data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of triangle \\( ABC \\) is 1. Circle \\( \\omega \\) is tangent to side \\( BC \\) and the extensions of side \\( AB \\) at point \\( P \\) and side \\( AC \\) at point \\( Q \\). The line passing through the midpoints of \\( AB \\) and \\( AC \\) intersects the c... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 33600,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of triangle \\( \\mathrm{ABC} \\) is 1. A circle touches side \\( \\mathrm{AB} \\) at point \\( P \\) and the extension of side \\( \\mathrm{AC} \\) at point \\( Q \\). A line passing through the midpoints of sides \\( \\mathrm{AB} \\) and \\( \\mathrm{AC} \\) int... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 33601,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of triangle \\(ABC\\) is 1. A circle \\(\\omega\\) touches side \\(BC\\), the extension of side \\(AB\\) at point \\(P\\), and the extension of side \\(AC\\) at point \\(Q\\). A line passing through the midpoints of \\(AB\\) and \\(AC\\) intersects the circumcircl... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 33602,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of triangle \\(ABC\\) is 1. Circle \\(\\omega\\) is tangent to side \\(BC\\), the extension of side \\(AB\\) at point \\(P\\), and the extension of side \\(AC\\) at point \\(Q\\). A line passing through the midpoints of \\(AB\\) and \\(AC\\) intersects the circumc... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"index": 33603,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The perpendicular bisectors of the sides $AB$ and $CD$ of the rhombus $ABCD$ are drawn. It turned out that they divided the diagonal $AC$ into three equal parts. Find the altitude of the rhombus if $AB = 1$ ."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"index": 33604,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The perpendicular bisectors of the sides of triangle $ABC$ meet its circumcircle at points $A',$ $B',$ and $C',$ as shown. If the perimeter of triangle $ABC$ is 35 and the radius of the circumcircle is 8, then find the area of hexagon $AB'CA'BC'.$\n\n[asy]\nunitsize(2 cm);\n\n... | math | {
"style": "rule",
"ground_truth": "140"
} | {
"index": 33605,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The perpendicular bisectors of the sides of triangle $DEF$ meet its circumcircle at points $D'$, $E'$, and $F'$, respectively. If the perimeter of triangle $DEF$ is 42 and the radius of the circumcircle is 10, find the area of hexagon $DE'F'D'E'F$."
}
] | math | {
"style": "rule",
"ground_truth": "105"
} | {
"index": 33606,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The perpendicular bisectors of the sides of triangle $PQR$ meet its circumcircle at points $P',$ $Q',$ and $R',$ respectively. If the perimeter of triangle $PQR$ is 30 and the radius of the circumcircle is 7, then find the area of hexagon $PQ'RP'QR'.$"
}
] | math | {
"style": "rule",
"ground_truth": "105"
} | {
"index": 33607,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The perpendicular to the side $AB$ of the trapezoid $ABCD$, passing through its midpoint $K$, intersects the side $CD$ at point $L$. It is known that the area of quadrilateral $AKLD$ is five times greater than the area of quadrilateral $BKLC$. Given $CL=3$, $DL=15$, and $KC=4$,... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 33608,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The photographer wants to arrange three boys and three girls in a row such that a boy or a girl could be at each end, and the rest alternate in the middle, calculate the total number of possible arrangements."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 33609,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The physical education teacher lined up the class so that everyone was facing him. There are 12 people to the right of Kolya, 20 people to the left of Sasha, and 8 people to the right of Sasha. How many people are to the left of Kolya?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 33610,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The picture shows several circles connected by segments. Tanya chooses a natural number \\( n \\) and places different natural numbers in the circles so that the following property holds for all these numbers:\n\nIf numbers \\( a \\) and \\( b \\) are not connected by a segment... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 33611,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The picture shows the same die in three different positions. When the die is rolled, what is the probability of rolling a 'YES'?\n\nA) \\(\\frac{1}{3}\\)\nB) \\(\\frac{1}{2}\\)\nC) \\(\\frac{5}{9}\\)\nD) \\(\\frac{2}{3}\\)\nE) \\(\\frac{5}{6}\\)"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 33612,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The pie charts below indicate the percent of students who prefer golf, bowling, or tennis at East Junior High School and West Middle School. The total number of students at East is 2000 and at West, 2500. In the two schools combined, the percent of students who prefer tennis ... | math | {
"style": "rule",
"ground_truth": "32\\%"
} | {
"index": 33613,
"solution": "1. **Calculate the number of students who prefer tennis at East Junior High School:**\n - The total number of students at East Junior High School is 2000.\n - The percentage of students who prefer tennis at East Junior High School is 22%.\n - Therefore, the number of students wh... |
DeepScaleR | [
{
"role": "user",
"content": "The plane angle at the vertex of a regular triangular pyramid is $90^{\\circ}$. Find the ratio of the lateral surface area of the pyramid to the area of its base."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 33614,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to\n[asy]\nunitsize(3mm); defaultpen(linewidth(0.8pt));\npath p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0);\npath p2=(0,1)--(1,1)--(1,0);\... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 33615,
"solution": "1. **Identify the basic unit of tiling**: The tiling pattern consists of a large square divided into 9 smaller squares, each with side length $a$. The large square thus has a side length of $3a$.\n\n2. **Calculate the area of the large square**: The area of the large square is $(3a)^2... |
DeepScaleR | [
{
"role": "user",
"content": "The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to\n[asy] unitsize(3mm); defaultpen(linewidth(0.8pt)); path p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0); path p2=(0,1)--(1,1)--(1,0); pa... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 33616,
"solution": "1. **Understanding the Tiling Pattern**: The problem states that the plane is tiled by congruent squares and congruent pentagons. We need to determine the percentage of the plane covered by the pentagons.\n\n2. **Analyzing a Single Tile**: Consider a single tile in the tiling pattern.... |
DeepScaleR | [
{
"role": "user",
"content": "The planet Xavier follows an elliptical orbit with its sun at one focus. At its nearest point (perigee), it is 2 astronomical units (AU) from the sun, while at its furthest point (apogee) it is 12 AU away. When Xavier is midway along its orbit, as shown, how far is it from th... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 33617,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The planning committee at school has 10 members. Exactly four of these members are teachers. A four-person subcommittee with at least one member who is a teacher must be formed from the members of the planning committee. How many distinct subcommittees are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "195"
} | {
"index": 33618,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 33619,
"solution": "Let $a$ be the number of $2$-point shots, $b$ be the number of $3$-point shots, and $x$ be the number of free throws. We need to find $x$.\n\nGiven:\n1. The points from two-point shots equal the points from three-point shots, so $2a = 3b$.\n2. The number of free throws is one more tha... |
DeepScaleR | [
{
"role": "user",
"content": "The poetry lecture lasted 2 hours and $m$ minutes. The positions of the hour and minute hands on the clock at the end of the lecture are exactly swapped from their positions at the beginning of the lecture. If $[x]$ denotes the integer part of the decimal number $x$, find $[m]=... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"index": 33620,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The point $(1,1,1)$ is rotated $180^\\circ$ about the $y$-axis, then reflected through the $yz$-plane, reflected through the $xz$-plane, rotated $180^\\circ$ about the $y$-axis, and reflected through the $xz$-plane. Find the coordinates of the point now."
}
] | math | {
"style": "rule",
"ground_truth": "(-1,1,1)"
} | {
"index": 33621,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The point $A$ $(3,4)$ is reflected over the $x$-axis to $B$. Then $B$ is reflected over the line $y=x$ to $C$. What is the area of triangle $ABC$?"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 33622,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The point $O$ is the center of the circle circumscribed about $\\triangle ABC$, with $\\angle BOC = 120^{\\circ}$ and $\\angle AOB =\n140^{\\circ}$, as shown. What is the degree measure of $\\angle\nABC$?\n\n[asy]\npair A,B,C;\ndraw(Circle((0,0),20),linewidth(0.7));\nlabel(\"$O... | math | {
"style": "rule",
"ground_truth": "50^{\\circ}"
} | {
"index": 33623,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The point $P$ $(4,5)$ is reflected over the $y$-axis to $Q$. Then $Q$ is reflected over the line $y=-x$ to $R$. What is the area of triangle $PQR$?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 33624,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The point $P$ is inside of an equilateral triangle with side length $10$ so that the distance from $P$ to two of the sides are $1$ and $3$. Find the distance from $P$ to the third side."
}
] | math | {
"style": "rule",
"ground_truth": "5\\sqrt{3} - 4"
} | {
"index": 33625,
"solution": "\nGiven an equilateral triangle with side length \\(10\\) and a point \\(P\\) inside the triangle, we are required to find the distance from \\(P\\) to the third side, knowing the distances from \\(P\\) to the other two sides are \\(1\\) and \\(3\\).\n\nFirst, recall the area formula ... |
DeepScaleR | [
{
"role": "user",
"content": "The point \\( N \\) is the center of the face \\( ABCD \\) of the cube \\( ABCDEFGH \\). Also, \\( M \\) is the midpoint of the edge \\( AE \\). If the area of \\(\\triangle MNH\\) is \\( 13 \\sqrt{14} \\), what is the edge length of the cube?"
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{13}"
} | {
"index": 33626,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The point \\((p, q)\\) is on the line \\(y=\\frac{2}{5} x\\). Also, the area of the rectangle shown is 90. What is the value of \\(p\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 33627,
"solution": "Since \\((p, q)\\) lies on the line \\(y=\\frac{2}{5} x\\), then \\(q=\\frac{2}{5} p\\). The given rectangle has two sides on the axes, so has width \\(p\\) and height \\(q\\). Therefore, the area of the rectangle equals \\(p q=p \\cdot \\frac{2}{5} p=\\frac{2}{5} p^{2}\\). Since we a... |
DeepScaleR | [
{
"role": "user",
"content": "The point is chosen at random within the rectangle in the coordinate plane whose vertices are $(0, 0), (3030, 0), (3030, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\\tfrac{1}{3}$. Find $d$ to the nearest tenth."
}
] | math | {
"style": "rule",
"ground_truth": "0.3"
} | {
"index": 33628,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The point with coordinates $(6,-10)$ is the midpoint of the segment with one endpoint at $(8,0)$. Find the sum of the coordinates of the other endpoint."
}
] | math | {
"style": "rule",
"ground_truth": "-16"
} | {
"index": 33629,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The points $A$ , $B$ , $C$ , $D$ , and $E$ lie in one plane and have the following properties: $AB = 12, BC = 50, CD = 38, AD = 100, BE = 30, CE = 40$ . \n\nFind the length of the segment $ED$ ."
}
] | math | {
"style": "rule",
"ground_truth": "74"
} | {
"index": 33630,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The points $(-1,4)$ and $(2,-3)$ are adjacent vertices of a square. What is the area of the square?"
}
] | math | {
"style": "rule",
"ground_truth": "58"
} | {
"index": 33631,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The points $(0,0),(1,2),(2,1),(2,2)$ in the plane are colored red while the points $(1,0),(2,0),(0,1),(0,2)$ are colored blue. Four segments are drawn such that each one connects a red point to a blue point and each colored point is the endpoint of some segment. The smallest po... | math | {
"style": "rule",
"ground_truth": "305"
} | {
"index": 33632,
"solution": "If $(2,2)$ is connected to $(0,1)$ or $(1,0)$, then the other 6 points can be connected with segments of total length 3, which is minimal. This leads to a total length of $3+\\sqrt{5}$. On the other hand, if $(2,2)$ is connected to $(0,2)$ or $(0,2)$, then connecting the other points ... |
DeepScaleR | [
{
"role": "user",
"content": "The points $(0,0)\\,$, $(a,11)\\,$, and $(b,37)\\,$ are the vertices of an equilateral triangle. Find the value of $ab\\,$."
}
] | math | {
"style": "rule",
"ground_truth": "315"
} | {
"index": 33633,
"solution": "Consider the points on the complex plane. The point $b+37i$ is then a rotation of $60$ degrees of $a+11i$ about the origin, so:\n\\[(a+11i)\\left(\\mathrm{cis}\\,60^{\\circ}\\right) = (a+11i)\\left(\\frac 12+\\frac{\\sqrt{3}i}2\\right)=b+37i.\\]\n Equating the real and imaginary parts... |
DeepScaleR | [
{
"role": "user",
"content": "The points $(0,4)$ and $(1,3)$ lie on a circle whose center is on the $x$-axis. What is the radius of the circle?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 33634,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The points $(1, 3)$ and $(5, -1)$ are adjacent vertices of a square. What is the area of the square?"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 33635,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The points $(1, 7), (13, 16)$ and $(5, k)$, where $k$ is an integer, are vertices of a triangle. What is the sum of the values of $k$ for which the area of the triangle is a minimum?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 33636,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The points $(2, 5), (10, 9)$, and $(6, m)$, where $m$ is an integer, are vertices of a triangle. What is the sum of the values of $m$ for which the area of the triangle is a minimum?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 33637,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The points $(2, 9), (12, 14)$, and $(4, m)$, where $m$ is an integer, are vertices of a triangle. What is the sum of the values of $m$ for which the area of the triangle is a minimum?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 33638,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The points $(2,-3)$, $(4,3)$, and $(5, k/2)$ are on the same straight line. The value(s) of $k$ is (are):"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 33639,
"solution": "To determine the value(s) of $k$ such that the points $(2,-3)$, $(4,3)$, and $(5, k/2)$ lie on the same straight line, we need to ensure that the slope between any two pairs of these points is the same.\n\n1. **Calculate the slope between the points $(2,-3)$ and $(4,3)$:**\n The slo... |
DeepScaleR | [
{
"role": "user",
"content": "The points $(9, -5)$ and $(-3, -1)$ are the endpoints of a diameter of a circle. What is the sum of the coordinates of the center of the circle?"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 33640,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The points $(x, y)$ represented in this table lie on a straight line. The point $(28, t)$ lies on the same line. What is the value of $t?$ \\begin{tabular}{c|c}\n$x$ & $y$ \\\\ \\hline\n1 & 7 \\\\\n3 & 13 \\\\\n5 & 19 \\\\\n\\end{tabular}"
}
] | math | {
"style": "rule",
"ground_truth": "88"
} | {
"index": 33641,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The points $A = (3,-4,2),$ $B = (5,-8,5),$ $C = (4,-3,0),$ and $D = (6,-7,3)$ in space form a flat quadrilateral. Find the area of this quadrilateral."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{110}"
} | {
"index": 33642,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The points $A$, $B$ and $C$ lie on the surface of a sphere with center $O$ and radius $20$. It is given that $AB=13$, $BC=14$, $CA=15$, and that the distance from $O$ to $\\triangle ABC$ is $\\frac{m\\sqrt{n}}k$, where $m$, $n$, and $k$ are positive integers, $m$ and $k$ are re... | math | {
"style": "rule",
"ground_truth": "118"
} | {
"index": 33643,
"solution": "Let $D$ be the foot of the perpendicular from $O$ to the plane of $ABC$. By the Pythagorean Theorem on triangles $\\triangle OAD$, $\\triangle OBD$ and $\\triangle OCD$ we get:\n\\[DA^2=DB^2=DC^2=20^2-OD^2\\]\n It follows that $DA=DB=DC$, so $D$ is the circumcenter of $\\triangle ABC$... |
DeepScaleR | [
{
"role": "user",
"content": "The points $A=\\left(4, \\frac{1}{4}\\right)$ and $B=\\left(-5,-\\frac{1}{5}\\right)$ lie on the hyperbola $x y=1$. The circle with diameter $A B$ intersects this hyperbola again at points $X$ and $Y$. Compute $X Y$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\frac{401}{5}}"
} | {
"index": 33644,
"solution": "Let $A=(a, 1 / a), B=(b, \\underline{1 / b})$, and $X=(x, 1 / x)$. Since $X$ lies on the circle with diameter $\\overline{A B}$, we have $\\angle A X B=90^{\\circ}$. Thus, $\\overline{A X}$ and $\\overline{B X}$ are perpendicular, and so the product of their slopes must be -1 . We ded... |
DeepScaleR | [
{
"role": "user",
"content": "The points $B(1, 1)$, $I(2, 4)$ and $G(5, 1)$ are plotted in the standard rectangular coordinate system to form triangle $BIG$. Triangle $BIG$ is translated five units to the left and two units upward to triangle $B'I'G'$, in such a way that $B'$ is the image of $B$, $I'$ is th... | math | {
"style": "rule",
"ground_truth": "(-2,3)"
} | {
"index": 33645,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The points $P(3,-2), Q(3,1), R(7,1)$, and $S$ form a rectangle. What are the coordinates of $S$?"
}
] | math | {
"style": "rule",
"ground_truth": "(7,-2)"
} | {
"index": 33646,
"solution": "Since $P$ and $Q$ have the same $x$-coordinate, then side $PQ$ of the rectangle is vertical. This means that side $SR$ must also be vertical, and so the $x$-coordinate of $S$ is the same as the $x$-coordinate of $R$, which is 7. Since $Q$ and $R$ have the same $y$-coordinate, then sid... |
DeepScaleR | [
{
"role": "user",
"content": "The points $P,$ $Q,$ and $R$ are represented by the complex numbers $z,$ $(1 + i) z,$ and $2 \\overline{z},$ respectively, where $|z| = 1.$ When $P,$ $Q$, and $R$ are not collinear, let $S$ be the fourth vertex of the parallelogram $PQSR.$ What is the maximum distance between... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 33647,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The points $Q(1,-1), R(-1,0)$ and $S(0,1)$ are three vertices of a parallelogram. What could be the coordinates of the fourth vertex of the parallelogram?"
}
] | math | {
"style": "rule",
"ground_truth": "(-2,2)"
} | {
"index": 33648,
"solution": "We plot the first three vertices on a graph. We see that one possible location for the fourth vertex, $V$, is in the second quadrant. If $V S Q R$ is a parallelogram, then $S V$ is parallel and equal to $Q R$. To get from $Q$ to $R$, we go left 2 units and up 1 unit. Therefore, to get... |
DeepScaleR | [
{
"role": "user",
"content": "The points \\( K, L, M, N \\) are the centers of the circles inscribed in the faces \\( S A B, S A C, S B C, \\) and \\( A B C \\) of the tetrahedron \\( S A B C \\). It is known that \\( A B = S C = 5 \\), \\( A C = S B = 7 \\), \\( B C = S A = 8 \\). Find the volume of the te... | math | {
"style": "rule",
"ground_truth": "0.66"
} | {
"index": 33649,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The points \\( M \\) and \\( N \\) are chosen on the angle bisector \\( AL \\) of a triangle \\( ABC \\) such that \\( \\angle ABM = \\angle ACN = 23^\\circ \\). \\( X \\) is a point inside the triangle such that \\( BX = CX \\) and \\( \\angle BXC = 2\\angle BML \\). Find \\( ... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"index": 33650,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The points \\((2,3)\\) and \\((3, 7)\\) lie on a circle whose center is on the \\(x\\)-axis. What is the radius of the circle?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{1717}}{2}"
} | {
"index": 33651,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The points on this graph show the end-of-year sales amounts for each year. During what year after 1994 did sales increase the most number of dollars? Express your answer as a 4-digit year.\n\n[asy]unitsize(101/10);\nreal textsize = 10, largetext = 12;\ndraw((10,0)--(0,0)--(0,1... | math | {
"style": "rule",
"ground_truth": "1998"
} | {
"index": 33652,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polar coordinate equation of curve C is given by C: ρ² = $\\frac{12}{5 - \\cos(2\\theta)}$, and the parametric equations of line l are given by $\\begin{cases} x = 1 + \\frac{\\sqrt{2}}{2}t \\\\ y = \\frac{\\sqrt{2}}{2}t \\end{cases}$ (where t is the parameter).\n\n1. Write... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{3}}{3}"
} | {
"index": 33653,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polar equation of circle C is $\\rho = 2\\cos(θ + \\frac{π}{4})$. The parametric equation of line l is \n$$\n\\begin{cases}\n x= \\sqrt{2}t \\\\\n y= \\sqrt{2}t+4\\sqrt{2}\n\\end{cases}\n$$\n(where t is the parameter). A tangent is drawn from a point P on line l to circle... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{6}"
} | {
"index": 33654,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 33655,
"solution": "\nTo solve this problem, we need to determine how many of the nine positions for the additional square allow the resulting figure to be folded into a cube with one face missing. We start by understanding the structure of the given figure and the implications of adding a square at each... |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial\n$$g(x) = x^3 - x^2 - (m^2 + m) x + 2m^2 + 4m + 2$$is divisible by $x-4$ and all of its zeroes are integers. Find all possible values of $m$."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 33656,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial\n\\[ax^4 + bx^3 + cx^2 + dx + e = 0\\]has coefficients that are all integers, and has roots $-2,$ $5,$ $9,$ and $-1/3.$ If $e$ is a positive integer, then find its smallest possible value."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 33657,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial\n\\[px^4 + qx^3 + rx^2 + sx + t = 0\\]\nhas coefficients that are all integers, and roots $-3$, $4$, $6$, and $\\frac{1}{2}$. If $t$ is a positive integer, find its smallest possible value."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 33658,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $-400x^5+2660x^4-3602x^3+1510x^2+18x-90$ has five rational roots. Suppose you guess a rational number which could possibly be a root (according to the rational root theorem). What is the probability that it actually is a root?"
}
] | math | {
"style": "rule",
"ground_truth": "1/36"
} | {
"index": 33659,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $P$ is a quadratic with integer coefficients. For every positive integer $n$ , the integers $P(n)$ and $P(P(n))$ are relatively prime to $n$ . If $P(3) = 89$ , what is the value of $P(10)$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "859"
} | {
"index": 33660,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $f(x)=x^3-4\\sqrt{3}x^2+13x-2\\sqrt{3}$ has three real roots, $a$ , $b$ , and $c$ . Find $$ \\max\\{a+b-c,a-b+c,-a+b+c\\}. $$"
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3} + 2\\sqrt{2}"
} | {
"index": 33661,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $$P(x)=(1+x+x^2+\\ldots+x^{17})^2-x^{17}$$has 34 complex zeros of the form $z_k=r_k\\left[\\cos(2\\pi\\alpha_k)\n+i\\sin(2\\pi\\alpha_k)\\right]$, $k=1,2,3,\\ldots,34$, with $0<\\alpha_1\\le\\alpha_2\\le\\alpha_3\\le\\dots\\le\\alpha_{34}<1$ and $r_k>0$. Find $\\... | math | {
"style": "rule",
"ground_truth": "\\frac{159}{323}"
} | {
"index": 33662,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $(x+y)^9$ is expanded in decreasing powers of $x$. The second and third terms have equal values \nwhen evaluated at $x=p$ and $y=q$, where $p$ and $q$ are positive numbers whose sum is one. What is the value of $p$?"
}
] | math | {
"style": "rule",
"ground_truth": "4/5"
} | {
"index": 33663,
"solution": "1. **Identify the Terms in the Expansion**: \n The polynomial $(x+y)^9$ can be expanded using the binomial theorem, which states that:\n \\[\n (x+y)^n = \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k\n \\]\n For $n=9$, the second term (where $k=1$) is:\n \\[\n \\binom{9}{1} x^{... |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $(x+y)^{10}$ is expanded in decreasing powers of $x$. The second and third terms have equal values when evaluated at $x=p$ and $y=q$, where $p$ and $q$ are positive numbers whose sum of $p$ plus twice $q$ equals one. Determine the value of $p$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{13}"
} | {
"index": 33664,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $-5x^2-xy^4+2^6xy+3$ has terms, and the degree of this polynomial is . The coefficient of the highest degree term is ."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 33665,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $1-x+x^2-x^3+\\cdots+x^{16}-x^{17}$ may be written in the form $a_0+a_1y+a_2y^2+\\cdots +a_{16}y^{16}+a_{17}y^{17}$, where $y=x+1$ and the $a_i$'s are constants. Find the value of $a_2$."
}
] | math | {
"style": "rule",
"ground_truth": "816"
} | {
"index": 33666,
"solution": "Using the geometric series formula, $1 - x + x^2 + \\cdots - x^{17} = \\frac {1 - x^{18}}{1 + x} = \\frac {1-x^{18}}{y}$. Since $x = y - 1$, this becomes $\\frac {1-(y - 1)^{18}}{y}$. We want $a_2$, which is the coefficient of the $y^3$ term in $-(y - 1)^{18}$ (because the $y$ in the ... |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $2x^3 + bx + 7$ has a factor of the form $x^2 + px + 1.$ Find $b.$"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{45}{2}"
} | {
"index": 33667,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $P(x) = 2x^3 + ax^2 + bx + c$ has the property that the mean of its zeros, the product of its zeros, and the sum of the coefficients are all equal. The $y$-intercept of the graph of $y = P(x)$ is 8. What is $b$?"
}
] | math | {
"style": "rule",
"ground_truth": "-38"
} | {
"index": 33668,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $P(x) = x^3 + ax^2 + bx +c$ has the property that the mean of its zeros, the product of its zeros, and the sum of its coefficients are all equal. If the $y$-intercept of the graph of $y=\nP(x)$ is 2, what is $b$?"
}
] | math | {
"style": "rule",
"ground_truth": "-11"
} | {
"index": 33669,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $P(x)$ is a monic, quartic polynomial with real coefficients, and two of its roots are $\\cos \\theta + i \\sin \\theta$ and $\\sin \\theta + i \\cos \\theta,$ where $0 < \\theta < \\frac{\\pi}{4}.$ When the four roots of $P(x)$ are plotted in the complex plane,... | math | {
"style": "rule",
"ground_truth": "1 + \\sqrt{3}"
} | {
"index": 33670,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $P(x)$ is cubic. What is the largest value of $k$ for which the polynomials $Q_1(x) = x^2 + (k-29)x - k$ and $Q_2(x) = 2x^2+ (2k-43)x + k$ are both factors of $P(x)$?"
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 33671,
"solution": "We can see that $Q_1$ and $Q_2$ must have a root in common for them to both be factors of the same cubic.\n Let this root be $a$.\n We then know that $a$ is a root of $Q_{2}(x)-2Q_{1}(x) = 2x^{2}+2kx-43x+k-2x^{2}-2kx+58x+2k = 15x+3k = 0$ , so $x = \\frac{-k}{5}$.\n We then know that $... |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $P(x)=(1+x+x^2+\\cdots+x^{17})^2-x^{17}$ has $34$ complex roots of the form $z_k = r_k[\\cos(2\\pi a_k)+i\\sin(2\\pi a_k)], k=1, 2, 3,\\ldots, 34,$ with $0 < a_1 \\le a_2 \\le a_3 \\le \\cdots \\le a_{34} < 1$ and $r_k>0.$ Given that $a_1 + a_2 + a_3 + a_4 + a_5 ... | math | {
"style": "rule",
"ground_truth": "482"
} | {
"index": 33672,
"solution": "We see that the expression for the polynomial $P$ is very difficult to work with directly, but there is one obvious transformation to make: sum the geometric series:\n\\begin{align*} P(x) &= \\left(\\frac{x^{18} - 1}{x - 1}\\right)^2 - x^{17} = \\frac{x^{36} - 2x^{18} + 1}{x^2 - 2x + ... |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $Q(x) = 3x^3 + dx^2 + ex + f$ has the property that the mean of its zeros, the product of its zeros, and the sum of the coefficients are all equal. The $y$-intercept of the graph of $y = Q(x)$ is 9. What is $e$?"
}
] | math | {
"style": "rule",
"ground_truth": "-42"
} | {
"index": 33673,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $Q(x)=x^3-21x+35$ has three different real roots. Find real numbers $a$ and $b$ such that the polynomial $x^2+ax+b$ cyclically permutes the roots of $Q$, that is, if $r$, $s$ and $t$ are the roots of $Q$ (in some order) then $P(r)=s$, $P(s)=t$ and $P(t)=r$."
}
... | math | {
"style": "rule",
"ground_truth": "a = 2, b = -14"
} | {
"index": 33674,
"solution": "\nGiven the polynomial \\( Q(x) = x^3 - 21x + 35 \\), which has three different real roots, we need to find real numbers \\( a \\) and \\( b \\), such that the polynomial \\( P(x) = x^2 + ax + b \\) cyclically permutes the roots of \\( Q \\). Let the roots of \\( Q \\) be \\( r, s, \\... |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $\\frac{1}{5}{x^2}{y^{|m|}}-(m+1)y+\\frac{1}{7}$ is a cubic binomial in terms of $x$ and $y$. Find the value of $m$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 33675,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $ax^4 + bx^3 + 32x^2 - 16x + 6$ has a factor of $3x^2 - 2x + 1.$ Find the ordered pair $(a,b).$"
}
] | math | {
"style": "rule",
"ground_truth": "(18,-24)"
} | {
"index": 33676,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $f(x) = x^3 + x^2 + 2x + 3$ has three distinct roots. Let $g(x) = x^3+bx^2+cx+d$ be a cubic polynomial with leading coefficient $1$ such that the roots of $g(x)$ are the squares of the roots of $f(x)$. Find the ordered triple $(b,c,d)$."
}
] | math | {
"style": "rule",
"ground_truth": "(3,-2,-9)"
} | {
"index": 33677,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $f(x)$ satisfies $f(x + 1) - f(x) = 6x + 4.$ Find the leading coefficient of $f(x).$"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 33678,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $f(x)=x^4+ax^3+bx^2+cx+d$ has real coefficients, and $f(2i)=f(2+i)=0$. What is $a+b+c+d$?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 33679,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $f(x)=x^{2007}+17x^{2006}+1$ has distinct zeroes $r_1,\\ldots,r_{2007}$. A polynomial $P$ of degree $2007$ has the property that\n\\[P\\left(r_j+\\dfrac{1}{r_j}\\right)=0\\]for $j=1,\\ldots,2007$. Determine the value of $\\frac{P(1)}{P(-1)}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{289}{259}"
} | {
"index": 33680,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $f(x)=x^{3}-3 x^{2}-4 x+4$ has three real roots $r_{1}, r_{2}$, and $r_{3}$. Let $g(x)=x^{3}+a x^{2}+b x+c$ be the polynomial which has roots $s_{1}, s_{2}$, and $s_{3}$, where $s_{1}=r_{1}+r_{2} z+r_{3} z^{2}$, $s_{2}=r_{1} z+r_{2} z^{2}+r_{3}, s_{3}=r_{1} z^{2}... | math | {
"style": "rule",
"ground_truth": "-26"
} | {
"index": 33681,
"solution": "Note that $z=e^{\\frac{2 \\pi}{3} i}=\\cos \\frac{2 \\pi}{3}+i \\sin \\frac{2 \\pi}{3}$, so that $z^{3}=1$ and $z^{2}+z+1=0$. Also, $s_{2}=s_{1} z$ and $s_{3}=s_{1} z^{2}$. Then, the sum of the coefficients of $g(x)$ is $g(1)=\\left(1-s_{1}\\right)\\left(1-s_{2}\\right)\\left(1-s_{3}\... |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $f(z)=az^{2018}+bz^{2017}+cz^{2016}$ has real coefficients not exceeding $2019$, and $f\\left(\\tfrac{1+\\sqrt{3}i}{2}\\right)=2015+2019\\sqrt{3}i$. Find the remainder when $f(1)$ is divided by $1000$."
}
] | math | {
"style": "rule",
"ground_truth": "53"
} | {
"index": 33682,
"solution": "We have $\\frac{1+\\sqrt{3}i}{2} = \\omega$ where $\\omega = e^{\\frac{i\\pi}{3}}$ is a primitive 6th root of unity. Then we have\n\\begin{align*} f(\\omega) &= a\\omega^{2018} + b\\omega^{2017} + c\\omega^{2016}\\\\ &= a\\omega^2 + b\\omega + c \\end{align*}\n We wish to find $f(1) =... |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $x^3 - 2004 x^2 + mx + n$ has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How many values of $n$ are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "250500"
} | {
"index": 33683,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $x^3 - ax^2 + bx - 2310$ has three positive integer roots. What is the smallest possible value of $a$?"
}
] | math | {
"style": "rule",
"ground_truth": "88"
} | {
"index": 33684,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $x^3 -ax^2 + bx -2010$ has three positive integer roots. What is the smallest possible value of $a$?"
}
] | math | {
"style": "rule",
"ground_truth": "78"
} | {
"index": 33685,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $x^8 - 1$ is factored as\n\\[x^8 - 1 = p_1(x) p_2(x) \\dotsm p_k(x),\\]where each factor $p_i(x)$ is a non-constant polynomial with real coefficients. Find the largest possible value of $k.$"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 33686,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $x^{101} + Ax + B$ is divisible by $x^2 + x + 1$ for some real numbers $A$ and $B.$ Find $A + B.$"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 33687,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial $x^{3}-3 x^{2}+1$ has three real roots $r_{1}, r_{2}$, and $r_{3}$. Compute $\\sqrt[3]{3 r_{1}-2}+\\sqrt[3]{3 r_{2}-2}+\\sqrt[3]{3 r_{3}-2}$."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 33688,
"solution": "Let $r$ be a root of the given polynomial. Then $$r^{3}-3 r^{2}+1=0 \\Longrightarrow r^{3}-3 r^{2}+3 r-1=3 r-2 \\Longrightarrow r-1=\\sqrt[3]{3 r-2}$$ Now by Vieta the desired value is $r_{1}+r_{2}+r_{3}-3=3-3=0$."
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial \\( f(x) \\) satisfies the equation \\( f(x) - f(x-2) = (2x-1)^{2} \\) for all \\( x \\). Find the sum of the coefficients of \\( x^{2} \\) and \\( x \\) in \\( f(x) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"index": 33689,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial \\( f(x)=x^{2007}+17 x^{2006}+1 \\) has distinct zeroes \\( r_{1}, \\ldots, r_{2007} \\). A polynomial \\( P \\) of degree 2007 has the property that \\( P\\left(r_{j}+\\frac{1}{r_{j}}\\right)=0 \\) for \\( j=1, \\ldots, 2007 \\). Determine the value of \\( P(1) ... | math | {
"style": "rule",
"ground_truth": "289/259"
} | {
"index": 33690,
"solution": "For some constant \\( k \\), we have \\( P(z)=k \\prod_{j=1}^{2007}\\left(z-\\left(r_{j}+\\frac{1}{r_{j}}\\right)\\right) \\). Now writing \\( \\omega^{3}=1 \\) with \\( \\omega \\neq 1 \\), we have \\( \\omega^{2}+\\omega=-1 \\). Then \\( P(1) / P(-1)=\\frac{k \\prod_{j=1}^{2007}\\le... |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial \\( g(x) = x^4 + ax^3 + bx^2 + cx + d \\) has real coefficients, with the roots \\( 3i \\) and \\( 1+2i \\). Calculate the sum of the coefficients \\( a + b + c + d \\)."
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"index": 33691,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial \\( p(x) = x^2 - 3x + 1 \\) has zeros \\( r \\) and \\( s \\). A quadratic polynomial \\( q(x) \\) has a leading coefficient of 1 and zeros \\( r^3 \\) and \\( s^3 \\). Find \\( q(1) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-16"
} | {
"index": 33692,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial \\( x^8 - 4x^7 + 7x^6 + \\cdots + a_0 \\) has all its roots positive and real numbers. Find the possible values for \\( a_0 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{256}"
} | {
"index": 33693,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial \\( x^{103} + Cx + D \\) is divisible by \\( x^2 + 2x + 1 \\) for some real numbers \\( C \\) and \\( D \\). Find \\( C + D \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 33694,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial \\( x^{2n} + 1 + (x+1)^{2n} \\) cannot be divided by \\( x^2 + x + 1 \\) under the condition that \\( n \\) is equal to:"
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 33695,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial equation \\[x^3 + bx + c = 0,\\]where $b$ and $c$ are rational numbers, has $3-\\sqrt{7}$ as a root. It also has an integer root. What is it?"
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"index": 33696,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial equation \\[x^3 + bx + c = 0,\\]where $b$ and $c$ are rational numbers, has $5-\\sqrt{2}$ as a root. It also has an integer root. What is it?"
}
] | math | {
"style": "rule",
"ground_truth": "-10"
} | {
"index": 33697,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial equation \\[x^4 + dx^2 + ex + f = 0,\\] where \\(d\\), \\(e\\), and \\(f\\) are rational numbers, has \\(3 - \\sqrt{5}\\) as a root. It also has two integer roots. Find the fourth root."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"index": 33698,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The polynomial sequence is defined as follows: \\( f_{0}(x)=1 \\) and \\( f_{n+1}(x)=\\left(x^{2}-1\\right) f_{n}(x)-2x \\) for \\( n=0,1,2, \\ldots \\). Find the sum of the absolute values of the coefficients of \\( f_{6}(x) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "190"
} | {
"index": 33699,
"solution": ""
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.