data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "The number 2017 has 7 ones and 4 zeros in binary representation. When will the nearest year come where the binary representation of the year has no more ones than zeros? (Enter the year.)"
}
] | math | {
"style": "rule",
"ground_truth": "2048"
} | {
"index": 33300,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number 21 is the sum of two consecutive integers 10 and 11. What is the largest number of positive, consecutive integers whose sum is 21?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 33301,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number 210 is the product of two consecutive positive integers and is also the product of three consecutive integers. What is the sum of those five integers?"
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"index": 33302,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number 236! ends in how many zeros when expressed as an integer?"
}
] | math | {
"style": "rule",
"ground_truth": "57"
} | {
"index": 33303,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number 25 is expressed as the sum of positive integers \\(x_{1}, x_{2}, \\cdots, x_{k}\\), where \\(k \\leq 25\\). What is the maximum value of the product of \\(x_{1}, x_{2}, x_{3}, \\cdots\\), and \\(x_{k}\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "8748"
} | {
"index": 33304,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number 3003 is the only number known to appear eight times in Pascal's triangle, at positions $\\binom{3003}{1},\\binom{3003}{3002},\\binom{a}{2},\\binom{a}{a-2},\\binom{15}{b},\\binom{15}{15-b},\\binom{14}{6},\\binom{14}{8}$. Compute $a+b(15-b)$."
}
] | math | {
"style": "rule",
"ground_truth": "128"
} | {
"index": 33305,
"solution": "We first solve for $a$. Note that $3003=3 \\cdot 7 \\cdot 11 \\cdot 13$. We have $3003=\\binom{a}{2}=\\frac{a(a-1)}{2} \\approx \\frac{a^{2}}{2}$. This means we can estimate $a \\approx \\sqrt{3003 \\cdot 2}$, so $a$ is a little less than 80. Furthermore, $11 \\mid 2 \\cdot 3003=a(a-1... |
DeepScaleR | [
{
"role": "user",
"content": "The number 5.6 may be expressed uniquely (ignoring order) as a product $\\underline{a} \\cdot \\underline{b} \\times \\underline{c} . \\underline{d}$ for digits $a, b, c, d$ all nonzero. Compute $\\underline{a} \\cdot \\underline{b}+\\underline{c} . \\underline{d}$."
}
] | math | {
"style": "rule",
"ground_truth": "5.1"
} | {
"index": 33306,
"solution": "We want $\\overline{a b} \\times \\overline{c d}=560=2^{4} \\times 5 \\times 7$. To avoid a zero digit, we need to group the 5 with the 7 to get 3.5 and 1.6 , and our answer is $3.5+1.6=5.1$."
} |
DeepScaleR | [
{
"role": "user",
"content": "The number 519 is formed using the digits 5, 1, and 9. The three digits of this number are rearranged to form the largest possible and then the smallest possible three-digit numbers. What is the difference between these largest and smallest numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "792"
} | {
"index": 33307,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number 74 can be factored as 2(37), so 74 is said to have two distinct prime factors. How many distinct prime factors does 210 have?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 33308,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number 770 is written on a blackboard. Melody repeatedly performs moves, where a move consists of subtracting either 40 or 41 from the number on the board. She performs moves until the number is not positive, and then she stops. Let $N$ be the number of sequences of moves t... | math | {
"style": "rule",
"ground_truth": "318"
} | {
"index": 33309,
"solution": "Notice that if we use the 41 move nine times or less, we will have to make a total of $\\left\\lceil\\frac{770}{40}\\right\\rceil=20$ moves, and if we use it ten times or more, we will have to make a total of $\\left\\lfloor\\frac{770}{40}\\right\\rfloor=19$ moves. So, doing casework ... |
DeepScaleR | [
{
"role": "user",
"content": "The number \\( 20! = 1 \\cdot 2 \\cdot \\ldots \\cdot 20 = 2432902008176640000 \\) has 41,040 natural divisors. How many of them are odd?"
}
] | math | {
"style": "rule",
"ground_truth": "2160"
} | {
"index": 33310,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number \\( A \\) in decimal notation has the form \\( A = \\overline{7a631b} \\), where \\( a \\) and \\( b \\) are non-zero digits. The number \\( B \\) is obtained by summing all the six-digit, distinct numbers derived from \\( A \\) by cyclic permutations of its digits (... | math | {
"style": "rule",
"ground_truth": "796317"
} | {
"index": 33311,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number \\( C \\) is defined as the sum of all the positive integers \\( n \\) such that \\( n-6 \\) is the second largest factor of \\( n \\). What is the value of \\( 11C \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "308"
} | {
"index": 33312,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number \\( N = 3^{16} - 1 \\) has a divisor of 193. It also has some divisors between 75 and 85 inclusive. What is the sum of these divisors?"
}
] | math | {
"style": "rule",
"ground_truth": "247"
} | {
"index": 33313,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number \\( N \\) has the smallest positive divisor 1, the second largest positive divisor \\( k \\), and the third largest positive divisor \\( m \\). Moreover, \\( k^k + m^m = N \\). What is \\( N \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "260"
} | {
"index": 33314,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number \\( N \\) is of the form \\( p^{\\alpha} q^{\\beta} r^{\\gamma} \\), where \\( p, q, r \\) are prime numbers, and \\( p q - r = 3, p r - q = 9 \\). Additionally, the numbers \\( \\frac{N}{p}, \\frac{N}{q}, \\frac{N}{r} \\) respectively have 20, 12, and 15 fewer divis... | math | {
"style": "rule",
"ground_truth": "857500"
} | {
"index": 33315,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number \\( a \\) is a root of the equation \\( x^{11} + x^{7} + x^{3} = 1 \\). Specify all natural values of \\( n \\) for which the equality \\( a^{4} + a^{3} = a^{n} + 1 \\) holds."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 33316,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number \\( n \\) has exactly six divisors (including 1 and itself). These divisors are arranged in ascending order. It turns out that the third divisor is seven times larger than the second one, and the fourth divisor is 10 larger than the third one. What is \\( n \\)?"
}... | math | {
"style": "rule",
"ground_truth": "2891"
} | {
"index": 33317,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number \\( x \\) is such that \\( \\log _{2}\\left(\\log _{4} x\\right) + \\log _{4}\\left(\\log _{8} x\\right) + \\log _{8}\\left(\\log _{2} x\\right) = 1 \\). Find the value of the expression \\( \\log _{4}\\left(\\log _{2} x\\right) + \\log _{8}\\left(\\log _{4} x\\right... | math | {
"style": "rule",
"ground_truth": "0.87"
} | {
"index": 33318,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number \\( x \\) is such that among the four numbers \\( x - \\sqrt{2} \\), \\( x - \\frac{1}{x} \\), \\( x + \\frac{1}{x} \\), \\( x^2 + 2 \\sqrt{2} \\), exactly one is not an integer. Find all such \\( x \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} - 1"
} | {
"index": 33319,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number \\(abcde\\) has five distinct digits, each different from zero. When this number is multiplied by 4, the result is a five-digit number \\(edcba\\), which is the reverse of \\(abcde\\). What is the value of \\(a + b + c + d + e\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 33320,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number \\(n\\) is a three-digit positive integer and is the product of the three factors \\(x\\), \\(y\\), and \\(5x+2y\\), where \\(x\\) and \\(y\\) are integers less than 10 and \\((5x+2y)\\) is a composite number. What is the largest possible value of \\(n\\) given these... | math | {
"style": "rule",
"ground_truth": "336"
} | {
"index": 33321,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number halfway between $\\dfrac{1}{8}$ and $\\dfrac{1}{3}$ is\nA) $\\dfrac{11}{48}$\nB) $\\dfrac{11}{24}$\nC) $\\dfrac{5}{24}$\nD) $\\dfrac{1}{4}$\nE) $\\dfrac{1}{5}$"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{11}{48}"
} | {
"index": 33322,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number halfway between $\\frac{1}{6}$ and $\\frac{1}{4}$ is"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{24}"
} | {
"index": 33323,
"solution": "To find the number halfway between two numbers, we calculate the average of the two numbers. The formula for the average of two numbers $a$ and $b$ is:\n\\[\n\\text{Average} = \\frac{a + b}{2}\n\\]\n\nGiven the numbers $\\frac{1}{6}$ and $\\frac{1}{4}$, we first find a common denomina... |
DeepScaleR | [
{
"role": "user",
"content": "The number halfway between $\\frac{1}{8}$ and $\\frac{1}{10}$ is"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"index": 33324,
"solution": "\n#### Step-by-step Calculation:\n\n1. **Identify the numbers between which the midpoint is to be found:**\n Given numbers are $\\frac{1}{8}$ and $\\frac{1}{10}$.\n\n2. **Convert fractions to a common denominator:**\n \\[\n \\frac{1}{8} = \\frac{5}{40}, \\quad \\frac{1}{10} = \\... |
DeepScaleR | [
{
"role": "user",
"content": "The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, $30 = 6 \\times 5$. What is the missing number in the top row?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 33325,
"solution": "Let's analyze the problem step by step, using the given information and setting up equations accordingly.\n\n1. **Understanding the structure**: The number in each box is the product of the numbers in the two boxes that touch it in the row above. We are given that $30 = 6 \\times 5$.\... |
DeepScaleR | [
{
"role": "user",
"content": "The number obtained from the last two nonzero digits of $70!$ is equal to $n$. Calculate the value of $n$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 33326,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number obtained from the last two nonzero digits of $80!$ is equal to $n$. Find the value of $n$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 33327,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number obtained from the last two nonzero digits of $90!$ is equal to $n$. What is $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 33328,
"solution": "1. **Count the number of factors of 10 in $90!$:**\n The number of factors of 10 in $90!$ is determined by the number of factors of 5, as there are more factors of 2 than 5. We calculate this using the formula for the number of factors of a prime $p$ in $n!$:\n \\[\n \\left\\lfl... |
DeepScaleR | [
{
"role": "user",
"content": "The number of 4-digit integers with distinct digits, whose first and last digits' absolute difference is 2, is between 1000 and 9999."
}
] | math | {
"style": "rule",
"ground_truth": "840"
} | {
"index": 33329,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of books issued from the library to readers constitutes $\\frac{1}{16}$ of the number of books on the shelves. After transferring 2000 books from the library to the reading room, the number of books absent from the shelves became $\\frac{1}{15}$ of the number of book... | math | {
"style": "rule",
"ground_truth": "544000"
} | {
"index": 33330,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of cans in the layers of a display in a supermarket form an arithmetic sequence. The bottom layer has 28 cans; the next layer has 25 cans and so on until there is one can at the top of the display. How many cans are in the entire display?"
}
] | math | {
"style": "rule",
"ground_truth": "145"
} | {
"index": 33331,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of children in the families $A$, $B$, $C$, $D$, and $E$ are as shown in the table below:\n\n| | $A$ | $B$ | $C$ | $D$ | $E$ |\n|---------|-----|-----|-----|-----|-----|\n| Boys | $0$ | $1$ | $0$ | $1$ | $1$ |\n| Girls | $0$ | $0$ | $1$ | $1$ | $2$ |\n\n$... | math | {
"style": "rule",
"ground_truth": "\\frac{6}{5}"
} | {
"index": 33332,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of children in the families $A$, $B$, $C$, $D$, and $E$ are shown in the table below:\n\n| | $A$ | $B$ | $C$ | $D$ | $E$ |\n|---------|-----|-----|-----|-----|-----|\n| Boys | $0$ | $1$ | $0$ | $1$ | $1$ |\n| Girls | $0$ | $0$ | $1$ | $1$ | $2$ |\n\n$(1)... | math | {
"style": "rule",
"ground_truth": "\\frac{6}{5}"
} | {
"index": 33333,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of circular pipes with an inside diameter of $1$ inch which will carry the same amount of water as a pipe with an inside diameter of $6$ inches is:"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 33334,
"solution": "1. **Assumption of Equal Height**: We assume that all pipes have the same height, which allows us to focus on comparing their cross-sectional areas to determine their water carrying capacities.\n\n2. **Cross-Sectional Area of the Larger Pipe**:\n - The diameter of the larger pipe is... |
DeepScaleR | [
{
"role": "user",
"content": "The number of degrees in the measures of the interior angles of a convex pentagon are five consecutive integers. What is the number of degrees in the largest angle?"
}
] | math | {
"style": "rule",
"ground_truth": "110^\\circ"
} | {
"index": 33335,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of diagonals of a regular polygon is subtracted from the number of sides of the polygon and the result is zero. What is the number of sides of this polygon?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 33336,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of diagonals that can be drawn in a polygon of 100 sides is:"
}
] | math | {
"style": "rule",
"ground_truth": "4850"
} | {
"index": 33337,
"solution": "To find the number of diagonals in a polygon, we use the formula:\n\\[\n\\frac{n(n-3)}{2}\n\\]\nwhere $n$ is the number of sides of the polygon. This formula arises because each vertex of the polygon can connect to $n-3$ other vertices to form a diagonal (it cannot connect to itself o... |
DeepScaleR | [
{
"role": "user",
"content": "The number of different arrangements of $6$ rescue teams to $3$ disaster sites, where site $A$ has at least $2$ teams and each site is assigned at least $1$ team."
}
] | math | {
"style": "rule",
"ground_truth": "360"
} | {
"index": 33338,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of different arrangements possible for 6 acts, if the original sequence of the 4 acts remains unchanged."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 33339,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of digits in $4^{16}5^{25}$ (when written in the usual base $10$ form) is"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 33340,
"solution": "1. **Rewrite the expression using properties of exponents:** \n The given expression is $4^{16}5^{25}$. We can express $4$ as $2^2$, so:\n \\[\n 4^{16} = (2^2)^{16} = 2^{32}\n \\]\n Therefore, the expression becomes:\n \\[\n 4^{16}5^{25} = 2^{32}5^{25}\n \\]\n\n2. **C... |
DeepScaleR | [
{
"role": "user",
"content": "The number of distinct even numbers that can be formed using the digits 0, 1, 2, and 3."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 33341,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of distinct ordered pairs $(x,y)$ where $x$ and $y$ have positive integral values satisfying the equation $x^4y^4-10x^2y^2+9=0$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 33342,
"solution": "1. **Rewrite the given equation**: Start by rewriting the equation $x^4y^4 - 10x^2y^2 + 9 = 0$ in terms of a new variable $a = (xy)^2$. This gives us:\n \\[\n a^2 - 10a + 9 = 0\n \\]\n\n2. **Factorize the quadratic equation**: The quadratic equation in $a$ can be factorized as:\... |
DeepScaleR | [
{
"role": "user",
"content": "The number of distinct pairs $(x,y)$ of real numbers satisfying both of the following equations: \n\\[x=x^2+y^2\\] \\[y=2xy\\] \nis"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 33343,
"solution": "We are given the system of equations:\n1. \\(x = x^2 + y^2\\)\n2. \\(y = 2xy\\)\n\nWe will analyze this system by considering different cases.\n\n#### Case 1: \\(y = 0\\)\nSubstituting \\(y = 0\\) into the first equation:\n\\[ x = x^2 + 0^2 \\]\n\\[ x = x^2 \\]\n\\[ x^2 - x = 0 \\]\n\... |
DeepScaleR | [
{
"role": "user",
"content": "The number of distinct pairs of integers $(x, y)$ such that $0<x<y$ and $\\sqrt{1984}=\\sqrt{x}+\\sqrt{y}$ is"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 33344,
"solution": "1. **Simplify the given equation**: We start by simplifying $\\sqrt{1984}$. We factorize 1984 as $1984 = 64 \\times 31 = 8^2 \\times 31$. Therefore, $\\sqrt{1984} = \\sqrt{8^2 \\times 31} = 8\\sqrt{31}$.\n\n2. **Rewrite the equation**: The equation $\\sqrt{1984} = \\sqrt{x} + \\sqrt{y... |
DeepScaleR | [
{
"role": "user",
"content": "The number of distinct points common to the curves $x^2+4y^2=1$ and $4x^2+y^2=4$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 33345,
"solution": "1. **Identify the curves**: The given equations are $x^2 + 4y^2 = 1$ and $4x^2 + y^2 = 4$. These represent an ellipse and another ellipse respectively.\n\n2. **Substitute variables**: Let $a = x^2$ and $b = y^2$. This substitution transforms the equations into:\n \\[\n a + 4b = 1 ... |
DeepScaleR | [
{
"role": "user",
"content": "The number of distinct points common to the graphs of $x^2+y^2=9$ and $y^2=9$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 33346,
"solution": "1. **Identify the equations**: We are given two equations:\n - Circle: \\(x^2 + y^2 = 9\\)\n - Horizontal lines: \\(y^2 = 9\\)\n\n2. **Solve the second equation**: To find the values of \\(y\\), we solve \\(y^2 = 9\\):\n \\[\n y = \\pm 3\n \\]\n\n3. **Substitute \\(y\\) valu... |
DeepScaleR | [
{
"role": "user",
"content": "The number of distinct points in the $xy$-plane common to the graphs of $(x+y-5)(2x-3y+5)=0$ and $(x-y+1)(3x+2y-12)=0$ is"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 33347,
"solution": "To find the number of distinct points common to the graphs of $(x+y-5)(2x-3y+5)=0$ and $(x-y+1)(3x+2y-12)=0$, we need to solve the systems of equations formed by each factor being zero.\n\n#### System 1: $x+y=5$ and $x-y=-1$\n1. Adding these two equations:\n \\[\n (x+y) + (x-y) = ... |
DeepScaleR | [
{
"role": "user",
"content": "The number of elderly employees in a sample of 32 young employees from a workplace with a total of 430 employees, 160 of whom are young and the number of middle-aged employees is twice the number of elderly employees, can be found by determining the ratio of young employees in ... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 33348,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of elderly employees in the sampled group is what fraction of the number of young employees in the sampled group if the sampled group consists of the employees of a certain unit, which has a total of 430 employees, among which there are 160 young employees, and the n... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 33349,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of elements in a finite set $P$ is denoted as $\\text{card}(P)$. It is known that $\\text{card}(M) = 10$, $A \\subseteq M$, $B \\subseteq M$, $A \\cap B = \\emptyset$, and $\\text{card}(A) = 2$, $\\text{card}(B) = 3$. If the set $X$ satisfies $A \\subseteq X \\subset... | math | {
"style": "rule",
"ground_truth": "256"
} | {
"index": 33350,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of four-digit even numbers formed without repeating digits from the numbers $2$, $0$, $1$, $7$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 33351,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of gallons of coffee a mathematician drinks on any given day is inversely proportional to how much sleep he gets the night before. On Monday, he got 9 hours of sleep and drank 2 gallons of coffee. On Tuesday, he got 6 hours of sleep. How many gallons of coffee did he... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 33352,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of games won by five baseball teams are displayed on a chart. The team names are missing, and we have several clues to help identify them:\n\n1. The Sharks won fewer games than the Raptors.\n2. The Royals won more games than the Dragons, but fewer games than the Knig... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 33353,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of games won by five basketball teams is shown in a bar chart. The teams' names are not displayed. The following clues provide information about the teams:\n\n1. The Hawks won more games than the Falcons.\n2. The Warriors won more games than the Knights, but fewer ga... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 33354,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of games won by five cricket teams is displayed in a chart, but the team names are missing. Use the clues below to determine how many games the Hawks won:\n\n1. The Hawks won fewer games than the Falcons.\n2. The Raiders won more games than the Wolves, but fewer game... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 33355,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of games won by five softball teams are displayed in the graph. However, the names of the teams are missing. The following clues provide information about the teams:\n\n1. The Tigers won more games than the Eagles.\n\n2. The Patriots won more games than the Cubs, but... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 33356,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of games won by six basketball teams are displayed in the graph, but the names of the teams are missing. The following clues provide information about the teams:\n\n1. The Hawks won more games than the Falcons.\n2. The Warriors won more games than the Knights, but fe... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 33357,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of games won by six volleyball teams are displayed in a graph, but the names of the teams are missing. The following clues provide information about the teams:\n\n1. The Falcons won more games than the Hawks.\n2. The Warriors won more games than the Knights but fewer... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 33358,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of geese in a flock increases so that the difference between the populations in year $n+2$ and year $n$ is directly proportional to the population in year $n+1$. If the populations in the years $1994$, $1995$, and $1997$ were $39$, $60$, and $123$, respectively, then... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"index": 33359,
"solution": "1. **Define Variables:**\n Let $x$ be the population in $1996$, and let $k$ be the constant of proportionality.\n\n2. **Set Up Equations Based on Given Information:**\n - For $n = 1994$, the population difference between $1996$ and $1994$ is directly proportional to the population... |
DeepScaleR | [
{
"role": "user",
"content": "The number of inches in the perimeter of a square is equal to the number of square inches in its area. Find the length, in inches, of a side of the square."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 33360,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of inches in the perimeter of an equilateral triangle equals the number of square inches in the area of its circumscribed circle. What is the radius, in inches, of the circle? Express your answer in terms of pi and in simplest radical form."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{\\pi}"
} | {
"index": 33361,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of increasing sequences of positive integers $a_1 \\le a_2 \\le a_3 \\le \\cdots \\le a_{10} \\le 2007$ such that $a_i-i$ is even for $1\\le i \\le 10$ can be expressed as ${m \\choose n}$ for some positive integers $m > n$. Compute the remainder when $m$ is divided ... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 33362,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of integer points inside the triangle $OAB$ (where $O$ is the origin) formed by the line $y=2x$, the line $x=100$, and the x-axis is $\\qquad$."
}
] | math | {
"style": "rule",
"ground_truth": "9801"
} | {
"index": 33363,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of integer solutions for the inequality \\( |x| < 3 \\pi \\) is ( )."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 33364,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of integer solutions to the inequality $\\log_{3}|x-2| < 2$."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 33365,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of integers \\(N\\) from 1 to 1990 for which \\(\\frac{N^{2}+7}{N+4}\\) is not a reduced fraction is:"
}
] | math | {
"style": "rule",
"ground_truth": "86"
} | {
"index": 33366,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of integers between 1 and 100 that are not divisible by 2, 3, or 5 is \\( (\\ \\ \\ )."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 33367,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of intersection points between the graphs of the functions \\( y = \\sin x \\) and \\( y = \\log_{2021} |x| \\) is:"
}
] | math | {
"style": "rule",
"ground_truth": "1286"
} | {
"index": 33368,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of lattice points (points with integer coordinates) inside the region bounded by the right branch of the hyperbola \\( x^{2}-y^{2}=1 \\) and the line \\( x=100 \\), excluding the boundary, is \\(\\qquad\\) ."
}
] | math | {
"style": "rule",
"ground_truth": "9800"
} | {
"index": 33369,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of minutes in a week is closest to:"
}
] | math | {
"style": "rule",
"ground_truth": "10000"
} | {
"index": 33370,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of natural numbers from 1 to 1992 that are multiples of 3, but not multiples of 2 or 5, is\n\n(Ninth \"Jinyun Cup\" Middle School Mathematics Invitational Competition, 1992)"
}
] | math | {
"style": "rule",
"ground_truth": "266"
} | {
"index": 33371,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of natural numbers from 1 to 1992 that are multiples of 3, but not multiples of 2 or 5, is calculated."
}
] | math | {
"style": "rule",
"ground_truth": "266"
} | {
"index": 33372,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of ordered pairs of integers $(m,n)$ for which $mn \\ge 0$ and\n\n$m^3 + n^3 + 99mn = 33^3$\nis equal to"
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 33373,
"solution": "1. **Recall the Factorization Identity**: We start by recalling the identity for the sum of cubes:\n \\[\n x^3 + y^3 + z^3 - 3xyz = \\frac{1}{2} \\cdot (x + y + z) \\cdot ((x - y)^2 + (y - z)^2 + (z - x)^2)\n \\]\n This identity holds for any real numbers $x, y, z$.\n\n2. **Ap... |
DeepScaleR | [
{
"role": "user",
"content": "The number of ounces of water needed to reduce $9$ ounces of shaving lotion containing $50$ % alcohol to a lotion containing $30$ % alcohol is:"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 33374,
"solution": "1. **Identify the amount of alcohol in the original lotion**: \n The original shaving lotion is 9 ounces with 50% alcohol. Therefore, the amount of alcohol in the lotion is:\n \\[\n \\frac{50}{100} \\times 9 = \\frac{9}{2} \\text{ ounces}\n \\]\n\n2. **Set up the equation for... |
DeepScaleR | [
{
"role": "user",
"content": "The number of points common to the graphs of\n$(x-y+2)(3x+y-4)=0$ and $(x+y-2)(2x-5y+7)=0$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 33375,
"solution": "1. **Identify the equations of the lines**: \n From the given equations, using the Zero Product Property, we have:\n - From $(x-y+2)(3x+y-4)=0$:\n - $x-y+2=0 \\Rightarrow y = x - 2$\n - $3x+y-4=0 \\Rightarrow y = -3x + 4$\n - From $(x+y-2)(2x-5y+7)=0$:\n - $x+y-2=0 \... |
DeepScaleR | [
{
"role": "user",
"content": "The number of points equidistant from a circle and two parallel tangents to the circle is:"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 33376,
"solution": "To solve this problem, we need to understand the geometric configuration and the properties of points equidistant from a circle and two parallel tangents.\n\n1. **Understanding the Configuration**:\n - Consider a circle with center $O$ and radius $r$.\n - Let there be two parallel... |
DeepScaleR | [
{
"role": "user",
"content": "The number of positive integer pairs $(a,b)$ that have $a$ dividing $b$ and $b$ dividing $2013^{2014}$ can be written as $2013n+k$ , where $n$ and $k$ are integers and $0\\leq k<2013$ . What is $k$ ? Recall $2013=3\\cdot 11\\cdot 61$ ."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 33377,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of positive integers $k$ for which the equation\n\\[kx-12=3k\\]has an integer solution for $x$ is"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 33378,
"solution": "We start by rearranging the given equation:\n\\[ kx - 12 = 3k. \\]\n\n1. Isolate $x$ on one side:\n \\[ kx - 3k = 12. \\]\n \\[ k(x - 3) = 12. \\]\n \\[ x - 3 = \\frac{12}{k}. \\]\n \\[ x = \\frac{12}{k} + 3. \\]\n\n2. For $x$ to be an integer, $\\frac{12}{k}$ must also be an ... |
DeepScaleR | [
{
"role": "user",
"content": "The number of positive integers from 1 to 2002 that contain exactly one digit 0."
}
] | math | {
"style": "rule",
"ground_truth": "414"
} | {
"index": 33379,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of positive integers less than $1000$ divisible by neither $5$ nor $7$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "686"
} | {
"index": 33380,
"solution": "1. **Count the integers divisible by 5:**\n The number of positive integers less than $1000$ that are divisible by $5$ can be calculated using the floor function:\n \\[\n \\left\\lfloor \\frac{999}{5} \\right\\rfloor = 199\n \\]\n This is because $999$ is the largest number ... |
DeepScaleR | [
{
"role": "user",
"content": "The number of real roots of the equation $\\frac{x}{100} = \\sin x$ is: \n(32nd United States of America Mathematical Olympiad, 1981)"
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"index": 33381,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of real values of $x$ satisfying the equation $2^{2x^2 - 7x + 5} = 1$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 33382,
"solution": "1. **Identify the Equation Form**: The given equation is $2^{2x^2 - 7x + 5} = 1$. We know that any number raised to the power of zero equals one, i.e., $a^0 = 1$ for any $a \\neq 0$. Therefore, we need $2x^2 - 7x + 5 = 0$.\n\n2. **Solve the Quadratic Equation**: The quadratic equation... |
DeepScaleR | [
{
"role": "user",
"content": "The number of recommendation plans the principal can make for a certain high school with 4 students and 3 universities can accept at most 2 students from that school is to be determined."
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"index": 33383,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of revolutions of a wheel, with fixed center and with an outside diameter of $6$ feet, required to cause a point on the rim to go one mile is:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{880}{\\pi}"
} | {
"index": 33384,
"solution": "1. **Identify the radius of the wheel**: Given the diameter of the wheel is $6$ feet, the radius $r$ is half of the diameter:\n \\[\n r = \\frac{6}{2} = 3 \\text{ feet}\n \\]\n\n2. **Calculate the circumference of the wheel**: The circumference $C$ of a circle is given by the fo... |
DeepScaleR | [
{
"role": "user",
"content": "The number of right-angled triangles with integer side lengths $a$ and $b$ (where $b < 2011$), and hypotenuse length $b + 1$ is ."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 33385,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of roots satisfying the equation $\\sqrt{5 - x} = x\\sqrt{5 - x}$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 33386,
"solution": "1. Start by solving the equation given:\n \\[\\sqrt{5 - x} = x\\sqrt{5 - x}\\]\n\n2. Factor out the common term $\\sqrt{5 - x}$:\n \\[\\sqrt{5 - x}(x - 1) = 0\\]\n\n3. From the factored equation, we have two cases to consider:\n - Case 1: $\\sqrt{5 - x} = 0$\n - Case 2: $x - 1... |
DeepScaleR | [
{
"role": "user",
"content": "The number of rounds of golf played by each golfer of an amateur golf association is shown in the chart below. What is the average number of rounds played by each golfer? Express your answer to the nearest whole number. [asy]\nsize(150);\ndraw((0,7)--(0,0)--(10,0));\nfor(int i ... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 33387,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of scalene triangles having all sides of integral lengths, and perimeter less than $13$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 33388,
"solution": "To find the number of scalene triangles with integral side lengths and a perimeter less than 13, we need to consider the properties of scalene triangles and the triangle inequality theorem. A scalene triangle has all sides of different lengths, and the sum of the lengths of any two si... |
DeepScaleR | [
{
"role": "user",
"content": "The number of sets $A$ that satisfy $\\{1, 2\\} \\subset A \\subseteq \\{1, 2, 3, 4, 5, 6\\}$ must be determined."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 33389,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of sets of two or more consecutive positive integers whose sum is 100 is"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 33390,
"solution": "1. **Formulate the sum of consecutive integers**: Let's consider a sequence of $n$ consecutive integers starting from $a$. The integers are $a, a+1, a+2, \\ldots, a+n-1$. The sum $S$ of these integers can be calculated using the formula for the sum of an arithmetic series:\n \\[\n ... |
DeepScaleR | [
{
"role": "user",
"content": "The number of significant digits in the measurement of the side of a square whose computed area is $1.1025$ square inches to the nearest ten-thousandth of a square inch is:"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 33391,
"solution": "To find the number of significant digits in the measurement of the side of a square, we start by considering the given area of the square, which is $1.1025$ square inches. This value is given to the nearest ten-thousandth of a square inch, indicating precision in the measurement.\n\n1... |
DeepScaleR | [
{
"role": "user",
"content": "The number of six-digit even numbers formed by 1, 2, 3, 4, 5, 6 without repeating any digit and with neither 1 nor 3 adjacent to 5 can be calculated."
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 33392,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of solution-pairs in the positive integers of the equation $3x+5y=501$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 33393,
"solution": "1. **Rewrite the equation and express $y$ in terms of $x$:**\n Given the equation $3x + 5y = 501$, we can express $y$ in terms of $x$:\n \\[\n y = \\frac{501 - 3x}{5}\n \\]\n For $y$ to be a positive integer, $501 - 3x$ must be a positive multiple of $5$.\n\n2. **Determine t... |
DeepScaleR | [
{
"role": "user",
"content": "The number of solutions in positive integers of $2x+3y=763$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "127"
} | {
"index": 33394,
"solution": "1. **Formulate the Equation**: We start with the Diophantine equation given:\n \\[\n 2x + 3y = 763\n \\]\n\n2. **Find a Particular Solution**: We need to find one particular solution $(x_0, y_0)$ in integers. By trial and error or using the extended Euclidean algorithm, we find:... |
DeepScaleR | [
{
"role": "user",
"content": "The number of solutions of $2^{2x}-3^{2y}=55$, in which $x$ and $y$ are integers, is:\n\\[\\textbf{(A)} \\ 0 \\qquad\\textbf{(B)} \\ 1 \\qquad \\textbf{(C)} \\ 2 \\qquad\\textbf{(D)} \\ 3\\qquad \\textbf{(E)} \\ \\text{More than three, but finite}\\]"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 33395,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of solutions to \\{1,~2\\} \\subseteq~X~\\subseteq~\\{1,~2,~3,~4,~5\\}, where $X$ is a subset of \\{1,~2,~3,~4,~5\\} is"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 33396,
"solution": "1. **Identify the Problem Requirements:**\n The problem requires us to find the number of subsets $X$ of the set $\\{1, 2, 3, 4, 5\\}$ such that $\\{1, 2\\} \\subseteq X$. This means every subset $X$ must include both elements 1 and 2.\n\n2. **Determine the Remaining Elements:**\n ... |
DeepScaleR | [
{
"role": "user",
"content": "The number of solutions to the equation $\\sin |x| = |\\cos x|$ in the closed interval $[-10\\pi, 10\\pi]$ is __."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 33397,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of students in Carlos' graduating class is more than 100 and fewer than 200 and is 2 less than a multiple of 4, 3 less than a multiple of 5, and 4 less than a multiple of 6. How many students are in Carlos' graduating class?"
}
] | math | {
"style": "rule",
"ground_truth": "182"
} | {
"index": 33398,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number of students in Jakob's graduating class is more than 100 and fewer than 200 and is 2 less than a multiple of 4, 3 less than a multiple of 5, and 4 less than a multiple of 6. How many students are in Jakob's graduating class?"
}
] | math | {
"style": "rule",
"ground_truth": "122"
} | {
"index": 33399,
"solution": ""
} |
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