\\overline{BD}$ and $\\overline{AE}$ intersect at point $O$. Furthermore, $\\overline{CE}$ and $\\overline{AD}$ are parallel. If $AD=2$ and $BD=3$, determine the area of $\\triangle AOB$.",
"role": "user"
},
{
"content": "Given triangle \\( \\triangle ABC ... | math | ai2-adapt-dev/openmath-2-math | null |
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