data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Charlie is planning to construct a boundary around a rectangular playground using exactly 380 feet of fencing. Regulations specify that the length of the boundary must be at least 100 feet and the width must be at least 60 feet. Charlie wants to maximize the area enclosed by th... | math | {
"style": "rule",
"ground_truth": "9000"
} | {
"index": 6100,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Charlyn walks completely around the boundary of a square whose sides are each 5 km long. From any point on her path she can see exactly 1 km horizontally in all directions. What is the area of the region consisting of all points Charlyn can see during her walk, expressed in squ... | math | {
"style": "rule",
"ground_truth": "39"
} | {
"index": 6101,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cheburashka spent his money to buy as many mirrors from Galya's store as Gena bought from Shapoklyak's store. If Gena were buying from Galya, he would have 27 mirrors, and if Cheburashka were buying from Shapoklyak, he would have 3 mirrors. How many mirrors would Gena and Chebu... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 6102,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Chelsea goes to La Verde's at MIT and buys 100 coconuts, each weighing 4 pounds, and 100 honeydews, each weighing 5 pounds. She wants to distribute them among \\( n \\) bags, so that each bag contains at most 13 pounds of fruit. What is the minimum \\( n \\) for which this is p... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"index": 6103,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Chen, Ruan, Lu, Tao, and Yang did push-ups. It is known that Chen, Lu, and Yang together averaged 40 push-ups per person, Ruan, Tao, and Chen together averaged 28 push-ups per person, and Ruan, Lu, Tao, and Yang together averaged 33 push-ups per person. How many push-ups did Ch... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 6104,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Chess piece called *skew knight*, if placed on the black square, attacks all the gray squares. \n\n\n\nWhat is the largest number of such knights that can be placed on the $8\\times 8$ chessboard without them att... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 6105,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Chester is traveling from Hualien to Lugang, Changhua, to participate in the Hua Luogeng Golden Cup Mathematics Competition. Before setting off, his father checked the car's odometer, which read a palindromic number of 69,696 kilometers (a palindromic number remains the same wh... | math | {
"style": "rule",
"ground_truth": "82.2"
} | {
"index": 6106,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Chester traveled from Hualien to Lukang in Changhua to participate in the Hua Luogeng Gold Cup Math Competition. Before leaving, his father checked the car’s odometer, which displayed a palindromic number of 69,696 kilometers (a palindromic number reads the same forward and bac... | math | {
"style": "rule",
"ground_truth": "82.2"
} | {
"index": 6107,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Chewbacca has 20 pieces of cherry gum and 30 pieces of grape gum. Some of the pieces are in complete packs, while others are loose. Each complete pack has exactly $x$ pieces of gum. If Chewbacca loses one pack of cherry gum, then the ratio of the number of pieces of cherry gu... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 6108,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Chewbacca has 25 pieces of orange gum and 35 pieces of apple gum. Some of the pieces are in complete packs, while others are loose. Each complete pack has exactly $y$ pieces of gum. If Chewbacca loses two packs of orange gum, then the ratio of the number of pieces of orange gum... | math | {
"style": "rule",
"ground_truth": "\\frac{15}{4}"
} | {
"index": 6109,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Chicks hatch on the night from Sunday to Monday. For two weeks, a chick sits with its beak open, during the third week it silently grows feathers, and during the fourth week it flies out of the nest. Last week, there were 20 chicks in the nest sitting with their beaks open, and... | math | {
"style": "rule",
"ground_truth": "165"
} | {
"index": 6110,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Children get in for half the price of adults. The price for $8$ adult tickets and $7$ child tickets is $42$. If a group buys more than $10$ tickets, they get an additional $10\\%$ discount on the total price. Calculate the cost of $10$ adult tickets and $8$ child tickets."
}
... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"index": 6111,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "China Unicom charges for mobile phone calls with two types of packages: Package $A$ (monthly fee of $15$ yuan, call fee of $0.1 yuan per minute) and Package $B$ (monthly fee of $0$ yuan, call fee of $0.15 yuan per minute). Let $y_{1}$ represent the monthly bill for Package $A$ ... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"index": 6112,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "China has become the world's largest electric vehicle market. Electric vehicles have significant advantages over traditional vehicles in ensuring energy security and improving air quality. After comparing a certain electric vehicle with a certain fuel vehicle, it was found that... | math | {
"style": "rule",
"ground_truth": "5000"
} | {
"index": 6113,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "China was the first country in the world to use negative numbers. Li Heng, in the book \"Fa Jing\" written during the Warring States period, already used negative numbers. If the year 500 BC is written as $-500$ years, then the year 2024 AD should be written as ______ years."
... | math | {
"style": "rule",
"ground_truth": "+2024"
} | {
"index": 6114,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "China's space station has entered the formal construction phase. The Tianhe core module, Wentian experimental module, and Mengtian experimental module will all dock in 2022, forming a \"T\" shaped structure. During the construction phase of the Chinese space station, there are ... | math | {
"style": "rule",
"ground_truth": "450"
} | {
"index": 6115,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Chinese mathematician Hua Luogeng saw a brain teaser in a magazine that the passenger next to him was reading while on a trip abroad: find the cube root of $59319$. Hua Luogeng blurted out the answer, astonishing everyone. They quickly asked about the calculation's mystery. Do ... | math | {
"style": "rule",
"ground_truth": "58"
} | {
"index": 6116,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Chloe and Zoe are both students in Ms. Demeanor's math class. Last night, they each solved half of the problems in their homework assignment alone and then solved the other half together. Chloe had correct answers to only $80\\%$ of the problems she solved alone, but overall $8... | math | {
"style": "rule",
"ground_truth": "93"
} | {
"index": 6117,
"solution": "1. **Define Variables:**\n Let $t$ be the total number of problems in the homework assignment. Let $x$ be the number of problems that Chloe and Zoe solved correctly together.\n\n2. **Calculate Chloe's Correct Answers:**\n Chloe solved half of the problems alone and got $80\\%$ of t... |
DeepScaleR | [
{
"role": "user",
"content": "Chloe chooses a real number uniformly at random from the interval $[ 0,2017 ]$. Independently, Laurent chooses a real number uniformly at random from the interval $[ 0 , 4034 ]$. What is the probability that Laurent's number is greater than Chloe's number?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 6118,
"solution": "1. **Define the random variables:**\n Let $x$ be the number chosen by Chloe, uniformly at random from the interval $[0, 2017]$. Thus, $0 \\leq x \\leq 2017$.\n Let $y$ be the number chosen by Laurent, uniformly at random from the interval $[0, 4034]$. Thus, $0 \\leq y \\leq 4034$.\... |
DeepScaleR | [
{
"role": "user",
"content": "Chloe chooses a real number uniformly at random from the interval $[0, 2017]$. Independently, Laurent chooses a real number uniformly at random from the interval $[0, 4034]$. What is the probability that Laurent's number is greater than Chloe's number?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 6119,
"solution": "1. **Define the problem in terms of geometric probability**: Let $x$ represent the number chosen by Chloe and $y$ represent the number chosen by Laurent. We are interested in finding the probability that $y > x$.\n\n2. **Set up the coordinate system**: Consider a coordinate plane where... |
DeepScaleR | [
{
"role": "user",
"content": "Chloe chooses a real number uniformly at random from the interval $[0, 3000]$. Independently, Max chooses a real number uniformly at random from the interval $[0, 4500]$. Determine the probability that Max's number is at least twice as large as Chloe's number."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"index": 6120,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Chloe is baking muffins for a school event. If she divides the muffins equally among 8 of her friends, she'll have 3 muffins left over. If she divides the muffins equally among 5 of her friends, she'll have 2 muffins left over. Assuming Chloe made fewer than 60 muffins, what is... | math | {
"style": "rule",
"ground_truth": "118"
} | {
"index": 6121,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Chloe wants to buy a hoodie that costs $\\$32.75$. She empties her wallet and finds she only has three $\\$10$ bills, eight quarters, and a pile of dimes. What is the minimum number of dimes that must be in her pile so she can pay for the hoodie?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 6122,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Chloe wants to buy a jacket that costs $45.50$. She has two $20$ bills, five quarters, a few nickels, and a pile of dimes in her wallet. What is the minimum number of dimes she needs if she also has six nickels?"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 6123,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Chloe wants to purchase a jacket that costs $\\$45.50$. She checks her purse and finds she has four $\\$10$ bills, ten quarters, and some nickels and dimes. What is the minimum number of dimes that she must have if she has 15 nickels?"
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 6124,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Choose $3$ different numbers from the $5$ numbers $0$, $1$, $2$, $3$, $4$ to form a three-digit even number."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 6125,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Choose $4-4$: Parameter Equation Lecture. In the plane rectangular coordinate system $xOy$, with $O$ as the pole and the non-negative half-axis of $x$ as the polar axis, establish a polar coordinate system. The polar coordinates of point $P$ are $(2\\sqrt{3}, \\dfrac{\\pi}{6})$... | math | {
"style": "rule",
"ground_truth": "\\dfrac{11\\sqrt{5}}{10} - 1"
} | {
"index": 6126,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Choose $n$ numbers from the 2017 numbers $1, 2, \\cdots, 2017$ such that the difference between any two chosen numbers is a composite number. What is the maximum value of $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "505"
} | {
"index": 6127,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Choose any $2$ numbers from $-5$, $-3$, $-1$, $2$, and $4$. Let the maximum product obtained be denoted as $a$, and the minimum quotient obtained be denoted as $b$. Then the value of $\\frac{a}{b}$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{15}{4}"
} | {
"index": 6128,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Choose any four distinct digits $w, x, y, z$ and form the four-digit numbers $wxyz$ and $zyxw$. What is the greatest common divisor of the numbers of the form $wxyz + zyxw + wxyz \\cdot zyxw$?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 6129,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Choose one digit from 0, 2, 4, and two digits from 1, 3, 5 to form a three-digit number without repeating digits. The total number of different three-digit numbers that can be formed is ( )\n\nA 36 B 48 C 52 D 54"
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 6130,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Choose one digit from the set {0, 2} and two different digits from the set {1, 3, 5} to form a three-digit number without any repeating digits. The total number of such odd three-digit numbers is _________."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 6131,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Choose one of the following conditions from (1) $a\\sin \\left(B+C\\right)+c\\sin C-b\\sin B=2a\\sin C\\sin B$, (2) $\\frac{cosB}{cosC}+\\frac{b}{c-\\sqrt{2}a}=0$, (3) $2a^{2}=(a^{2}+b^{2}-c^{2})(1+\\tan C)$, and fill in the blank in the question below, and answer the correspon... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{2}}{5}"
} | {
"index": 6132,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Choose one of the following three conditions:①$a_{2}=60$, ②the sum of binomial coefficients is $64$, ③the maximum term of the binomial coefficients is the $4$th term. Fill in the blank below. Given ${(1-2x)}^{n}={a}_{0}+{a}_{1}x+{a}_{2}{x}^{2}+…+{a}_{n}{x}^{n}(n∈{N}_{+})$,_____... | math | {
"style": "rule",
"ground_truth": "63"
} | {
"index": 6133,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Choose one of the three conditions in $①$ $ac=\\sqrt{3}$, $②$ $c\\sin A=3$, $③$ $c=\\sqrt{3}b$, and supplement it in the following question. If the triangle in the question exists, find the value of $c$; if the triangle in the question does not exist, explain the reason.<br/>Qu... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 6134,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Choose one of the three conditions: ①$ac=\\sqrt{3}$, ②$c\\sin A=3$, ③$c=\\sqrt{3}b$, and supplement it in the following question. If the triangle in the question exists, find the value of $c$; if the triangle in the question does not exist, explain the reason.<br/>Question: Doe... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 6135,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Choose positive integers $b_1, b_2, \\dotsc$ satisfying\n\\[1=\\frac{b_1}{1^2} > \\frac{b_2}{2^2} > \\frac{b_3}{3^2} > \\frac{b_4}{4^2} > \\dotsb\\]\nand let $r$ denote the largest real number satisfying $\\tfrac{b_n}{n^2} \\geq r$ for all positive integers $n$. What are the po... | math | {
"style": "rule",
"ground_truth": "0 \\leq r \\leq \\frac{1}{2}"
} | {
"index": 6136,
"solution": "\nLet \\( r \\) denote the largest real number satisfying \\(\\frac{b_n}{n^2} \\geq r\\) for all positive integers \\( n \\), where \\( b_1, b_2, \\dotsc \\) are positive integers satisfying\n\\[\n1 = \\frac{b_1}{1^2} > \\frac{b_2}{2^2} > \\frac{b_3}{3^2} > \\frac{b_4}{4^2} > \\dotsb\n... |
DeepScaleR | [
{
"role": "user",
"content": "Choose six out of the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 to fill in the blanks below, so that the equation is true. Each blank is filled with a single digit, and no two digits are the same. \n\n$\\square + \\square \\square = \\square \\square \\square$. \n\nWhat is the la... | math | {
"style": "rule",
"ground_truth": "105"
} | {
"index": 6137,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Choose the largest of the following sums, and express it as a fraction in simplest form:\n$$\\frac{1}{4} + \\frac{1}{5}, \\ \\ \\frac{1}{4} + \\frac{1}{6}, \\ \\ \\frac{1}{4} + \\frac{1}{3}, \\ \\ \\frac{1}{4} + \\frac{1}{8}, \\ \\ \\frac{1}{4} + \\frac{1}{7}$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{12}"
} | {
"index": 6138,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Choose three digits from the odd numbers 1, 3, 5, 7, 9 and two digits from the even numbers 2, 4, 6, 8 to form a five-digit number with no repeating digits, such that the odd and even digits alternate. How many such five-digit numbers can be formed?"
}
] | math | {
"style": "rule",
"ground_truth": "720"
} | {
"index": 6139,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Choose two different numbers from the set of numbers {1, 2, ..., 8, 9}, and find the probability that their product is an odd number. (Express the result as a numerical value)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{18}"
} | {
"index": 6140,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Chords $A B$ and $C D$ of a circle are perpendicular and intersect at a point $P$. If $A P=6, B P=12$, and $C D=22$, find the area of the circle."
}
] | math | {
"style": "rule",
"ground_truth": "130 \\pi"
} | {
"index": 6141,
"solution": "Let $O$ be the center of the circle and let $M$ be the midpoint of segment $A B$ and let $N$ be the midpoint of segment $C D$. Since quadrilateral $O M P N$ is a rectangle we have that $O N=M P=A M-A P=3$ so $$O C=\\sqrt{O N^{2}+N C^{2}}=\\sqrt{9+121}=\\sqrt{130}$$ Hence the desired ar... |
DeepScaleR | [
{
"role": "user",
"content": "Chords $\\overline{A B}$ and $\\overline{C D}$ of circle $\\omega$ intersect at $E$ such that $A E=8, B E=2, C D=10$, and $\\angle A E C=90^{\\circ}$. Let $R$ be a rectangle inside $\\omega$ with sides parallel to $\\overline{A B}$ and $\\overline{C D}$, such that no point in t... | math | {
"style": "rule",
"ground_truth": "26+6 \\sqrt{17}"
} | {
"index": 6142,
"solution": "By power of a point, $(C E)(E D)=(A E)(E B)=16$, and $C E+E D=C D=$ 10. Thus $C E, E D$ are 2, 8. Without loss of generality, assume $C E=8$ and $D E=2$. Assume our circle is centered at the origin, with points $A=(-3,5), B=(-3,-5), C=(5,-3)$, $D=(-5,-3)$, and the equation of the circl... |
DeepScaleR | [
{
"role": "user",
"content": "Chords \\(AB\\) and \\(CD\\) of a circle with center \\(O\\) both have a length of 5. The extensions of segments \\(BA\\) and \\(CD\\) beyond points \\(A\\) and \\(D\\) intersect at point \\(P\\), where \\(DP=13\\). The line \\(PO\\) intersects segment \\(AC\\) at point \\(L\\)... | math | {
"style": "rule",
"ground_truth": "13/18"
} | {
"index": 6143,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Chris and Paul each rent a different room of a hotel from rooms $1-60$. However, the hotel manager mistakes them for one person and gives \"Chris Paul\" a room with Chris's and Paul's room concatenated. For example, if Chris had 15 and Paul had 9, \"Chris Paul\" has 159. If the... | math | {
"style": "rule",
"ground_truth": "\\frac{153}{1180}"
} | {
"index": 6144,
"solution": "There are $60 \\cdot 59=3540$ total possible outcomes, and we need to count the number of these which concatenate into a number at most 60. Of these, $9 \\cdot 8$ result from both Chris and Paul getting one-digit room numbers. If Chris gets a two-digit number, then he must get a number... |
DeepScaleR | [
{
"role": "user",
"content": "Chris received a mark of $50 \\%$ on a recent test. Chris answered 13 of the first 20 questions correctly. Chris also answered $25 \\%$ of the remaining questions on the test correctly. If each question on the test was worth one mark, how many questions in total were on the tes... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 6145,
"solution": "Suppose that there were $n$ questions on the test. Since Chris received a mark of $50 \\%$ on the test, then he answered $\\frac{1}{2} n$ of the questions correctly. We know that Chris answered 13 of the first 20 questions correctly and then $25 \\%$ of the remaining questions. Since t... |
DeepScaleR | [
{
"role": "user",
"content": "Christina draws a pair of concentric circles. She draws chords $\\overline{DE}$, $\\overline{EF}, \\ldots$ of the larger circle, each chord being tangent to the smaller circle. If $m\\angle DEF = 85^\\circ$, how many segments will she draw before returning to her starting point... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 6146,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Christine must buy at least $45$ fluid ounces of milk at the store. The store only sells milk in $200$ milliliter bottles. If there are $33.8$ fluid ounces in $1$ liter, then what is the smallest number of bottles that Christine could buy? (You may use a calculator on this pro... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 6147,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Chubby makes nonstandard checkerboards that have $31$ squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard?"
}
] | math | {
"style": "rule",
"ground_truth": "481"
} | {
"index": 6148,
"solution": "1. **Understanding the Checkerboard Pattern**: The checkerboard is $31 \\times 31$ with alternating red and black squares. Each corner square is black. Since $31$ is odd, each row and each column starts and ends with a black square.\n\n2. **Counting Black Squares in a $30 \\times 30$ B... |
DeepScaleR | [
{
"role": "user",
"content": "Chuck the llama is tied to the corner of a $2\\text{ m}$ by $3\\text{ m}$ shed on a $3\\text{ m}$ leash. How much area (in square meters) does Chuck have in which to play if he can go only around the outside of the shed? [asy]\ndraw((0,0)--(15,0)--(15,10)--(0,10)--cycle,black+l... | math | {
"style": "rule",
"ground_truth": "7\\pi"
} | {
"index": 6149,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cinderella and her fairy godmother released a collection of seven new models of crystal slippers. The storybook heroines held a presentation of the collection for some guests: the audience had to state which slippers they liked. The guests wrote in a survey which models they co... | math | {
"style": "rule",
"ground_truth": "128"
} | {
"index": 6150,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cindy leaves school at the same time every day. If she cycles at \\(20 \\ \\text{km/h}\\), she arrives home at 4:30 in the afternoon. If she cycles at \\(10 \\ \\text{km/h}\\), she arrives home at 5:15 in the afternoon. At what speed, in \\(\\text{km/h}\\), must she cycle to ar... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 6151,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cindy leaves school at the same time every day. If she cycles at \\(20 \\ \\text{km/h}\\), she arrives home at 4:30 in the afternoon. If she cycles at \\(10 \\ \\text{km/h}\\), she arrives home at 5:15 in the afternoon. Determine the speed, in \\(\\text{km/h}\\), at which she m... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 6152,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cindy wants to arrange her coins into $X$ piles, each consisting of the same number of coins, $Y$. Each pile will have more than one coin and no pile will have all the coins. If there are 16 possible values for $Y$ given all of the restrictions, what is the smallest number of c... | math | {
"style": "rule",
"ground_truth": "131072"
} | {
"index": 6153,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been had she worked the problem correctly?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 6154,
"solution": "1. **Understanding the problem**: Cindy was supposed to subtract 3 from a number $x$ and then divide by 9. Instead, she subtracted 9 and divided by 3, resulting in 43.\n\n2. **Setting up the equation for Cindy's incorrect operation**: \n \\[\n \\frac{x - 9}{3} = 43\n \\]\n\n3. **... |
DeepScaleR | [
{
"role": "user",
"content": "Cindy wishes to arrange her coins into $X$ piles, each consisting of the same number of coins, $Y$. Each pile will have more than one coin and no pile will have all the coins. If there are 13 possible values for $Y$ given all of the restrictions, what is the smallest number of ... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 6155,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circle $\\Gamma$ has diameter $\\overline{AB}$ with $AB = 6$ . Point $C$ is constructed on line $AB$ so that $AB = BC$ and $A \\neq C$ . Let $D$ be on $\\Gamma$ so that $\\overleftrightarrow{CD}$ is tangent to $\\Gamma$ . Compute the distance from line $\\o... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"index": 6156,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circle $\\Gamma$ with radius $1$ is centered at point $A$ on the circumference of circle $\\omega$ with radius $7$ . Suppose that point $P$ lies on $\\omega$ with $AP=4$ . Determine the product of the distances from $P$ to the two intersections of $\\omega$ an... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 6157,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circle $\\omega_1$ with radius 3 is inscribed in a strip $S$ having border lines $a$ and $b$ . Circle $\\omega_2$ within $S$ with radius 2 is tangent externally to circle $\\omega_1$ and is also tangent to line $a$ . Circle $\\omega_3$ within $S$ is tangent ... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{8}"
} | {
"index": 6158,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circle $o$ contains the circles $m$ , $p$ and $r$ , such that they are tangent to $o$ internally and any two of them are tangent between themselves. The radii of the circles $m$ and $p$ are equal to $x$ . The circle $r$ has radius $1$ and passes through the c... | math | {
"style": "rule",
"ground_truth": "8/9"
} | {
"index": 6159,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circle $A$ has its center at $A(4, 4)$ and has a radius of 4 units. Circle $B$ has its center at $B(12, 4)$ and has a radius of 4 units. What is the area of the gray region bound by the circles and the $x$-axis? Express your answer in terms of $\\pi$. [asy]\nimport olympiad; si... | math | {
"style": "rule",
"ground_truth": "32 - 8\\pi"
} | {
"index": 6160,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circle $A$ has radius $100$. Circle $B$ has an integer radius $r<100$ and remains internally tangent to circle $A$ as it rolls once around the circumference of circle $A$. The two circles have the same points of tangency at the beginning and end of circle $B$'s trip. How many p... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 6161,
"solution": "1. **Calculate the circumference of circle A**: \n The radius of circle A is given as $100$. Therefore, its circumference is:\n \\[\n C_A = 2\\pi \\times 100 = 200\\pi\n \\]\n\n2. **Expression for the circumference of circle B**:\n Circle B has radius $r$, and its circumferen... |
DeepScaleR | [
{
"role": "user",
"content": "Circle $A$ is in the interior of circle $B$. The diameter of circle $B$ is 16 cm. What is the diameter of circle $A$ for which the ratio of the shaded area to the area of circle $A$ is 3:1?\n\n[asy]size(101);\nimport graph;\npair B = (0,0);\npair A = (1,-3);\nfilldraw(Circle(B,... | math | {
"style": "rule",
"ground_truth": "8\\text{ cm}"
} | {
"index": 6162,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circle $A$ is tangent to circle $B$ at one point, and the center of circle $A$ lies on the circumference of circle $B$. The area of circle $A$ is $16\\pi$ square units. Find the area of circle $B$."
}
] | math | {
"style": "rule",
"ground_truth": "64\\pi"
} | {
"index": 6163,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circle $B$ has its center at $(-6, 2)$ and a radius of $10$ units. What is the sum of the $y$-coordinates of the two points on circle $B$ that are also on the $y$-axis?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 6164,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circle $C$ has its center at $C(5, 5)$ and has a radius of 3 units. Circle $D$ has its center at $D(14, 5)$ and has a radius of 3 units. What is the area of the gray region bound by the circles and the $x$-axis?\n```asy\nimport olympiad; size(150); defaultpen(linewidth(0.8));\n... | math | {
"style": "rule",
"ground_truth": "45 - \\frac{9\\pi}{2}"
} | {
"index": 6165,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circle $C$ has radius 6 cm. How many square centimeters are in the area of the largest possible inscribed triangle having one side as a diameter of circle $C$?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 6166,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circle $C$ with radius 2 has diameter $\\overline{AB}$. Circle D is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circle $C$, externally tangent to circle $D$, and tangent to $\\overline{AB}$. The radius of circle $D$ is three times the radius of ... | math | {
"style": "rule",
"ground_truth": "254"
} | {
"index": 6167,
"solution": "We use the notation of Solution 1 for triangle $\\triangle DEC$ \\[\\sin \\angle EDC = \\frac {EF}{DE} = \\frac {1}{4} \\implies \\cos \\angle EDC = \\frac {\\sqrt{15}}{4}.\\] We use Cosine Law for $\\triangle DEC$ and get: \\[(4r)^2 +(2 – 3r)^2 – 2 \\cdot 4r \\cdot (2 – 3r) \\cdot \\f... |
DeepScaleR | [
{
"role": "user",
"content": "Circle $C$ with radius 6 has diameter $\\overline{AB}$. Circle $D$ is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circle $C$, externally tangent to circle $D$, and tangent to $\\overline{AB}$. The radius of circle $D$ is twice the radius of circ... | math | {
"style": "rule",
"ground_truth": "186"
} | {
"index": 6168,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circle $C_0$ has radius $1$, and the point $A_0$ is a point on the circle. Circle $C_1$ has radius $r<1$ and is internally tangent to $C_0$ at point $A_0$. Point $A_1$ lies on circle $C_1$ so that $A_1$ is located $90^{\\circ}$ counterclockwise from $A_0$ on $C_1$. Circle $C_2$... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"index": 6169,
"solution": "Let the center of circle $C_i$ be $O_i$. Note that $O_0BO_1$ is a right triangle, with right angle at $B$. Also, $O_1B=\\frac{11}{60}O_0B$, or $O_0B = \\frac{60}{61}O_0O_1$. It is clear that $O_0O_1=1-r=\\frac{49}{60}$, so $O_0B=\\frac{60}{61}\\times\\frac{49}{60}=\\frac{49}{61}$. Our ... |
DeepScaleR | [
{
"role": "user",
"content": "Circle $C_1$ has its center $O$ lying on circle $C_2$. The two circles meet at $X$ and $Y$. Point $Z$ in the exterior of $C_1$ lies on circle $C_2$ and $XZ=13$, $OZ=11$, and $YZ=7$. What is the radius of circle $C_1$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{30}"
} | {
"index": 6170,
"solution": "1. **Identify the relationship between circles and triangles**: \n - Circle $C_2$ is the circumcircle for both $\\triangle XOZ$ and $\\triangle OYZ$.\n - The center of circle $C_1$, denoted as $O$, lies on circle $C_2$.\n - $X$ and $Y$ are the points where circles $C_1$ and $C_2$... |
DeepScaleR | [
{
"role": "user",
"content": "Circle $I$ passes through the center of, and is tangent to, circle $II$. The area of circle $I$ is $4$ square inches. \nThen the area of circle $II$, in square inches, is:"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 6171,
"solution": "1. **Understanding the Relationship Between the Circles**:\n Given that Circle $I$ is tangent to Circle $II$ and passes through the center of Circle $II$, we can deduce that the diameter of Circle $I$ is equal to the radius of Circle $II$. This is because the point of tangency to the... |
DeepScaleR | [
{
"role": "user",
"content": "Circle $O$ has chord $A B$. A circle is tangent to $O$ at $T$ and tangent to $A B$ at $X$ such that $A X=2 X B$. What is \\frac{A T}{B T}?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 6172,
"solution": "Let $T X$ meet circle $O$ again at $Y$. Since the homothety centered at $T$ takes $X$ to $Y$ also takes $A B$ to the tangent line of circle $O$ passing through $Y$, we have $Y$ is the midpoint of arc $A B$. This means that $\\angle A T Y=\\angle Y T B$. By the Angle Bisector Theorem, $... |
DeepScaleR | [
{
"role": "user",
"content": "Circle $O$ is located on the coordinate plane with center at $(2,3)$. One endpoint of a diameter is at $(-1,-1)$. What are the coordinates of the other endpoint of this diameter? Express your answer as an ordered pair."
}
] | math | {
"style": "rule",
"ground_truth": "(5,7)"
} | {
"index": 6173,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circle $T$ has a circumference of $12\\pi$ inches, and segment $XY$ is a diameter. If the measure of angle $TXZ$ is $60^{\\circ}$, what is the length, in inches, of segment $XZ$?\n\n[asy]\nsize(150);\ndraw(Circle((0,0),13),linewidth(1));\ndraw((-12,-5)--(-5,-12)--(12,5)--cycle,... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 6174,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circle $T$ has its center at point $T(-2,6)$. Circle $T$ is reflected across the $y$-axis and then translated 8 units down. What are the coordinates of the image of the center of circle $T$?"
}
] | math | {
"style": "rule",
"ground_truth": "(2, -2)"
} | {
"index": 6175,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circle $\\Gamma$ is the incircle of $\\triangle ABC$ and is also the circumcircle of $\\triangle XYZ$. The point $X$ is on $\\overline{BC}$, point $Y$ is on $\\overline{AB}$, and the point $Z$ is on $\\overline{AC}$. If $\\angle A=40^\\circ$, $\\angle B=60^\\circ$, and $\\ang... | math | {
"style": "rule",
"ground_truth": "120^\\circ"
} | {
"index": 6176,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circle $\\Gamma$ is the incircle of $\\triangle ABC$ and is also the circumcircle of $\\triangle XYZ$. The point $X$ is on $\\overline{BC}$, the point $Y$ is on $\\overline{AB}$, and the point $Z$ is on $\\overline{AC}$. If $\\angle A=40^\\circ$, $\\angle B=60^\\circ$, and $\... | math | {
"style": "rule",
"ground_truth": "60^\\circ"
} | {
"index": 6177,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circle $\\Omega$ has radius 13. Circle $\\omega$ has radius 14 and its center $P$ lies on the boundary of circle $\\Omega$. Points $A$ and $B$ lie on $\\Omega$ such that chord $A B$ has length 24 and is tangent to $\\omega$ at point $T$. Find $A T \\cdot B T$."
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 6178,
"solution": "Let $M$ be the midpoint of chord $A B$; then $A M=B M=12$ and Pythagoras on triangle $A M O$ gives $M O=5$. Note that $\\angle A O M=\\angle A O B / 2=\\angle A P B=\\angle A P T+\\angle T P B$ or $\\tan (\\angle A O M)=\\tan (\\angle A P T+\\angle T P B)$. Applying the tangent additio... |
DeepScaleR | [
{
"role": "user",
"content": "Circle $\\Omega$ has radius 5. Points $A$ and $B$ lie on $\\Omega$ such that chord $A B$ has length 6. A unit circle $\\omega$ is tangent to chord $A B$ at point $T$. Given that $\\omega$ is also internally tangent to $\\Omega$, find $A T \\cdot B T$."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 6179,
"solution": "Let $M$ be the midpoint of chord $A B$ and let $O$ be the center of $\\Omega$. Since $A M=B M=3$, Pythagoras on triangle $A M O$ gives $O M=4$. Now let $\\omega$ be centered at $P$ and say that $\\omega$ and $\\Omega$ are tangent at $Q$. Because the diameter of $\\omega$ exceeds 1, poi... |
DeepScaleR | [
{
"role": "user",
"content": "Circle $\\omega$ has radius 5 and is centered at $O$. Point $A$ lies outside $\\omega$ such that $OA=13$. The two tangents to $\\omega$ passing through $A$ are drawn, and points $B$ and $C$ are chosen on them (one on each tangent), such that line $BC$ is tangent to $\\omega$ an... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 6180,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circle $\\omega$ is inscribed in rhombus $H M_{1} M_{2} T$ so that $\\omega$ is tangent to $\\overline{H M_{1}}$ at $A, \\overline{M_{1} M_{2}}$ at $I, \\overline{M_{2} T}$ at $M$, and $\\overline{T H}$ at $E$. Given that the area of $H M_{1} M_{2} T$ is 1440 and the area of $E... | math | {
"style": "rule",
"ground_truth": "540"
} | {
"index": 6181,
"solution": "First, from equal tangents, we know that $T E=T M$. As the sides of a rhombus are also equal, this gives from SAS similarity that $E M T \\sim T H M_{2}$. Further, the ratio of their areas is $\\frac{405}{1440 / 2}=\\frac{9}{16}$. This means that $T E=T M=\\frac{3}{4} H T$. Then, we ge... |
DeepScaleR | [
{
"role": "user",
"content": "Circle I is externally tangent to Circle II and passes through the center of Circle II. Given that the area of Circle I is increased to 16 square inches, determine the area of Circle II, in square inches."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 6182,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circle inscribed in square $ABCD$ , is tangent to sides $AB$ and $CD$ at points $M$ and $K$ respectively. Line $BK$ intersects this circle at the point $L, X$ is the midpoint of $KL$ . Find the angle $\\angle MXK $ ."
}
] | math | {
"style": "rule",
"ground_truth": "135"
} | {
"index": 6183,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circles $C_1$ and $C_2$ intersect at points $X$ and $Y$ . Point $A$ is a point on $C_1$ such that the tangent line with respect to $C_1$ passing through $A$ intersects $C_2$ at $B$ and $C$ , with $A$ closer to $B$ than $C$ , such that $2016 \\cdot AB ... | math | {
"style": "rule",
"ground_truth": "2017"
} | {
"index": 6184,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circles $A$ and $B$ each have a radius of 1 and are tangent to each other. Circle $C$ has a radius of 2 and is tangent to the midpoint of $\\overline{AB}.$ What is the area inside circle $C$ but outside circle $A$ and circle $B?$\nA) $1.16$\nB) $3 \\pi - 2.456$\nC) $4 \\pi - 4.... | math | {
"style": "rule",
"ground_truth": "4 \\pi - 4.912"
} | {
"index": 6185,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circles $A, B,$ and $C$ each have radius 1. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\\overline{AB}.$ What is the area inside circle $C$ but outside circle $A$ and circle $B?$"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 6186,
"solution": "1. **Identify the Configuration**: Circles $A$, $B$, and $C$ each have a radius of 1. Circles $A$ and $B$ are tangent to each other, and circle $C$ is tangent to the midpoint $M$ of line segment $\\overline{AB}$. \n\n2. **Positioning of Circle $C$**: Since circle $C$ is tangent to the ... |
DeepScaleR | [
{
"role": "user",
"content": "Circles $A,B$, and $C$ are externally tangent to each other and internally tangent to circle $D$. Circles $B$ and $C$ are congruent. Circle $A$ has radius 1 and passes through the center of $D$. What is the radius of circle $B$?\n\n[asy]unitsize(1cm);\npair A,B,C,D;\nA=(-1,0);\... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{9}"
} | {
"index": 6187,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circles $C_1$ and $C_2$ are externally tangent, and they are both internally tangent to circle $C_3.$ The radii of $C_1$ and $C_2$ are 3 and 9, respectively, and the centers of the three circles are all collinear. A chord of $C_3$ is also a common external tangent of $C_1$ and ... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 6188,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circles $C_1$ and $C_2$ are externally tangent, and they are both internally tangent to circle $C_3.$ The radii of $C_1$ and $C_2$ are 4 and 10, respectively, and the centers of the three circles are all collinear. A chord of $C_3$ is also a common external tangent of $C_1$ and... | math | {
"style": "rule",
"ground_truth": "405"
} | {
"index": 6189,
"solution": "Call our desired length $x$. Note for any $X$ on $\\overline{AB}$ and $Y$ on $\\overline{O_1O_2}$ such that $\\overline{XY}\\perp\\overline{AB}$ that the function $f$ such that $f(\\overline{O_1Y})=\\overline{XY}$ is linear. Since $(0,4)$ and $(14,10)$, we can quickly interpolate that ... |
DeepScaleR | [
{
"role": "user",
"content": "Circles $C_{1}, C_{2}, C_{3}$ have radius 1 and centers $O, P, Q$ respectively. $C_{1}$ and $C_{2}$ intersect at $A, C_{2}$ and $C_{3}$ intersect at $B, C_{3}$ and $C_{1}$ intersect at $C$, in such a way that $\\angle A P B=60^{\\circ}, \\angle B Q C=36^{\\circ}$, and $\\angle ... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 6190,
"solution": "Using a little trig, we have $B C=2 \\sin 18, A C=2 \\sin 36$, and $A B=2 \\sin 30$ (see left diagram). Call these $a, b$, and $c$, respectively. By the law of cosines, $b^{2}=a^{2}+c^{2}-2 a c \\cos A B C$, therefore $\\cos A B C=\\frac{\\sin ^{2} 18+\\sin ^{2} 30-\\sin ^{2} 36}{2 \\s... |
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": 6191,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circles $\\mathcal{C}_1, \\mathcal{C}_2,$ and $\\mathcal{C}_3$ have their centers at (0,0), (12,0), and (24,0), and have radii 1, 2, and 4, respectively. Line $t_1$ is a common internal tangent to $\\mathcal{C}_1$ and $\\mathcal{C}_2$ and has a positive slope, and line $t_2$ is... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 6192,
"solution": "Call the centers $O_1, O_2, O_3$, the points of tangency $r_1, r_2, s_1, s_2$ (with $r$ on $t_1$ and $s$ on $t_2$, and $s_2$ on $\\mathcal{C}_2$), and the intersection of each common internal tangent to the X-axis $r, s$. $\\triangle O_1r_1r \\sim \\triangle O_2r_2r$ since both triangl... |
DeepScaleR | [
{
"role": "user",
"content": "Circles $\\mathcal{C}_{1}$ and $\\mathcal{C}_{2}$ intersect at two points, one of which is $(9,6)$, and the product of the radii is $68$. The x-axis and the line $y = mx$, where $m > 0$, are tangent to both circles. It is given that $m$ can be written in the form $a\\sqrt {b}/c... | math | {
"style": "rule",
"ground_truth": "282"
} | {
"index": 6193,
"solution": "Let the smaller angle between the $x$-axis and the line $y=mx$ be $\\theta$. Note that the centers of the two circles lie on the angle bisector of the angle between the $x$-axis and the line $y=mx$. Also note that if $(x,y)$ is on said angle bisector, we have that $\\frac{y}{x}=\\tan{\... |
DeepScaleR | [
{
"role": "user",
"content": "Circles $\\mathcal{P}$ and $\\mathcal{Q}$ have radii $1$ and $4$, respectively, and are externally tangent at point $A$. Point $B$ is on $\\mathcal{P}$ and point $C$ is on $\\mathcal{Q}$ so that line $BC$ is a common external tangent of the two circles. A line $\\ell$ through $... | math | {
"style": "rule",
"ground_truth": "129"
} | {
"index": 6194,
"solution": "Let $P$ and $Q$ be the centers of circles $\\mathcal{P}$ and $\\mathcal{Q}$ , respectively.\n Let $M$ be midpoint $BC, \\beta = \\angle ACB.$\n Upper diagram shows that\n$\\sin 2\\beta = \\frac {4}{5}$ and $AC = 2 AB.$ Therefore $\\cos 2\\beta = \\frac {3}{5}.$\n Let $CH\\perp l, BH'\\... |
DeepScaleR | [
{
"role": "user",
"content": "Circles $\\omega_1$ and $\\omega_2$ intersect at points $X$ and $Y$. Line $\\ell$ is tangent to $\\omega_1$ and $\\omega_2$ at $A$ and $B$, respectively, with line $AB$ closer to point $X$ than to $Y$. Circle $\\omega$ passes through $A$ and $B$ intersecting $\\omega_1$ again a... | math | {
"style": "rule",
"ground_truth": "270"
} | {
"index": 6195,
"solution": "$AB^2 = 4 AM^2 =2x(2x+ 2 XY) =(XP - XY) (XP + XY) = XP^2 - XY^2 = XC \\cdot XD - XY^2 = 67 \\cdot 37 - 47^2 = \\boxed{270}.$\nvladimir.shelomovskii@gmail.com, vvsss\n ~MathProblemSolvingSkills.com"
} |
DeepScaleR | [
{
"role": "user",
"content": "Circles $\\omega_1$ and $\\omega_2$ intersect at two points $P$ and $Q,$ and their common tangent line closer to $P$ intersects $\\omega_1$ and $\\omega_2$ at points $A$ and $B,$ respectively. The line parallel to $AB$ that passes through $P$ intersects $\\omega_1$ and $\\omega... | math | {
"style": "rule",
"ground_truth": "033"
} | {
"index": 6196,
"solution": "Notice that line $\\overline{PQ}$ is the radical axis of circles $\\omega_1$ and $\\omega_2$. By the radical axis theorem, we know that the tangents of any point on line $\\overline{PQ}$ to circles $\\omega_1$ and $\\omega_2$ are equal. Therefore, line $\\overline{PQ}$ must pass throug... |
DeepScaleR | [
{
"role": "user",
"content": "Circles $\\omega_1$ and $\\omega_2$ with radii $961$ and $625$, respectively, intersect at distinct points $A$ and $B$. A third circle $\\omega$ is externally tangent to both $\\omega_1$ and $\\omega_2$. Suppose line $AB$ intersects $\\omega$ at two points $P$ and $Q$ such that... | math | {
"style": "rule",
"ground_truth": "672"
} | {
"index": 6197,
"solution": "Suppose we label the points as shown here. By radical axis, the tangents to $\\omega$ at $D$ and $E$ intersect on $AB$. Thus $PDQE$ is harmonic, so the tangents to $\\omega$ at $P$ and $Q$ intersect at $X \\in DE$. Moreover, $OX \\parallel O_1O_2$ because both $OX$ and $O_1O_2$ are per... |
DeepScaleR | [
{
"role": "user",
"content": "Circles $\\omega_1$, $\\omega_2$, and $\\omega_3$ each have radius $4$ and are placed in the plane so that each circle is externally tangent to the other two. Points $P_1$, $P_2$, and $P_3$ lie on $\\omega_1$, $\\omega_2$, and $\\omega_3$ respectively such that $P_1P_2=P_2P_3=P... | math | {
"style": "rule",
"ground_truth": "552"
} | {
"index": 6198,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Circles $\\omega_1$, $\\omega_2$, and $\\omega_3$ each have radius $6$ and are placed in the plane so that each circle is externally tangent to the other two. Points $Q_1$, $Q_2$, and $Q_3$ lie on $\\omega_1$, $\\omega_2$, and $\\omega_3$ respectively such that triangle $\\tria... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 6199,
"solution": ""
} |
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