data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "The complex number $(3 \\operatorname{cis} 18^\\circ)(-2\\operatorname{cis} 37^\\circ)$ is expressed in polar form as $r \\operatorname{cis} \\theta,$ where $r > 0$ and $0^\\circ \\le \\theta < 360^\\circ.$ Enter the ordered pair $(r, \\theta).$ Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "(6,235^\\circ)"
} | {
"split": "train",
"index": 4900,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many polynomial functions $f$ of degree $\\ge 1$ satisfy\n$f(x^2)=[f(x)]^2=f(f(x))$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 4901,
"solution": "1. **Assume the polynomial form**: Let $f(x) = \\sum_{k=0}^{n} a_{k} x^{k}$ be a polynomial of degree $n \\geq 1$ satisfying the given conditions $f(x^2)=[f(x)]^2=f(f(x))$.\n\n2. **Analyze the highest degree terms**: \n - In $f(x^2)$, replace $x$ with $x^2$ to get... |
deepscaler | [
{
"role": "user",
"content": "Given two sets\n$$\n\\begin{array}{l}\nA=\\{(x, y) \\mid |x|+|y|=a, a>0\\}, \\\\\nB=\\{(x, y) \\mid |xy|+1=|x|+|y|\\}.\n\\end{array}\n$$\n\nIf \\( A \\cap B \\) is the set of vertices of a regular octagon in the plane, determine the value of \\( a \\). Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "2 + \\sqrt{2}"
} | {
"split": "train",
"index": 4902,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all values of $x$ which satisfy\n\\[\\frac{6}{\\sqrt{x - 8} - 9} + \\frac{1}{\\sqrt{x - 8} - 4} + \\frac{7}{\\sqrt{x - 8} + 4} + \\frac{12}{\\sqrt{x - 8} + 9} = 0.\\]Enter all the solutions, separated by commas. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "17,44"
} | {
"split": "train",
"index": 4903,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The volume of the top portion of the water tower is equal to the volume of a sphere, which can be calculated using the formula $V = \\frac{4}{3}\\pi r^3$. The top portion of the real tower has a volume of 50,000 liters, so we can solve for the radius:\n\n$\\frac{4}{3}\\pi r^3 =... | math | {
"style": "rule",
"ground_truth": "0.95"
} | {
"split": "train",
"index": 4904,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the roots of $\\tan^2 x - 8\\tan x + \\sqrt{2} = 0$ that are between $x=0$ and $x=2\\pi$ radians. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"split": "train",
"index": 4905,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the lateral side \\(CD\\) of trapezoid \\(ABCD (AD \\parallel BC)\\), point \\(M\\) is marked. A perpendicular \\(AH\\) is dropped from vertex \\(A\\) to segment \\(BM\\). It turns out that \\(AD = HD\\). Find the length of segment \\(AD\\), given that \\(BC = 16\\), \\(CM =... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 4906,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Last year, the East Sea Crystal City World's business hall made a profit of 3 million yuan. At the beginning of this year, it relocated to the new Crystal City business hall, expanding its scope of operations. To achieve higher profits, it is necessary to increase advertising e... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 4907,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the domain of the function f(x) is R, f(2x-2) is an even function, f(x-3)+f(-x+1)=0 when x∈[-2,-1], f(x)=\\frac{1}{a^x}-ax-4 where a>0 and a≠1, and f(-2)=4, find Σ from k=1 to 19 of |f(k)|. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 4908,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Randomly color the four vertices of a tetrahedron with two colors, red and yellow. The probability that \"three vertices on the same face are of the same color\" is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{5}{8}"
} | {
"split": "train",
"index": 4909,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the four-digit number that is a perfect square, where the thousands digit is the same as the tens digit, and the hundreds digit is 1 greater than the units digit. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8281"
} | {
"split": "train",
"index": 4910,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Inside a square with side length 8, four congruent equilateral triangles are drawn such that each triangle shares one side with a side of the square and each has a vertex at one of the square's vertices. What is the side length of the largest square that can be inscribed in the... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"split": "train",
"index": 4911,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate\n\\[i^{14762} + i^{14763} + i^{14764} + i^{14765}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 4912,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If Greg rolls four fair six-sided dice, what is the probability that he rolls more 1's than 6's? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{421}{1296}"
} | {
"split": "train",
"index": 4913,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has an inradius of $5$ and a circumradius of $16$. If $2\\cos{B} = \\cos{A} + \\cos{C}$, then the area of triangle $ABC$ can be expressed as $\\frac{a\\sqrt{b}}{c}$, where $a, b,$ and $c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is... | math | {
"style": "rule",
"ground_truth": "141"
} | {
"split": "train",
"index": 4914,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the equations $\\left(625\\right)^{0.24}$ and $\\left(625\\right)^{0.06}$, find the value of their product. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5^{6/5}"
} | {
"split": "train",
"index": 4915,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system, the equation of circle C is $x^2 + y^2 - 4x = 0$, and its center is point C. Consider the polar coordinate system with the origin as the pole and the non-negative half of the x-axis as the polar axis. Curve $C_1: \\rho = -4\\sqrt{3}\\sin\\the... | math | {
"style": "rule",
"ground_truth": "1:2"
} | {
"split": "train",
"index": 4916,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Marisa has two identical cubical dice labeled with the numbers \\(\\{1,2,3,4,5,6\\}\\). However, the two dice are not fair, meaning that they can land on each face with different probability. Marisa rolls the two dice and calculates their sum. Given that the sum is 2 with proba... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 4917,
"solution": "Let \\(p_{i}\\) be the probability that the dice lands on the number \\(i\\). The problem gives that \\(p_{1}^{2}=0.04, p_{6}^{2}=0.01\\), so we have \\(p_{1}=0.2, \\quad p_{6}=0.1, \\quad p_{2}+p_{3}+p_{4}+p_{5}=0.7\\). We are asked to maximize $$2\\left(p_{1} p_{6... |
deepscaler | [
{
"role": "user",
"content": "How many unique numbers can you get by multiplying two or more distinct members of the set $\\{1,2,3,5,7\\}$ together? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 4918,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For the function $f(x) = x - 2 - \\ln x$, we know that $f(3) = 1 - \\ln 3 < 0$, $f(4) = 2 - \\ln 4 > 0$. Using the bisection method to find the approximate value of the root of $f(x)$ within the interval $(3, 4)$, we first calculate the function value $f(3.5)$. Given that $\\ln... | math | {
"style": "rule",
"ground_truth": "3.25"
} | {
"split": "train",
"index": 4919,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For distinct real numbers $x$ and $y$, let $M(x,y)$ be the larger of $x$ and $y$ and let $m(x,y)$ be the smaller of $x$ and $y$. If $a<b<c<d<e$, then\n$M(M(a,m(b,c)),m(d,m(a,e)))=$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "b"
} | {
"split": "train",
"index": 4920,
"solution": "To solve the problem, we need to evaluate the expression $M(M(a,m(b,c)),m(d,m(a,e)))$ step by step using the definitions of $M(x,y)$ and $m(x,y)$, and the given inequalities $a<b<c<d<e$.\n\n1. **Evaluate $m(b,c)$ and $m(a,e)$:**\n - Since $b < c$, $m(b,c) = b$.\n ... |
deepscaler | [
{
"role": "user",
"content": "If $r$ is the remainder when each of the numbers $1059$, $1417$, and $2312$ is divided by $d$, where $d$ is an integer greater than $1$, then $d-r$ equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 4921,
"solution": "1. We start with the given congruences:\n \\[\n 1059 \\equiv r \\pmod{d} \\quad \\text{(i)}\n \\]\n \\[\n 1417 \\equiv r \\pmod{d} \\quad \\text{(ii)}\n \\]\n \\[\n 2312 \\equiv r \\pmod{d} \\quad \\text{(iii)}\n \\]\n\n2. Subtract congruence (i) f... |
deepscaler | [
{
"role": "user",
"content": "A cyclist initially traveled at a speed of 20 km/h. After covering one-third of the distance, the cyclist looked at the clock and decided to increase the speed by 20%. With the new speed, the cyclist traveled the remaining part of the distance. What is the average speed of the ... | math | {
"style": "rule",
"ground_truth": "22.5"
} | {
"split": "train",
"index": 4922,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any integer $n$, define $\\lfloor n\\rfloor$ as the greatest integer less than or equal to $n$. For any positive integer $n$, let $$f(n)=\\lfloor n\\rfloor+\\left\\lfloor\\frac{n}{2}\\right\\rfloor+\\left\\lfloor\\frac{n}{3}\\right\\rfloor+\\cdots+\\left\\lfloor\\frac{n}{n}... | math | {
"style": "rule",
"ground_truth": "55"
} | {
"split": "train",
"index": 4923,
"solution": "55 Notice that, for fixed $a,\\lfloor n / a\\rfloor$ counts the number of integers $b \\in$ $\\{1,2, \\ldots, n\\}$ which are divisible by $a$; hence, $f(n)$ counts the number of pairs $(a, b), a, b \\in$ $\\{1,2, \\ldots, n\\}$ with $b$ divisible by $a$. For any fi... |
deepscaler | [
{
"role": "user",
"content": "In the star shown, the sum of the four integers along each straight line is to be the same. Five numbers have been entered. The five missing numbers are 19, 21, 23, 25, and 27. Which number is represented by \\( q \\)? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 4924,
"solution": "Suppose that the sum of the four integers along each straight line equals \\( S \\). Then \\( S=9+p+q+7=3+p+u+15=3+q+r+11=9+u+s+11=15+s+r+7 \\). Thus, \\( 5S = (9+p+q+7)+(3+p+u+15)+(3+q+r+11)+(9+u+s+11)+(15+s+r+7) = 2p+2q+2r+2s+2u+90 \\). Since \\( p, q, r, s, \\) a... |
deepscaler | [
{
"role": "user",
"content": "Given fixed points \\( A(3,0) \\), \\( B(0,4) \\), and point \\( P \\) on the incircle of triangle \\( \\triangle AOB \\) (where \\( O \\) is the origin), find the maximum value of \\( |PA|^2 + |PB|^2 + |PO|^2 \\). Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 4925,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the equation \\[\\sqrt{x^2+y^2} + |y-1| = 3\\]consists of portions of two different parabolas. Compute the distance between the vertices of the parabolas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 4926,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a trapezoid with $AB \\parallel CD$ and $\\angle D=90^{\\circ}$. Suppose that there is a point $E$ on $CD$ such that $AE=BE$ and that triangles $AED$ and $CEB$ are similar, but not congruent. Given that $\\frac{CD}{AB}=2014$, find $\\frac{BC}{AD}$. Let's think ste... | math | {
"style": "rule",
"ground_truth": "\\sqrt{4027}"
} | {
"split": "train",
"index": 4927,
"solution": "Let $M$ be the midpoint of $AB$. Let $AM=MB=ED=a, ME=AD=b$, and $AE=BE=c$. Since $\\triangle BEC \\sim \\triangle DEA$, but $\\triangle BEC$ is not congruent to $\\triangle DAE$, we must have $\\triangle BEC \\sim \\triangle DEA$. Thus, $BC / BE=AD / DE=b / a$, so $... |
deepscaler | [
{
"role": "user",
"content": "If a four-digit number is called a \"good number\" when its unit digit is 1 and it has exactly three identical digits, then how many \"good numbers\" are there among the four-digit numbers formed by the digits 1, 2, 3, and 4 with repetitions? Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 4928,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Some bugs are sitting on squares of $10\\times 10$ board. Each bug has a direction associated with it **(up, down, left, right)**. After 1 second, the bugs jump one square in **their associated**direction. When the bug reaches the edge of the board, the associated direction r... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 4929,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three faucets fill a 100-gallon tub in 6 minutes. How long, in seconds, does it take six faucets to fill a 25-gallon tub? Assume that all faucets dispense water at the same rate. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 4930,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\\times 5$ square array of dots? (Two rectangles are different if they do not share all four vertices.) Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 4931,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A volunteer organizes a spring sports event and wants to form a vibrant and well-trained volunteer team. They plan to randomly select 3 people from 4 male volunteers and 3 female volunteers to serve as the team leader. The probability of having at least one female volunteer as ... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{17}"
} | {
"split": "train",
"index": 4932,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f$ be a function for which $f\\left(\\dfrac{x}{3}\\right) = x^2 + x + 1$. Find the sum of all values of $z$ for which $f(3z) = 7$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1/9"
} | {
"split": "train",
"index": 4933,
"solution": "1. **Identify the function and equation:** Given the function $f\\left(\\frac{x}{3}\\right) = x^2 + x + 1$, we need to find the sum of all values of $z$ for which $f(3z) = 7$.\n\n2. **Substitute and form a new equation:** Since $f\\left(\\frac{x}{3}\\right) = x^2 + ... |
deepscaler | [
{
"role": "user",
"content": "The equation $x^2-kx-12=0$ has only integer solutions for certain positive integers $k$. What is the sum of all such values of $k$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 4934,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $2.4 \\times 10^{8}$ is doubled, what is the result? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4.8 \\times 10^{8}"
} | {
"split": "train",
"index": 4935,
"solution": "When $2.4 \\times 10^{8}$ is doubled, the result is $2 \\times 2.4 \\times 10^{8}=4.8 \\times 10^{8}$."
} |
deepscaler | [
{
"role": "user",
"content": "Positive integers \\( d, e, \\) and \\( f \\) are chosen such that \\( d < e < f \\), and the system of equations \n\\[ 2x + y = 2010 \\quad \\text{and} \\quad y = |x-d| + |x-e| + |x-f| \\]\nhas exactly one solution. What is the minimum value of \\( f \\)? Let's think step by s... | math | {
"style": "rule",
"ground_truth": "1006"
} | {
"split": "train",
"index": 4936,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A candy company makes 5 colors of jellybeans, which come in equal proportions. If I grab a random sample of 5 jellybeans, what is the probability that I get exactly 2 distinct colors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{12}{125}"
} | {
"split": "train",
"index": 4937,
"solution": "There are $\\binom{5}{2}=10$ possible pairs of colors. Each pair of colors contributes $2^{5}-2=30$ sequences of beans that use both colors. Thus, the answer is $10 \\cdot 30 / 5^{5}=12 / 125$."
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest five-digit positive integer such that it is not a multiple of 11, and any number obtained by deleting some of its digits is also not divisible by 11. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "98765"
} | {
"split": "train",
"index": 4938,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A string of digits is defined to be similar to another string of digits if it can be obtained by reversing some contiguous substring of the original string. For example, the strings 101 and 110 are similar, but the strings 3443 and 4334 are not. (Note that a string is always si... | math | {
"style": "rule",
"ground_truth": "1126"
} | {
"split": "train",
"index": 4939,
"solution": "We first count the number of substrings that one could pick to reverse to yield a new substring. If we insert two dividers into the sequence of 50 digits, each arrangement of 2 dividers among the 52 total objects specifies a substring that is contained between the t... |
deepscaler | [
{
"role": "user",
"content": "Given 40 feet of fencing, what is the greatest possible number of square feet in the area of a rectangular pen enclosed by the fencing? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 4940,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be a positive integer. Brothers Michael and Kylo each select a positive integer less than or equal to $N$, independently and uniformly at random. Let $p_{N}$ denote the probability that the product of these two integers has a units digit of 0. The maximum possible value... | math | {
"style": "rule",
"ground_truth": "2800"
} | {
"split": "train",
"index": 4941,
"solution": "For $k \\in\\{2,5,10\\}$, let $q_{k}=\\frac{\\lfloor N / k\\rfloor}{N}$ be the probability that an integer chosen uniformly at random from $[N]$ is a multiple of $k$. Clearly, $q_{k} \\leq \\frac{1}{k}$, with equality iff $k$ divides $N$. The product of $p_{1}, p_{2... |
deepscaler | [
{
"role": "user",
"content": "In the circle \\(x^{2} + y^{2} \\leq R^{2}\\), the two-dimensional probability density is \\(f(x, y) = C\\left(R - \\sqrt{x^{2} + y^{2}}\\right)\\); outside the circle \\(f(x, y) = 0\\). Find: a) the constant \\(C\\); b) the probability that a random point \\((X, Y)\\) falls wi... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 4942,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest possible value of $x$ in the simplified form $x=\\frac{a+b\\sqrt{c}}{d}$ if $\\frac{5x}{6}+1=\\frac{3}{x}$, where $a,b,c,$ and $d$ are integers. What is $\\frac{acd}{b}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-55"
} | {
"split": "train",
"index": 4943,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To calculate $41^2$, Tom mentally computes $40^2$ and adds a number. To find $39^2$, Tom subtracts a number from $40^2$. What number does he add to calculate $41^2$ and subtract to calculate $39^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "79"
} | {
"split": "train",
"index": 4944,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute \\[\\lfloor \\sqrt{1} \\rfloor + \\lfloor \\sqrt{2} \\rfloor + \\lfloor \\sqrt{3} \\rfloor + \\cdots + \\lfloor \\sqrt{25} \\rfloor.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 4945,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $k$ for the ellipse $\\frac{x^2}{k+8} + \\frac{y^2}{9} = 1$ with an eccentricity of $\\frac{1}{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{5}{4}"
} | {
"split": "train",
"index": 4946,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The probability that Class A will be assigned exactly 2 of the 8 awards, with each of the 4 classes (A, B, C, and D) receiving at least 1 award is $\\qquad$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{7}"
} | {
"split": "train",
"index": 4947,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mary and Sally were once the same height. Since then, Sally grew \\( 20\\% \\) taller and Mary's height increased by half as many centimetres as Sally's height increased. Sally is now 180 cm tall. How tall, in cm, is Mary now? Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "165"
} | {
"split": "train",
"index": 4948,
"solution": "Suppose that Sally's original height was \\( s \\) cm. Since Sally grew \\( 20\\% \\) taller, her new height is \\( 1.2s \\) cm. Since Sally is now 180 cm tall, then \\( 1.2s=180 \\) or \\( s=\\frac{180}{1.2}=150 \\). Thus, Sally grew \\( 180-150=30 \\) cm. Since Ma... |
deepscaler | [
{
"role": "user",
"content": "A student at Harvard named Kevin was counting his stones by 11. He messed up $n$ times and instead counted 9s and wound up at 2007. How many values of $n$ could make this limerick true? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 4949,
"solution": "The mathematical content is that $9 n+11 k=2007$, for some nonnegative integers $n$ and $k$. As $2007=9 \\cdot 223, k$ must be divisible by 9. Using modulo 11, we see that $n$ is 3 more than a multiple of 11. Thus, the possibilities are $n=223,212,201, \\ldots, 3$, ... |
deepscaler | [
{
"role": "user",
"content": "In rectangle $ABCD,$ $P$ is a point on side $\\overline{BC}$ such that $BP = 16$ and $CP = 8.$ If $\\tan \\angle APD = 3,$ then find $AB.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 4950,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the expression \\(1.6 \\frac{\\left(\\frac{1}{a}+\\frac{1}{b}-\\frac{2 c}{a b}\\right)(a+b+2 c)}{\\frac{1}{a^{2}}+\\frac{1}{b^{2}}+\\frac{2}{a b}-\\frac{4 c^{2}}{a^{2} b^{2}}}\\) given that \\(a = 7.4\\) and \\(b = \\frac{5}{37}\\). Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "1.6"
} | {
"split": "train",
"index": 4951,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Johny's father tells him: \"I am twice as old as you will be seven years from the time I was thrice as old as you were\". What is Johny's age? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 4952,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "George purchases a sack of apples, a bunch of bananas, a cantaloupe, and a carton of dates for $ \\$ 20$. If a carton of dates costs twice as much as a sack of apples and the price of a cantaloupe is equal to the price of a sack of apples minus a bunch of bananas, how much woul... | math | {
"style": "rule",
"ground_truth": "\\$ 5"
} | {
"split": "train",
"index": 4953,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Joe and JoAnn each bought 12 ounces of coffee in a 16-ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe's c... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{6}"
} | {
"split": "train",
"index": 4954,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = |3\\{x\\} - 1.5|$, where $\\{x\\}$ denotes the fractional part of $x$. Find the smallest positive integer $n$ such that the equation \\[nf(xf(x)) = 2x\\] has at least $1000$ real solutions. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "250"
} | {
"split": "train",
"index": 4955,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of the first $n$ terms of the sequence $\\{a\\_n\\}$ is $S\\_n=3n^{2}+8n$, and $\\{b\\_n\\}$ is an arithmetic sequence with $a\\_n=b\\_n+b_{n+1}$:\n\n1. Find the general term formula for the sequence $\\{b\\_n\\}$.\n2. Find the maximum value of $c\\_n=\\frac{... | math | {
"style": "rule",
"ground_truth": "\\frac{87}{2}"
} | {
"split": "train",
"index": 4956,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a, b,c$ and $d$ be real numbers such that $a + b + c + d = 2$ and $ab + bc + cd + da + ac + bd = 0$.\nFind the minimum value and the maximum value of the product $abcd$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0\\frac{1}{16}"
} | {
"split": "train",
"index": 4957,
"solution": "\nGiven the conditions:\n\\[ a + b + c + d = 2 \\]\n\\[ ab + bc + cd + da + ac + bd = 0, \\]\nwe are required to find the minimum and maximum values of the product \\( abcd \\).\n\n### Step 1: Consider the Polynomial Approach\n\nWe associate the real numbers \\( a, ... |
deepscaler | [
{
"role": "user",
"content": "Given the curve $C:\\begin{cases}x=2\\cos a \\\\ y= \\sqrt{3}\\sin a\\end{cases} (a$ is the parameter) and the fixed point $A(0,\\sqrt{3})$, ${F}_1,{F}_2$ are the left and right foci of this curve, respectively. With the origin $O$ as the pole and the positive half-axis of $x$ ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{12\\sqrt{3}}{13}"
} | {
"split": "train",
"index": 4958,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In \\(\\triangle ABC\\), \\(D\\) is the midpoint of \\(BC\\), \\(E\\) is the foot of the perpendicular from \\(A\\) to \\(BC\\), and \\(F\\) is the foot of the perpendicular from \\(D\\) to \\(AC\\). Given that \\(BE = 5\\), \\(EC = 9\\), and the area of \\(\\triangle ABC\\) is... | math | {
"style": "rule",
"ground_truth": "\\frac{6 \\sqrt{37}}{5}"
} | {
"split": "train",
"index": 4959,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the application problem by setting up equations:<br/>A gift manufacturing factory receives an order for a batch of teddy bears and plans to produce them in a certain number of days. If they produce $20$ teddy bears per day, they will be $100$ short of the order. If they p... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 4960,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the singing scores 9.4, 8.4, 9.4, 9.9, 9.6, 9.4, 9.7, calculate the average and variance of the remaining data after removing the highest and lowest scores. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.016"
} | {
"split": "train",
"index": 4961,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\left(x+y\\right)^{2}=1$ and $\\left(x-y\\right)^{2}=49$, find the values of $x^{2}+y^{2}$ and $xy$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-12"
} | {
"split": "train",
"index": 4962,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $\\frac {x^{2}}{16}- \\frac {y^{2}}{9}=1$, and a chord AB with a length of 6 connected to the left focus F₁, calculate the perimeter of △ABF₂ (F₂ being the right focus). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 4963,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\arccos (\\sin 3).$ All functions are in radians. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 - \\frac{\\pi}{2}"
} | {
"split": "train",
"index": 4964,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(3000,0),(3000,2000),$ and $(0,2000)$. What is the probability that $x > 5y$? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{20}"
} | {
"split": "train",
"index": 4965,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Trevor and Edward play a game in which they take turns adding or removing beans from a pile. On each turn, a player must either add or remove the largest perfect square number of beans that is in the heap. The player who empties the pile wins. For example, if Trevor goes first ... | math | {
"style": "rule",
"ground_truth": "0, 5, 20, 29, 45, 80, 101, 116, 135, 145, 165, 173, 236, 257, 397, 404, 445, 477, 540, 565, 580, 629, 666, 836, 845, 885, 909, 944, 949, 954, 975"
} | {
"split": "train",
"index": 4966,
"solution": "The correct answers are 0 (worth imaginary points), 5 (worth 0 points), 20 (4 points), 29, 45 (5 points), 80 (6 points), 101, 116, 135, 145, 165, 173 (7 points), 236, 257 (8 points), 397, 404, 445, 477, 540, 565, 580, 629, 666 (9 points), 836, 845, 885, 909, 944, 94... |
deepscaler | [
{
"role": "user",
"content": "How many ordered pairs $(m,n)$ of positive integers are solutions to\n\\[\\frac{4}{m}+\\frac{2}{n}=1?\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 4967,
"solution": "1. Start with the given equation:\n \\[\n \\frac{4}{m} + \\frac{2}{n} = 1\n \\]\n\n2. Multiply both sides by $mn$ to eliminate the denominators:\n \\[\n 4n + 2m = mn\n \\]\n\n3. Rearrange the equation to bring all terms to one side:\n \\[\n mn - 4n -... |
deepscaler | [
{
"role": "user",
"content": "The diagonals of trapezoid \\(ABCD\\) intersect at point \\(M\\). The areas of triangles \\(ABM\\) and \\(CDM\\) are 18 and 50 units, respectively. What is the area of the trapezoid? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 4968,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.293"
} | {
"split": "train",
"index": 4969,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the eight-term sequence $A,B,C,D,E,F,G,H$, the value of $C$ is $5$ and the sum of any three consecutive terms is $30$. What is $A+H$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 4970,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The yearly changes in the population census of a town for four consecutive years are, respectively, 25% increase, 25% increase, 25% decrease, 25% decrease. The net change over the four years, to the nearest percent, is: Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "-12"
} | {
"split": "train",
"index": 4971,
"solution": "1. **Understanding the percentage changes**: \n - A 25% increase in population means the population becomes 125% of its original, which can be expressed as a multiplication by $\\frac{5}{4}$.\n - A 25% decrease means the population becomes 75% of its previous am... |
deepscaler | [
{
"role": "user",
"content": "Given the parabola $C: x^{2}=2py\\left(p \\gt 0\\right)$ with focus $F$, and the minimum distance between $F$ and a point on the circle $M: x^{2}+\\left(y+4\\right)^{2}=1$ is $4$.<br/>$(1)$ Find $p$;<br/>$(2)$ If point $P$ lies on $M$, $PA$ and $PB$ are two tangents to $C$ with... | math | {
"style": "rule",
"ground_truth": "20\\sqrt{5}"
} | {
"split": "train",
"index": 4972,
"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"
} | {
"split": "train",
"index": 4973,
"solution": "1. **Identify the number of rings**: \n The top ring has an outside diameter of $20$ cm, and the bottom ring has an outside diameter of $3$ cm. Each subsequent ring's outside diameter decreases by $1$ cm. Therefore, the number of rings is $20 - 3 + 1 = 18$ rings.\... |
deepscaler | [
{
"role": "user",
"content": "How many candies were in the bag before the first day if a group of friends eat candies over five days as follows: On the first day, they eat \\( \\frac{1}{2} \\) of the candies, on the second day \\( \\frac{2}{3} \\) of the remaining, on the third day \\( \\frac{3}{4} \\) of t... | math | {
"style": "rule",
"ground_truth": "720"
} | {
"split": "train",
"index": 4974,
"solution": "We work backwards through the given information. At the end, there is 1 candy remaining. Since \\( \\frac{5}{6} \\) of the candies are removed on the fifth day, this 1 candy represents \\( \\frac{1}{6} \\) of the candies left at the end of the fourth day. Thus, ther... |
deepscaler | [
{
"role": "user",
"content": "The rectangle $ABCD^{}_{}$ below has dimensions $AB^{}_{} = 12 \\sqrt{3}$ and $BC^{}_{} = 13 \\sqrt{3}$. Diagonals $\\overline{AC}$ and $\\overline{BD}$ intersect at $P^{}_{}$. If triangle $ABP^{}_{}$ is cut out and removed, edges $\\overline{AP}$ and $\\overline{BP}$ are joine... | math | {
"style": "rule",
"ground_truth": "594"
} | {
"split": "train",
"index": 4975,
"solution": "Let $X$ be the apex of the pyramid and $M$ be the midpoint of $\\overline{CD}$. We find the side lengths of $\\triangle XMP$.\n$MP = \\frac{13\\sqrt3}{2}$. $PX$ is half of $AC$, which is $\\frac{\\sqrt{3\\cdot13^2+3\\cdot12^2}}{2} = \\frac{\\sqrt{939}}{2}$. To find ... |
deepscaler | [
{
"role": "user",
"content": "The logarithm of $27\\sqrt[4]{9}\\sqrt[3]{9}$ to the base $3$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\frac{1}{6}"
} | {
"split": "train",
"index": 4976,
"solution": "1. **Rewrite the expression using properties of exponents and logarithms:**\n The given expression is $27\\sqrt[4]{9}\\sqrt[3]{9}$. We know that $27 = 3^3$, $\\sqrt[4]{9} = 9^{1/4}$, and $\\sqrt[3]{9} = 9^{1/3}$. Since $9 = 3^2$, we can rewrite these as:\n \\[\n... |
deepscaler | [
{
"role": "user",
"content": "Let \\( A \\) be the set of real numbers \\( x \\) satisfying the inequality \\( x^{2} + x - 110 < 0 \\) and \\( B \\) be the set of real numbers \\( x \\) satisfying the inequality \\( x^{2} + 10x - 96 < 0 \\). Suppose that the set of integer solutions of the inequality \\( x^... | math | {
"style": "rule",
"ground_truth": "71"
} | {
"split": "train",
"index": 4977,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from 3 pears and 8 ounces of orange juice from 2 oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. What per... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 4978,
"solution": "1. **Determine the amount of juice from pears and oranges:**\n - Miki extracts 8 ounces of pear juice from 3 pears. Therefore, the amount of pear juice per pear is:\n \\[\n \\frac{8 \\text{ ounces}}{3 \\text{ pears}} = \\frac{8}{3} \\text{ ounces per pear}... |
deepscaler | [
{
"role": "user",
"content": "If $g(2x - 5) = 3x + 9$ for all real $x$, then what is $g(1)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 4979,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circle $O$ has chord $A B$. A circle is tangent to $O$ at $T$ and tangent to $A B$ at $X$ such that $A X=2 X B$. What is \\frac{A T}{B T}? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 4980,
"solution": "Let $T X$ meet circle $O$ again at $Y$. Since the homothety centered at $T$ takes $X$ to $Y$ also takes $A B$ to the tangent line of circle $O$ passing through $Y$, we have $Y$ is the midpoint of arc $A B$. This means that $\\angle A T Y=\\angle Y T B$. By the Angle... |
deepscaler | [
{
"role": "user",
"content": "Find the largest real number \\(x\\) such that\n\\[ \n\\frac{x^{2} + x - 1 + \\left|x^{2} - (x - 1)\\right|}{2} = 35x - 250. \n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 4981,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\tan x+\\tan y=25$ and $\\cot x + \\cot y=30$, what is $\\tan(x+y)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 4982,
"solution": "Since $\\cot$ is the reciprocal function of $\\tan$:\n$\\cot x + \\cot y = \\frac{1}{\\tan x} + \\frac{1}{\\tan y} = \\frac{\\tan x + \\tan y}{\\tan x \\cdot \\tan y} = 30$\n Thus, $\\tan x \\cdot \\tan y = \\frac{\\tan x + \\tan y}{30} = \\frac{25}{30} = \\frac{5}{... |
deepscaler | [
{
"role": "user",
"content": "Given that a basketball player has a 40% chance of scoring with each shot, estimate the probability that the player makes exactly two out of three shots using a random simulation method. The simulation uses a calculator to generate a random integer between 0 and 9, with 1, 2, 3... | math | {
"style": "rule",
"ground_truth": "\\frac {1}{4}"
} | {
"split": "train",
"index": 4983,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the difference between $1000_7$ and $666_7$ in base $7$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1_7"
} | {
"split": "train",
"index": 4984,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given that $b= \\sqrt {2}$, $c=3$, $B+C=3A$.\n(1) Find the length of side $a$;\n(2) Find the value of $\\sin (B+ \\frac {3π}{4})$. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{10}"
} | {
"split": "train",
"index": 4985,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line has a slope of $-7$ and contains the point $(3,0)$. The equation of this line can be written in the form $y = mx+b$. What is the value of $m+b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 4986,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a number \\\\(x\\\\) randomly selected from the interval \\\\(\\left[-\\frac{\\pi}{4}, \\frac{2\\pi}{3}\\right]\\\\), find the probability that the function \\\\(f(x)=3\\sin\\left(2x- \\frac{\\pi}{6}\\right)\\\\) is not less than \\\\(0\\\\). Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "\\frac{6}{11}"
} | {
"split": "train",
"index": 4987,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest four-digit number that is divisible by 4? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9996"
} | {
"split": "train",
"index": 4988,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circles $\\omega_1$ and $\\omega_2$ intersect at points $X$ and $Y$. Line $\\ell$ is tangent to $\\omega_1$ and $\\omega_2$ at $A$ and $B$, respectively, with line $AB$ closer to point $X$ than to $Y$. Circle $\\omega$ passes through $A$ and $B$ intersecting $\\omega_1$ again a... | math | {
"style": "rule",
"ground_truth": "270"
} | {
"split": "train",
"index": 4989,
"solution": "$AB^2 = 4 AM^2 =2x(2x+ 2 XY) =(XP - XY) (XP + XY) = XP^2 - XY^2 = XC \\cdot XD - XY^2 = 67 \\cdot 37 - 47^2 = \\boxed{270}.$\nvladimir.shelomovskii@gmail.com, vvsss\n ~MathProblemSolvingSkills.com"
} |
deepscaler | [
{
"role": "user",
"content": "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? Let's think step by... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{15}"
} | {
"split": "train",
"index": 4990,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the tens digit of the smallest six-digit positive integer that is divisible by each of $10,11,12,13,14$, and 15? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 4991,
"solution": "Among the list $10,11,12,13,14,15$, the integers 11 and 13 are prime. Also, $10=2 \\times 5$ and $12=2 \\times 2 \\times 3$ and $14=2 \\times 7$ and $15=3 \\times 5$. For an integer $N$ to be divisible by each of these six integers, $N$ must include at least two fac... |
deepscaler | [
{
"role": "user",
"content": "How many non-empty subsets $S$ of $\\{1, 2, 3, \\ldots, 10\\}$ satisfy the following two properties?\n1. No two consecutive integers belong to $S$.\n2. If $S$ contains $k$ elements, then $S$ contains no number less than $k$. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "143"
} | {
"split": "train",
"index": 4992,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The five integers $2, 5, 6, 9, 14$ are arranged into a different order. In the new arrangement, the sum of the first three integers is equal to the sum of the last three integers. What is the middle number in the new arrangement? Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 4993,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, $c$, with $b=6$, $c=10$, and $\\cos C=-\\frac{2}{3}$. \n$(1)$ Find $\\cos B$; \n$(2)$ Find the height on side $AB$. Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "\\frac{20 - 4\\sqrt{5}}{5}"
} | {
"split": "train",
"index": 4994,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest possible value of the expression $$\\frac{(2a+3b)^2 + (b-c)^2 + (2c-a)^2}{b^2}$$ where \\( b > a > c \\) are real numbers, and \\( b \\neq 0 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 4995,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the remainder when \\(1 + 5 + 5^2 + \\cdots + 5^{1002}\\) is divided by \\(500\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 4996,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $O$ and $H$ be the circumcenter and orthocenter of triangle $ABC$, respectively. Let $a$, $b$, and $c$ denote the side lengths, and let $R$ denote the circumradius. Find $OH^2$ if $R = 7$ and $a^2 + b^2 + c^2 = 29$. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "412"
} | {
"split": "train",
"index": 4997,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $|\\overrightarrow{a}|=5$, $|\\overrightarrow{b}|=3$, and $\\overrightarrow{a}\\cdot\\overrightarrow{b}=-12$, find the projection of vector $\\overrightarrow{a}$ onto vector $\\overrightarrow{b}$. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 4998,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid $ABCD$ , $AD \\parallel BC$ and $\\angle ABC + \\angle CDA = 270^{\\circ}$ . Compute $AB^2$ given that $AB \\cdot \\tan(\\angle BCD) = 20$ and $CD = 13$ .\n\n*Proposed by Lewis Chen* Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "260"
} | {
"split": "train",
"index": 4999,
"solution": ""
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.