data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Circles \\(P\\), \\(Q\\), and \\(R\\) are externally tangent to each other and internally tangent to circle \\(S\\). Circles \\(Q\\) and \\(R\\) are congruent. Circle \\(P\\) has radius 2 and passes through the center of \\(S\\). What is the radius of circle \\(Q\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{16}{9}"
} | {
"index": 6200,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circles \\(\\omega_{1}\\) and \\(\\omega_{2}\\) intersect at points \\(A\\) and \\(B\\). Segment \\(PQ\\) is tangent to \\(\\omega_{1}\\) at \\(P\\) and to \\(\\omega_{2}\\) at \\(Q\\), and \\(A\\) is closer to \\(PQ\\) than \\(B\\). Point \\(X\\) is on \\(\\omega_{1}\\) such t... | math | {
"style": "rule",
"ground_truth": "2 - \\sqrt{3}"
} | {
"index": 6201,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circles centered at $A$ and $B$ each have radius 2, as shown. Point $O$ is the midpoint of $\\overline{AB}$, and $OA=2\\sqrt{2}$. Segments $OC$ and $OD$ are tangent to the circles centered at $A$ and $B$, respectively, and $\\overline{EF}$ is a common tangent. What is the are... | math | {
"style": "rule",
"ground_truth": "8\\sqrt{2}-4-\\pi"
} | {
"index": 6202,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circles of diameter 1 inch and 3 inches have the same center. The smaller circle is painted red, and the portion outside the smaller circle and inside the larger circle is painted blue. What is the ratio of the blue-painted area to the red-painted area?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 6203,
"solution": "1. **Calculate the area of the smaller circle (red area):**\n The diameter of the smaller circle is $1$ inch, so its radius is $\\frac{1}{2}$ inch. The area of a circle is given by the formula $A = \\pi r^2$. Therefore, the area of the smaller circle is:\n \\[\n A_{\\text{red}} =... |
DeepScaleR | [
{
"role": "user",
"content": "Circles of radii $5, 5, 8,$ and $\\frac mn$ are mutually externally tangent, where $m$ and $n$ are relatively prime positive integers. Find $m + n.$"
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 6204,
"solution": "We may also use Descartes' theorem, $k_4=k_1+k_2+k_3\\pm 2\\sqrt{k_1k_2+k_2k_3+k_3k_1}$ where each of $k_i$ is the curvature of a circle with radius $r_i$, and the curvature is defined as $k_i=\\frac{1}{r_i}$. The larger solution for $k_4$ will give the curvature of the circle external... |
DeepScaleR | [
{
"role": "user",
"content": "Circles of radius $3$ and $6$ are externally tangent to each other and are internally tangent to a circle of radius $9$ . The circle of radius $9$ has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 6205,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circles of radius $3$ and $6$ are externally tangent to each other and are internally tangent to a circle of radius $9$. The circle of radius $9$ has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord.\n[asy] pointpen... | math | {
"style": "rule",
"ground_truth": "224"
} | {
"index": 6206,
"solution": "We label the points as following: the centers of the circles of radii $3,6,9$ are $O_3,O_6,O_9$ respectively, and the endpoints of the chord are $P,Q$. Let $A_3,A_6,A_9$ be the feet of the perpendiculars from $O_3,O_6,O_9$ to $\\overline{PQ}$ (so $A_3,A_6$ are the points of tangency). ... |
DeepScaleR | [
{
"role": "user",
"content": "Circles of radius 2 and 3 are externally tangent and are circumscribed by a third circle, as shown in the figure. Find the area of the shaded region. Express your answer in terms of $\\pi$.\n\n[asy]\nfill(Circle((-1,0),5),gray(0.7));\nfill(Circle((-3,0),3),white);\nfill(circle(... | math | {
"style": "rule",
"ground_truth": "12\\pi"
} | {
"index": 6207,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circles of radius 3 and 4 are externally tangent and are circumscribed by a third circle. Find the area of the shaded region. Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "24\\pi"
} | {
"index": 6208,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circles of radius 4 and 5 are externally tangent and are circumscribed by a third circle. Calculate the area of the region outside the smaller circles but inside the larger circle."
}
] | math | {
"style": "rule",
"ground_truth": "40\\pi"
} | {
"index": 6209,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circles of radius 4 and 5 are externally tangent and are circumscribed by a third circle. Find the area of the shaded region. Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "40\\pi"
} | {
"index": 6210,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circles with centers $(2,4)$ and $(14,9)$ have radii $4$ and $9$, respectively. The equation of a common external tangent to the circles can be written in the form $y=mx+b$ with $m>0$. What is $b$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{912}{119}"
} | {
"index": 6211,
"solution": "1. **Identify the centers and radii of the circles**: \n - Circle 1 has center $(2,4)$ and radius $4$.\n - Circle 2 has center $(14,9)$ and radius $9$.\n\n2. **Calculate the slope of the line connecting the centers**:\n - The slope of the line $L_1$ connecting $(2,4)$ and $(14,9)... |
DeepScaleR | [
{
"role": "user",
"content": "Circles with centers $A$, $B$, and $C$ each have radius $r$, where $1 < r < 2$. The distance between each pair of centers is $2$. If $B'$ is the point of intersection of circle $A$ and circle $C$ which is outside circle $B$, and if $C'$ is the point of intersection of circle $A... | math | {
"style": "rule",
"ground_truth": "1+\\sqrt{3(r^2-1)}"
} | {
"index": 6212,
"solution": "1. **Setting up the problem**: We are given three circles with centers at $A$, $B$, and $C$, each having radius $r$ where $1 < r < 2$. The distance between each pair of centers is $2$. We need to find the length of the line segment $B'C'$, where $B'$ and $C'$ are points of intersection... |
DeepScaleR | [
{
"role": "user",
"content": "Circles with centers $P, Q$ and $R$, having radii $1, 2$ and $3$, respectively, lie on the same side of line $l$ and are tangent to $l$ at $P', Q'$ and $R'$, respectively, with $Q'$ between $P'$ and $R'$. The circle with center $Q$ is externally tangent to each of the other two... | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}-\\sqrt{2}"
} | {
"index": 6213,
"solution": "To solve this problem, we need to determine the positions of the centers $P$, $Q$, and $R$ of the circles relative to each other and then calculate the area of triangle $PQR$.\n\n1. **Positioning the Circles:**\n - Since the circles are tangent to line $l$ and each other, we can dete... |
DeepScaleR | [
{
"role": "user",
"content": "Circles with centers at $(5,5)$ and $(20,15)$ are both tangent to the $x$-axis. What is the distance between the closest points of the two circles?"
}
] | math | {
"style": "rule",
"ground_truth": "5 \\sqrt{13} - 20"
} | {
"index": 6214,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circles with centers of $(2,2)$ and $(17,10)$ are both tangent to the $x$-axis. What is the distance between the closest points of the two circles?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 6215,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circles with radii $1$, $2$, and $3$ are mutually externally tangent. What is the area of the triangle determined by the points of tangency?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6}{5}"
} | {
"index": 6216,
"solution": "1. **Identify the Triangle Formed by the Centers**: Let $O_1, O_2, O_3$ be the centers of the circles with radii $1, 2, 3$ respectively. The circles are mutually externally tangent, so the distance between any two centers is the sum of the radii of the two circles. Thus, $O_1O_2 = 1 + ... |
DeepScaleR | [
{
"role": "user",
"content": "Circular arcs of radius 3 inches form a continuous pattern as shown. What is the area, in square inches, of the shaded region in a 2-foot length of this pattern? Each arc completes half of a circle."
}
] | math | {
"style": "rule",
"ground_truth": "18\\pi"
} | {
"index": 6217,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cirlce $\\Omega$ is inscribed in triangle $ABC$ with $\\angle BAC=40$ . Point $D$ is inside the angle $BAC$ and is the intersection of exterior bisectors of angles $B$ and $C$ with the common side $BC$ . Tangent form $D$ touches $\\Omega$ in $E$ . FInd $\\an... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"index": 6218,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cities $A$, $B$, $C$, $D$, and $E$ are connected by roads $\\widetilde{AB}$, $\\widetilde{AD}$, $\\widetilde{AE}$, $\\widetilde{BC}$, $\\widetilde{BD}$, $\\widetilde{CD}$, and $\\widetilde{DE}$. How many different routes are there from $A$ to $B$ that use each road exactly once... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 6219,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "City A has 2 attractions, $A$ and $B$, while City B has 3 attractions, $C$, $D$, and $E$. When randomly selecting attractions to visit, find the probability of the following events:\n1. Selecting exactly 1 attraction in City A.\n2. Selecting exactly 2 attractions in the same ci... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"index": 6220,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle?"
}
] | math | {
"style": "rule",
"ground_truth": "143"
} | {
"index": 6221,
"solution": "1. **Understanding the problem**: Claire calculated the sum of the interior angles of a convex polygon as $2017^\\circ$. However, she missed one angle. We need to find the measure of this forgotten angle.\n\n2. **Using the formula for the sum of interior angles**: The sum of the interi... |
DeepScaleR | [
{
"role": "user",
"content": "Claire begins with 40 sweets. Amy gives one third of her sweets to Beth, Beth gives one third of all the sweets she now has to Claire, and then Claire gives one third of all the sweets she now has to Amy. Given that all the girls end up having the same number of sweets, determi... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 6222,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Class 2 of the second grade has 42 students, including $n$ male students. They are numbered from 1 to $n$. During the winter vacation, student number 1 called 3 students, student number 2 called 4 students, student number 3 called 5 students, ..., and student number $n$ called ... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 6223,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Class 2-5 planted 142 trees. Class 2-3 planted 18 fewer trees than Class 2-5. How many trees did Class 2-3 plant? How many trees did the two classes plant in total?"
}
] | math | {
"style": "rule",
"ground_truth": "266"
} | {
"index": 6224,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Class 5(2) has 28 female students, which is 6 more than the male students. The ratio of female to male students is ____, and the percentage of male students in the whole class is ____."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{25}"
} | {
"index": 6225,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Class A and Class B each send 2 students to participate in the grade math competition. The probability of each participating student passing the competition is 0.6, and the performance of the participating students does not affect each other. Find: \n(1) The probability that t... | math | {
"style": "rule",
"ground_truth": "0.9744"
} | {
"index": 6226,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Claudia has 12 coins, each of which is a 5-cent coin or a 10-cent coin. There are exactly 17 different values that can be obtained as combinations of one or more of her coins. How many 10-cent coins does Claudia have?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 6227,
"solution": "1. **Define Variables:**\n Let $x$ be the number of 5-cent coins Claudia has, and let $y$ be the number of 10-cent coins. Since Claudia has 12 coins in total, we have:\n \\[\n x + y = 12\n \\]\n\n2. **Express Total Value Range:**\n The smallest value Claudia can make is 5 cen... |
DeepScaleR | [
{
"role": "user",
"content": "Climbing the first flight of stairs takes Jimmy 20 seconds, and each following flight takes 5 seconds more than the preceding one. How many total seconds does it take to climb the first five flights of stairs?"
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 6228,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are each $\\frac {1}{3}$. The probability that Club Truncator will finish the season with more wins than... | math | {
"style": "rule",
"ground_truth": "341"
} | {
"index": 6229,
"solution": "At first, it wins $6$ games, only one way\n Secondly, it wins $5$ games, the other game can be either win or loss, there are $\\binom{6}{5}\\cdot 2=12$ ways\n Thirdly, it wins $4$ games, still the other two games can be either win or loss, there are $\\binom{6}{4}\\cdot 2^2=60$ ways\n ... |
DeepScaleR | [
{
"role": "user",
"content": "Club Truncator is now in a soccer league with four other teams, each of which it plays once. In any of its 4 matches, the probabilities that Club Truncator will win, lose, or tie are $\\frac{1}{3}$, $\\frac{1}{3}$, and $\\frac{1}{3}$ respectively. The probability that Club Trun... | math | {
"style": "rule",
"ground_truth": "112"
} | {
"index": 6230,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Coach Grunt is preparing the 5-person starting lineup for his basketball team, the Grunters. There are 12 players on the team. Two of them, Ace and Zeppo, are league All-Stars, so they'll definitely be in the starting lineup. How many different starting lineups are possible?... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 6231,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Coach Randall is preparing a 6-person starting lineup for her soccer team, the Rangers, which has 15 players. Among the players, three are league All-Stars (Tom, Jerry, and Spike), and they are guaranteed to be in the starting lineup. Additionally, the lineup must include at le... | math | {
"style": "rule",
"ground_truth": "55"
} | {
"index": 6232,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Coins are arranged in a row from left to right. It is known that two of them are fake, they lie next to each other, the left one weighs 9 grams, the right one weighs 11 grams, and all the remaining are genuine and each weighs 10 grams. The coins are weighed on a balance scale, ... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 6233,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Color the vertices of a quadrilateral pyramid so that the endpoints of each edge are different colors. If there are only 5 colors available, what is the total number of distinct coloring methods?"
}
] | math | {
"style": "rule",
"ground_truth": "420"
} | {
"index": 6234,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Come up with at least one three-digit number PAU (all digits are different), such that \\((P + A + U) \\times P \\times A \\times U = 300\\). (Providing one example is sufficient)"
}
] | math | {
"style": "rule",
"ground_truth": "235"
} | {
"index": 6235,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Company $W$'s product $p$ produced by line $D$ follows a normal distribution $N(80, 0.25)$ in terms of size. $400$ randomly selected products were tested from the current production line, and the size distribution is summarized in the table below:\n\n| Product Size ($mm$) | $[7... | math | {
"style": "rule",
"ground_truth": "\\frac{57}{4}"
} | {
"index": 6236,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Comparing two rectangular parallelepiped bars, it was found that the length, width, and height of the second bar are each 1 cm greater than those of the first bar, and the volume and total surface area of the second bar are 18 cm³ and 30 cm² greater, respectively, than those of... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 6237,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Complete the conversion between the following number systems: $101101_{(2)} = \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_{(10)}\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_{(7)}$"
}
] | math | {
"style": "rule",
"ground_truth": "63_{(7)}"
} | {
"index": 6238,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Complete the following questions: \n$(1)$ Calculate: $(\\sqrt{8}-\\sqrt{\\frac{1}{2}})\\div \\sqrt{2}$. \n$(2)$ Calculate: $2\\sqrt{3}\\times (\\sqrt{12}-3\\sqrt{75}+\\frac{1}{3}\\sqrt{108})$. \n$(3)$ Given $a=3+2\\sqrt{2}$ and $b=3-2\\sqrt{2}$, find the value of the algebra... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 6239,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Complete the table below, discover the patterns of square roots and cube roots, and apply the patterns to solve the problem.\n\n| $x$ | $\\ldots $ | $0.064$ | $0.64$ | $64$ | $6400$ | $64000$ | $\\ldots $ |\n|---------|-----------|---------|--------|-------|--------|------... | math | {
"style": "rule",
"ground_truth": "208.879"
} | {
"index": 6240,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Completely factor the expression: $$x^8-256$$"
}
] | math | {
"style": "rule",
"ground_truth": "(x^4+16)(x^2+4)(x+2)(x-2)"
} | {
"index": 6241,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Completely factor the following expression: $9x^2+3x$."
}
] | math | {
"style": "rule",
"ground_truth": "3x(3x+1)"
} | {
"index": 6242,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Completely factor the following expression: \\[(15x^3+80x-5)-(-4x^3+4x-5).\\]"
}
] | math | {
"style": "rule",
"ground_truth": "19x(x^2+4)"
} | {
"index": 6243,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Completely factor the following expression: \\[(6a^3+92a^2-7)-(-7a^3+a^2-7)\\]"
}
] | math | {
"style": "rule",
"ground_truth": "13a^2(a+7)"
} | {
"index": 6244,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Completely factor the following expression: \\[(9x^5+25x^3-4)-(x^5-3x^3-4).\\]"
}
] | math | {
"style": "rule",
"ground_truth": "4x^3(2x^2+7)"
} | {
"index": 6245,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Complex number $\\omega$ satisfies $\\omega^{5}=2$. Find the sum of all possible values of $\\omega^{4}+\\omega^{3}+\\omega^{2}+\\omega+1$."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 6246,
"solution": "The value of $\\omega^{4}+\\omega^{3}+\\omega^{2}+\\omega+1=\\frac{\\omega^{5}-1}{\\omega-1}=\\frac{1}{\\omega-1}$. The sum of these values is therefore the sum of $\\frac{1}{\\omega-1}$ over the five roots $\\omega$. Substituting $z=\\omega-1$, we have that $(z+1)^{5}=2$, so $z^{5}+5 ... |
DeepScaleR | [
{
"role": "user",
"content": "Complex numbers $a, b, c$ form an equilateral triangle with side length 18 in the complex plane. If $|a+b+c|=36$, find $|b c+c a+a b|$."
}
] | math | {
"style": "rule",
"ground_truth": "432"
} | {
"index": 6247,
"solution": "Using basic properties of vectors, we see that the complex number $d=\\frac{a+b+c}{3}$ is the center of the triangle. From the given, $|a+b+c|=36 \\Longrightarrow|d|=12$. Then, let $a^{\\prime}=a-d, b^{\\prime}=b-d$, and $c^{\\prime}=c-d$. Due to symmetry, $\\left|a^{\\prime}+b^{\\prim... |
DeepScaleR | [
{
"role": "user",
"content": "Complex numbers $a,$ $b,$ $c$ form an equilateral triangle with side length 18 in the complex plane. If $|a + b + c| = 36,$ find $|ab + ac + bc|.$"
}
] | math | {
"style": "rule",
"ground_truth": "432"
} | {
"index": 6248,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Complex numbers $a,$ $b,$ and $c$ are zeros of a polynomial $P(z) = z^3 + qz + r,$ and $|a|^2 + |b|^2 + |c|^2 = 250.$ The points corresponding to $a,$ $b,$ and $c$ in the complex plane are the vertices of a right triangle with hypotenuse $h.$ Find $h^2.$"
}
] | math | {
"style": "rule",
"ground_truth": "375"
} | {
"index": 6249,
"solution": "As noted in the previous solutions, $a+b+c = 0$. Let $a = a_1+a_2 i$, $b = b_1+b_2 i$, $c = c_1+c_2 i$ and we have $a_1 + b_1 + c_1 = a_2 + b_2 + c_2 = 0$. Then the given $|a|^2 + |b|^2 + |c|^2 = 250$ translates to $\\sum_{} ( {a_1}^2 + {a_2}^2 ) = 250.$ Note that in a right triangle, ... |
DeepScaleR | [
{
"role": "user",
"content": "Complex numbers $d$, $e$, and $f$ are zeros of a polynomial $Q(z) = z^3 + sz^2 + tz + u$, and $|d|^2 + |e|^2 + |f|^2 = 300$. The points corresponding to $d$, $e$, and $f$ in the complex plane are the vertices of an equilateral triangle. Find the square of the length of each sid... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"index": 6250,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Complex numbers $p$, $q$, and $r$ are zeros of a polynomial $Q(z) = z^4 - 2z^3 + sz + t,$ where $s, t \\in \\mathbb{C}$. It is given that $|p|^2 + |q|^2 + |r|^2 = 300$. The points corresponding to $p$, $q$, and $r$ in the complex plane form an equilateral triangle. Find the squ... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"index": 6251,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Complex numbers $p$, $q$, and $r$ form an equilateral triangle with side length 24 in the complex plane. If $|p + q + r| = 48$, find $|pq + pr + qr|$."
}
] | math | {
"style": "rule",
"ground_truth": "768"
} | {
"index": 6252,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Complex numbers $p, q, r$ form an equilateral triangle with side length 24 in the complex plane. If $|p + q + r| = 48,$ find $|pq + pr + qr|.$"
}
] | math | {
"style": "rule",
"ground_truth": "768"
} | {
"index": 6253,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Complex numbers $p,$ $q,$ $r$ form an equilateral triangle with side length 24 in the complex plane. If $|p + q + r| = 48,$ find $|pq + pr + qr|.$ These complex numbers have been translated by the same complex number $z$ compared to their original positions on the origin."
}
... | math | {
"style": "rule",
"ground_truth": "768"
} | {
"index": 6254,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Complex numbers $p,$ $q,$ and $r$ are zeros of a polynomial $P(z) = z^3 + sz + t,$ and $|p|^2 + |q|^2 + |r|^2 = 325.$ The points corresponding to $p,$ $q,$ and $r$ in the complex plane form a right triangle with right angle at $q.$ Find the square of the hypotenuse, $h^2,$ of t... | math | {
"style": "rule",
"ground_truth": "487.5"
} | {
"index": 6255,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Complex numbers $p,$ $q,$ and $r$ are zeros of a polynomial $Q(z) = z^3 + sz^2 + tz + u,$ and $|p|^2 + |q|^2 + |r|^2 = 360.$ The points corresponding to $p,$ $q,$ and $r$ in the complex plane are the vertices of a right triangle with hypotenuse $k.$ Find $k^2.$"
}
] | math | {
"style": "rule",
"ground_truth": "540"
} | {
"index": 6256,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Complex numbers $z_1,$ $z_2,$ and $z_3$ are zeros of a polynomial $Q(z) = z^3 + pz + s,$ where $|z_1|^2 + |z_2|^2 + |z_3|^2 = 300$. The points corresponding to $z_1,$ $z_2,$ and $z_3$ in the complex plane are the vertices of a right triangle with the right angle at $z_3$. Find ... | math | {
"style": "rule",
"ground_truth": "450"
} | {
"index": 6257,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Complex numbers \\( a \\), \\( b \\), and \\( c \\) form an equilateral triangle with side length 24 in the complex plane. If \\( |a + b + c| = 48 \\), find \\( |ab + ac + bc| \\)."
}
] | math | {
"style": "rule",
"ground_truth": "768"
} | {
"index": 6258,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Complex numbers \\( p, q, r \\) form an equilateral triangle with a side length of 24 in the complex plane. If \\( |p + q + r| = 48 \\), find \\( |pq + pr + qr| \\)."
}
] | math | {
"style": "rule",
"ground_truth": "768"
} | {
"index": 6259,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Complex numbers \\(a\\), \\(b\\), \\(c\\) form an equilateral triangle with side length 24 in the complex plane. If \\(|a + b + c| = 48\\), find \\(|ab + ac + bc|\\)."
}
] | math | {
"style": "rule",
"ground_truth": "768"
} | {
"index": 6260,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Complex numbers are often used when dealing with alternating current (AC) circuits. In the equation $V = IZ$, $V$ is voltage, $I$ is current, and $Z$ is a value known as impedance. If $V = 1-i$ and $Z=1+3i$, find $I$. Express your answer as a complex number in the form $a+bi$,... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{5} - \\frac{2}{5}i"
} | {
"index": 6261,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Comprehensive exploration: When two algebraic expressions containing square roots are multiplied together and the product does not contain square roots, we call these two expressions rationalizing factors of each other. For example, $\\sqrt{2}+1$ and $\\sqrt{2}-1$, $2\\sqrt{3}+... | math | {
"style": "rule",
"ground_truth": "1011"
} | {
"index": 6262,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\n$3(1+3(1+3(1+3(1+3(1+3(1+3(1+3(1+3(1+3)))))))))$"
}
] | math | {
"style": "rule",
"ground_truth": "88572"
} | {
"index": 6263,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\n$\\frac{1622^2-1615^2}{1629^2-1608^2}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 6264,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n$$\n\\int_{1}^{2} \\frac{9x+4}{x^{5}+3x^{2}+x} \\, dx."
}
] | math | {
"style": "rule",
"ground_truth": "\\ln \\frac{80}{23}"
} | {
"index": 6265,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n$$\n\\int_{1}^{\\sqrt{3}} x^{2 x^{2}+1}+\\ln \\left(x^{2 x^{2 x^{2}+1}}\\right) d x. \\text{ }$$"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 6266,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n$$\n\\lim _{h \\rightarrow 0} \\frac{\\sin \\left(\\frac{\\pi}{3}+4 h\\right)-4 \\sin \\left(\\frac{\\pi}{3}+3 h\\right)+6 \\sin \\left(\\frac{\\pi}{3}+2 h\\right)-4 \\sin \\left(\\frac{\\pi}{3}+h\\right)+\\sin \\left(\\frac{\\pi}{3}\\right)}{h^{4}}\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"index": 6267,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n$$\\sum_{k=1}^{1000} k(\\lceil \\log_{\\sqrt{2}}{k}\\rceil- \\lfloor\\log_{\\sqrt{2}}{k} \\rfloor).$$"
}
] | math | {
"style": "rule",
"ground_truth": "499477"
} | {
"index": 6268,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n$$\\sum_{k=1}^{2000} k(\\lceil \\log_{2}{k}\\rceil- \\lfloor\\log_{2}{k} \\rfloor).$$"
}
] | math | {
"style": "rule",
"ground_truth": "1998953"
} | {
"index": 6269,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n$$\\sum_{k=1}^{2000} k(\\lceil \\log_{\\sqrt{3}}{k}\\rceil - \\lfloor\\log_{\\sqrt{3}}{k} \\rfloor).$$"
}
] | math | {
"style": "rule",
"ground_truth": "1999907"
} | {
"index": 6270,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\n\\begin{vmatrix} \\cos 1 & \\cos 2 & \\cos 3 \\\\ \\cos 4 & \\cos 5 & \\cos 6 \\\\ \\cos 7 & \\cos 8 & \\cos 9 \\end{vmatrix}\n.\\]All the angles are in radians."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 6271,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\n\\frac{(1 + 23) \\left( 1 + \\dfrac{23}{2} \\right) \\left( 1 + \\dfrac{23}{3} \\right) \\dotsm \\left( 1 + \\dfrac{23}{25} \\right)}{(1 + 25) \\left( 1 + \\dfrac{25}{2} \\right) \\left( 1 + \\dfrac{25}{3} \\right) \\dotsm \\left( 1 + \\dfrac{25}{23} \\right)}.\n\... | math | {
"style": "rule",
"ground_truth": "600"
} | {
"index": 6272,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\n\\frac{(1 + 23) \\left( 1 + \\dfrac{23}{2} \\right) \\left( 1 + \\dfrac{23}{3} \\right) \\dotsm \\left( 1 + \\dfrac{23}{25} \\right)}{(1 + 27) \\left( 1 + \\dfrac{27}{2} \\right) \\left( 1 + \\dfrac{27}{3} \\right) \\dotsm \\left( 1 + \\dfrac{27}{21} \\right)}.\n\... | math | {
"style": "rule",
"ground_truth": "421200"
} | {
"index": 6273,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\n\\left( 1 + \\sin \\frac {\\pi}{12} \\right) \\left( 1 + \\sin \\frac {5\\pi}{12} \\right) \\left( 1 + \\sin \\frac {7\\pi}{12} \\right) \\left( 1 + \\sin \\frac {11\\pi}{12} \\right).\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"index": 6274,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\n\\left( 1 - \\sin \\frac {\\pi}{8} \\right) \\left( 1 - \\sin \\frac {3\\pi}{8} \\right) \\left( 1 - \\sin \\frac {5\\pi}{8} \\right) \\left( 1 - \\sin \\frac {7\\pi}{8} \\right).\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 6275,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\n\\log_2 \\left( \\prod_{a=1}^{2015} \\prod_{b=1}^{2015} (1+e^{2\\pi i a b/2015}) \\right)\n\\]\nHere $i$ is the imaginary unit (that is, $i^2=-1$)."
}
] | math | {
"style": "rule",
"ground_truth": "13725"
} | {
"index": 6276,
"solution": "The answer is $13725$.\nWe first claim that if $n$ is odd, then $\\prod_{b=1}^{n} (1+e^{2\\pi i ab/n}) = 2^{\\gcd(a,n)}$. To see this, write $d = \\gcd(a,n)$ and $a = da_1$, $n=dn_1$ with $\\gcd(a_1,n_1) = 1$. Then\n$a_1, 2a_1,\\dots,n_1 a_1$ modulo $n_1$ is a permutation of $1,2,\\dot... |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\n\\sin^2 0^\\circ + \\sin^2 10^\\circ + \\sin^2 20^\\circ + \\dots + \\sin^2 180^\\circ.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 6277,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\n\\sum_{n = 1}^\\infty \\frac{1}{(n+1)(n + 3)}.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{12}"
} | {
"index": 6278,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\n\\sum_{n = 1}^\\infty \\frac{1}{n(n + 3)}.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 6279,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\n\\sum_{n=1}^{\\infty} \\frac{4n + 1}{(4n - 1)^3 (4n + 3)^3}.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{432}"
} | {
"index": 6280,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[ e^{2 \\pi i/17} + e^{4 \\pi i/17} + e^{6 \\pi i/17} + \\dots + e^{32 \\pi i/17}. \\]"
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 6281,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\begin{pmatrix} 0 & c & -b \\\\ -c & 0 & a \\\\ b & -a & 0 \\end{pmatrix} \\begin{pmatrix} a^2 & ab & ac \\\\ ab & b^2 & bc \\\\ ac & bc & c^2 \\end{pmatrix}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix}"
} | {
"index": 6282,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\begin{pmatrix} 1 & 1 & -2 \\\\ 0 & 4 & -3 \\\\ -1 & 4 & 3 \\end{pmatrix} \\begin{pmatrix} 2 & -2 & 0 \\\\ 1 & 0 & -3 \\\\ 4 & 0 & 0 \\end{pmatrix}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -5 & -2 & -3 \\\\ -8 & 0 & -12 \\\\ 14 & 2 & -12 \\end{pmatrix}"
} | {
"index": 6283,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\begin{vmatrix} -5 & 3 \\\\ 4 & -4 \\end{vmatrix}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 6284,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\begin{vmatrix} 1 & -3 & 3 \\\\ 0 & 5 & -1 \\\\ 4 & -2 & 1 \\end{vmatrix}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "-45"
} | {
"index": 6285,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\begin{vmatrix} 1 & \\cos (a - b) & \\cos a \\\\ \\cos(a - b) & 1 & \\cos b \\\\ \\cos a & \\cos b & 1 \\end{vmatrix}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 6286,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\begin{vmatrix} 2 & 0 & -1 \\\\ 7 & 4 & -3 \\\\ 2 & 2 & 5 \\end{vmatrix}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "46"
} | {
"index": 6287,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\begin{vmatrix} 7 & 3 \\\\ -1 & 2 \\end{vmatrix}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 6288,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\cos^2 0^\\circ + \\cos^2 1^\\circ + \\cos^2 2^\\circ + \\dots + \\cos^2 90^\\circ.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{91}{2}"
} | {
"index": 6289,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\cos^6 0^\\circ + \\cos^6 1^\\circ + \\cos^6 2^\\circ + \\dots + \\cos^6 90^\\circ.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{229}{8}"
} | {
"index": 6290,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\csc \\frac{\\pi}{14} - 4 \\cos \\frac{2 \\pi}{7}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 6291,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\frac{(1 + 17) \\left( 1 + \\dfrac{17}{2} \\right) \\left( 1 + \\dfrac{17}{3} \\right) \\dotsm \\left( 1 + \\dfrac{17}{19} \\right)}{(1 + 19) \\left( 1 + \\dfrac{19}{2} \\right) \\left( 1 + \\dfrac{19}{3} \\right) \\dotsm \\left( 1 + \\dfrac{19}{17} \\right)}.\\]"
... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 6292,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "373"
} | {
"index": 6293,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\frac{(10^4+400)(26^4+400)(42^4+400)(58^4+400)}{(2^4+400)(18^4+400)(34^4+400)(50^4+400)}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "962"
} | {
"index": 6294,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\frac{1}{2^{1990}} \\sum_{n = 0}^{995} (-3)^n \\binom{1990}{2n}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 6295,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\frac{1}{\\cos 80^\\circ} - \\frac{\\sqrt{3}}{\\sin 80^\\circ}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 6296,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\frac{2 + 6}{4^{100}} + \\frac{2 + 2 \\cdot 6}{4^{99}} + \\frac{2 + 3 \\cdot 6}{4^{98}} + \\dots + \\frac{2 + 98 \\cdot 6}{4^3} + \\frac{2 + 99 \\cdot 6}{4^2} + \\frac{2 + 100 \\cdot 6}{4}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 6297,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\frac{5}{3^2 \\cdot 7^2} + \\frac{9}{7^2 \\cdot 11^2} + \\frac{13}{11^2 \\cdot 15^2} + \\dotsb.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{72}"
} | {
"index": 6298,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute\n\\[\\frac{\\lfloor \\sqrt[4]{1} \\rfloor \\cdot \\lfloor \\sqrt[4]{3} \\rfloor \\cdot \\lfloor \\sqrt[4]{5} \\rfloor \\dotsm \\lfloor \\sqrt[4]{2015} \\rfloor}{\\lfloor \\sqrt[4]{2} \\rfloor \\cdot \\lfloor \\sqrt[4]{4} \\rfloor \\cdot \\lfloor \\sqrt[4]{6} \\rfloor \\... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{16}"
} | {
"index": 6299,
"solution": ""
} |
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