RockToken/qwen3_30b_a3b_to_4b_onpolicy_5k_src35k-40k_cont2
Text Generation • 4B • Updated • 27
prompt_messages listlengths 1 1 |
|---|
[
{
"from": "human",
"value": "Two circles, one with a radius of $2$ and the other with a radius of $8$, are internally tangent. A third circle is tangent to the first two and to one of their common internal tangents. What is the radius of the third circle? [asy]\ndraw((-3,0)--(15,0));\ndraw(Circle((2,2),2),l... |
[
{
"from": "human",
"value": "In a certain country, there are 10 cities connected by a network of roads. Each city has exactly 4 roads leading out of it, and no road connects a city to itself. The number of roads that connect two cities directly is either 0 or 1. A traveler starts in one of the cities and tr... |
[
{
"from": "human",
"value": "Consider the graph of $y = \\sin (ax) + \\sin (bx),$ where $a$ and $b$ are positive constants. Suppose that for some positive integer $n,$ the graph has $n$ times as many $x$-intercepts as it has local maxima in the interval $[0,2\\pi].$ Find the smallest possible value of $n.... |
[
{
"from": "human",
"value": "What is the largest positive integer that is not the denominator of any fraction whose decimal representation is a repeating decimal with a single repeating digit?"
}
] |
[
{
"from": "human",
"value": "Let $\\triangle ABC$ be a right triangle with integer side lengths such that the area of the circle inscribed within its boundaries is 121. Given that the perimeter of $\\triangle ABC$ is 840, what is its area?"
}
] |
[
{
"from": "human",
"value": "Given the functions:\n\n$$\\begin{array}{ccc}\nf(x) & = & x^3 - 2x^2 + 5x + 1\\\\\ng(x) & = & 2x^2 + mx + 3\n\\end{array}$$\n\nIf the graphs of $f(x)$ and $g(x)$ intersect at $x = 2$ and the area between their curves from $x = 0$ to $x = 2$ is $4$, what is the value of $m$?"
}... |
[
{
"from": "human",
"value": "Find the value of $x$ that satisfies $\\frac{\\log(2x+1)}{\\log(x+2)}=\\frac{\\log 2}{\\log 3}$. Express your answer as a common fraction."
}
] |
[
{
"from": "human",
"value": "The sum of the areas of two rectangles is 60 square units. The length of the first rectangle is 5 units longer than its width, and the length of the second rectangle is 7 units longer than its width. If both rectangles have integer dimensions, what is the sum of all possible val... |
[
{
"from": "human",
"value": "The product of three different numbers is 240. The sum of the two smaller numbers is 8, and the difference between the two larger numbers is 5. What is the value of the largest number?"
}
] |
[
{
"from": "human",
"value": "On a 12-hour analog clock, the hour hand is between the 9 and the 10, and the minute hand is on the 6. If the angle between the hour and minute hands is 150 degrees, what is the time on the clock?"
}
] |
[
{
"from": "human",
"value": "Let $f(x)$ and $g(x)$ be two functions with $f(x)=\\ds\\sqrt{x^2+a^2}$ and $g(x)=\\ds\\sqrt{x^2+b^2}$ for some positive constants $a$ and $b$. If $f(x)$ and $g(x)$ are tangent to each other at the point $(c,c)$, determine the value of $\\ds\\sqrt{\\frac{a^2+b^2}{2}}$."
}
] |
[
{
"from": "human",
"value": "Given nine distinct positive integers, we want to place them into the cells of a 3x3 grid in such a way that the product of the numbers in each row or column is as small as possible. What is the smallest possible value of the sum of these six products?"
}
] |
[
{
"from": "human",
"value": "Solve for $y$: $\\dfrac{1}{y+1} + \\dfrac{1}{y+2} + \\dfrac{1}{y+3} = \\dfrac{7}{6}$."
}
] |
[
{
"from": "human",
"value": "The number of positive divisors of two positive integers is 36 and 6 is their least common multiple. Determine the sum of the two numbers."
}
] |
[
{
"from": "human",
"value": "The two acute angles of an isosceles triangle are such that one is triple the other.\nIf the difference between the two acute angles is 30 degrees, find the difference between the squares of the two acute angles."
}
] |
[
{
"from": "human",
"value": "Find all the solutions to\n\\[\\sqrt{x+1}+\\sqrt{2x-1}+\\sqrt{3x-2}=5.\\]Enter all the solutions, separated by commas."
}
] |
[
{
"from": "human",
"value": "Consider a cube with side length 1. Let M be a point on the side BC such that BM=2CM. Let N be the point on the side DE such that DN=2EN. The plane through points B, M, and N intersects the side BF at point P, where F is the vertex opposite B. Find the area of the pentagon BPNMD... |
[
{
"from": "human",
"value": "Find all real values of $x$ that satisfy\n\n$$\\frac{1}{x^2+4} < \\frac{2}{x} - \\frac{5}{3}.$$\n\n(Give your answer in interval notation.)"
}
] |
[
{
"from": "human",
"value": "Consider the polynomial $Q(w) = w^6 + (2\\sqrt{2} + 3)w^3 - (2\\sqrt{2} + 5)$. Find the maximum perimeter among all the 6-sided polygons in the complex plane whose vertices are the zeros of $Q(w)$."
}
] |
[
{
"from": "human",
"value": "Find all values of $b$ for which the equation $\\sqrt{x^2 + b^2} = x^2 - 3x + 2$ has exactly two solutions for $x$."
}
] |
[
{
"from": "human",
"value": "Let $f(n)$ denote the sum of the squares of the base $n$ digits of $1992$. For example,\n\n$$f(5)=1^2+4^2+3^2+2^2=30.$$Let $S=\\{n>3 \\mid f(n)=f(n+1)\\}$. Compute $\\sum_{n\\in S} n$."
}
] |
[
{
"from": "human",
"value": "Let $m$ be the least positive integer $n$ such that\n$\\dbinom{n}{2},\\dbinom{n}{3},\\dbinom{n}{4}$\nform an arithmetic sequence. Find the sum\n$\\dbinom{m}{2}+\\dbinom{m}{3}+\\dbinom{m}{4}$."
}
] |
[
{
"from": "human",
"value": "A map of the town that Annie, Barbara, and Charlie live in can be represented by the Cartesian plane. Annie is located at $(6,-20)$ and Barbara is located at $(1, 14)$. They agree to meet at the closest point that is equidistant from their current locations and walk upwards toge... |
[
{
"from": "human",
"value": "Let $p$, $q$, and $r$ be distinct primes such that $pq+r$, $pr+q$, and $qr+p$ are all perfect squares. What is the smallest possible value of $p+q+r$?"
}
] |
[
{
"from": "human",
"value": "Consider three functions $f(x)$, $g(x)$, and $h(x)$ such that\n\\[\\frac{f(x)g(x)}{g(x) + h(x)} + \\frac{g(x)h(x)}{h(x) + f(x)} + \\frac{h(x)f(x)}{f(x) + g(x)} = 1\\]and\n\\[\\frac{g(x)h(x)}{g(x) + h(x)} + \\frac{h(x)f(x)}{h(x) + f(x)} + \\frac{f(x)g(x)}{f(x) + g(x)} = 2.\\]Find... |
[
{
"from": "human",
"value": "Let $f(x) = \\sqrt{x} - 2$, and for $n \\ge 2$, define $f_n(x) = f(f_{n-1}(x))$. Find the value of $x$ that satisfies $f_{1234}(x) = 2 - \\sqrt{x}$."
}
] |
[
{
"from": "human",
"value": "A five-digit integer $n$ and the five-digit integer obtained by rotating the digits of $n$ by two positions to the right (i.e., the third digit becomes the first digit, the fourth digit becomes the second digit, and so on) are both perfect squares. If $n$ is divisible by 11, wha... |
[
{
"from": "human",
"value": "Let $P(x) = 8x^3+16x^2-7x+4$ and $Q(x) = 2x+5$. Determine the value of $k$ such that $P(x) + k \\cdot Q(x)$ has exactly two distinct real roots."
}
] |
[
{
"from": "human",
"value": "Ava is trying to find the circumcenter $Q$ of $\\triangle DEF$, where $D$ is at the origin, $E$ is at $(6,3)$, and $F$ is at $(2,-5)$ (the circumcenter is the point equidistant from the vertices of a triangle). She thinks that the point is at $Q = (3,1)$, and computes the distan... |
[
{
"from": "human",
"value": "In a quadrilateral $ABCD$, the diagonals $AC$ and $BD$ intersect at point $E$, and $\\angle ABE = 3 \\angle CDE$. The measures of the sides $AB$, $BC$, $CD$, and $DA$ are $4$, $5$, $6$, and $7$ respectively. Find the measure of $\\angle BCD$ in degrees."
}
] |
[
{
"from": "human",
"value": "In the country of Marr, there are three types of coins: gold, silver, and copper. Each coin has a distinct weight, and the weights of the coins are in the ratio 1:2:3. A good stack of coins is one in which no gold coin is placed on top of a silver coin, no silver coin is placed ... |
[
{
"from": "human",
"value": "What is the largest possible value of $x$ such that $\\sqrt{x}$ is an integer and $x^3 + x^2 + x + 1$ is a perfect square?"
}
] |
[
{
"from": "human",
"value": "Three positive integers have an arithmetic mean of 26 and median of 27. If the median is 5 less than the largest number, what is the smallest number?\n\nAnother problem inspired by this one:\n\nFour positive integers have a harmonic mean of 12 and a median of 15. If the median i... |
[
{
"from": "human",
"value": "Problem:\nA polynomial function is defined by $f(x) = (3x+4)(7x+1)(13x+6)(2x+9)$. If $x$ is a multiple of 15336, determine the greatest common divisor of $f(x)$ and $x$.\n\nAnother problem:\nLet $f(x) = x^4 + 3x^3 + 7x^2 + 13x + 6$. Find the smallest positive integer $x$ such th... |
[
{
"from": "human",
"value": "Find the smallest positive integer $n$ such that $\\frac{n^3+2n^2+3n+1}{n+1}$ is a perfect square."
}
] |
[
{
"from": "human",
"value": "How many of the positive integers from 1 to 600, inclusive, can be expressed as the sum of two squares in exactly one way? (For example, 10 can be expressed as 1^2+3^2 and 50 can be expressed as 5^2+5^2, but 20 cannot be expressed as the sum of two squares in exactly one way sin... |
[
{
"from": "human",
"value": "Let ABCD be a regular tetrahedron. Its vertices are on a certain sphere. E, F, G, H are the midpoints of the edges AB, BC, CD, DA. Find the ratio of the volume of tetrahedron EFGH to the volume of ABCD."
}
] |
[
{
"from": "human",
"value": "Let $q(x) = x^5 + x^4 + x^3 + x^2 + x + 1$. The polynomial $p(x)q(x)$ is divided by $(x^3 + 1)(x^4 + x^2 + 1)$. Find the remainder when the absolute value of the constant term of the remainder is divided by 1000."
}
] |
[
{
"from": "human",
"value": "In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 12 cm and 20 cm respectively. Points $E$ and $F$ are on $AB$ and $CD$ respectively, and they divide these segments into ratios of $2:3$. The line segment $EF$ intersects the line $AD$ at point $G$. If $AG:GD =... |
[
{
"from": "human",
"value": "There exist real numbers $x,$ $y,$ and $z$ such that\n\\[2 \\cos{\\frac{\\pi}{7}} = x + y \\cos{\\frac{2\\pi}{7}} + z \\cos{\\frac{3\\pi}{7}}.\\]\nFind $x^2 + y^2 + z^2.$"
}
] |
[
{
"from": "human",
"value": "Find the smallest positive integer $a$ such that $a^2+1$ is a perfect cube and $a^3-1$ is a perfect square."
}
] |
[
{
"from": "human",
"value": "Two identical right circular cones are given. The vertex of one of them is placed on the base of the other. The height of the resulting body is 18, and the distance from the base of the lower cone to the vertex of the upper cone is 11. Determine the area of the circle formed by ... |
[
{
"from": "human",
"value": "Given that the polynomial $x^4 - 6x^3 + 16x^2 - 25x + 10$ can be factored as $(x^2 - 2x + k)q(x) + x + a,$ determine the ordered pair $(k,a)$ so that $q(x)$ is a quadratic polynomial with no linear term."
}
] |
[
{
"from": "human",
"value": "In a rectangular prism, the volume is divided into 8 congruent smaller rectangular prisms by three planes parallel to the faces of the original prism. The volume of each smaller prism is $4\\pi$ cubic units. If the height of the original prism is twice the width, what is the rat... |
[
{
"from": "human",
"value": "Let $a$ and $b$ be two real numbers such that\n$0 < a < b < 1$. The numbers $a$ and $b$ satisfy the following\nproperty: when $a$ and $b$ are expressed as decimals in base 10,\nwith no leading zero, the digits of $a$ are a permutation of the\ndigits of $b$. Determine the smalles... |
[
{
"from": "human",
"value": "Point $Q$ lies on the diagonal $BD$ of square $ABCD$ with $BQ > DQ$. Let $O_{3}$ and $O_{4}$ be the circumcenters of triangles $ABQ$ and $CDQ$ respectively. Given that $AB = 16$ and $\\angle O_{3}QO_{4} = 135^{\\circ}$, then $BQ = \\sqrt{c} + \\sqrt{d}$, where $c$ and $d$ are po... |
[
{
"from": "human",
"value": "At a local bookstore, 10 fiction books and 5 non-fiction books are to be placed on a shelf in a way that the number of fiction books between any two non-fiction books is the same. What is the largest number of different arrangements that can be made?"
}
] |
[
{
"from": "human",
"value": "Find the smallest positive integer $n$ such that the difference between its cube and its triple is equal to the square of its smallest prime factor."
}
] |
[
{
"from": "human",
"value": "The sum of three numbers $a$, $b$, and $c$ is 270. When the middle number $b$ is divided by 3, the result is $m$. The value $m$ is obtained by adding 10 to the smallest number $a$. This value $m$ also results from subtracting 5 from the largest number $c$. What is the sum of the... |
[
{
"from": "human",
"value": "A choir consists of 4 boys and 6 girls, and they want to line up for a performance. The requirement is that at least 2 girls must stand next to each other, and no boy can stand between two girls. How many ways can the choir arrange themselves to meet these conditions?"
}
] |
[
{
"from": "human",
"value": "Let there be two circles A and B externally tangent to each other and to a given circle C at the ends of its diameter. Circle A is twice as large as circle B. Draw a tangent line to circle C at point P, such that P lies on the line connecting the centers of circles A and B. Find... |
[
{
"from": "human",
"value": "The following system of equations is given:\n\n$$\n\\begin{cases}\n\\sqrt{a-3b}+\\sqrt{b-3a}=\\sqrt{a+b} \\\\\n\\sqrt{b-a}+\\sqrt{b+a}=2\n\\end{cases}\n$$\n\nFind a + b."
}
] |
[
{
"from": "human",
"value": "Let $p$ be a prime number such that $p^2-2p-1$ is divisible by 11. Find the smallest positive integer $k$ such that $p^2-3p-k$ is divisible by 13."
}
] |
[
{
"from": "human",
"value": "A theatre company has 12 actors, 4 of whom are male leads, 4 female leads, and 4 supporting actors. They will perform a play in which there is a scene where all the male leads must be on stage at the same time, all the female leads must be on stage at the same time, but no suppo... |
[
{
"from": "human",
"value": "Consider the system of equations\n\\begin{align*}\nx^2 + y^2 - 6xy + 2 &= 0, \\\\\nx^2y^2 - 3x^2 + 3y^2 - 1 &= 0.\n\\end{align*}\nFind $x+y$."
}
] |
[
{
"from": "human",
"value": "Let $a,$ $b,$ and $c$ be the roots of the equation $x^3 + 24x^2 - 45x + 11 = 0.$ Find the value of $\\frac{a^2}{a + bc} + \\frac{b^2}{b + ac} + \\frac{c^2}{c + ab}.$"
}
] |
[
{
"from": "human",
"value": "Find all values of $x$ which satisfy\n\\[\\frac{3}{\\sqrt{x - 5} - 7} + \\frac{2}{\\sqrt{x - 5} - 3} + \\frac{9}{\\sqrt{x - 5} + 3} + \\frac{15}{\\sqrt{x - 5} + 7} = 0.\\]Enter all the solutions, separated by commas."
}
] |
[
{
"from": "human",
"value": "Let $ABCD$ be a tetrahedron with a right dihedral angle at the edge $BC$. The areas of faces $ABC$ and $BCD$ are $75$ and $60$, respectively. If $\\sin\\angle BAC=\\frac{4}{5}$ and $BC=12$, find the volume of the tetrahedron."
}
] |
[
{
"from": "human",
"value": "Find the smallest positive integer n for which the product of n, n + 1, and n + 2 is a perfect cube."
}
] |
[
{
"from": "human",
"value": "Find the sum of all the solutions of $2^x + 2^{\\frac{1}{x}} = 10$."
}
] |
[
{
"from": "human",
"value": "What is the sum of all three-digit positive integers whose cubes end with the digits 27 and whose tens digit is equal to the sum of the other two digits?"
}
] |
[
{
"from": "human",
"value": "A sequence of four real numbers forms a harmonic progression with a first term of 12. If 3 is subtracted from the second term, 2 is added to the third term, and 6 is added to the fourth term, the four resulting numbers form an arithmetic progression. What is the largest possible... |
[
{
"from": "human",
"value": "A group of eight friends are planning a group picture. They want to stand in two rows of four, with the constraint that certain pairs of friends are not allowed to stand next to each other. Specifically, Sarah and Emily cannot stand next to each other, and neither can Jack and B... |
[
{
"from": "human",
"value": "Let $a,$ $b,$ and $c$ be complex numbers such that $|a|=|b|=|c|=1$ and $|a+b+c|=2$. If $z=ab+ac+bc$, then determine the largest possible value of $|z-3|$."
}
] |
[
{
"from": "human",
"value": "Let $a$, $b$, and $c$ be positive integers such that $a^2+b^2+c^2=3001$ and $a<b<c$. What is the largest possible value of the sum $a+b+c$?"
}
] |
[
{
"from": "human",
"value": "In a rectangular garden, five identical rectangular flowerbeds are arranged in a row, leaving no space between them. The flowerbeds are then rearranged to form a large rectangular flowerbed with the same perimeter as the original five flowerbeds. What is the ratio of the total a... |
[
{
"from": "human",
"value": "Triangle $DEF$ has $DF = 550$ and $EF = 375$. Points $M$ and $N$ are located on $\\overline{DF}$ and $\\overline{DE}$ respectively so that $DM = FM$, and $\\overline{FN}$ is the angle bisector of angle $F$. Let $Q$ be the point of intersection of $\\overline{EM}$ and $\\overline... |
[
{
"from": "human",
"value": "The digits of a three-digit number $ABC$ are rotated to form two new three-digit numbers, $BCA$ and $CAB$. If $A$, $B$, and $C$ are distinct digits, what prime number must be a factor of the difference between the largest and smallest of the three numbers?"
}
] |
[
{
"from": "human",
"value": "Suppose that $g(x)$ is a quadratic function satisfying the equation $g(x) = (g^{-1}(x))^2 + 3$. Given that $g(0) = 2$ and $g(1) = 4$, find $g(2)$."
}
] |
[
{
"from": "human",
"value": "In triangle $ABC,$ angles $A$ and $B$ are congruent. What is the measure, in degrees, of the vertex angle if the triangle is rotated clockwise by $200^\\circ$ and then reflected over the line through the vertex angle and the midpoint of the base, resulting in the image of the ve... |
[
{
"from": "human",
"value": "How many different parallelograms have integer side lengths and area 36?"
}
] |
[
{
"from": "human",
"value": "Determine the value of $a$ such that the polynomial $Q(x) = x^4 - ax^2 + 3x - 2$ has a factor of $x^2 + 2x + 1$."
}
] |
[
{
"from": "human",
"value": "Let $g(x) = x^3 - 2x + 1.$ Find\n\\[g \\left( \\frac{1}{1001} \\right) \\cdot g \\left( \\frac{2}{1001} \\right) \\cdot g \\left( \\frac{3}{1001} \\right) \\dotsm g \\left( \\frac{1000}{1001} \\right).\\]"
}
] |
[
{
"from": "human",
"value": "A sequence of real numbers $a_n$ is defined by\n$a_n = 2020 - 2021 \\operatorname{sign}(a_{n-1}+a_{n-2})$\nfor $n \\geq 3,$ where $a_1$ and $a_2$ are given real numbers.\nDetermine the number of distinct values that $a_{10}$ can take."
}
] |
[
{
"from": "human",
"value": "Evaluate\n\\[\\int_{0}^{\\infty} \\frac{x^2 + x + 2}{x^4 + x^3 + 2x^2 + x + 1} dx.\\]"
}
] |
[
{
"from": "human",
"value": "Let $h(x) = Ax^2 + Bx + C$ and $k(x) = Cx^2 + Bx + A$, where $A \\neq C$. If $h(k(x)) - k(h(x)) = 2(C - A)$, what is $A + C$?"
}
] |
[
{
"from": "human",
"value": "A bookshelf contains 5 shelves, and each shelf can hold any number of books of the same color. There are 5 different colors of books, and the number of books of each color is unlimited. If the books are stacked horizontally on the shelves and the colors of the books on adjacent ... |
[
{
"from": "human",
"value": "A round table has 5 red chairs and 5 blue chairs, arranged randomly. Alice and Bob are playing a game where they take turns sitting in a chair, with Alice going first. The game ends when one of them sits in a chair of the same color as the previous chair. Let $p$ be the probabil... |
[
{
"from": "human",
"value": "A circle is inscribed in a square with side length 24 cm. Find the diameter of the circle if it circumscribes a triangle with an area of 30 cm²."
}
] |
[
{
"from": "human",
"value": "In quadrilateral $ABCD$, $AC$ is a diameter of the circumcircle. Angles $ACB$ and $ADC$ are both equal to $45^\\circ$. Let $M$ be the midpoint of $\\overline{BC}$ and $N$ the midpoint of $\\overline{AD}$. If angle $MAN$ is equal to $20^\\circ$, find the measure of angle $BND$."
... |
[
{
"from": "human",
"value": "In the figure, $ABCD$ is a convex quadrilateral with $AB = 20$, $BC = 24$, and $CD = 13$. Points $P$ and $Q$ are the midpoints of sides $AB$ and $CD$, respectively. If the area of triangle $APQ$ is 42 square units, what is the area of triangle $BCQ$?"
}
] |
[
{
"from": "human",
"value": "Factor $(x^3 + 2x^2 - 5x + 3)(x^2 + 4x - 7) - (2x^2 + 5x - 11)$ as the product of a linear polynomial and a cubic polynomial."
}
] |
[
{
"from": "human",
"value": "A very small, nimble, and quick spider named Sindy decided to spin a web with five radial threads. Each thread has five nodes, so there are six nodes including the center of the web. She labeled each of the nodes with a different nonnegative integer with a maximum of 10. Then... |
[
{
"from": "human",
"value": "Let $a,$ $b,$ $c$ be distinct, nonzero real numbers such that\n\\[\\sqrt{a} + \\frac{1}{\\sqrt{b}} = \\sqrt{b} + \\frac{1}{\\sqrt{c}} = \\sqrt{c} + \\frac{1}{\\sqrt{a}}.\\]Find the value of $|a^2b^2c^2|$"
}
] |
[
{
"from": "human",
"value": "In a triangular prism, the base is an equilateral triangle with a side length of 10 cm, and the height is 12 cm. A plane passing through one of the base's vertices and parallel to the opposite edge divides the prism into two parts of equal volume. What is the angle between this ... |
[
{
"from": "human",
"value": "The Green Valley Golf Course has 18 holes, and the total par is 72. In a two-day tournament, each player plays each hole twice. Lily played 15 holes on the first day and scored 70, and played the remaining 21 holes on the second day and scored 90. In contrast, Iris did not play ... |
[
{
"from": "human",
"value": "In quadrilateral $ABCD$, $AB = 17$, $BC = 15$, and $AD = 19$. Point $E$ is on $\\overline{CD}$ with $DE = 8$. Point $F$ is on $\\overline{AD}$ such that $\\angle BAE = \\angle FAD$. Find $AF.$"
}
] |
[
{
"from": "human",
"value": "Let $f(x)=2x^4+5x^3-4x^2+x$. If $r$ is a real root of $f(x)$, show that there exists a real root $s$ of $f(x)$ such that $|r-s|\\leq 1$. (Assume the existence of at least one real root.)"
}
] |
[
{
"from": "human",
"value": "Find the smallest possible value of $d$ where $a<b<c<d$ are four integers such that $a,b,c,d$ is an arithmetic progression, and $a,d,b,c$ is a harmonic progression."
}
] |
[
{
"from": "human",
"value": "In a convex quadrilateral $ABCD$, the diagonals $AC$ and $BD$ intersect at point $O$. The triangle $AOB$ is isosceles with base $AB$ and the triangle $COD$ is isosceles with base $CD$. If $AB = 24$ and $CD = 16$, what is the perimeter of quadrilateral $ABCD$?"
}
] |
[
{
"from": "human",
"value": "Find all the solutions to\n\\[\\sqrt[5]{3x - 2} + \\sqrt[5]{2x + 3} = 2 \\sqrt[5]{x^2}.\\]Enter all the solutions, separated by commas."
}
] |
[
{
"from": "human",
"value": "A group of 10 friends want to play a game of cards, but they only have 4 decks available. Each deck can be played with 3-6 players. How many different combinations of players can be formed to play the game, considering that each player can only play with one deck?"
}
] |
[
{
"from": "human",
"value": "Determine $b$ such that the polynomial $x^4+bx^2+16x-8$ has at least one irrational root."
}
] |
[
{
"from": "human",
"value": "Consider the polynomial\n$$P(x) = (x^2 + ax + b)(x^2 + cx + d)(x^2 + ex + f),$$\nwhere $a, b, c, d, e, f$ are real numbers. Suppose that $P(x)$ has exactly one real root and four complex roots such that the sum of the real parts of the complex roots is equal to the real root. Fi... |
[
{
"from": "human",
"value": "A rectangular solid has two congruent faces whose diagonals are 10 cm and 13 cm. Find the maximum possible length of the diagonal of the third face, assuming it is also an integer."
}
] |
[
{
"from": "human",
"value": "Problem:\nGiven that $3-2\\sqrt{3}$ is a root of the equation $x^4+ax^3+bx^2+cx+13=0$, and that $a$, $b$, and $c$ are integers, compute the sum of all possible values of $a$.\n\nNote: This problem requires a different approach than the original, as it involves a quartic equation... |
[
{
"from": "human",
"value": "A right-angled triangle $ABC$ is drawn inside another right-angled triangle $DEF$, where $A$ lies on $DE$, $C$ lies on $DF$, and $B$ lies on $EF$. The area of $\\triangle ABC$ is equal to one-third of the area of $\\triangle DEF$. If $\\angle EFD = 60^{\\circ}$ and $\\angle FDE ... |
[
{
"from": "human",
"value": "What is the sum of all possible values of $x$ such that $x^2 + 3x - 5$ is strictly between $\\frac{1}{x}$ and $\\frac{2}{x}$?"
}
] |
[
{
"from": "human",
"value": "Let $p(x)$ be a monic quadratic polynomial such that there exist distinct prime numbers $a$, $b$, and $c$ with the property that $p(a)=b$, $p(b)=c$, and $p(c)=a$. Find the largest possible value of $a+b+c$."
}
] |
[
{
"from": "human",
"value": "Given a regular hexagon $ABCDEF$ of side length 6, draw a circle centered at $A$ with radius 3, and a circle centered at $C$ with radius 4. If the circles intersect at $P$ and $Q$, and the distance from $P$ to $\\overline{AC}$ is equal to the distance from $Q$ to $\\overline{AC}... |
End of preview. Expand in Data Studio
No dataset card yet
- Downloads last month
- 29
Models trained or fine-tuned on RockToken/openthoughts_prompt_math_5k_src35k-40k