data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "A cheesecake is shaped like a $3 \\times 3 \\times 3$ cube and is decorated such that only the top and three of the vertical faces are covered with icing. The cake is cut into $27$ smaller cubes, each $1 \\times 1 \\times 1$ inch. Calculate the number of the smaller cubes that ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 1100,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cheetah takes strides of 2 meters each, while a fox takes strides of 1 meter each. The time it takes for the cheetah to run 2 strides is the same as the time it takes for the fox to run 3 strides. If the distance between the cheetah and the fox is 30 meters, how many meters m... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 1101,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A chemistry student conducted an experiment: starting with a bottle filled with syrup solution, the student poured out one liter of liquid, refilled the bottle with water, then poured out one liter of liquid again, and refilled the bottle with water once more. As a result, the ... | math | {
"style": "rule",
"ground_truth": "1.2"
} | {
"index": 1102,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A chess piece called \"the four-liner\" attacks two vertical and two horizontal squares adjacent to the square on which it stands. What is the maximum number of non-attacking four-liners that can be placed on a $10 \\times 10$ board?"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 1103,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A chess team has $26$ members. However, only $16$ members attended the last meeting: half of the girls attended but all of the boys attended. How many boys are on the chess team?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 1104,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A chess team has $26$ members. However, only $16$ members attended the last meeting: half of the girls attended but all of the boys attended. How many girls are on the chess team?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 1105,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A chess tournament is held with the participation of boys and girls. The girls are twice as many as boys. Each player plays against each other player exactly once. By the end of the tournament, there were no draws and the ratio of girl winnings to boy winnings was $7/9$ . How ... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 1106,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A chessboard has its squares labeled according to the rule $\\frac{1}{c_i + r_j}$, where $c_i$ is the column number and $r_j$ is the row number. Eight squares are to be chosen such that there is exactly one chosen square in each row and each column. Find the minimum sum of the ... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{9}"
} | {
"index": 1107,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A chessboard of size $8 \\times 8$ is considered.\n\nHow many ways are there to place 6 rooks such that no two rooks are ever on the same row or column?"
}
] | math | {
"style": "rule",
"ground_truth": "564480"
} | {
"index": 1108,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A chessboard’s squares are labeled with numbers as follows:\n\n[asy]\nunitsize(0.8 cm);\n\nint i, j;\n\nfor (i = 0; i <= 8; ++i) {\n draw((i,0)--(i,8));\n draw((0,i)--(8,i));\n}\n\nfor (i = 0; i <= 7; ++i) {\nfor (j = 0; j <= 7; ++j) {\n label(\"$\\frac{1}{\" + string(9 - i ... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{9}"
} | {
"index": 1109,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A chest of gold coins is divided among 10 pirates where the kth pirate takes k/10 of the remaining coins. Determine the smallest number of coins initially in the chest such that each pirate gets a positive whole number of coins and find the number of coins the 10th pirate recei... | math | {
"style": "rule",
"ground_truth": "362880"
} | {
"index": 1110,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A child builds towers using identically shaped cubes of different colors. Determine the number of different towers with a height of 6 cubes that can be built with 3 yellow cubes, 3 purple cubes, and 2 orange cubes (Two cubes will be left out)."
}
] | math | {
"style": "rule",
"ground_truth": "350"
} | {
"index": 1111,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A child builds towers using identically shaped cubes of different colors. How many different towers with a height 8 cubes can the child build with 2 red cubes, 3 blue cubes, and 4 green cubes? (One cube will be left out.)"
}
] | math | {
"style": "rule",
"ground_truth": "1260"
} | {
"index": 1112,
"solution": "1. **Identify the Total Number of Cubes and Colors**: The child has a total of $2$ red cubes, $3$ blue cubes, and $4$ green cubes, making a total of $9$ cubes. However, the tower must be built with only $8$ cubes, so one cube will be left out.\n\n2. **Calculate the Total Arrangements o... |
DeepScaleR | [
{
"role": "user",
"content": "A child gave Carlson 111 candies. They ate some of them right away, 45% of the remaining candies went to Carlson for lunch, and a third of the candies left after lunch were found by Freken Bok during cleaning. How many candies did she find?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 1113,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A child has a set of 96 distinct blocks. Each block is one of 2 materials (plastic, wood), 3 sizes (small, medium, large), 4 colors (blue, green, red, yellow), and 4 shapes (circle, hexagon, square, triangle). How many blocks in the set differ from the 'plastic medium red circl... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 1114,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A child lines up $2020^2$ pieces of bricks in a row, and then remove bricks whose positions are square numbers (i.e. the 1st, 4th, 9th, 16th, ... bricks). Then he lines up the remaining bricks again and remove those that are in a 'square position'. This process is repeated un... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 1115,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A child spends their time drawing pictures of Native Americans (referred to as \"Indians\") and Eskimos. Each drawing depicts either a Native American with a teepee or an Eskimo with an igloo. However, the child sometimes makes mistakes and draws a Native American with an igloo... | math | {
"style": "rule",
"ground_truth": "7/8"
} | {
"index": 1116,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A child who does not understand English tries to spell the word \"hello\" using cards with the letters \"e\", \"o\", \"h\", \"l\", \"l\". How many possible incorrect arrangements can there be if the cards cannot be laid horizontally or upside down?"
}
] | math | {
"style": "rule",
"ground_truth": "59"
} | {
"index": 1117,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A child whose age is between 13 and 19 writes his own age after his father's age, creating a four-digit number. The absolute difference between their ages is subtracted from this new number to obtain 4289. What is the sum of their ages?\n\n(Note: From the 22nd Annual USA High S... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"index": 1118,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A child's wading pool contains 200 gallons of water. If water evaporates at the rate of 0.5 gallons per day and no other water is added or removed, how many gallons of water will be in the pool after 30 days?"
}
] | math | {
"style": "rule",
"ground_truth": "185"
} | {
"index": 1119,
"solution": "1. **Identify the initial amount of water**: The pool initially contains 200 gallons of water.\n\n2. **Determine the rate of evaporation**: Water evaporates at a rate of 0.5 gallons per day.\n\n3. **Calculate the total evaporation over 30 days**: Multiply the daily evaporation rate by ... |
DeepScaleR | [
{
"role": "user",
"content": "A chocolate bar consists of \\(5 \\times 8\\) square pieces. The bar is broken along lines dividing the pieces until there are 40 individual pieces. How many times must the bar be broken?"
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"index": 1120,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A chocolate bar originally weighed 400 grams and cost 150 rubles. Recently, to save money, the manufacturer reduced the weight of the bar to 300 grams and increased its price to 180 rubles. By what percentage did the manufacturer's revenue increase?"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 1121,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A chocolate bar weighed 250 g and cost 50 rubles. Recently, for cost-saving purposes, the manufacturer reduced the weight of the bar to 200 g and increased its price to 52 rubles. By what percentage did the manufacturer's income increase?"
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 1122,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A choir director must select a group of singers from among his $6$ tenors and $8$ basses. The only requirements are that the difference between the number of tenors and basses must be a multiple of $4$, and the group must have at least one singer. Let $N$ be the number of diffe... | math | {
"style": "rule",
"ground_truth": "95"
} | {
"index": 1123,
"solution": "1. **Define Variables:**\n Let $t$ be the number of tenors and $b$ be the number of basses selected. The conditions are:\n - $t - b$ is a multiple of $4$.\n - At least one singer is selected, i.e., $(t, b) \\neq (0, 0)$.\n\n2. **Combinatorial Choices:**\n The number of ways to ... |
DeepScaleR | [
{
"role": "user",
"content": "A chord $AB$ that makes an angle of $\\frac{\\pi}{6}$ with the horizontal passes through the left focus $F_1$ of the hyperbola $x^{2}- \\frac{y^{2}}{3}=1$. \n$(1)$ Find $|AB|$; \n$(2)$ Find the perimeter of $\\triangle F_{2}AB$ ($F_{2}$ is the right focus)."
}
] | math | {
"style": "rule",
"ground_truth": "3+3\\sqrt{3}"
} | {
"index": 1124,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A chord AB of length 6 passes through the left focus F<sub>1</sub> of the hyperbola $$\\frac {x^{2}}{16}- \\frac {y^{2}}{9}=1$$. Find the perimeter of the triangle ABF<sub>2</sub> (where F<sub>2</sub> is the right focus)."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 1125,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A chord PQ of the left branch of the hyperbola $x^2 - y^2 = 4$ passes through its left focus $F_1$, and the length of $|PQ|$ is 7. If $F_2$ is the right focus of the hyperbola, then the perimeter of $\\triangle PF_2Q$ is."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 1126,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A chord intercepted on the circle $x^{2}+y^{2}=4$ by the line $\\begin{cases} x=2-\\frac{1}{2}t \\\\ y=-1+\\frac{1}{2}t \\end{cases} (t\\text{ is the parameter})$ has a length of $\\_\\_\\_\\_\\_\\_\\_.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{14}"
} | {
"index": 1127,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A chord is drawn on a circle by choosing two points uniformly at random along its circumference. This is done two more times to obtain three total random chords. The circle is cut along these three lines, splitting it into pieces. The probability that one of the pieces is a tri... | math | {
"style": "rule",
"ground_truth": "115"
} | {
"index": 1128,
"solution": "Instead of choosing three random chords, we instead first choose 6 random points on the circle and then choosing a random pairing of the points into 3 pairs with which to form chords. If the chords form a triangle, take a chord $C$. Any other chord $C^{\\prime}$ must have its endpoints... |
DeepScaleR | [
{
"role": "user",
"content": "A chord of a circle is perpendicular to a radius at the midpoint of the radius. The ratio of the area of the larger of the two regions into which the chord divides the circle to the smaller can be expressed in the form $\\frac{a\\pi+b\\sqrt{c}}{d\\pi-e\\sqrt{f}},$ where $a, b, ... | math | {
"style": "rule",
"ground_truth": "592"
} | {
"index": 1129,
"solution": "Let $r$ be the length of the radius of the circle. A right triangle is formed by half of the chord, half of the radius (since the chord bisects it), and the radius. Thus, it is a $30^\\circ$ - $60^\\circ$ - $90^\\circ$ triangle, and the area of two such triangles is $2 \\cdot \\frac{1}... |
DeepScaleR | [
{
"role": "user",
"content": "A chord of length √3 divides a circle of radius 1 into two arcs. R is the region bounded by the chord and the shorter arc. What is the largest area of a rectangle that can be drawn in R?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"index": 1130,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A chord which is the perpendicular bisector of a radius of length 12 in a circle, has length"
}
] | math | {
"style": "rule",
"ground_truth": "12\\sqrt{3}"
} | {
"index": 1131,
"solution": "1. **Draw the Diagram**: Consider a circle with center \\( O \\) and a radius \\( OA \\) of length 12. Let \\( CD \\) be the chord that is the perpendicular bisector of \\( OA \\). Since \\( CD \\) bisects \\( OA \\) perpendicularly, \\( OC \\) and \\( OD \\) are both equal to half of ... |
DeepScaleR | [
{
"role": "user",
"content": "A chunk fell out of a dictionary. The first page of the chunk has the number 213, and the number of the last page is written using the same digits in a different order. How many pages are in the chunk that fell out?"
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 1132,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A church rings its bells every 15 minutes, the school rings its bells every 20 minutes and the day care center rings its bells every 25 minutes. If they all ring their bells at noon on the same day, at what time will they next all ring their bells together? (Answer in the form... | math | {
"style": "rule",
"ground_truth": "05\\!:\\!00"
} | {
"index": 1133,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cinema has 21 rows of seats, with 26 seats in each row. How many seats are there in total in this cinema?"
}
] | math | {
"style": "rule",
"ground_truth": "546"
} | {
"index": 1134,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cinema is showing four animated movies: Toy Story, Ice Age, Shrek, and Monkey King. The ticket prices are 50 yuan, 55 yuan, 60 yuan, and 65 yuan, respectively. Each viewer watches at least one movie and at most two movies. However, due to time constraints, viewers cannot watc... | math | {
"style": "rule",
"ground_truth": "1792"
} | {
"index": 1135,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle $C$ is defined by the equation $x^2 + y^2 = 1$. After the transformation $\\begin{cases} x' = 2x \\\\ y' = \\sqrt{2}y \\end{cases}$, we obtain the curve $C_1$. Establish a polar coordinate system with the coordinate origin as the pole and the positive half of the $x$-a... | math | {
"style": "rule",
"ground_truth": "\\frac{12\\sqrt{2}}{5}"
} | {
"index": 1136,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle $U$ has a circumference of $18\\pi$ inches, and segment $AB$ is a diameter. If the measure of angle $UAV$ is $45^{\\circ}$, what is the length, in inches, of segment $AV$?"
}
] | math | {
"style": "rule",
"ground_truth": "9\\sqrt{2 - \\sqrt{2}}"
} | {
"index": 1137,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle $\\Gamma$ with center $O$ has radius 1. Consider pairs $(A, B)$ of points so that $A$ is inside the circle and $B$ is on its boundary. The circumcircle $\\Omega$ of $O A B$ intersects $\\Gamma$ again at $C \\neq B$, and line $A C$ intersects $\\Gamma$ again at $X \\neq... | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\pi}{4}"
} | {
"index": 1138,
"solution": "We claim that $(A, B)$ is techy if and only if $O A=A B$. Note that $O X$ is tangent to the circle $(O B C)$ if and only if $O X$ is perpendicular to the angle bisector of $\\angle B O C$, since $O B=O C$. Thus $(A, B)$ is techy if and only if $O X$ is parallel to $B C$. Now since $O C... |
DeepScaleR | [
{
"role": "user",
"content": "A circle $\\omega$ is circumscribed around the triangle $ABC$. A circle $\\omega_{1}$ is tangent to the line $AB$ at point $A$ and passes through point $C$, and a circle $\\omega_{2}$ is tangent to the line $AC$ at point $A$ and passes through point $B$. A tangent to circle $\\... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 1139,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle $\\omega_{1}$ of radius 15 intersects a circle $\\omega_{2}$ of radius 13 at points $P$ and $Q$. Point $A$ is on line $P Q$ such that $P$ is between $A$ and $Q$. $R$ and $S$ are the points of tangency from $A$ to $\\omega_{1}$ and $\\omega_{2}$, respectively, such that... | math | {
"style": "rule",
"ground_truth": "14+\\sqrt{97}"
} | {
"index": 1140,
"solution": "Let their point of intersection be $X$. Using the Pythagorean theorem, the fact that $P Q=24$, and our knowledge of the radii of the circles, we can compute that $O_{1} X=9$ and $O_{2} X=5$, so $O_{1} O_{2}=14$. Let $S O_{1}$ and $R O_{2}$ meet at $Y$. Then $S A R Y$ is a square, say o... |
DeepScaleR | [
{
"role": "user",
"content": "A circle \\( K \\) goes through the vertices \\( A \\) and \\( C \\) of triangle \\( ABC \\). The center of circle \\( K \\) lies on the circumcircle of triangle \\( ABC \\). Circle \\( K \\) intersects side \\( AB \\) at point \\( M \\). Find the angle \\( BAC \\) if \\( AM : ... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 1141,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle and two distinct lines are drawn on a sheet of paper. What is the largest possible number of points of intersection of these figures?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 1142,
"solution": "To determine the largest possible number of points of intersection involving a circle and two distinct lines, we need to consider the maximum number of intersections each pair of figures can have:\n\n1. **Intersections between a line and a circle**: A line can intersect a circle at mos... |
DeepScaleR | [
{
"role": "user",
"content": "A circle can be circumscribed around the quadrilateral $ABCD$. Additionally, $AB = 3$, $BC = 4$, $CD = 5$, and $AD = 2$. \n\nFind $AC$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\frac{299}{11}}"
} | {
"index": 1143,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle can be inscribed in a trapezoid with bases of lengths 3 and 5, and a circle can also be circumscribed about it. Calculate the area of the pentagon formed by the radii of the inscribed circle that are perpendicular to the legs of the trapezoid, its smaller base, and the... | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{15}}{2}"
} | {
"index": 1144,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle centered at $A$ with a radius of $1$ and a circle centered at $B$ with a radius of $4$ are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. What is the radius of the third circle? [asy]\ndraw((-3,0)--... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"index": 1145,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle centered at $O$ has radius 1 and contains the point $A$. Segment $AB$ is tangent to the circle at $A$ and $\\angle\nAOB=\\theta$. If point $C$ lies on $\\overline{OA}$ and $\\overline{BC}$ bisects $\\angle ABO$, then express $OC$ in terms of $s$ and $c,$ where $s = \\s... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{1 + s}"
} | {
"index": 1146,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle centered at $O$ is circumscribed about $\\triangle ABC$ as follows: [asy]\npair pA, pB, pC, pO;\npO = (0, 0);\npA = pO + dir(-20);\npB = pO + dir(90);\npC = pO + dir(190);\ndraw(pA--pB--pC--pA);\ndraw(pO--pA);\ndraw(pO--pB);\ndraw(pO--pC);\nlabel(\"$O$\", pO, S);\nlabe... | math | {
"style": "rule",
"ground_truth": "50^\\circ"
} | {
"index": 1147,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle constructed on the side $AD$ of the parallelogram $ABCD$ as its diameter passes through the midpoint of diagonal $AC$ and intersects side $AB$ at point $M$. Find the ratio $AM: AB$ if $AC = 3BD$."
}
] | math | {
"style": "rule",
"ground_truth": "4/5"
} | {
"index": 1148,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle has 2017 distinct points $A_{1}, \\ldots, A_{2017}$ marked on it, and all possible chords connecting pairs of these points are drawn. A line is drawn through the point $A_{1}$, which does not pass through any of the points $A_{2}, \\ldots A_{2017}$. Find the maximum po... | math | {
"style": "rule",
"ground_truth": "1018080"
} | {
"index": 1149,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle has a radius of $\\log_{10}{(a^2)}$ and a circumference of $\\log_{10}{(b^4)}$. What is $\\log_{a}{b}$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\pi"
} | {
"index": 1150,
"solution": "1. **Identify the given values:** \n The radius of the circle is given as $r = \\log_{10}(a^2)$ and the circumference is given as $C = \\log_{10}(b^4)$.\n\n2. **Use logarithmic properties to simplify expressions:** \n Using the power rule of logarithms, we can simplify these expr... |
DeepScaleR | [
{
"role": "user",
"content": "A circle has a radius of 15 units. Suppose a chord in this circle bisects a radius perpendicular to the chord and the distance from the center of the circle to the chord is 9 units. What is the area of the smaller segment cut off by the chord?\nA) $100 \\pi$\nB) $117.29$\nC) $1... | math | {
"style": "rule",
"ground_truth": "117.29"
} | {
"index": 1151,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle has a radius of 3 units. A line segment of length 3 units is tangent to the circle at its midpoint. Determine the area of the region consisting of all such line segments.\nA) $1.5\\pi$\nB) $2.25\\pi$\nC) $3\\pi$\nD) $4.5\\pi$"
}
] | math | {
"style": "rule",
"ground_truth": "2.25\\pi"
} | {
"index": 1152,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle has a radius of 3 units. There are many line segments of length 4 units that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments.\nA) $3\\pi$\nB) $5\\pi$\nC) $4\\pi$\nD) $7\\pi$\nE) $6\\pi$"
}
] | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"index": 1153,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle has a radius of 6. What is the area of the smallest square that can entirely contain this circle, and what is the circumference of the circle?"
}
] | math | {
"style": "rule",
"ground_truth": "12\\pi"
} | {
"index": 1154,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle has a radius of six inches. Two parallel chords $CD$ and $EF$ are drawn such that the distance from the center of the circle to chord $CD$ is three inches, and the distance to chord $EF$ is four inches. How long is chord $CD$ and $EF$? Express your answer in the simple... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{5}"
} | {
"index": 1155,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle has a radius of three inches. The distance from the center of the circle to chord $CD$ is two inches. How many inches long is chord $CD$? Express your answer in simplest radical form."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"index": 1156,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle has an area of $16\\pi$ square units. What are the lengths of the circle's diameter and circumference, in units?"
}
] | math | {
"style": "rule",
"ground_truth": "8\\pi"
} | {
"index": 1157,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle has an area of $M\\text{ cm}^2$ and a circumference of $N\\text{ cm}$. If $\\dfrac{M}{N}=20$, what is the radius of the circle, in cm?"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 1158,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle has an area of $\\pi$ square units. What is the length of the circle's diameter, in units?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 1159,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle has center $(-10, -4)$ and has radius $13$. Another circle has center $(3, 9)$ and radius $\\sqrt{65}$. The line passing through the two points of intersection of the two circles has equation $x+y=c$. What is $c$?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 1160,
"solution": "1. **Identify the equations of the circles**: \n The first circle has center $(-10, -4)$ and radius $13$. Its equation is:\n \\[\n (x+10)^2 + (y+4)^2 = 169.\n \\]\n The second circle has center $(3, 9)$ and radius $\\sqrt{65}$. Its equation is:\n \\[\n (x-3)^2 + (y-9)^2 =... |
DeepScaleR | [
{
"role": "user",
"content": "A circle has its center at $(2,0)$ with a radius of 2, and another circle has its center at $(5,0)$ with a radius of 1. A line is tangent to both circles in the first quadrant. The $y$-intercept of this line is closest to:"
}
] | math | {
"style": "rule",
"ground_truth": "$2 \\sqrt{2}$"
} | {
"index": 1161,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle has radius $52$ and center $O$ . Points $A$ is on the circle, and point $P$ on $\\overline{OA}$ satisfies $OP = 28$ . Point $Q$ is constructed such that $QA = QP = 15$ , and point $B$ is constructed on the circle so that $Q$ is on $\\overline{OB}$ .... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 1162,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle having center $(0,k)$, with $k>6$, is tangent to the lines $y=x$, $y=-x$ and $y=6$. What is the radius of this circle?"
}
] | math | {
"style": "rule",
"ground_truth": "6\\sqrt{2}+6"
} | {
"index": 1163,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle having radius $r_{1}$ centered at point $N$ is tangent to a circle of radius $r_{2}$ centered at $M$. Let $l$ and $j$ be the two common external tangent lines to the two circles. A circle centered at $P$ with radius $r_{2}$ is externally tangent to circle $N$ at the po... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 1164,
"solution": "Suppose the lines are parallel. Draw the other tangent line to $N$ and $P$ - since $M$ and $P$ have the same radius, it is tangent to all three circles. Let $j$ and $k$ meet circle $N$ at $A$ and $B$, respectively. Then by symmetry we see that $\\angle A N M=\\angle M N P=\\angle P N B... |
DeepScaleR | [
{
"role": "user",
"content": "A circle inscribed in a square has two chords as shown in a pair. It has radius 2, and $P$ bisects $T U$. The chords' intersection is where? Answer the question by giving the distance of the point of intersection from the center of the circle."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2} - 2"
} | {
"index": 1165,
"solution": "The point lies between $X$ and $Q$. Then $M N X Q$ is a parallelogram. For, $O B \\| N M$ by homothety at $C$ and $P M \\| N X$ because $M N X P$ is an isoceles trapezoid. It follows that $Q X=M N$. Considering that the center of the circle together with points $M, C$, and $N$ determin... |
DeepScaleR | [
{
"role": "user",
"content": "A circle inscribed in an isosceles trapezoid divides its lateral side into segments equal to 4 and 9. Find the area of the trapezoid."
}
] | math | {
"style": "rule",
"ground_truth": "156"
} | {
"index": 1166,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle inscribed in triangle \\( ABC \\) divides median \\( BM \\) into three equal parts. Find the ratio \\( BC: CA: AB \\)."
}
] | math | {
"style": "rule",
"ground_truth": "5:10:13"
} | {
"index": 1167,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle inscribed in triangle \\( ABC \\) touches side \\( AB \\) at point \\( M \\), and \\( AM = 1 \\), \\( BM = 4 \\). Find \\( CM \\) given that \\( \\angle BAC = 120^\\circ \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{273}"
} | {
"index": 1168,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle intersects the $y$ -axis at two points $(0, a)$ and $(0, b)$ and is tangent to the line $x+100y = 100$ at $(100, 0)$ . Compute the sum of all possible values of $ab - a - b$ ."
}
] | math | {
"style": "rule",
"ground_truth": "10000"
} | {
"index": 1169,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle intersects the sides $AC$ and $CB$ of an isosceles triangle $ACB$ at points $P$ and $Q$ respectively, and is circumscribed around triangle $ABQ$. The segments $AQ$ and $BP$ intersect at point $D$ such that $AQ: AD = 4:3$. Find the area of triangle $DQB$ if the area of ... | math | {
"style": "rule",
"ground_truth": "9/2"
} | {
"index": 1170,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle is circumscribed about an equilateral triangle with side lengths of $12$ units each. Calculate the area of the circle, and then find the perimeter of the triangle."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 1171,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle is circumscribed about an equilateral triangle with side lengths of $9$ units each. What is the area of the circle, in square units? Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "27\\pi"
} | {
"index": 1172,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle is circumscribed around $ABCD$ as follows: [asy]\npair pA, pB, pC, pD, pO;\npO = (0, 0);\npA = pO + dir(-40);\npB = pO + dir(40);\npC = pO + dir(130);\npD = pO + dir(190);\ndraw(pA--pB--pC--pA);\ndraw(pA--pD--pC--pA);\nlabel(\"$A$\", pA, SE);\nlabel(\"$B$\", pB, NE);\n... | math | {
"style": "rule",
"ground_truth": "110^\\circ"
} | {
"index": 1173,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle is circumscribed around a unit square \\(ABCD\\), and a point \\(M\\) is selected on the circle.\n\nWhat is the maximum value that the product \\(MA \\cdot MB \\cdot MC \\cdot MD\\) can take?"
}
] | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"index": 1174,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle is circumscribed around an isosceles triangle whose two congruent angles have degree measure $x$. Two points are chosen independently and uniformly at random on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is $\\... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 1175,
"solution": "Because we know that we have an isosceles triangle with angles of $x$ (and we know that x is an inscribed angle), that means that the arc that is intercepted by this angle is $2x$. We form this same conclusion for the other angle $x$, and $180-2x$. Therefore we get $3$ arcs, namely, $2... |
DeepScaleR | [
{
"role": "user",
"content": "A circle is constructed on the side $BC$ of triangle $ABC$ as its diameter, and it intersects segment $AB$ at point $D$. Find the ratio of the areas of triangles $ABC$ and $BCD$, given that $AC = 15$, $BC = 20$, and $\\angle ABC = \\angle ACD$."
}
] | math | {
"style": "rule",
"ground_truth": "25/16"
} | {
"index": 1176,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle is divided into seven arcs such that the sum of any two adjacent arcs does not exceed $103^\\circ$. Determine the largest possible value of $A$ such that, in any such division, each of the seven arcs contains at least $A^\\circ$."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"index": 1177,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle is divided into six equal sections. Each section is to be coloured with a single colour so that three sections are red, one is blue, one is green, and one is yellow. Two circles have the same colouring if one can be rotated to match the other. How many different colour... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 1178,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle is divided into two segments by a chord equal to the side of a regular inscribed triangle. Determine the ratio of the areas of these segments."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4\\pi - 3\\sqrt{3}}{8\\pi + 3\\sqrt{3}}"
} | {
"index": 1179,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smaller circle to the area of the larger square?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{8}"
} | {
"index": 1180,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle is inscribed in a circular sector with an arc that measures $60^{\\circ}$. Find the ratio of the area of this circle to the area of the sector."
}
] | math | {
"style": "rule",
"ground_truth": "2/3"
} | {
"index": 1181,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle is inscribed in a convex quadrilateral \\(ABCD\\) with its center at point \\(O\\), and \\(AO=OC\\). Additionally, \\(BC=5\\), \\(CD=12\\), and \\(\\angle DAB\\) is a right angle.\n\nFind the area of the quadrilateral \\(ABCD\\)."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 1182,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle is inscribed in a regular hexagon. A smaller hexagon has two non-adjacent sides coinciding with the sides of the larger hexagon and the remaining vertices touching the circle. What percentage of the area of the larger hexagon is the area of the smaller hexagon? Assume ... | math | {
"style": "rule",
"ground_truth": "25\\%"
} | {
"index": 1183,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle is inscribed in a right triangle. The point of tangency divides the hypotenuse into two segments measuring 6 cm and 7 cm. Calculate the area of the triangle."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 1184,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle is inscribed in a right triangle. The point of tangency divides the hypotenuse into two segments of lengths $6 \\mathrm{~cm}$ and $7 \\mathrm{~cm}$. Calculate the area of the triangle."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 1185,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle is inscribed in a square, and within this circle, a smaller square is inscribed such that one of its sides coincides with a side of the larger square and two vertices lie on the circle. Calculate the percentage of the area of the larger square that is covered by the sm... | math | {
"style": "rule",
"ground_truth": "50\\%"
} | {
"index": 1186,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle is inscribed in a square, then a square is inscribed in this circle. Following this, a regular hexagon is inscribed in the smaller circle and finally, a circle is inscribed in this hexagon. What is the ratio of the area of the smallest circle to the area of the origina... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\pi}{32}"
} | {
"index": 1187,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle is inscribed in a triangle with side lengths $8, 13$, and $17$. Let the segments of the side of length $8$, made by a point of tangency, be $r$ and $s$, with $r<s$. What is the ratio $r:s$?"
}
] | math | {
"style": "rule",
"ground_truth": "1:3"
} | {
"index": 1188,
"solution": "1. **Label the Triangle and Points of Tangency**: Consider triangle $ABC$ with $AB = 17$, $AC = 13$, and $BC = 8$. Let $L$, $J$, and $K$ be the points of tangency on sides $BC$, $AC$, and $AB$, respectively.\n\n2. **Assign Variables to Segments**: Let $AK = x$. By the property of tange... |
DeepScaleR | [
{
"role": "user",
"content": "A circle is inscribed in a triangle, and the points of tangency with the sides of the triangle are connected to each other. A new circle is inscribed in the resulting triangle, and the points of tangency with its sides are the vertices of a third triangle, which has the same an... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 1189,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle is inscribed in quadrilateral $ABCD$, tangent to $\\overline{AB}$ at $P$ and to $\\overline{CD}$ at $Q$. Given that $AP=19$, $PB=26$, $CQ=37$, and $QD=23$, find the square of the radius of the circle."
}
] | math | {
"style": "rule",
"ground_truth": "647"
} | {
"index": 1190,
"solution": "Just use the area formula for tangential quadrilaterals. The numbers are really big. A terrible problem to work on ($a, b, c,$ and $d$ are the tangent lengths, not the side lengths). \\[A = \\sqrt{(a+b+c+d)(abc+bcd+cda+dab)} = 105\\sqrt{647}\\] $r^2=\\frac{A^2}{a+b+c+d} = \\boxed{647}$... |
DeepScaleR | [
{
"role": "user",
"content": "A circle is inscribed in quadrilateral $EFGH$, tangent to $\\overline{EF}$ at $R$ and to $\\overline{GH}$ at $S$. Given that $ER=24$, $RF=31$, $GS=40$, and $SH=29$, find the square of the radius of the circle."
}
] | math | {
"style": "rule",
"ground_truth": "945"
} | {
"index": 1191,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle is inscribed in quadrilateral $EFGH$, tangent to $\\overline{EF}$ at $R$ and to $\\overline{GH}$ at $S$. Given that $ER=25$, $RF=35$, $GS=40$, and $SH=20$, and that the circle is also tangent to $\\overline{EH}$ at $T$ with $ET=45$, find the square of the radius of the... | math | {
"style": "rule",
"ground_truth": "3600"
} | {
"index": 1192,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle is inscribed in quadrilateral \\( ABCD \\), tangent to \\( \\overline{AB} \\) at \\( P \\) and to \\( \\overline{CD} \\) at \\( Q \\). Given that \\( AP=15 \\), \\( PB=35 \\), \\( CQ=45 \\), and \\( QD=25 \\), find the square of the radius of the circle."
}
] | math | {
"style": "rule",
"ground_truth": "160"
} | {
"index": 1193,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle is inscribed in trapezoid \\( PQRS \\).\n\nIf \\( PS = QR = 25 \\) cm, \\( PQ = 18 \\) cm, and \\( SR = 32 \\) cm, what is the length of the diameter of the circle?"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 1194,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle is tangent to both branches of the hyperbola $x^{2}-20y^{2}=24$ as well as the $x$-axis. Compute the area of this circle."
}
] | math | {
"style": "rule",
"ground_truth": "504\\pi"
} | {
"index": 1195,
"solution": "Invert about the unit circle centered at the origin. $\\omega$ turns into a horizontal line, and the hyperbola turns into the following: $$\\begin{aligned} \\frac{x^{2}}{\\left(x^{2}+y^{2}\\right)^{2}}-\\frac{20y^{2}}{\\left(x^{2}+y^{2}\\right)^{2}}=24 & \\Longrightarrow x^{2}-20y^{2}=... |
DeepScaleR | [
{
"role": "user",
"content": "A circle is tangent to sides \\( AB \\) and \\( AD \\) of rectangle \\( ABCD \\) and intersects side \\( DC \\) at a single point \\( F \\) and side \\( BC \\) at a single point \\( E \\).\n\nFind the area of trapezoid \\( AFCB \\) if \\( AB = 32 \\), \\( AD = 40 \\), and \\( B... | math | {
"style": "rule",
"ground_truth": "1180"
} | {
"index": 1196,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle is tangent to the extensions of two sides \\(AB\\) and \\(AD\\) of a square \\(ABCD\\), and the point of tangency cuts off a segment of length \\(6 - 2\\sqrt{5}\\) cm from vertex \\(A\\). Two tangents are drawn to this circle from point \\(C\\). Find the side length of... | math | {
"style": "rule",
"ground_truth": "(\\sqrt{5} - 1)(2\\sqrt{2} - \\sqrt{5} + 1)"
} | {
"index": 1197,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle is tangent to the lines $4x - 3y = 30$ and $4x - 3y = -10.$ The center of the circle lies on the line $2x + y = 0.$ Find the center of the circle."
}
] | math | {
"style": "rule",
"ground_truth": "(1,-2)"
} | {
"index": 1198,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle is tangent to the sides of an angle at points $A$ and $B$. The distance from a point $C$ on the circle to the line $AB$ is 6. Find the sum of the distances from point $C$ to the sides of the angle, given that one of these distances is nine times smaller than the other.... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 1199,
"solution": ""
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.