data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "A number is divisible by $9$ if the sum of its digits is divisible by $9.$ For example, the number $19\\,836$ is divisible by $9$ but $19\\,825$ is not.\n\nIf $D\\,767\\,E89$ is divisible by $9,$ where $D$ and $E$ each represent a single digit, what is the sum of all possible v... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 2400,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number is formed using the digits 1, 2, ..., 9. Any digit can be used more than once, but adjacent digits cannot be the same. Once a pair of adjacent digits has occurred, that pair, in that order, cannot be used again. How many digits are in the largest such number?"
}
] | math | {
"style": "rule",
"ground_truth": "73"
} | {
"index": 2401,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number is formed using the digits \\(1, 2, \\ldots, 9\\). Any digit can be used more than once, but adjacent digits cannot be the same. Once a pair of adjacent digits has occurred, that pair, in that order, cannot be used again. How many digits are in the largest such number?... | math | {
"style": "rule",
"ground_truth": "73"
} | {
"index": 2402,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number is randomly selected from the interval $[-π, π]$. Calculate the probability that the value of the function $y = \\cos x$ falls within the range $[-\\frac{\\sqrt{3}}{2}, \\frac{\\sqrt{3}}{2}]$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 2403,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number is reduced by 5 times and then increased by 20 times to get 40. What is this number?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 2404,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number is the product of four prime numbers. What is this number if the sum of the squares of the four prime numbers is 476?"
}
] | math | {
"style": "rule",
"ground_truth": "1989"
} | {
"index": 2405,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number like 45132 is called a \"wave number,\" which means the tens and thousands digits are both larger than their respective neighboring digits. What is the probability of forming a non-repeating five-digit \"wave number\" using the digits 1, 2, 3, 4, 5?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{15}"
} | {
"index": 2406,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number of linked rings, each $1$ cm thick, are hanging on a peg. The top ring has an outside diameter of $20$ cm. The outside diameter of each of the outer rings is $1$ cm less than that of the ring above it. The bottom ring has an outside diameter of $3$ cm. What is the dist... | math | {
"style": "rule",
"ground_truth": "173"
} | {
"index": 2407,
"solution": "1. **Identify the sequence of inside diameters**: The outside diameter of the top ring is $20$ cm, and each subsequent ring has an outside diameter $1$ cm less than the ring above it. Since each ring is $1$ cm thick, the inside diameter of each ring is $2$ cm less than its outside diam... |
DeepScaleR | [
{
"role": "user",
"content": "A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of $8^{\\text{th}}$-graders to $6^{\\text{th}}$-graders is $5:3$, and the the ratio of $8^{\\text{th}}$-graders to $7^{\\text{th}}$-graders is $8:5$. What is the smallest... | math | {
"style": "rule",
"ground_truth": "89"
} | {
"index": 2408,
"solution": "\n#### Step 1: Establish the ratios\nWe are given two ratios:\n- The ratio of $8^\\text{th}$-graders to $6^\\text{th}$-graders is $5:3$.\n- The ratio of $8^\\text{th}$-graders to $7^\\text{th}$-graders is $8:5$.\n\n#### Step 2: Equalize the number of $8^\\text{th}$-graders\nTo combine ... |
DeepScaleR | [
{
"role": "user",
"content": "A number of tourists want to take a cruise, and it is required that the number of people on each cruise ship is the same. If each cruise ship carries 12 people, there will be 1 person left who cannot board. If one cruise ship leaves empty, then all tourists can be evenly distri... | math | {
"style": "rule",
"ground_truth": "169"
} | {
"index": 2409,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number of trucks with the same capacity were requested to transport cargo from one place to another. Due to road issues, each truck had to carry 0.5 tons less than planned, which required 4 additional trucks. The mass of the transported cargo was at least 55 tons but did not ... | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"index": 2410,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number which when divided by $10$ leaves a remainder of $9$, when divided by $9$ leaves a remainder of $8$, by $8$ leaves a remainder of $7$, etc., down to where, when divided by $2$, it leaves a remainder of $1$, is:"
}
] | math | {
"style": "rule",
"ground_truth": "2519"
} | {
"index": 2411,
"solution": "1. **Understanding the Problem:**\n The problem states that a number $n$ has specific remainders when divided by integers from $10$ down to $2$. Specifically, when $n$ is divided by any integer $k$ (where $10 \\geq k \\geq 2$), it leaves a remainder of $k-1$.\n\n2. **Formulating the ... |
DeepScaleR | [
{
"role": "user",
"content": "A number's cube is a five-digit number; the first digit is 4, and the fifth digit is 6. What is this number?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 2412,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A numerical sequence is defined by the conditions: \\( a_{1} = 1 \\), \\( a_{n+1} = a_{n} + \\left\\lfloor \\sqrt{a_{n}} \\right\\rfloor \\).\n\nHow many perfect squares are there among the first terms of this sequence that do not exceed \\( 1{,}000{,}000 \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 2413,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A numerical sequence is defined by the conditions: \\( a_{1}=1 \\), \\( a_{n+1}=a_{n}+\\left \\lfloor \\sqrt{a_{n}} \\right \\rfloor \\).\n\nHow many perfect squares are among the first terms of this sequence that do not exceed 1,000,000?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 2414,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A numerical value is assigned to each letter of the alphabet. The value of a word is determined by adding up the numerical values of each of its letters. The value of SET is 2, the value of HAT is 7, the value of TASTE is 3, and the value of MAT is 4. What is the value of the w... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 2415,
"solution": "From the given information, we know that $S+E+T=2$, $H+A+T=7$, $T+A+S+T+E=3$, and $M+A+T=4$. Since $T+A+S+T+E=3$ and $S+E+T=2$, then $T+A=3-2=1$. Since $H+A+T=7$ and $T+A=1$, then $H=7-1=6$. Since $M+A+T=4$ and $H=7$, then $M+(A+T)+H=4+6=10$. Therefore, the value of the word MATH is 10... |
DeepScaleR | [
{
"role": "user",
"content": "A numismatist has 100 identical-looking coins, knowing that among them are 30 genuine coins and 70 counterfeit coins. The masses of all genuine coins are the same, while the masses of all counterfeit coins are different, with each counterfeit coin being heavier than a genuine c... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 2416,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A nut has the shape of a regular hexagonal prism. Each lateral face of the nut is painted in one of three colors: white, red, or blue, with adjacent faces painted in different colors. How many different nut paintings are possible? (It is not necessary to use all three colors in... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 2417,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is $\\frac{1}{2}$ foot from the top face. The second cut is $\\frac{1}{3}$ foot below the first cut, and the third cut is $\\frac{1}{17}$ foot below the second cut. F... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 2418,
"solution": "1. **Understanding the Problem**: We have a cube with a volume of 1 cubic foot, and it is cut into four pieces by three cuts parallel to the top face. The cuts are made at $\\frac{1}{2}$ foot, $\\frac{1}{3}$ foot, and $\\frac{1}{17}$ foot below each preceding cut. The pieces are then r... |
DeepScaleR | [
{
"role": "user",
"content": "A one-meter gas pipe has rusted in two places. Determine the probability that all three resulting parts can be used as connections to gas stoves, if the regulations stipulate that the stove should not be closer than 25 cm to the main gas pipe."
}
] | math | {
"style": "rule",
"ground_truth": "9/16"
} | {
"index": 2419,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A one-meter gas pipe has rusted through in two places. Determine the probability that all three resulting parts can be used as connectors to gas stoves, given that according to regulations, the stove must not be closer than 25 cm to the main gas pipe."
}
] | math | {
"style": "rule",
"ground_truth": "1/16"
} | {
"index": 2420,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A package of milk with a volume of 1 liter cost 60 rubles. Recently, for the purpose of economy, the manufacturer reduced the package volume to 0.9 liters and increased its price to 81 rubles. By what percentage did the manufacturer's revenue increase?"
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 2421,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A package of seeds was passed around a table. The first person took 1 seed, the second person took 2 seeds, the third took 3 seeds, and so forth, with each subsequent person taking one more seed than the previous one. It is known that during the second round a total of 100 more... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 2422,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A packet of seeds was passed around the table. The first person took 1 seed, the second took 2 seeds, the third took 3 seeds, and so on: each subsequent person took one more seed than the previous one. It is known that on the second round, a total of 100 more seeds were taken t... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 2423,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A paint brush is swept along both diagonals of a square to produce the symmetric painted area, as shown. Half the area of the square is painted. What is the ratio of the side length of the square to the brush width?"
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}+2"
} | {
"index": 2424,
"solution": "\nLet's denote the side length of the square as $s$ and the width of the brush as $w$. We are given that half the area of the square is painted, so the painted area is $\\frac{s^2}{2}$.\n\n#### Step 1: Analyze the geometry\nThe brush sweeps along both diagonals of the square. Each diag... |
DeepScaleR | [
{
"role": "user",
"content": "A painting $18$\" X $24$\" is to be placed into a wooden frame with the longer dimension vertical. The wood at the top and bottom is twice as wide as the wood on the sides. If the frame area equals that of the painting itself, the ratio of the smaller to the larger dimension of... | math | {
"style": "rule",
"ground_truth": "2:3"
} | {
"index": 2425,
"solution": "1. **Define the variables:**\n Let the width of the frame on the sides be $x$ inches. Then, the width of the frame on the top and bottom is $2x$ inches, since it is twice as wide.\n\n2. **Determine the overall dimensions of the framed painting:**\n - The width of the framed paintin... |
DeepScaleR | [
{
"role": "user",
"content": "A painting $20$\" X $30$\" is to be framed where the frame border at the top and bottom is three times as wide as the frame on the sides. If the total area of the frame (not including the painting) equals twice the area of the painting itself, find the ratio of the smaller dime... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 2426,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A pair of dice is rolled twice. What is the probability that the sum of the numbers facing up is 5?\n\nA) $\\frac{1}{9}$\nB) $\\frac{1}{4}$\nC) $\\frac{1}{36}$\nD) 97"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"index": 2427,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A pair of fair 8-sided dice are rolled, yielding numbers $a$ and $b$. Determine the probability that both digits $a$ and $b$ as well as the two-digit number $ab$ formed by them are all divisible by 4."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"index": 2428,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A pair of natural numbers is called \"good\" if one of the numbers is divisible by the other. The numbers from 1 to 30 are divided into 15 pairs. What is the maximum number of good pairs that could be formed?"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 2429,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A pair of standard $6$-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"index": 2430,
"solution": "1. **Understanding the problem**: We need to find the probability that the area of a circle is less than its circumference, given that the diameter $d$ of the circle is determined by the sum of two 6-sided dice.\n\n2. **Relating area and circumference**: The formula for the circumferen... |
DeepScaleR | [
{
"role": "user",
"content": "A pair of standard $6$-sided dice is rolled to determine the side length of a square. What is the probability that the numerical value of the area of the square is less than the numerical value of the perimeter?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"index": 2431,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A palindrome between $1000$ and $10000$ is chosen at random. What is the probability that it is divisible by $7$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"index": 2432,
"solution": "1. **Identify the form of the palindrome**: A four-digit palindrome can be expressed in the form $\\overline{abba}$, where $a$ and $b$ are digits, and $a \\neq 0$ to ensure it is a four-digit number.\n\n2. **Total number of palindromes**: Since $a$ can be any digit from 1 to 9 (9 choic... |
DeepScaleR | [
{
"role": "user",
"content": "A palindrome between $10000$ and $100000$ is chosen at random. What is the probability that it is divisible by $11$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{10}"
} | {
"index": 2433,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base 10 with no leading zeros. A 6-digit palindrome $n$ is chosen uniformly at random. What is the probability that $\\frac{n}{11}$ is also a palindrome?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{30}"
} | {
"index": 2434,
"solution": "To solve this problem, we first need to understand the structure of a 6-digit palindrome and then determine how many of these palindromes are divisible by 11 and also form a palindrome when divided by 11.\n\n1. **Structure of a 6-digit palindrome**: \n A 6-digit palindrome can be rep... |
DeepScaleR | [
{
"role": "user",
"content": "A palindrome is a number that reads the same forward and backward. How many three-digit perfect squares are palindromes?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 2435,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A palindrome is a number that reads the same forward and backward. What is the smallest 5-digit palindrome in base 2 that can be expressed as a 3-digit palindrome in a different base? Give your response in base 2."
}
] | math | {
"style": "rule",
"ground_truth": "10001_2"
} | {
"index": 2436,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A palindrome is a number that reads the same forwards and backwards, such as 3003. How many positive four-digit integers are palindromes?"
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 2437,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A palindrome is a number that reads the same from left to right and right to left. For example, the numbers 333 and 4884 are palindromes. It is known that a three-digit number \\( x \\) is a palindrome. When 22 is added to it, a four-digit number is obtained, which is also a pa... | math | {
"style": "rule",
"ground_truth": "979"
} | {
"index": 2438,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A palindrome is a number, word, or text that reads the same backward as forward. How much time in a 24-hour day display palindromes on a clock, showing time from 00:00:00 to 23:59:59?"
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 2439,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A palindrome is a positive integer that reads the same backwards as forwards, such as 82328. What is the smallest 5 -digit palindrome that is a multiple of 99 ?"
}
] | math | {
"style": "rule",
"ground_truth": "54945"
} | {
"index": 2440,
"solution": "Write the number as $X Y Z Y X$. This is the same as $10000 X+1000 Y+100 Z+10 Y+X=$ $99(101 X+10 Y+Z)+20 Y+2 X+Z$. We thus want $20 Y+2 X+Z$ to be a multiple of 99 , with $X$ as small as possible. This expression cannot be larger than $20 \\cdot 9+2 \\cdot 9+9=207$, and it is greater t... |
DeepScaleR | [
{
"role": "user",
"content": "A palindrome is a positive integer whose digits are the same when read forwards or backwards. For example, 2002 is a palindrome. What is the smallest number which can be added to 2002 to produce a larger palindrome?"
}
] | math | {
"style": "rule",
"ground_truth": "110"
} | {
"index": 2441,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A palindrome is a string that does not change when its characters are written in reverse order. Let S be a 40-digit string consisting only of 0's and 1's, chosen uniformly at random out of all such strings. Let $E$ be the expected number of nonempty contiguous substrings of $S$... | math | {
"style": "rule",
"ground_truth": "113"
} | {
"index": 2442,
"solution": "Note that $S$ has $41-n$ contiguous substrings of length $n$, so we see that the expected number of palindromic substrings of length $n$ is just $(41-n) \\cdot 2^{-\\lfloor n / 2\\rfloor}$. By linearity of expectation, $E$ is just the sum of this over all $n$ from 1 to 40. However, it ... |
DeepScaleR | [
{
"role": "user",
"content": "A palindrome is an integer that reads the same forward and backward, such as 1221. What percent of the palindromes between 1000 and 2000 contain at least one digit 7?"
}
] | math | {
"style": "rule",
"ground_truth": "10\\%"
} | {
"index": 2443,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A palindrome is an integer that reads the same forward and backward, such as 3663. What percent of the palindromes between 100 and 500 contain at least one 5?"
}
] | math | {
"style": "rule",
"ground_truth": "10\\%"
} | {
"index": 2444,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A palindromic number is a number that reads the same when the order of its digits is reversed. What is the difference between the largest and smallest five-digit palindromic numbers that are both multiples of 45?"
}
] | math | {
"style": "rule",
"ground_truth": "9090"
} | {
"index": 2445,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A paper equilateral triangle $ABC$ has side length $12$. The paper triangle is folded so that vertex $A$ touches a point on side $\\overline{BC}$ a distance $9$ from point $B$. The length of the line segment along which the triangle is folded can be written as $\\frac{m\\sqrt{p... | math | {
"style": "rule",
"ground_truth": "113"
} | {
"index": 2446,
"solution": "Let $P$ and $Q$ be the points on $\\overline{AB}$ and $\\overline{AC}$, respectively, where the paper is folded.\n Let $D$ be the point on $\\overline{BC}$ where the folded $A$ touches it.\n Let $a$, $b$, and $x$ be the lengths $AP$, $AQ$, and $PQ$, respectively.\n We have $PD = a$, $Q... |
DeepScaleR | [
{
"role": "user",
"content": "A paper equilateral triangle $ABC$ has side length 12. The paper triangle is folded so that vertex $A$ touches a point on side $\\overline{BC}$ a distance 9 from point $B$. Find the square of the length of the line segment along which the triangle is folded.\n\n[asy]\nimport cs... | math | {
"style": "rule",
"ground_truth": "\\frac{59319}{1225}"
} | {
"index": 2447,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A paper equilateral triangle of side length 2 on a table has vertices labeled \\(A\\), \\(B\\), and \\(C\\). Let \\(M\\) be the point on the sheet of paper halfway between \\(A\\) and \\(C\\). Over time, point \\(M\\) is lifted upwards, folding the triangle along segment \\(BM\... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{6}"
} | {
"index": 2448,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A paper triangle with sides of lengths $3,4,$ and $5$ inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease?"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{15}{8}$"
} | {
"index": 2449,
"solution": "1. **Identify the Triangle Type**: Given the side lengths $3, 4, 5$, we recognize $\\triangle ABC$ as a right triangle (by the Pythagorean theorem, $3^2 + 4^2 = 5^2$).\n\n2. **Folding Point A to Point B**: When the triangle is folded such that point $A$ falls on point $B$, the crease w... |
DeepScaleR | [
{
"role": "user",
"content": "A paperboy delivers newspapers to 10 houses along Main Street. Wishing to save effort, he doesn't always deliver to every house, but to avoid being fired he never misses three consecutive houses. Compute the number of ways the paperboy could deliver papers in this manner."
}
... | math | {
"style": "rule",
"ground_truth": "504"
} | {
"index": 2450,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A paperboy delivers newspapers to 11 houses along Main Street. He may skip delivering to any house, but he must not skip three consecutive houses to avoid losing his job. Determine the number of possible delivery sequences available to the paperboy under these conditions."
}
... | math | {
"style": "rule",
"ground_truth": "927"
} | {
"index": 2451,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parabola and an ellipse share a focus, and the directrix of the parabola is the line containing the minor axis of the ellipse. The parabola and ellipse intersect at two points. Given that the equation of the ellipse is $\\frac{x^2}{25} + \\frac{y^2}{9} = 1,$ find the distance... | math | {
"style": "rule",
"ground_truth": "\\tfrac{4\\sqrt{14}}{3}"
} | {
"index": 2452,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parabola has focus $F$ and vertex $V$ , where $VF = 1$ 0. Let $AB$ be a chord of length $100$ that passes through $F$ . Determine the area of $\\vartriangle VAB$ ."
}
] | math | {
"style": "rule",
"ground_truth": "100\\sqrt{10}"
} | {
"index": 2453,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parabola has focus $(3,3)$ and directrix $3x + 7y = 21.$ Express the equation of the parabola in the form\n\\[ax^2 + bxy + cy^2 + dx + ey + f = 0,\\]where $a,$ $b,$ $c,$ $d,$ $e,$ $f$ are integers, $a$ is a positive integer, and $\\gcd(|a|,|b|,|c|,|d|,|e|,|f|) = 1.$"
}
] | math | {
"style": "rule",
"ground_truth": "49x^2 - 42xy + 9y^2 - 222x - 54y + 603 = 0"
} | {
"index": 2454,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parabola has the following optical properties: a light ray passing through the focus of the parabola and reflecting off the parabola will result in a light ray parallel to the axis of symmetry of the parabola; conversely, an incident light ray parallel to the axis of symmetry... | math | {
"style": "rule",
"ground_truth": "\\frac{25}{4}"
} | {
"index": 2455,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parabola has vertex $V = (0,0)$ and focus $F = (0,1).$ Let $P$ be a point in the first quadrant, lying on the parabola, so that $PF = 101.$ Find $P.$"
}
] | math | {
"style": "rule",
"ground_truth": "(20,100)"
} | {
"index": 2456,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parabola is given by the equation $y^{2}=4x$. Let $F$ be its focus, and let $A$ and $B$ be the points where a line passing through $F$ intersects the parabola. Let $O$ be the origin of the coordinate system. If $|AF|=3$, find the area of $\\triangle AOB$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{2}}{2}"
} | {
"index": 2457,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parabola with equation $y = x^2 + bx + c$ passes through the points $(2,3)$ and $(4,3)$. What is $c$?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 2458,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parabola with equation $y=ax^2+bx+c$ is reflected about the $x$-axis. The parabola and its reflection are translated horizontally five units in opposite directions to become the graphs of $y=f(x)$ and $y=g(x)$, respectively. Which of the following describes the graph of $y=(f... | math | {
"style": "rule",
"ground_truth": "\\text{(D)}"
} | {
"index": 2459,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parabola with equation $y=x^2+bx+c$ passes through the points $(-1,-11)$ and $(3,17)$. What is $c$?"
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"index": 2460,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parabola, given by the equation $y^{2}=2px (p > 0)$, has a focus that lies on the line $l$. This line intersects the parabola at two points, $A$ and $B$. A circle with the chord $AB$ as its diameter has the equation $(x-3)^{2}+(y-2)^{2}=16$. Find the value of $p$."
}
] | math | {
"style": "rule",
"ground_truth": "p = 2"
} | {
"index": 2461,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parabolic arch has a height of $16$ inches and a span of $40$ inches. The height, in inches, of the arch at the point $5$ inches from the center $M$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 2462,
"solution": "1. **Identify the shape and equation of the arch**: Given that the arch is parabolic, we can model it using a quadratic equation of the form:\n \\[\n y = ax^2 + k\n \\]\n where \\(y\\) is the height of the arch at a horizontal distance \\(x\\) from the center, \\(a\\) is a cons... |
DeepScaleR | [
{
"role": "user",
"content": "A parallelogram $ABCD$ has a base $\\overline{BC}$ of 6 units and a height from $A$ to line $BC$ of 3 units. Extend diagonal $AC$ beyond $C$ to point $E$ such that $\\overline{CE}$ is 2 units, making $ACE$ a right triangle at $C$. Find the area of the combined shape formed by p... | math | {
"style": "rule",
"ground_truth": "18 + 3\\sqrt{5}"
} | {
"index": 2463,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parallelogram $ABCD$ is inscribed in an ellipse $\\frac{x^2}{4}+\\frac{y^2}{2}=1$. The slope of line $AB$ is $k_1=1$. Calculate the slope of line $AD$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 2464,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parallelogram $ABCD$ is inscribed in the ellipse $\\frac{x^{2}}{4}+y^{2}=1$, where the slope of the line $AB$ is $k_{1}=1$. Determine the slope of the line $AD$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{4}"
} | {
"index": 2465,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parallelogram has 2 sides of length 20 and 15. Given that its area is a positive integer, find the minimum possible area of the parallelogram."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 2466,
"solution": "The area of the parallelogram can be made arbitrarily small, so the smallest positive integer area is 1."
} |
DeepScaleR | [
{
"role": "user",
"content": "A parallelogram has a base of 6 cm and a height of 20 cm. Its area is \\_\\_\\_\\_\\_\\_ square centimeters. If both the base and the height are tripled, its area will increase by \\_\\_\\_\\_\\_\\_ times, resulting in \\_\\_\\_\\_\\_\\_ square centimeters."
}
] | math | {
"style": "rule",
"ground_truth": "1080"
} | {
"index": 2467,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parallelogram has adjacent sides of lengths $s$ units and $2s$ units forming a 45-degree angle. The area of the parallelogram is $8\\sqrt 2$ square units. What is the value of $s$? Express your answer in simplest radical form."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 2468,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parallelogram has its diagonals making an angle of \\(60^{\\circ}\\) with each other. If two of its sides have lengths 6 and 8, find the area of the parallelogram."
}
] | math | {
"style": "rule",
"ground_truth": "14\\sqrt{3}"
} | {
"index": 2469,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parallelogram has side lengths of 10, $12x-2$, $5y+5$, and $4$. What is the value of $x+y$?\n\n[asy]draw((0,0)--(24,0)--(30,20)--(6,20)--cycle);\nlabel(\"$12x-2$\",(15,0),S);\nlabel(\"10\",(3,10),W);\nlabel(\"$5y+5$\",(18,20),N);\nlabel(\"4\",(27,10),E);\n[/asy]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"index": 2470,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parallelogram has side lengths of 10, 12, $10y-2$, and $4x+6$. Determine the value of $x+y$."
}
] | math | {
"style": "rule",
"ground_truth": "2.7"
} | {
"index": 2471,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parallelogram in the coordinate plane has vertices at points (2,1), (7,1), (5,6), and (10,6). Calculate the sum of the perimeter and the area of the parallelogram."
}
] | math | {
"style": "rule",
"ground_truth": "35 + 2\\sqrt{34}"
} | {
"index": 2472,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parallelogram is defined by the vectors $\\begin{pmatrix} 3 \\\\ 0 \\\\ 4 \\end{pmatrix}$ and $\\begin{pmatrix} 1 \\\\ 2 \\\\ 3 \\end{pmatrix}$. Determine the cosine of the angle $\\theta$ between the diagonals of this parallelogram."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-11}{3\\sqrt{69}}"
} | {
"index": 2473,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parallelogram is generated by the vectors $\\begin{pmatrix} 2 \\\\ 1\\\\ 1 \\end{pmatrix}$ and $\\begin{pmatrix} 1 \\\\ -1 \\\\ - 1 \\end{pmatrix}.$\n\n[asy]\nunitsize(0.4 cm);\n\npair A, B, C, D;\n\nA = (0,0);\nB = (7,2);\nC = (1,3);\nD = B + C;\n\ndraw(A--B,Arrow(6));\ndraw... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 2474,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parallelogram is generated by the vectors $\\begin{pmatrix} 3 \\\\ 1 \\\\ 1 \\end{pmatrix}$ and $\\begin{pmatrix} 1 \\\\ -1 \\\\ -1 \\end{pmatrix}$. Find the cosine of the angle $\\theta$ between the diagonals of the parallelogram."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{3}}{3}"
} | {
"index": 2475,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parallelogram-shaped paper WXYZ with an area of 7.17 square centimeters is placed on another parallelogram-shaped paper EFGH, as shown in the diagram. The intersection points A, C, B, and D are formed, and AB // EF and CD // WX. What is the area of the paper EFGH in square ce... | math | {
"style": "rule",
"ground_truth": "7.17"
} | {
"index": 2476,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parametric graph is defined by:\n\\[\nx = \\cos t + \\frac{t}{3}, \\quad y = \\sin t.\n\\]\nDetermine the number of times the graph intersects itself between \\(x = 3\\) and \\(x = 45\\)."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 2477,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parametric graph is given by\n\\begin{align*}\nx &= \\cos t + \\frac{t}{2}, \\\\\ny &= \\sin t.\n\\end{align*}How many times does the graph intersect itself between $x = 1$ and $x = 40$?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 2478,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A park is in the shape of a regular hexagon $2$ km on a side. Starting at a corner, Alice walks along the perimeter of the park for a distance of $5$ km. How many kilometers is she from her starting point?"
}
] | math | {
"style": "rule",
"ground_truth": "$\\sqrt{13}$"
} | {
"index": 2479,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A park is in the shape of a regular hexagon $2$ km on a side. Starting at a corner, Alice walks along the perimeter of the park for a distance of $5$ km. How many kilometers is she from her starting point?\n$\\textbf{(A)}\\ \\sqrt{13}\\qquad \\textbf{(B)}\\ \\sqrt{14}\\qquad \\... | math | {
"style": "rule",
"ground_truth": "\\sqrt{13}"
} | {
"index": 2480,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parking lot consists of 2012 parking spots equally spaced in a line, numbered 1 through 2012. One by one, 2012 cars park in these spots under the following procedure: the first car picks from the 2012 spots uniformly randomly, and each following car picks uniformly randomly a... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2062300}"
} | {
"index": 2481,
"solution": "We see that for 1 to be the last spot, 2 must be picked first (with probability $\\frac{1}{n}$ ), after which spot $n$ is picked. Then, cars from 3 to $n-1$ will be picked until there are only gaps of 1 or 2 remaining. At this point, each of the remaining spots (including spot 1) is pi... |
DeepScaleR | [
{
"role": "user",
"content": "A parking lot has 16 spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers chose spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. What is the probability that she is... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{28}"
} | {
"index": 2482,
"solution": "To solve this problem, we need to calculate the probability that there are at least two adjacent empty spaces among the 16 spaces after 12 cars have parked. Each car occupies one space, and they choose their spaces randomly.\n\n#### Step 1: Total ways to park 12 cars in 16 spaces\nThe ... |
DeepScaleR | [
{
"role": "user",
"content": "A parking lot has 20 spaces in a row. Fifteen cars arrive, each of which requires one parking space, and their drivers choose spaces at random from among the available spaces. Uncle Ben arrives in his RV, which requires 3 adjacent spaces. What is the probability that he is able... | math | {
"style": "rule",
"ground_truth": "\\frac{232}{323}"
} | {
"index": 2483,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A parking lot in Flower Town is a square with $7 \\times 7$ cells, each of which can accommodate a car. The parking lot is enclosed by a fence, and one of the corner cells has an open side (this is the gate). Cars move along paths that are one cell wide. Neznaika was asked to p... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 2484,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A part of a book has fallen out. The number of the first fallen page is 387, and the number of the last page consists of the same digits but in a different order. How many sheets fell out of the book?"
}
] | math | {
"style": "rule",
"ground_truth": "176"
} | {
"index": 2485,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A particle begins at a point P on the parabola y = x^2 - 2x - 8 where the y-coordinate is 8. It rolls along the parabola to the nearest point Q where the y-coordinate is -8. Calculate the horizontal distance traveled by the particle, defined as the absolute difference between t... | math | {
"style": "rule",
"ground_truth": "\\sqrt{17} - 1"
} | {
"index": 2486,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A particle is located on the coordinate plane at $(5,0)$. Define a ''move'' for the particle as a counterclockwise rotation of $\\frac{\\pi}{4}$ radians about the origin followed by a translation of $10$ units in the positive $x$-direction. Find the particle's position after $... | math | {
"style": "rule",
"ground_truth": "(-5 \\sqrt{2}, 5 + 5 \\sqrt{2})"
} | {
"index": 2487,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A particle is located on the coordinate plane at $(5,0)$. Define a move for the particle as a counterclockwise rotation of $\\pi/4$ radians about the origin followed by a translation of $10$ units in the positive $x$-direction. Given that the particle's position after $150$ mov... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 2488,
"solution": "Let $T:\\begin{pmatrix}x\\\\y\\end{pmatrix}\\rightarrow R(\\frac{\\pi}{4})\\begin{pmatrix}x\\\\y\\end{pmatrix}+\\begin{pmatrix}10\\\\0\\end{pmatrix}$. We assume that the rotation matrix $R(\\frac{\\pi}{4}) = R$ here. Then we have\n$T^{150}\\begin{pmatrix}5\\\\0\\end{pmatrix}=R(R(...R(R... |
DeepScaleR | [
{
"role": "user",
"content": "A particle is placed on the curve $y = x^3 - 3x^2 - x + 3$ at a point $P$ whose $y$-coordinate is $5$. It is allowed to roll along the curve until it reaches the nearest point $Q$ whose $y$-coordinate is $-2$. Compute the horizontal distance traveled by the particle.\nA) $|\\sq... | math | {
"style": "rule",
"ground_truth": "|\\sqrt{6} - \\sqrt{3}|"
} | {
"index": 2489,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A particle is placed on the parabola $y = x^2- x -6$ at a point $P$ whose $y$-coordinate is $6$. It is allowed to roll along the parabola until it reaches the nearest point $Q$ whose $y$-coordinate is $-6$. The horizontal distance traveled by the particle (the numerical value o... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 2490,
"solution": "To solve this problem, we need to find the $x$-coordinates of points $P$ and $Q$ on the parabola $y = x^2 - x - 6$ where $y = 6$ and $y = -6$, respectively, and then calculate the horizontal distance between these points.\n\n1. **Finding $x$-coordinates for $P$ where $y = 6$:**\n \\[... |
DeepScaleR | [
{
"role": "user",
"content": "A particle moves in a straight line inside a square of side 1. It is reflected from the sides, but absorbed by the four corners. It starts from an arbitrary point \\( P \\) inside the square. Let \\( c(k) \\) be the number of possible starting directions from which it reaches a... | math | {
"style": "rule",
"ground_truth": "\\pi"
} | {
"index": 2491,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A particle moves in the Cartesian plane according to the following rules:\nFrom any lattice point $(a,b),$ the particle may only move to $(a+1,b), (a,b+1),$ or $(a+1,b+1).$\nThere are no right angle turns in the particle's path.\n How many different paths can the particle take ... | math | {
"style": "rule",
"ground_truth": "83"
} | {
"index": 2492,
"solution": "The length of the path (the number of times the particle moves) can range from $l = 5$ to $9$; notice that $d = 10-l$ gives the number of diagonals. Let $R$ represent a move to the right, $U$ represent a move upwards, and $D$ to be a move that is diagonal. Casework upon the number of d... |
DeepScaleR | [
{
"role": "user",
"content": "A particle moves in the Cartesian plane according to the following rules:\nFrom any lattice point $(a,b),$ the particle may only move to $(a+1,b), (a,b+1),$ or $(a+1,b+1).$\nThere are no right angle turns in the particle's path.\nHow many different paths can the particle take f... | math | {
"style": "rule",
"ground_truth": "83"
} | {
"index": 2493,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A particle moves so that it is at $(2t + 7, 4t - 13)$ at time $t.$ Find the speed of the particle, measured in unit of distance per unit of time."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{5}"
} | {
"index": 2494,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A particle moves so that its speed for the second and subsequent miles varies inversely as the integral number of miles already traveled. For each subsequent mile the speed is constant. If the second mile is traversed in $2$ hours, then the time, in hours, needed to traverse th... | math | {
"style": "rule",
"ground_truth": "2(n-1)"
} | {
"index": 2495,
"solution": "1. **Define the relationship**: Given that the speed $s_n$ for the $n^\\text{th}$ mile varies inversely as the number of miles already traveled, we can express this relationship as:\n \\[ s_n = \\frac{k}{d} \\]\n where $d$ is the number of miles already traveled and $k$ is a consta... |
DeepScaleR | [
{
"role": "user",
"content": "A particle moves through the first quadrant as follows. During the first minute it moves from the origin to $(1,0)$. Thereafter, it continues to follow the directions indicated in the figure, going back and forth between the positive x and y axes, moving one unit of distance pa... | math | {
"style": "rule",
"ground_truth": "(44,35)"
} | {
"index": 2496,
"solution": "1. **Understanding the Movement Pattern**: The particle starts at the origin and moves in a pattern that encloses squares of increasing size. Each square is $n \\times n$ where $n$ starts from 1 and increases by 1 for each new square. The movement pattern is such that the particle move... |
DeepScaleR | [
{
"role": "user",
"content": "A particle projected vertically upward reaches, at the end of $t$ seconds, an elevation of $s$ feet where $s = 160 t - 16t^2$. The highest elevation is:"
}
] | math | {
"style": "rule",
"ground_truth": "400"
} | {
"index": 2497,
"solution": "1. **Identify the type of function:** The function given for elevation $s$ in terms of time $t$ is $s = 160t - 16t^2$. This is a quadratic function in the standard form $s = at^2 + bt + c$ where $a = -16$, $b = 160$, and $c = 0$.\n\n2. **Determine the vertex:** The vertex of a parabola... |
DeepScaleR | [
{
"role": "user",
"content": "A particle starts at $(0,0,0)$ in three-dimensional space. Each second, it randomly selects one of the eight lattice points a distance of $\\sqrt{3}$ from its current location and moves to that point. What is the probability that, after two seconds, the particle is a distan... | math | {
"style": "rule",
"ground_truth": "3/8"
} | {
"index": 2498,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A particle starts from the origin on the number line, and at each step, it can move either 1 unit in the positive direction or 1 unit in the negative direction. After 10 steps, if the distance between the particle and the origin is 4, then the total number of distinct ways the ... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 2499,
"solution": ""
} |
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