data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first 1000 terms of a geometric sequence is 300. The sum of the first 2000 terms is 570. Find the sum of the first 3000 terms."
}
] | math | {
"style": "rule",
"ground_truth": "813"
} | {
"index": 34500,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first 2011 terms of a geometric sequence is 200. The sum of the first 4022 terms is 380. Find the sum of the first 6033 terms."
}
] | math | {
"style": "rule",
"ground_truth": "542"
} | {
"index": 34501,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first 2015 digits of the decimal part of the repeating decimal \\(0.0142857\\) is $\\qquad$"
}
] | math | {
"style": "rule",
"ground_truth": "9065"
} | {
"index": 34502,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first 3 terms of a geometric sequence $\\{a_n\\}$ is 13, and the sum of the first 6 terms is 65. Find $S_{12}$."
}
] | math | {
"style": "rule",
"ground_truth": "1105"
} | {
"index": 34503,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first \\( n \\) terms of a geometric sequence \\(\\left\\{a_{n}\\right\\}\\) with all real numbers is \\( S_{n} \\). If \\( S_{10} = 10 \\) and \\( S_{30} = 70 \\), then \\( S_{40} \\) is equal to:"
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 34504,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first and the third of three consecutive odd integers is 152. What is the value of the second integer?"
}
] | math | {
"style": "rule",
"ground_truth": "76"
} | {
"index": 34505,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first and third of three consecutive integers is 118. What is the value of the second integer?"
}
] | math | {
"style": "rule",
"ground_truth": "59"
} | {
"index": 34506,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first eighty positive odd integers subtracted from the sum of the first eighty positive even integers is"
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 34507,
"solution": "1. **Identify the pattern in the sequence of odd and even integers:**\n - The $n$-th odd integer can be represented as $2n - 1$.\n - The $n$-th even integer can be represented as $2n$.\n\n2. **Calculate the difference between the $n$-th even and $n$-th odd integer:**\n - The dif... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first fifty positive odd integers subtracted from the sum of the first fifty positive even integers, each decreased by 3, calculate the result."
}
] | math | {
"style": "rule",
"ground_truth": "-100"
} | {
"index": 34508,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first four terms of an arithmetic progression, as well as the sum of the first nine terms, are natural numbers. Additionally, the first term \\( b_{1} \\) of this progression satisfies the inequality \\( b_{1} \\leq \\frac{3}{4} \\). What is the greatest possible... | math | {
"style": "rule",
"ground_truth": "11/15"
} | {
"index": 34509,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first four terms of an arithmetic progression, as well as the sum of the first seven terms, are natural numbers. Furthermore, its first term \\(a_1\\) satisfies the inequality \\(a_1 \\leq \\frac{2}{3}\\). What is the greatest value that \\(a_1\\) can take?"
}
... | math | {
"style": "rule",
"ground_truth": "9/14"
} | {
"index": 34510,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first four terms of an arithmetic sequence is $10$. If the fifth term is $5$, what is the sixth term?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 34511,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first n terms of a geometric sequence $\\{a_n\\}$ is $S_n$. If for any positive integer $k$, it holds that $a_k = \\lim_{n\\to\\infty}(S_n - S_k)$, then the common ratio $q = \\boxed{\\frac{1}{2}}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 34512,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first n terms of the sequence $-1, 4, -7, 10, \\ldots, (-1)^{n}(3n-2)$ is given by $S_{n}$. Calculate $S_{11}+S_{20}$."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 34513,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first n terms of the sequence $\\{a_n\\}$ is $S_n$. If the terms of the sequence $\\{a_n\\}$ are arranged according to the following rule: $$\\frac {1}{2}, \\frac {1}{3}, \\frac {2}{3}, \\frac {1}{4}, \\frac {2}{4}, \\frac {3}{4}, \\frac {1}{5}, \\frac {2}{5}, \\... | math | {
"style": "rule",
"ground_truth": "\\frac{6}{7}"
} | {
"index": 34514,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first n terms of the sequence {a_n} is S_n = n^2 + n + 1, and b_n = (-1)^n a_n (n ∈ N^*). Determine the sum of the first 50 terms of the sequence {b_n}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 34515,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first thirteen terms of an arithmetic progression is $50\\%$ of the sum of the last thirteen terms of this progression. The sum of all terms of this progression, excluding the first three terms, is to the sum of all terms excluding the last three terms in the rat... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 34516,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first three terms of an arithmetic progression, as well as the sum of the first six terms, are natural numbers. Additionally, its first term \\( d_{1} \\) satisfies the inequality \\( d_{1} \\geqslant \\frac{1}{2} \\). What is the smallest possible value that \\(... | math | {
"style": "rule",
"ground_truth": "5/9"
} | {
"index": 34517,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the following seven numbers is exactly 19: $a_1 = 2.56$ , $a_2 = 2.61$ , $a_3 = 2.65$ , $a_4 = 2.71$ , $a_5 = 2.79$ , $a_6 = 2.82$ , $a_7 = 2.86$ . It is desired to replace each $a_i$ by an integer approximation $A_i$ , $1\\le i \\le 7$ , so that the sum of the $A_i$... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"index": 34518,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the following seven numbers is exactly 19: $a_1 = 2.56$, $a_2 = 2.61$, $a_3 = 2.65$, $a_4 = 2.71$, $a_5 = 2.79$, $a_6 = 2.82$, $a_7 = 2.86$. It is desired to replace each $a_i$ by an integer approximation $A_i$, $1\\le i \\le 7$, so that the sum of the $A_i$'s is als... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"index": 34519,
"solution": "If any of the approximations $A_i$ is less than 2 or more than 3, the error associated with that term will be larger than 1, so the largest error will be larger than 1. However, if all of the $A_i$ are 2 or 3, the largest error will be less than 1. So in the best case, we write 19 as ... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the greatest integer less than or equal to $x$ and the least integer greater than or equal to $x$ is $5$. The solution set for $x$ is"
}
] | math | {
"style": "rule",
"ground_truth": "\\{x \\mid 2 < x < 3\\}"
} | {
"index": 34520,
"solution": "1. **Understanding the Problem:**\n The problem asks us to find the set of values for $x$ such that the sum of the greatest integer less than or equal to $x$ (denoted $\\lfloor x \\rfloor$) and the least integer greater than or equal to $x$ (denoted $\\lceil x \\rceil$) equals 5.\n\... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the heights on the two equal sides of an isosceles triangle is equal to the height on the base. Find the sine of the base angle."
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{\\sqrt{15}}{4}$"
} | {
"index": 34521,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the house numbers on one side of a street from corner to corner is 117. What is the house number of the fifth house from the beginning of this section?"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 34522,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the interior numbers in the sixth row of Pascal's Triangle is 30. What is the sum of the interior numbers of the eighth row?"
}
] | math | {
"style": "rule",
"ground_truth": "126"
} | {
"index": 34523,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the largest number and the smallest number of a triple of positive integers $(x,y,z)$ is the power of the triple. Compute the sum of powers of all triples $(x,y,z)$ where $x,y,z \\leq 9$."
}
] | math | {
"style": "rule",
"ground_truth": "7290"
} | {
"index": 34524,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the lengths of all the edges of a cube is 60 cm. Find the number of cubic centimeters in the volume of the cube."
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"index": 34525,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the lengths of the twelve edges of a rectangular box is $140$, and the distance from one corner of the box to the farthest corner is $21$. What is the total surface area of the box?"
}
] | math | {
"style": "rule",
"ground_truth": "784"
} | {
"index": 34526,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the maximum and minimum values of the function $y=2\\sin \\left( \\frac{\\pi x}{6}- \\frac{\\pi}{3}\\right)$ where $(0\\leqslant x\\leqslant 9)$ is to be determined."
}
] | math | {
"style": "rule",
"ground_truth": "2-\\sqrt{3}"
} | {
"index": 34527,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the non-negative numbers \\(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}\\) is 1. Let \\(M\\) be the maximum of the quantities \\(a_{1} + a_{2} + a_{3}, a_{2} + a_{3} + a_{4}, a_{3} + a_{4} + a_{5}, a_{4} + a_{5} + a_{6}, a_{5} + a_{6} + a_{7}\\).\n\nHow small can... | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"index": 34528,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the numbers 300, 2020, and 10001 is"
}
] | math | {
"style": "rule",
"ground_truth": "12321"
} | {
"index": 34529,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the numerical coefficients in the binomial $(2a+2b)^7$ is $\\boxed{32768}$."
}
] | math | {
"style": "rule",
"ground_truth": "16384"
} | {
"index": 34530,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the numerical coefficients in the complete expansion of $(x^2 - 2xy + y^2)^7$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 34531,
"solution": "1. **Identify the Expression**: The given expression is $(x^2 - 2xy + y^2)^7$. \n\n2. **Recognize the Binomial Expansion**: Observe that $x^2 - 2xy + y^2$ can be rewritten as $(x-y)^2$. Therefore, the expression becomes $((x-y)^2)^7 = (x-y)^{14}$.\n\n3. **Apply the Binomial Theorem**:... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the numerical coefficients in the expansion of the binomial $(a+b)^6$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 34532,
"solution": "To find the sum of the numerical coefficients in the expansion of the binomial $(a+b)^6$, we can substitute $a = 1$ and $b = 1$ into the binomial. This simplifies the expression to just the sum of the coefficients, as each term in the expansion will have the form $\\binom{6}{k} a^{6-k... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the numerical coefficients of all the terms in the expansion of $(x-2y)^{18}$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 34533,
"solution": "1. **Identify the Polynomial**: We are given the polynomial $(x-2y)^{18}$ and asked to find the sum of the numerical coefficients of all the terms in its expansion.\n\n2. **Understanding the Sum of Coefficients**: The sum of the coefficients of a polynomial when all variables are repl... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the positive divisors of a positive integer of the form $2^i3^j$ is equal to $600$. What is $i + j$?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 34534,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the real values of $x$ satisfying the equality $|x+2|=2|x-2|$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "6\\frac{2}{3}"
} | {
"index": 34535,
"solution": "To solve the equation $|x+2| = 2|x-2|$, we need to consider the different cases for the absolute values based on the sign of the expressions inside them.\n\n#### Case 1: $x + 2 \\geq 0$ and $x - 2 \\geq 0$\nThis implies $x \\geq 2$. The equation becomes:\n\\[ x+2 = 2(x-2) \\]\n\\[ x+2... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the series $\\frac{3}{4} + \\frac{5}{8} + \\frac{9}{16} + \\frac{17}{32} + \\frac{33}{64} + \\frac{65}{128} - 3.5$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-1}{128}"
} | {
"index": 34536,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the squares of four consecutive positive integers is 9340. What is the sum of the cubes of these four integers?"
}
] | math | {
"style": "rule",
"ground_truth": "457064"
} | {
"index": 34537,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the squares of the first ten binomial coefficients ${C}_{2}^{2}+{C}_{3}^{2}+{C}_{4}^{2}+\\cdots +{C}_{10}^{2}$ can be found."
}
] | math | {
"style": "rule",
"ground_truth": "165"
} | {
"index": 34538,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the squares of three consecutive positive integers is 7805. What is the sum of the cubes of the three original integers?"
}
] | math | {
"style": "rule",
"ground_truth": "398259"
} | {
"index": 34539,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the squares of two positive integers is 193. The product of the two integers is 84. What is the sum of the two integers?"
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 34540,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the squares of two positive integers is 90. The product of the two integers is 27. What is the sum of the two integers?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 34541,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the terms in an infinite geometric series is 15, and the sum of their squares is 45. Find the first term."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 34542,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the three largest natural divisors of a natural number \\( N \\) is 10 times the sum of its three smallest natural divisors. Find all possible values of \\( N \\)."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 34543,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of three consecutive one-digit, positive, odd integers is one-seventh of the product of the same three integers. What is the middle integer when the three integers are listed in ascending order?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 34544,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of three different numbers is 100. The two larger numbers differ by 8 and the two smaller numbers differ by 5. What is the value of the largest number?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{121}{3}"
} | {
"index": 34545,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of three different numbers is 67. The two larger numbers differ by 7 and the two smaller numbers differ by 3. What is the value of the largest number?"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 34546,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of three different numbers is 75. The two larger numbers differ by 5 and the two smaller numbers differ by 4. What is the value of the largest number?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{89}{3}"
} | {
"index": 34547,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of three numbers $a$, $b$, and $c$ is 99. If we increase $a$ by 6, decrease $b$ by 6 and multiply $c$ by 5, the three resulting numbers are equal. What is the value of $b$?"
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"index": 34548,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of three numbers $a, b$ and $c$ is 60. If we decrease $a$ by 7, we get the value $N$. If we increase $b$ by 7, we get the value $N$. If we multiply $c$ by 7, we also get the value $N$. What is the value of $N$?"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 34549,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of three numbers $x$ ,$y$, $z$ is 165. When the smallest number $x$ is multiplied by 7, the result is $n$. The value $n$ is obtained by subtracting 9 from the largest number $y$. This number $n$ also results by adding 9 to the third number $z$. What is the product of th... | math | {
"style": "rule",
"ground_truth": "64,\\!328"
} | {
"index": 34550,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of three numbers $x$, $y$, and $z$ is 150. If we increase $x$ by 7, decrease $y$ by 12 and multiply $z$ by 4, the three resulting numbers are equal. What is the value of $y$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{688}{9}"
} | {
"index": 34551,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of three numbers \\( a \\), \\( b \\), and \\( c \\) is 150. If we increase \\( a \\) by 10, decrease \\( b \\) by 5, and multiply \\( c \\) by 7, the three resulting numbers are equal. What is the value of \\( b \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "77.\\overline{3}"
} | {
"index": 34552,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of three numbers \\(a\\), \\(b\\), and \\(c\\) is 150. If we increase \\(a\\) by 10, decrease \\(b\\) by 3, and multiply \\(c\\) by 4, the three resulting numbers are equal. What is the value of \\(b\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{655}{9}"
} | {
"index": 34553,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of three numbers \\(x, y,\\) and \\(z\\) is 120. If we decrease \\(x\\) by 10, we get the value \\(M\\). If we increase \\(y\\) by 10, we also get the value \\(M\\). If we multiply \\(z\\) by 10, we also get the value \\(M\\). What is the value of \\(M\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{400}{7}"
} | {
"index": 34554,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of three numbers is $20$. The first is four times the sum of the other two. The second is seven times the third. What is the product of all three?"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 34555,
"solution": "\nLet the three numbers be $x$, $y$, and $z$. According to the problem, we have the following conditions:\n1. $x + y + z = 20$\n2. $x = 4(y + z)$\n3. $y = 7z$\n\nWe can use these equations to find the values of $x$, $y$, and $z$.\n\n#### Step 1: Substitute $y$ in terms of $z$ into the... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of three numbers is $98$. The ratio of the first to the second is $\\frac {2}{3}$, \nand the ratio of the second to the third is $\\frac {5}{8}$. The second number is:"
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 34556,
"solution": "1. Let the three numbers be $a$, $b$, and $c$. According to the problem, the sum of these numbers is given by:\n \\[ a + b + c = 98 \\]\n\n2. The ratio of the first number to the second number is $\\frac{2}{3}$, which can be expressed as:\n \\[ \\frac{a}{b} = \\frac{2}{3} \\]\n ... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of three positive numbers is 1, and none of the numbers is greater than twice any other number. What is the minimum product of the three numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "1/32"
} | {
"index": 34557,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of two angles of a triangle is $\\frac{6}{5}$ of a right angle, and one of these two angles is $30^{\\circ}$ larger than the other. What is the degree measure of the largest angle in the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 34558,
"solution": "1. **Identify the given information and the problem statement**: We are given that the sum of two angles of a triangle is $\\frac{6}{5}$ of a right angle, and one of these angles is $30^\\circ$ larger than the other. We need to find the degree measure of the largest angle in the trian... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of two fractions is $\\frac{11}{12}$ and their product is $\\frac{1}{6}$. What is the lesser of the two fractions? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 34559,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of two natural numbers and their greatest common divisor is equal to their least common multiple. Determine the ratio of the two numbers."
}
] | math | {
"style": "rule",
"ground_truth": "3 : 2"
} | {
"index": 34560,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of two natural numbers is $17402$. One of the two numbers is divisible by $10$. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "14238"
} | {
"index": 34561,
"solution": "1. **Identify the relationship between the two numbers**: Given that one number is divisible by $10$ and removing its units digit (which is $0$) gives the other number, we can denote the smaller number as $a$ and the larger number as $10a$.\n\n2. **Set up the equation for their sum**:... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of two nonzero natural numbers is 210, and their least common multiple is 1547. What is their product? $\\qquad$"
}
] | math | {
"style": "rule",
"ground_truth": "10829"
} | {
"index": 34562,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of two nonzero real numbers is $4$ times their product. What is the sum of the reciprocals of the two numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 34563,
"solution": "1. **Define the variables**: Let the two nonzero real numbers be $x$ and $y$.\n\n2. **Translate the given condition into an equation**: We are given that the sum of $x$ and $y$ is $4$ times their product. This can be written as:\n \\[\n x + y = 4xy\n \\]\n\n3. **Manipulate the e... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of two nonzero real numbers is 4 times their product. What is the sum of the reciprocals of the two numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 34564,
"solution": "Let the two nonzero real numbers be $x$ and $y$. According to the problem, the sum of these two numbers is 4 times their product. This can be expressed as:\n\\[ x + y = 4xy. \\]\n\nWe are asked to find the sum of the reciprocals of $x$ and $y$. Let's denote the reciprocals by $a = \\f... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of two numbers $x$ and $y$ is 399, and the value of the fraction $\\frac{x}{y}$ is 0.9. What is the value of $y - x$?"
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 34565,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of two numbers is $10$; their product is $20$. The sum of their reciprocals is:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 34566,
"solution": "Given the equations:\n1. \\(x + y = 10\\)\n2. \\(xy = 20\\)\n\nWe are asked to find the sum of the reciprocals of \\(x\\) and \\(y\\), which is \\(\\frac{1}{x} + \\frac{1}{y}\\).\n\nUsing the identity for the sum of reciprocals:\n\\[\n\\frac{1}{x} + \\frac{1}{y} = \\frac{x + y}{xy}\n\... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of two numbers is $12$ and their difference is $20$. What is the smaller number?"
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 34567,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of two numbers is $19$ and their difference is $5$. What is their product?"
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"index": 34568,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of two numbers is $30$. If we double the larger number, and subtract three times the smaller number, the result is 5. What is the positive difference between the two numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 34569,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of two numbers is $30$. Their difference is $4$. What is the larger of the two numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 34570,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of two numbers is $40$. If we triple the larger number and subtract four times the smaller number, the result is $10$. What is the positive difference between the two numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "8.57"
} | {
"index": 34571,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of two numbers is $45$. Their difference is $3$. What is the lesser of the two numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 34572,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of two numbers is $S$. Suppose $3$ is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "2S + 12"
} | {
"index": 34573,
"solution": "1. Let the two numbers be $a$ and $b$. According to the problem, their sum is $S$, so we have:\n \\[\n a + b = S\n \\]\n\n2. According to the problem, $3$ is added to each number, resulting in the numbers $a+3$ and $b+3$.\n\n3. Each of these new numbers is then doubled. Therefor... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of two numbers is 22. Their difference is 4. What is the greater of the two numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 34574,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of two numbers is 25 and their product is 126. What is the absolute value of the difference of the two numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 34575,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of two numbers is 50 and their difference is 6. What is their product?"
}
] | math | {
"style": "rule",
"ground_truth": "616"
} | {
"index": 34576,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of two numbers is 6. The difference of their squares is 12. What is the positive difference of the two numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 34577,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of two numbers is \\( t \\) and the positive difference between the squares of these two numbers is 208. What is the larger of the two numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "53"
} | {
"index": 34578,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of two positive integers $a$ and $b$ is 1001. What is the largest possible value of $\\gcd(a,b)$?"
}
] | math | {
"style": "rule",
"ground_truth": "143"
} | {
"index": 34579,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of two positive integers is 50 and their difference is 12. What is the value of the positive difference of the squares of the integers?"
}
] | math | {
"style": "rule",
"ground_truth": "600"
} | {
"index": 34580,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller number?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 34581,
"solution": "1. Let $a$ and $b$ be two positive numbers such that $a > b$. According to the problem, the sum of these two numbers is $5$ times their difference. We can express this relationship mathematically as:\n \\[\n a + b = 5(a - b)\n \\]\n\n2. Expanding and rearranging the equation:\n ... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of two prime numbers is $85$. What is the product of these two prime numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "166"
} | {
"index": 34582,
"solution": "1. **Identify the nature of the prime numbers**: The sum of two prime numbers is given as $85$, which is an odd number. Since the sum of two odd numbers is even and the sum of two even numbers is even, one of these primes must be even and the other must be odd (because the sum of an e... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of two sides of a rectangle is 11, and the sum of three sides is 19.5. Find the product of all possible distinct values of the perimeter of such a rectangle."
}
] | math | {
"style": "rule",
"ground_truth": "15400"
} | {
"index": 34583,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum other than $11$ which occurs with the same probability when all 8 dice are rolled is equal to what value."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 34584,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum to infinity of the terms of an infinite geometric progression is $6$. The sum of the first two terms is $4\\frac{1}{2}$. The first term of the progression is:"
}
] | math | {
"style": "rule",
"ground_truth": "9 or 3"
} | {
"index": 34585,
"solution": "1. **Identify the given information and the formula to use:**\n - The sum to infinity of the terms of an infinite geometric progression is given as $6$.\n - The sum of the first two terms is $4\\frac{1}{2}$, which can be written as $\\frac{9}{2}$.\n - The sequence can be express... |
DeepScaleR | [
{
"role": "user",
"content": "The sum to infinity of the terms of an infinite geometric progression is 10. The sum of the first two terms is 7. Compute the first term of the progression."
}
] | math | {
"style": "rule",
"ground_truth": "10\\left(1 + \\sqrt{\\frac{3}{10}}\\right)"
} | {
"index": 34586,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum\\[\\sum_{x=2}^{44} 2\\sin{x}\\sin{1}[1 + \\sec (x-1) \\sec (x+1)]\\]can be written in the form $\\sum_{n=1}^{4} (-1)^n \\frac{\\Phi(\\theta_n)}{\\Psi(\\theta_n)}$, where $\\Phi,\\, \\Psi$ are trigonometric functions and $\\theta_1,\\, \\theta_2, \\, \\theta_3, \\, \\the... | math | {
"style": "rule",
"ground_truth": "92"
} | {
"index": 34587,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 34588,
"solution": "Let the three numbers be $a$, $b$, and $c$, where $a < b < c$. We are given the sums of these numbers taken in pairs:\n1. $a + b = 12$\n2. $a + c = 17$\n3. $b + c = 19$\n\nWe aim to find the value of $b$, the middle number. To do this, we can manipulate the given equations to isolate ... |
DeepScaleR | [
{
"role": "user",
"content": "The supermarket sold two types of goods, both for a total of 660 yuan. One item made a profit of 10%, while the other suffered a loss of 10%. Express the original total price of these two items using a formula."
}
] | math | {
"style": "rule",
"ground_truth": "1333\\frac{1}{3}"
} | {
"index": 34589,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The surface area of a cube is 24. What is the volume of the cube?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 34590,
"solution": "A cube has six identical faces. If the surface area of a cube is 24, the area of each face is $\\frac{24}{6}=4$. Since each face of this cube is a square with area 4, the edge length of the cube is $\\sqrt{4}=2$. Thus, the volume of the cube is $2^{3}$ which equals 8."
} |
DeepScaleR | [
{
"role": "user",
"content": "The surface area of a particular sphere is $324\\pi\\text{ cm}^2$. What is the volume, in cubic centimeters, of the sphere? Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "972\\pi"
} | {
"index": 34591,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The surface area of a sphere with edge lengths 3, 4, and 5 on the rectangular solid is what?"
}
] | math | {
"style": "rule",
"ground_truth": "50\\pi"
} | {
"index": 34592,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The surface area of a sphere with radius $r$ is $4\\pi r^2$. Including the area of its circular base, what is the total surface area of a hemisphere with radius 6 cm? Express your answer in terms of $\\pi$.\n\n[asy]\n\nimport markers;\nsize(150);\nimport geometry;\ndraw((0,-7)-... | math | {
"style": "rule",
"ground_truth": "108\\pi"
} | {
"index": 34593,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The surface area of the circumscribed sphere of cube \\( K_1 \\) is twice the surface area of the inscribed sphere of cube \\( K_2 \\). Let \\( V_1 \\) denote the volume of the inscribed sphere of cube \\( K_1 \\), and \\( V_2 \\) denote the volume of the circumscribed sphere o... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{2}}{27}"
} | {
"index": 34594,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The surface area ratio of an equilateral cone to a sphere is 3:1. How many times larger is the volume of the equilateral cone compared to the volume of the sphere?"
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 34595,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The surface of a 3 x 3 x 3 Rubik's Cube consists of 54 cells. What is the maximum number of cells you can mark such that the marked cells do not share any vertices?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 34596,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The surface of a 3x3x3 Rubik's cube consists of 54 cells. What is the maximum number of cells that can be marked such that no two marked cells share a common vertex?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 34597,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The surface of a clock is circular, and on its circumference, there are 12 equally spaced points representing the hours. Calculate the total number of rectangles that can have these points as vertices."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 34598,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The symbol $R_k$ stands for an integer whose base-ten representation is a sequence of $k$ ones. For example, $R_3=111, R_5=11111$, etc. When $R_{24}$ is divided by $R_4$, the quotient $Q=R_{24}/R_4$ is an integer whose base-ten representation is a sequence containing only ones ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 34599,
"solution": "1. **Expression for $R_k$:** \n The number $R_k$ is defined as an integer consisting of $k$ ones in its decimal representation. We can express $R_k$ as:\n \\[\n R_k = 111\\ldots1 \\quad (\\text{$k$ times})\n \\]\n This can be written as a sum of powers of 10:\n \\[\n R_... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.