data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a King and the second card is a $\\heartsuit$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{52}"
} | {
"index": 36200,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a Queen and the second card is a $\\diamondsuit$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{289}{15068}"
} | {
"index": 36201,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace and the second card is a $\\spadesuit$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{52}"
} | {
"index": 36202,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace and the second card is a King?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{4}{663}"
} | {
"index": 36203,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cards are dealt at random from two standard decks of 104 cards mixed together. What is the probability that the first card drawn is an ace and the second card drawn is also an ace?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{7}{1339}"
} | {
"index": 36204,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cards are dealt from a deck of four red cards labeled $A$, $B$, $C$, $D$ and four green cards labeled $A$, $B$, $C$, $D$. A winning pair is two of the same color or two of the same letter. What is the probability of drawing a winning pair?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{7}"
} | {
"index": 36205,
"solution": "To solve this problem, we need to calculate the probability of drawing a winning pair, which is defined as either two cards of the same color or two cards of the same letter. We will use casework to consider these two scenarios separately and then combine the probabilities.\n\n#### To... |
DeepScaleR | [
{
"role": "user",
"content": "Two cards are dealt from a standard deck of 52 cards. What is the probability that the first card dealt is a $\\diamondsuit$ and the second card dealt is a $\\spadesuit$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{204}"
} | {
"index": 36206,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cards are dealt from a standard deck of 52 cards. What is the probability that the first card dealt is a $\\clubsuit$ and the second card dealt is a $\\heartsuit$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{204}"
} | {
"index": 36207,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cards are dealt from a standard deck of 52 cards. What is the probability that the first card dealt is a $\\heartsuit$ and the second card dealt is a $\\clubsuit$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{204}"
} | {
"index": 36208,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cards are dealt from a standard deck of 52 cards. What is the probability that the first card dealt is a $\\heartsuit$ and the second card dealt is a face card $\\clubsuit$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{204}"
} | {
"index": 36209,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cards are placed in each of three different envelopes, and cards 1 and 2 are placed in the same envelope. Calculate the total number of different arrangements."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 36210,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cars covered the same distance. The speed of the first car was constant and three times less than the initial speed of the second car. The second car traveled the first half of the journey without changing speed, then its speed was suddenly halved, then traveled with consta... | math | {
"style": "rule",
"ground_truth": "5/3"
} | {
"index": 36211,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cars start from the same location at the same time, moving in the same direction at a constant speed. Each car can carry a maximum of 24 barrels of gasoline, and each barrel of gasoline allows a car to travel 60km. Both cars must return to the starting point, but they do no... | math | {
"style": "rule",
"ground_truth": "360"
} | {
"index": 36212,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cars travel along a circular track $n$ miles long, starting at the same point. One car travels $25$ miles along the track in some direction. The other car travels $3$ miles along the track in some direction. Then the two cars are once again at the same point along the... | math | {
"style": "rule",
"ground_truth": "89"
} | {
"index": 36213,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cars, $A$ and $B$, depart from one city to another. In the first 5 minutes, they traveled the same distance. Then, due to an engine failure, $B$ had to reduce its speed to 2/5 of its original speed, and thus arrived at the destination 15 minutes after car $A$, which continu... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 36214,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cars, A and B, start from points A and B respectively and travel towards each other at the same time. They meet at point C after 6 hours. If car A maintains its speed and car B increases its speed by 5 km/h, they will meet 12 km away from point C. If car B maintains its spe... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 36215,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cars, Car A and Car B, start from point A and point B, respectively, and move towards each other simultaneously. They meet after 3 hours, after which Car A turns back towards point A and Car B continues moving forward. Once Car A reaches point A, it turns towards point B ag... | math | {
"style": "rule",
"ground_truth": "432"
} | {
"index": 36216,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cars, Car A and Car B, start simultaneously from points A and B, respectively, and move towards each other. After 6 hours, the distance they have traveled is 3/5 of the distance between points A and B. Car A travels at a speed of 42 km per hour, which is 1/7 less than Car B... | math | {
"style": "rule",
"ground_truth": "780"
} | {
"index": 36217,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cars, Car A and Car B, travel towards each other from cities A and B, which are 330 kilometers apart. Car A starts from city A first. After some time, Car B starts from city B. The speed of Car A is $\\frac{5}{6}$ of the speed of Car B. When the two cars meet, Car A has tra... | math | {
"style": "rule",
"ground_truth": "55"
} | {
"index": 36218,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cells in a \\(20 \\times 20\\) board are adjacent if they have a common edge (a cell is not considered adjacent to itself). What is the maximum number of cells that can be marked in a \\(20 \\times 20\\) board such that every cell is adjacent to at most one marked cell?"
... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 36219,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two chess players, A and B, are in the midst of a match. Player A needs to win 2 more games to be the final winner, while player B needs to win 3 more games. If each player has a probability of $\\frac{1}{2}$ to win any given game, then calculate the probability of player A bec... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{16}"
} | {
"index": 36220,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two children at a time can play pairball. For $90$ minutes, with only two children playing at time, five children take turns so that each one plays the same amount of time. The number of minutes each child plays is"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 36221,
"solution": "1. **Calculate Total Playing Time**: Each game involves two children playing at the same time. The total duration for which the game is played is $90$ minutes. However, since there are two children playing simultaneously, the total playing time in terms of \"child-minutes\" is doubled... |
DeepScaleR | [
{
"role": "user",
"content": "Two chords \\( PQ \\) and \\( PR \\) are drawn in a circle with diameter \\( PS \\). The point \\( T \\) lies on \\( PR \\) and \\( QT \\) is perpendicular to \\( PR \\). The angle \\(\\angle QPR = 60^\\circ\\), \\( PQ = 24 \\text{ cm} \\), and \\( RT = 3 \\text{ cm} \\). What ... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 36222,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two chords \\(AB\\) and \\(CD\\) of a circle with center \\(O\\) each have a length of 10. The extensions of segments \\(BA\\) and \\(CD\\) beyond points \\(A\\) and \\(D\\) respectively intersect at point \\(P\\), with \\(DP = 3\\). The line \\(PO\\) intersects segment \\(AC\\... | math | {
"style": "rule",
"ground_truth": "3/13"
} | {
"index": 36223,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two chords, $AB$ and $CD,$ meet inside a circle at $P.$ If $AP = 3$ and $CP = 8,$ then what is $\\frac{BP}{DP}$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}"
} | {
"index": 36224,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles $\\Gamma_{1}$ and $\\Gamma_{2}$ of radius 1 and 2, respectively, are centered at the origin. A particle is placed at $(2,0)$ and is shot towards $\\Gamma_{1}$. When it reaches $\\Gamma_{1}$, it bounces off the circumference and heads back towards $\\Gamma_{2}$. The ... | math | {
"style": "rule",
"ground_truth": "403"
} | {
"index": 36225,
"solution": "By symmetry, the particle must bounce off of $\\Gamma_{2}$ at points that make angles of $60^{\\circ}, 120^{\\circ}, 180^{\\circ}, 240^{\\circ}$, and $300^{\\circ}$ with the positive $x$-axis. Similarly, the particle must bounce off of $\\Gamma_{1}$ at points that make angles of $30^{... |
DeepScaleR | [
{
"role": "user",
"content": "Two circles \\( C_{1} \\) and \\( C_{2} \\) have their centers at the point \\( (3, 4) \\) and touch a third circle, \\( C_{3} \\). The center of \\( C_{3} \\) is at the point \\( (0, 0) \\) and its radius is 2. Find the sum of the radii of the two circles \\( C_{1} \\) and \\(... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 36226,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles \\( C_{1} \\) and \\( C_{2} \\) have their centers at the point \\( (3, 4) \\) and touch a third circle, \\( C_{3} \\). The center of \\( C_{3} \\) is at the point \\( (0, 0) \\) and its radius is 2. What is the sum of the radii of the two circles \\( C_{1} \\) and ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 36227,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles \\( C_{1} \\) and \\( C_{2} \\) touch each other externally and the line \\( l \\) is a common tangent. The line \\( m \\) is parallel to \\( l \\) and touches the two circles \\( C_{1} \\) and \\( C_{3} \\). The three circles are mutually tangent. If the radius of ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 36228,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles \\(C_{1}\\) and \\(C_{2}\\) touch each other externally and the line \\(l\\) is a common tangent. The line \\(m\\) is parallel to \\(l\\) and touches the two circles \\(C_{1}\\) and \\(C_{3}\\). The three circles are mutually tangent. If the radius of \\(C_{2}\\) is... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 36229,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles are centered at \\( (5,5) \\) and \\( (25,15) \\) respectively, each tangent to the \\( y \\)-axis. Find the distance between the closest points of these two circles."
}
] | math | {
"style": "rule",
"ground_truth": "10\\sqrt{5} - 20"
} | {
"index": 36230,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles are centered at the origin, as shown. The point $P(8,6)$ is on the larger circle and the point $S(0,k)$ is on the smaller circle. If $QR=3$, what is the value of $k$?\n\n[asy]\nunitsize(0.2 cm);\n\ndefaultpen(linewidth(.7pt)+fontsize(10pt));\ndotfactor=4;\ndraw(Ci... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 36231,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles are centred at the origin. The point $P(8,6)$ is on the larger circle and the point $S(0, k)$ is on the smaller circle. If $Q R=3$, what is the value of $k$?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 36232,
"solution": "We can determine the distance from $O$ to $P$ by dropping a perpendicular from $P$ to $T$ on the $x$-axis. We have $O T=8$ and $P T=6$, so by the Pythagorean Theorem, $O P^{2}=O T^{2}+P T^{2}=8^{2}+6^{2}=64+36=100$. Since $O P>0$, then $O P=\\sqrt{100}=10$. Therefore, the radius of th... |
DeepScaleR | [
{
"role": "user",
"content": "Two circles are drawn in a 12-inch by 14-inch rectangle. Each circle has a diameter of 6 inches. If the circles do not extend beyond the rectangular region, what is the greatest possible distance (in inches) between the centers of the two circles?"
}
] | math | {
"style": "rule",
"ground_truth": "10\\text{ inches}"
} | {
"index": 36233,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles are externally tangent to each other at point \\( A \\), and both are tangent to a third circle at points \\( B \\) and \\( C \\). The extension of chord \\( AB \\) of the first circle intersects the second circle at point \\( D \\), and the extension of chord \\( A... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 36234,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles are externally tangent. Lines $\\overline{PAB}$ and $\\overline{PA'B'}$ are common tangents with points $A$, $A'$ on the smaller circle and $B$, $B'$ on the larger circle. If $PA=AB=5$ and the radius of the larger circle is 3 times the radius of the smaller circle, ... | math | {
"style": "rule",
"ground_truth": "5\\pi"
} | {
"index": 36235,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles are inscribed in an angle of 60 degrees and they touch each other. The radius of the smaller circle is 24. What is the radius of the larger circle?"
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 36236,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles centered at \\( O_{1} \\) and \\( O_{2} \\) have radii 2 and 3 and are externally tangent at \\( P \\). The common external tangent of the two circles intersects the line \\( O_{1} O_{2} \\) at \\( Q \\). What is the length of \\( PQ \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 36237,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles have centers at $(3,5)$ and $(20,15)$. Both circles are tangent to the x-axis. Determine the distance between the closest points of the two circles."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{389} - 20"
} | {
"index": 36238,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles have centers at (1,3) and (4,1) respectively. A line is tangent to the first circle at point (4,6) and to the second circle at point (7,4). Find the slope of the tangent line at these points."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 36239,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles have centers that are \\( d \\) units apart, and each has a diameter \\( \\sqrt{d} \\). For any \\( d \\), let \\( A(d) \\) be the area of the smallest circle that contains both of these circles. Find \\( \\lim _{d \\rightarrow \\infty} \\frac{A(d)}{d^{2}} \\)."
}... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 36240,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles have radii 13 and 30, and their centers are 41 units apart. The line through the centers of the two circles intersects the smaller circle at two points; let $A$ be the one outside the larger circle. Suppose $B$ is a point on the smaller circle and $C$ a point on the... | math | {
"style": "rule",
"ground_truth": "12 \\sqrt{13}"
} | {
"index": 36241,
"solution": "$12 \\sqrt{13}$ Call the large circle's center $O_{1}$. Scale the small circle by a factor of 2 about $A$; we obtain a new circle whose center $O_{2}$ is at a distance of $41-13=28$ from $O_{1}$, and whose radius is 26. Also, the dilation sends $B$ to $C$, which thus lies on circles $... |
DeepScaleR | [
{
"role": "user",
"content": "Two circles have radius $2$ and $3$ , and the distance between their centers is $10$ . Let $E$ be the intersection of their two common external tangents, and $I$ be the intersection of their two common internal tangents. Compute $EI$ .\n\n(A *common external tangent* i... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 36242,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles have radius 5 and 26. The smaller circle passes through center of the larger one. What is the difference between the lengths of the longest and shortest chords of the larger circle that are tangent to the smaller circle?\n\n*Ray Li.*"
}
] | math | {
"style": "rule",
"ground_truth": "52 - 2\\sqrt{235}"
} | {
"index": 36243,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles have the same center $C.$ (Circles which have the same center are called concentric.) The larger circle has radius $10$ and the smaller circle has radius $6.$ Determine the area of the ring between these two circles. [asy]\nimport graph;\nfilldraw(circle((0,0),10), ... | math | {
"style": "rule",
"ground_truth": "64\\pi"
} | {
"index": 36244,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles have the same center O. Point X is the midpoint of segment OP. What is the ratio of the area of the circle with radius OX to the area of the circle with radius OP? Express your answer as a common fraction.\n\n[asy]\nimport graph;\ndraw(Circle((0,0),20));\ndraw(Circl... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 36245,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles lie outside regular hexagon $ABCDEF$. The first is tangent to $\\overline{AB}$, and the second is tangent to $\\overline{DE}$. Both are tangent to lines $BC$ and $FA$. What is the ratio of the area of the second circle to that of the first circle?"
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"index": 36246,
"solution": "1. **Assumption and Setup**: Assume without loss of generality (WLOG) that the regular hexagon $ABCDEF$ has a side length of 1. The first circle is tangent to $\\overline{AB}$ and the second circle is tangent to $\\overline{DE}$. Both circles are tangent to lines $BC$ and $FA$.\n\n2. ... |
DeepScaleR | [
{
"role": "user",
"content": "Two circles of radii 4 and 5 are externally tangent to each other and are circumscribed by a third circle. Find the area of the shaded region created in this way. Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "40\\pi"
} | {
"index": 36247,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles of radius $r$ are externally tangent to each other and internally tangent to the ellipse $x^2 + 4y^2 = 5$. Find $r$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{15}}{4}"
} | {
"index": 36248,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles of radius $r$ are externally tangent to each other and internally tangent to the ellipse $x^2 + 4y^2 = 8$. Find the value of $r$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"index": 36249,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles of radius $r$ are externally tangent to each other and internally tangent to the ellipse $x^2 + 5y^2 = 6,$ as shown below. Find $r.$\n[asy]\nsize(7cm);\ndraw(scale(sqrt(6), sqrt(6)/sqrt(5))* unitcircle);\ndraw((0,-1.5)--(0,1.7),EndArrow);\ndraw((-3,0)--(3,0),EndArro... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt6}{5}"
} | {
"index": 36250,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles of radius $s$ are externally tangent to each other and internally tangent to the ellipse $x^2 + 4y^2 = 8.$ Find $s.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\frac{3}{2}}"
} | {
"index": 36251,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles of radius 1 are to be constructed as follows. The center of circle $A$ is chosen uniformly and at random from the line segment joining $(0,0)$ and $(2,0)$. The center of circle $B$ is chosen uniformly and at random, and independently of the first choice, from the li... | math | {
"style": "rule",
"ground_truth": "\\frac {4 \\sqrt {3} - 3}{4}"
} | {
"index": 36252,
"solution": "To solve this problem, we need to determine the probability that the distance between the centers of circles $A$ and $B$ is less than or equal to 2 (since each circle has a radius of 1, and two circles intersect if the distance between their centers is less than or equal to the sum of... |
DeepScaleR | [
{
"role": "user",
"content": "Two circles of radius 10 cm overlap such that each circle passes through the center of the other, as shown. How long, in cm, is the common chord (dotted segment) of the two circles? Express your answer in simplest radical form.\n\n[asy]\ndraw(Circle((0,0),10),linewidth(1));\ndr... | math | {
"style": "rule",
"ground_truth": "10\\sqrt3"
} | {
"index": 36253,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles of radius 2 are centered at $(2,0)$ and at $(0,2)$. What is the area of the intersection of the interiors of the two circles? Express your answer in fully expanded form in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\pi-4"
} | {
"index": 36254,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles of radius 3 and 4 are internally tangent to a larger circle. The larger circle circumscribes both the smaller circles. Find the area of the shaded region surrounding the two smaller circles within the larger circle. Express your answer in terms of \\(\\pi\\)."
}
] | math | {
"style": "rule",
"ground_truth": "24\\pi"
} | {
"index": 36255,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles of radius 3 are centered at $(0,3)$ and at $(3,0)$. What is the area of the intersection of the interiors of these two circles? Express your answer in terms of $\\pi$ in its simplest form."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9\\pi - 18}{2}"
} | {
"index": 36256,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. Determine the area of their overlapping interiors. Express your answer in expanded form in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}\\pi - 9"
} | {
"index": 36257,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of the two circles? Express your answer in fully expanded form in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9\\pi - 18}{2}"
} | {
"index": 36258,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of these two circles?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9\\pi - 18}{2}"
} | {
"index": 36259,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of these two circles? Express your answer in fully expanded form in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9\\pi}{2} - 9"
} | {
"index": 36260,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points $A$ and $B$, as shown in the diagram. The distance $AB$ can be written in the form $\\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integer... | math | {
"style": "rule",
"ground_truth": "69"
} | {
"index": 36261,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles of radius \\( r \\) are externally tangent to each other and internally tangent to the ellipse \\( x^2 + 4y^2 = 8 \\). Find \\( r \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"index": 36262,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles of radius \\( r \\) are externally tangent to each other and internally tangent to the ellipse defined by \\( x^2 + 4y^2 = 5 \\). The centers of the circles are located along the x-axis. Find the value of \\( r \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{15}}{4}"
} | {
"index": 36263,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles of radius \\( r \\) touch each other. Additionally, each of them is externally tangent to a third circle of radius \\( R \\) at points \\( A \\) and \\( B \\) respectively.\n\nFind the radius \\( r \\), given that \\( AB = 12 \\) and \\( R = 8 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 36264,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles that share the same center have radii $10$ meters and $20$ meters. An aardvark runs along the path shown, starting at $A$ and ending at $K$. How many meters does the aardvark run?"
}
] | math | {
"style": "rule",
"ground_truth": "20\\pi + 40"
} | {
"index": 36265,
"solution": "To solve this problem, we need to calculate the total distance the aardvark runs along different segments of the circles and straight lines. We assume the path is symmetric and consists of arcs and radial segments.\n\n1. **Arc of the larger circle (radius = 20 meters):** \n The aar... |
DeepScaleR | [
{
"role": "user",
"content": "Two circles touch each other and the sides of two adjacent angles, one of which is $60^{\\circ}$. Find the ratio of the radii of the circles."
}
] | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"index": 36266,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles touch each other at a common point $A$. Through point $B$, which lies on their common tangent passing through $A$, two secants are drawn. One secant intersects the first circle at points $P$ and $Q$, and the other secant intersects the second circle at points $M$ an... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 36267,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles touch each other internally at point K. Chord \\( AB \\) of the larger circle touches the smaller circle at point L, with \\( A L = 10 \\). Find \\( B L \\) given that \\( A K : B K = 2 : 5 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 36268,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles touch each other internally at point K. The chord \\(A B\\) of the larger circle is tangent to the smaller circle at point \\(L\\), and \\(A L = 10\\). Find \\(B L\\) if \\(A K: B K = 2: 5\\)."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 36269,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles touch in $M$ , and lie inside a rectangle $ABCD$ . One of them touches the sides $AB$ and $AD$ , and the other one touches $AD,BC,CD$ . The radius of the second circle is four times that of the first circle. Find the ratio in which the common tangent of the ci... | math | {
"style": "rule",
"ground_truth": "1:1"
} | {
"index": 36270,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles with a radius of 15 cm overlap such that each circle passes through the center of the other. Determine the length of the common chord (dotted segment) in centimeters between these two circles. Express your answer in simplest radical form."
}
] | math | {
"style": "rule",
"ground_truth": "15\\sqrt{3}"
} | {
"index": 36271,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles with centers $A$ and $B$ intersect at points $X$ and $Y$ . The minor arc $\\angle{XY}=120$ degrees with respect to circle $A$ , and $\\angle{XY}=60$ degrees with respect to circle $B$ . If $XY=2$ , \ffind the area shared by the two circles."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10\\pi - 12\\sqrt{3}}{9}"
} | {
"index": 36272,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles with centers $A$ and $B$ and radii 2 and 1, respectively, are tangent to each other. Point $C$ lies on a line that is tangent to each of the circles and is at a distance of $\\frac{3 \\sqrt{3}}{2 \\sqrt{2}}$ from the midpoint of segment $AB$. Find the area $S$ of tr... | math | {
"style": "rule",
"ground_truth": "\\frac{15 \\sqrt{2}}{8}"
} | {
"index": 36273,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles with centers \\( M \\) and \\( N \\), lying on the side \\( AB \\) of triangle \\( ABC \\), are tangent to each other and intersect the sides \\( AC \\) and \\( BC \\) at points \\( A, P \\) and \\( B, Q \\) respectively. Additionally, \\( AM = PM = 2 \\) and \\( BN... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 36274,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles with equal radii intersect as shown. The area of the shaded region equals the sum of the areas of the two unshaded regions. If the area of the shaded region is $216\\pi$, what is the circumference of each circle?"
}
] | math | {
"style": "rule",
"ground_truth": "36\\pi"
} | {
"index": 36275,
"solution": "Suppose that the radius of each of the circles is $r$. Since the two circles are identical, then the two circles have equal area. Since the shaded area is common to the two circles, then the unshaded pieces of each circle have equal areas. Since the combined area of the unshaded regio... |
DeepScaleR | [
{
"role": "user",
"content": "Two circles with radii $\\sqrt{5}$ and $\\sqrt{2}$ intersect at point $A$. The distance between the centers of the circles is 3. A line through point $A$ intersects the circles at points $B$ and $C$ such that $A B = A C$ (point $B$ does not coincide with $C$). Find $A B$."
}
... | math | {
"style": "rule",
"ground_truth": "\\frac{6\\sqrt{5}}{5}"
} | {
"index": 36276,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles with radii of 4 and 5 are externally tangent to each other and are both circumscribed by a third circle. Find the area of the shaded region outside these two smaller circles but within the larger circle. Express your answer in terms of $\\pi$. Assume the configurati... | math | {
"style": "rule",
"ground_truth": "40\\pi"
} | {
"index": 36277,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles with radius $2$ and radius $4$ have a common center at P. Points $A, B,$ and $C$ on the larger circle are the vertices of an equilateral triangle. Point $D$ is the intersection of the smaller circle and the line segment $PB$ . Find the square of the are... | math | {
"style": "rule",
"ground_truth": "192"
} | {
"index": 36278,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles, both with the same radius $r$ , are placed in the plane without intersecting each other. A line in the plane intersects the first circle at the points $A,B$ and the other at points $C,D$ , so that $|AB|=|BC|=|CD|=14\\text{cm}$ . Another line intersects the cir... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 36279,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles, circle $A$ with radius 2 and circle $B$ with radius 1.5, are to be constructed with the following process: The center of circle $A$ is chosen uniformly and at random from the line segment joining $(0,0)$ to $(3,0)$. The center of circle $B$ is chosen uniformly and ... | math | {
"style": "rule",
"ground_truth": "0.96"
} | {
"index": 36280,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles, each of radius $4$, are drawn with centers at $(0, 20)$, and $(6, 12)$. A line passing through $(4,0)$ is such that the total area of the parts of the two circles to one side of the line is equal to the total area of the parts of the two circles to the other side o... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 36281,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles, each with a radius of 12 cm, overlap such that each circle passes through the center of the other. Determine the length, in cm, of the common chord formed by these circles. Express your answer in simplest radical form."
}
] | math | {
"style": "rule",
"ground_truth": "12\\sqrt{3}"
} | {
"index": 36282,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles, one of radius 5 inches, the other of radius 2 inches, are tangent at point P. Two bugs start crawling at the same time from point P, one crawling along the larger circle at $3\\pi$ inches per minute, the other crawling along the smaller circle at $2.5\\pi$ inches p... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 36283,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circles, one with radius 4 and the other with radius 5, are externally tangent to each other and are circumscribed by a third circle. Calculate the area of the shaded region formed between these three circles. Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "40\\pi"
} | {
"index": 36284,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circular poles, with diameters of 8 inches and 24 inches, touch each other at a single point. A wire is wrapped around them such that it goes around the entire configuration including a straight section tangential to both poles. Find the length of the shortest wire that suf... | math | {
"style": "rule",
"ground_truth": "16\\sqrt{3} + 32\\pi"
} | {
"index": 36285,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two circular tracks $\\alpha$ and $\\beta$ of the same radius are tangent to each other. A car $A$ travels clockwise on track $\\alpha$ and a car $B$ travels counterclockwise on track $\\beta$. At the start, cars $A$ and $B$ are on the same line with the center of track $\\alph... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 36286,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two classmates, A and B, live in the same neighborhood and leave the neighborhood gate at the same time to go to school. Initially, A travels at a speed of 40 meters per minute, while B travels at a speed of 60 meters per minute. After A has walked half the distance, they reali... | math | {
"style": "rule",
"ground_truth": "960"
} | {
"index": 36287,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two companies, A and B, each donated 60,000 yuan to the \"See a Just Cause Foundation\". It is known that the average donation per person in company B is 40 yuan more than that in company A, and the number of people in company A is 20% more than that in company B. What are the ... | math | {
"style": "rule",
"ground_truth": "250"
} | {
"index": 36288,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two complementary angles are in a ratio of $3:2$. What is the measure, in degrees, of the smaller angle?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 36289,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two concentric circles $\\omega, \\Omega$ with radii $8,13$ are given. $AB$ is a diameter of $\\Omega$ and the tangent from $B$ to $\\omega$ touches $\\omega$ at $D$ . What is the length of $AD$ ."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 36290,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two concentric circles are centered at point P. The sides of a 45 degree angle at P form an arc on the smaller circle that is the same length as an arc on the larger circle formed by the sides of a 36 degree angle at P. What is the ratio of the area of the smaller circle to th... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{25}"
} | {
"index": 36291,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two concentric circles have radii $1$ and $4$ . Six congruent circles form a ring where each of the six circles is tangent to the two circles adjacent to it as shown. The three lightly shaded circles are internally tangent to the circle with radius $4$ while the three da... | math | {
"style": "rule",
"ground_truth": "126"
} | {
"index": 36292,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two concentric circles have radii $1$ and $2$. Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 36293,
"solution": "1. **Identify the Problem Setup**: We have two concentric circles with radii $1$ and $2$. We need to find the probability that a randomly chosen chord on the outer circle intersects the inner circle.\n\n2. **Choosing a Reference Point**: Let $O$ be the center of both circles. Choose a... |
DeepScaleR | [
{
"role": "user",
"content": "Two concentric circles have radii $r$ and $R>r$. Three new circles are drawn so that they are each tangent to the big two circles and tangent to the other two new circles. Find $\\frac{R}{r}$."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 36294,
"solution": "The centers of the three new circles form a triangle. The diameter of the new circles is $R-r$, so the side length of the triangle is $R-r$. Call the center of the concentric circle $O$, two vertices of the triangle $A$ and $B$, and $A B$ 's midpoint $D$. $O A$ is the average $R$ and ... |
DeepScaleR | [
{
"role": "user",
"content": "Two concentric circles have radii of $12$ meters and $24$ meters respectively. A rabbit starts at point X and runs along a sixth of the circumference of the larger circle, followed by a straight line to the smaller circle, then along a third of the circumference of the smaller ... | math | {
"style": "rule",
"ground_truth": "16\\pi + 36"
} | {
"index": 36295,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two concentric circles have radii of 15 meters and 30 meters. An aardvark starts at point $A$ on the smaller circle and runs along the path that includes half the circumference of each circle and each of the two straight segments that connect the circumferences directly (radial... | math | {
"style": "rule",
"ground_truth": "45\\pi + 30"
} | {
"index": 36296,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two concentric circles have radii of 24 and 36 units, respectively. A shaded region is formed between these two circles. A new circle is to be drawn such that its diameter is equal to the area of the shaded region. What must the diameter of this new circle be? Express your answ... | math | {
"style": "rule",
"ground_truth": "720 \\pi"
} | {
"index": 36297,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two concentric circles have the same center, labeled $C$. The larger circle has a radius of $12$ units while the smaller circle has a radius of $7$ units. Determine the area of the ring formed between these two circles and also calculate the circumference of the larger circle."... | math | {
"style": "rule",
"ground_truth": "24\\pi"
} | {
"index": 36298,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two concentric circles with radii of 19 and 29 units bound a shaded region. A third circle will be drawn with area equal to that of the shaded area. What must the radius of the third circle be? Express your answer in simplest radical form.\n\n[asy]\nfilldraw(circle((0,0),29),gr... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{30}"
} | {
"index": 36299,
"solution": ""
} |
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