data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Find the number of solutions to\n\\[\\cos 4x + \\cos^2 3x + \\cos^3 2x + \\cos^4 x = 0\\]for $-\\pi \\le x \\le \\pi.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 29549,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two chords \\( PQ \\) and \\( PR \\) are drawn in a circle with diameter \\( PS \\). The point \\( T \\) lies on \\( PR \\) and \\( QT \\) is perpendicular to \\( PR \\). The angle \\(\\angle QPR = 60^\\circ\\), \\( PQ = 24 \\text{ cm} \\), and \\( RT = 3 \\text{ cm} \\). What ... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 29550,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $c = \\frac{2\\pi}{11}.$ What is the value of\n\\[\\frac{\\sin 3c \\cdot \\sin 6c \\cdot \\sin 9c \\cdot \\sin 12c \\cdot \\sin 15c}{\\sin c \\cdot \\sin 2c \\cdot \\sin 3c \\cdot \\sin 4c \\cdot \\sin 5c}?\\] Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 29551,
"solution": "1. **Define the constant and simplify the expression:**\n Let \\( c = \\frac{2\\pi}{11} \\). We need to evaluate:\n \\[\n \\frac{\\sin 3c \\cdot \\sin 6c \\cdot \\sin 9c \\cdot \\sin 12c \\cdot \\sin 15c}{\\sin c \\cdot \\sin 2c \\cdot \\sin 3c \\cdot \\sin 4... |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, if $a=2$, $c=2\\sqrt{3}$, and $\\angle A=30^\\circ$, then the area of $\\triangle ABC$ is equal to __________. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 29552,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "90 + 91 + 92 + 93 + 94 + 95 + 96 + 97 + 98 + 99 = Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "945"
} | {
"split": "train",
"index": 29553,
"solution": "We are given the sum of consecutive integers from $90$ to $99$. We can calculate this sum using the formula for the sum of an arithmetic series, or by simplifying the expression directly.\n\n1. **Identify the series**: The series is $90 + 91 + 92 + 93 + 94 + 95 + 9... |
deepscaler | [
{
"role": "user",
"content": "A nine-digit integer is formed by repeating a positive three-digit integer three times. For example, 123,123,123 or 456,456,456 are integers of this form. What is the greatest common divisor of all nine-digit integers of this form? Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "1001001"
} | {
"split": "train",
"index": 29554,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the complex number $z\\_1$ satisfies $((z\\_1-2)(1+i)=1-i)$, the imaginary part of the complex number $z\\_2$ is $2$, and $z\\_1z\\_2$ is a real number, find $z\\_2$ and $|z\\_2|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt {5}"
} | {
"split": "train",
"index": 29555,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle ABC$ have sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively. The areas of the equilateral triangles with side lengths $a$, $b$, $c$ are $S_{1}$, $S_{2}$, $S_{3}$ respectively. Given $S_{1}-S_{2}+S_{3}=\\frac{{\\sqrt{3}}}{2}$ and $\\sin B=\\frac{1... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 29556,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line segment is divided into three parts, $x$, $y$, and $z$, such that, $x < y < z$ and $x$ to $y$ is as $y$ to $z$. If all three parts combined form a segment of length $s$, and $x + y = z$, determine the value of the ratio $\\frac{x}{y}$.\nA) $\\frac{-1 - \\sqrt{5}}{2}$\nB)... | math | {
"style": "rule",
"ground_truth": "\\frac{-1 + \\sqrt{5}}{2}"
} | {
"split": "train",
"index": 29557,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Set $A$ consists of $m$ consecutive integers whose sum is $2m$, and set $B$ consists of $2m$ consecutive integers whose sum is $m.$ The absolute value of the difference between the greatest element of $A$ and the greatest element of $B$ is $99$. Find $m.$ Let's think step by st... | math | {
"style": "rule",
"ground_truth": "201"
} | {
"split": "train",
"index": 29558,
"solution": "Note that since set $A$ has $m$ consecutive integers that sum to $2m$, the middle integer (i.e., the median) must be $2$. Therefore, the largest element in $A$ is $2 + \\frac{m-1}{2}$.\n Further, we see that the median of set $B$ is $0.5$, which means that the \"mi... |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $AB = 5$, $AC = 7$, $BC = 9$, and $D$ lies on $\\overline{BC}$ such that $\\overline{AD}$ bisects $\\angle BAC$. Find $\\cos \\angle BAD$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{0.45}"
} | {
"split": "train",
"index": 29559,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Earl and Bob start their new jobs on the same day. Earl's work schedule is to work for 3 days followed by 1 day off, while Bob's work schedule is to work for 7 days followed by 3 days off. In the first 1000 days, how many days off do they have in common? Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 29560,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Terrell usually lifts two 20-pound weights 12 times. If he uses two 15-pound weights instead, how many times must Terrell lift them in order to lift the same total weight? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 29561,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the batch of rice contains 1512 bushels and a sample of 216 grains contains 27 grains of wheat, calculate the approximate amount of wheat mixed in this batch of rice. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "189"
} | {
"split": "train",
"index": 29562,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sequence $\\{a_n\\}$ is a geometric sequence, and the sequence $\\{b_n\\}$ is an arithmetic sequence. If $a_1-a_6-a_{11}=-3\\sqrt{3}$ and $b_1+b_6+b_{11}=7\\pi$, then the value of $\\tan \\frac{b_3+b_9}{1-a_4-a_3}$ is ______. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "-\\sqrt{3}"
} | {
"split": "train",
"index": 29563,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three real numbers $a,b,$ and $c$ satisfy the equations $a+b+c=2$, $ab+ac+bc=-7$ and $abc=-14$. What is the largest of the three numbers? Express your answer in simplest radical form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{7}"
} | {
"split": "train",
"index": 29564,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "By multiplying a natural number by the number that is one greater than it, the product takes the form $ABCD$, where $A, B, C, D$ are different digits. Starting with the number that is 3 less, the product takes the form $CABD$. Starting with the number that is 30 less, the produ... | math | {
"style": "rule",
"ground_truth": "8372"
} | {
"split": "train",
"index": 29565,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Hooligan Vasya likes to run on the escalator in the subway. He runs down twice as fast as he runs up. If the escalator is not working, it takes Vasya 6 minutes to run up and down. If the escalator is moving downward, it takes him 13.5 minutes to run up and down. How many second... | math | {
"style": "rule",
"ground_truth": "324"
} | {
"split": "train",
"index": 29566,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x)=\\cos x+\\cos (x+ \\frac {π}{2}).$\n(1) Find $f( \\frac {π}{12})$;\n(2) Suppose $α$ and $β∈(- \\frac {π}{2},0)$, $f(α+ \\frac {3π}{4})=- \\frac {3 \\sqrt {2}}{5}$, $f( \\frac {π}{4}-β)=- \\frac {5 \\sqrt {2}}{13}$, find $\\cos (α+β)$. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "\\frac {16}{65}"
} | {
"split": "train",
"index": 29567,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "**\nEvaluate \n\\[\n\\frac{3}{\\log_8{3000^4}} + \\frac{4}{\\log_9{3000^4}},\n\\]\ngiving your answer as a fraction in lowest terms.\n\n** Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 29568,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the digits $1$ through $7$ , one can form $7!=5040$ numbers by forming different permutations of the $7$ digits (for example, $1234567$ and $6321475$ are two such permutations). If the $5040$ numbers are then placed in ascending order, what is the $2013^{\\... | math | {
"style": "rule",
"ground_truth": "3546127"
} | {
"split": "train",
"index": 29569,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the values $1256, 2561, 5612, 6125$, calculate the sum. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15554"
} | {
"split": "train",
"index": 29570,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the length of side $XY$ in the triangle below.\n\n[asy]\nunitsize(1inch);\npair X,Y,Z;\nX = (0,0);\nY= (2,0);\nZ = (0,sqrt(3));\ndraw (X--Y--Z--X,linewidth(0.9));\ndraw(rightanglemark(Y,X,Z,3));\nlabel(\"$X$\",X,S);\nlabel(\"$Y$\",Y,S);\nlabel(\"$Z$\",Z,N);\nlabel(\"$12$\"... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 29571,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The remainder can be defined for all real numbers $x$ and $y$ with $y \\neq 0$ by $\\text{rem} (x ,y)=x-y\\left \\lfloor \\frac{x}{y} \\right \\rfloor$ where $\\left \\lfloor \\tfrac{x}{y} \\right \\rfloor$ denotes the greatest integer less than or equal to $\\tfrac{x}{y}$. Wha... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{40}"
} | {
"split": "train",
"index": 29572,
"solution": "1. **Definition of Remainder**: The remainder function $\\text{rem}(x, y)$ for real numbers $x$ and $y$ (with $y \\neq 0$) is defined as:\n \\[\n \\text{rem}(x, y) = x - y \\left\\lfloor \\frac{x}{y} \\right\\rfloor\n \\]\n where $\\left\\lfloor \\cdot \\ri... |
deepscaler | [
{
"role": "user",
"content": "Two watermelons and one banana together weigh 8100 grams, and two watermelons and three bananas together weigh 8300 grams, calculate the weight of one watermelon and one banana. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 29573,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Multiply $(x^4 +18 x^2 + 324) (x^2-18)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x^6-5832"
} | {
"split": "train",
"index": 29574,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Since 2021, the \"Study Strong Country\" app has launched a \"Four-Person Match\" answer module. The rules are as follows: Users need to answer two rounds of questions in the \"Four-Person Match\". At the beginning of each round, the system will automatically match 3 people to ... | math | {
"style": "rule",
"ground_truth": "3.3"
} | {
"split": "train",
"index": 29575,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the equation $\\frac{m}{x-3}-\\frac{1}{3-x}=2$ has a positive root with respect to $x$, then the value of $m$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 29576,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain year the price of gasoline rose by $20\\%$ during January, fell by $20\\%$ during February, rose by $25\\%$ during March, and fell by $x\\%$ during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the near... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 29577,
"solution": "1. **Assume the initial price**: Let's assume the initial price of gasoline at the beginning of January is $P_0 = 100$ dollars.\n\n2. **Price after January's increase**: The price increased by 20% in January. Therefore, the new price at the end of January, $P_1$, i... |
deepscaler | [
{
"role": "user",
"content": "A domino is a 1-by-2 or 2-by-1 rectangle. A domino tiling of a region of the plane is a way of covering it (and only it) completely by nonoverlapping dominoes. For instance, there is one domino tiling of a 2-by-1 rectangle and there are 2 tilings of a 2-by-2 rectangle (one cons... | math | {
"style": "rule",
"ground_truth": "89"
} | {
"split": "train",
"index": 29578,
"solution": "The number of tilings of a 2-by-$n$, rectangle is the $n$th Fibonacci number $F_{n}$, where $F_{0}=F_{1}=1$ and $F_{n}=F_{n-1}+F_{n-1}$ for $n \\geq 2$. (This is not hard to show by induction.) The answer is 89."
} |
deepscaler | [
{
"role": "user",
"content": "Compute the following sum:\n\\[\n\\frac{1}{2^{2024}} \\sum_{n = 0}^{1011} (-3)^n \\binom{2024}{2n}.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 29579,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "During a journey, the distance read on the odometer was 450 miles. On the return trip, using snow tires for the same distance, the reading was 440 miles. If the original wheel radius was 15 inches, find the increase in the wheel radius, correct to the nearest hundredth of an in... | math | {
"style": "rule",
"ground_truth": "0.34"
} | {
"split": "train",
"index": 29580,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $C$ be a unit cube and let $p$ denote the orthogonal projection onto the plane. Find the maximum area of $p(C)$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 29581,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An airplane has three sections: First Class (24 seats), Business Class ($25\\%$ of the total number of seats) and Economy ($\\frac{2}{3}$ of the total number of seats). How many seats does the plane have? Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "288"
} | {
"split": "train",
"index": 29582,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(a, b, c, d, e\\) be positive integers. Their sum is 2345. Let \\(M = \\max (a+b, b+c, c+d, d+e)\\). Find the smallest possible value of \\(M\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "782"
} | {
"split": "train",
"index": 29583,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points A, B, C, and D lie along a line, in that order. If $AB:AC=1:5$, and $BC:CD=2:1$, what is the ratio $AB:CD$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1:2"
} | {
"split": "train",
"index": 29584,
"solution": "Suppose that $AB=x$ for some $x>0$. Since $AB:AC=1:5$, then $AC=5x$. This means that $BC=AC-AB=5x-x=4x$. Since $BC:CD=2:1$ and $BC=4x$, then $CD=2x$. Therefore, $AB:CD=x:2x=1:2$."
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be an acute triangle. Let $DAC,EAB$, and $FBC$ be isosceles triangles exterior to $ABC$, with $DA=DC, EA=EB$, and $FB=FC$, such that\n\\[\n\\angle ADC = 2\\angle BAC, \\quad \\angle BEA= 2 \\angle ABC, \\quad\n\\angle CFB = 2 \\angle ACB.\n\\]\nLet $D'$ be the inter... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 29585,
"solution": "\nConsider the given configuration of triangle \\( ABC \\) with the constructed isosceles triangles \\( \\triangle DAC \\), \\( \\triangle EAB \\), and \\( \\triangle FBC \\). Each of these triangles is constructed externally such that:\n- \\( \\angle ADC = 2\\angl... |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be a positive integer. At most how many distinct unit vectors can be selected in $\\mathbb{R}^{n}$ such that from any three of them, at least two are orthogonal? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2n"
} | {
"split": "train",
"index": 29586,
"solution": "Solution 1. $2 n$ is the maximal number. An example of $2 n$ vectors in the set is given by a basis and its opposite vectors. In the rest of the text we prove that it is impossible to have $2 n+1$ vectors in the set. Consider the Gram matrix $A$ with entries $a_{i ... |
deepscaler | [
{
"role": "user",
"content": "How many positive integers less than 100, which are also not divisible by 3, have an even number of positive divisors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 29587,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_{1}=3$, and for $n \\geq 1$, let $a_{n+1}=(n+1) a_{n}-n$. Find the smallest $m \\geq 2005$ such that $a_{m+1}-1 \\mid a_{m}^{2}-1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2010"
} | {
"split": "train",
"index": 29588,
"solution": "We will show that $a_{n}=2 \\cdot n!+1$ by induction. Indeed, the claim is obvious for $n=1$, and $(n+1)(2 \\cdot n!+1)-n=2 \\cdot(n+1)!+1$. Then we wish to find $m \\geq 2005$ such that $2(m+1)!\\mid 4(m!)^{2}+4 m$ !, or dividing by $2 \\cdot m$ !, we want $m+1 \\... |
deepscaler | [
{
"role": "user",
"content": "Find $1273 + 120 \\div 60 - 173$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1102"
} | {
"split": "train",
"index": 29589,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of 3-digit positive integers whose digits multiply to 30. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 29590,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be a triangle with $A B=8, B C=15$, and $A C=17$. Point $X$ is chosen at random on line segment $A B$. Point $Y$ is chosen at random on line segment $B C$. Point $Z$ is chosen at random on line segment $C A$. What is the expected area of triangle $X Y Z$ ? Let's thi... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 29591,
"solution": "Let $\\mathbb{E}(X)$ denote the expected value of $X$, and let $[S]$ denote the area of $S$. Then $$\\begin{aligned} \\mathbb{E}([\\triangle X Y Z]) & =\\mathbb{E}([\\triangle A B C]-[\\triangle X Y B]-[\\triangle Z Y C]-[\\triangle X B Z]) \\\\ & =[\\triangle A B ... |
deepscaler | [
{
"role": "user",
"content": "Given a cube with an edge length of 1, find the surface area of the smaller sphere that is tangent to the larger sphere and the three faces of the cube. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7\\pi - 4\\sqrt{3}\\pi"
} | {
"split": "train",
"index": 29592,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $\\left(\\sqrt{(\\sqrt3)^3}\\right)^4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 29593,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "500"
} | {
"split": "train",
"index": 29594,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $-π < x < 0$, $\\sin x + \\cos x = \\frac{1}{5}$,\n(1) Find the value of $\\sin x - \\cos x$;\n(2) Find the value of $\\frac{3\\sin^2 \\frac{x}{2} - 2\\sin \\frac{x}{2}\\cos \\frac{x}{2} + \\cos^2 \\frac{x}{2}}{\\tan x + \\frac{1}{\\tan x}}$. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "-\\frac{132}{125}"
} | {
"split": "train",
"index": 29595,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the expansion of $(x^{4}+y^{2}+\\frac{1}{2xy})^{7}$, the constant term is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{105}{16}"
} | {
"split": "train",
"index": 29596,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $y$ if $y$ is positive and $y \\cdot \\lfloor y \\rfloor = 132$. Express your answer as a decimal. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 29597,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the original number be expressed as $x$. When the decimal point of $x$ is moved one place to the right, the resulting number can be expressed as $100x$. According to the given information, we have the equation $100x = x + 34.65$. Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "3.85"
} | {
"split": "train",
"index": 29598,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the number of ordered pairs of integers $(x, y)$ with $1 \\leq x < y \\leq 200$ such that $i^{x} + i^{y}$ is a real number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3651"
} | {
"split": "train",
"index": 29599,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$Q$ is the point of intersection of the diagonals of one face of a cube whose edges have length 2 units. Calculate the length of $QR$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"split": "train",
"index": 29600,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $\\left(\\frac{1+i}{\\sqrt{2}}\\right)^{46}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-i"
} | {
"split": "train",
"index": 29601,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We flip a fair coin 10 times. What is the probability that we get heads in exactly 8 of the 10 flips? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{45}{1024}"
} | {
"split": "train",
"index": 29602,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that in triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and $\\left(a+b\\right)\\left(\\sin A-\\sin B\\right)=\\left(c-b\\right)\\sin C$ with $a=2$, find the maximum area of triangle $\\triangle ABC$... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 29603,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kate rode her bicycle for 30 minutes at a speed of 16 mph, then walked for 90 minutes at a speed of 4 mph. What was her overall average speed in miles per hour? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 29604,
"solution": "To find Kate's overall average speed, we first need to calculate the total distance she traveled and the total time she spent traveling.\n\n1. **Calculate the distance traveled while biking:**\n Kate biked at a speed of 16 mph for 30 minutes. Since speed is in mi... |
deepscaler | [
{
"role": "user",
"content": "The hypotenuse of a right triangle whose legs are consecutive even numbers is 50 units. What is the sum of the lengths of the two legs? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 29605,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among the following propositions, the correct ones are __________.\n\n(1) The regression line $\\hat{y}=\\hat{b}x+\\hat{a}$ always passes through the center of the sample points $(\\bar{x}, \\bar{y})$, and at least through one sample point;\n\n(2) After adding the same constant... | math | {
"style": "rule",
"ground_truth": "(2)(6)(7)"
} | {
"split": "train",
"index": 29606,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a passenger travels from Moscow to St. Petersburg by a regular train, it will take him 10 hours. If he takes the express train, which he has to wait for more than 2.5 hours, he will arrive 3 hours earlier than the regular train. Find the ratio of the speeds of the express tr... | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"split": "train",
"index": 29607,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A_0=(0,0)$. Distinct points $A_1,A_2,\\dots$ lie on the $x$-axis, and distinct points $B_1,B_2,\\dots$ lie on the graph of $y=\\sqrt{x}$. For every positive integer $n,\\ A_{n-1}B_nA_n$ is an equilateral triangle. What is the least $n$ for which the length $A_0A_n\\geq100$... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 29608,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Buses from Dallas to Houston leave every hour on the hour. Buses from Houston to Dallas leave every hour on the half hour. The trip from one city to the other takes $5$ hours. Assuming the buses travel on the same highway, how many Dallas-bound buses does a Houston-bound bus pa... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 29609,
"solution": "1. **Understanding the Problem**: A Houston-bound bus leaves at 12:30 PM and takes 5 hours to reach Houston, i.e., it arrives at 5:30 PM. Dallas-bound buses leave every hour on the hour and also take 5 hours. We need to determine how many Dallas-bound buses the Hou... |
deepscaler | [
{
"role": "user",
"content": "Many of the students in M. Gamache's class brought a skateboard or a bicycle to school yesterday. The ratio of the number of skateboards to the number of bicycles was $7:4$. There were 12 more skateboards than bicycles. How many skateboards and bicycles were there in total? Let... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 29610,
"solution": "Since the ratio of the number of skateboards to the number of bicycles was $7:4$, then the numbers of skateboards and bicycles can be written in the form $7k$ and $4k$ for some positive integer $k$. Since the difference between the numbers of skateboards and bicycl... |
deepscaler | [
{
"role": "user",
"content": "Given a pyramid \\( S-ABCD \\) with a square base where each side measures 2, and \\( SD \\perp \\) plane \\( ABCD \\) and \\( SD = AB \\). Determine the surface area of the circumscribed sphere of the pyramid \\( S-ABCD \\). Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "12\\pi"
} | {
"split": "train",
"index": 29611,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area in square feet of a square with a perimeter of 32ft. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 29612,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the values 1432, 3214, 2143, and 4321, calculate their sum. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11110"
} | {
"split": "train",
"index": 29613,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has $AB=40,AC=31,$ and $\\sin{A}=\\frac{1}{5}$. This triangle is inscribed in rectangle $AQRS$ with $B$ on $\\overline{QR}$ and $C$ on $\\overline{RS}$. Find the maximum possible area of $AQRS$. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "744"
} | {
"split": "train",
"index": 29614,
"solution": "Note that if angle $BAC$ is obtuse, it would be impossible for the triangle to inscribed in a rectangle. This can easily be shown by drawing triangle ABC, where $A$ is obtuse. Therefore, angle A is acute. Let angle $CAS=n$ and angle $BAQ=m$. Then, $\\overline{AS}=3... |
deepscaler | [
{
"role": "user",
"content": "If $f^{-1}(g(x))=x^2-4$ and $g$ has an inverse, find $g^{-1}(f(10))$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{14}"
} | {
"split": "train",
"index": 29615,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An infinite geometric series has a first term of \\( 416 \\) and a sum of \\( 3120 \\). What is its common ratio? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{84}{97}"
} | {
"split": "train",
"index": 29616,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$|2+i^{2}+2i^{2}|=$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 29617,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 5 girls sitting in a row on five chairs, and opposite them, on five chairs, there are 5 boys sitting. It was decided that the boys would switch places with the girls. In how many ways can this be done? Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "14400"
} | {
"split": "train",
"index": 29618,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You roll a fair 12-sided die repeatedly. The probability that all the prime numbers show up at least once before seeing any of the other numbers can be expressed as a fraction \\( \\frac{p}{q} \\) in lowest terms. What is \\( p+q \\)? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "793"
} | {
"split": "train",
"index": 29619,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five balls are numbered 1 through 5 and placed in a bowl. Josh will randomly choose a ball from the bowl, look at its number and then put it back into the bowl. Then Josh will again randomly choose a ball from the bowl and look at its number. What is the probability that... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 29620,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convert the point $(\\rho,\\theta,\\phi) = \\left( 3, \\frac{5 \\pi}{12}, 0 \\right)$ in spherical coordinates to rectangular coordinates. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(0,0,3)"
} | {
"split": "train",
"index": 29621,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer $n$ is picante if $n$ ! ends in the same number of zeroes whether written in base 7 or in base 8 . How many of the numbers $1,2, \\ldots, 2004$ are picante? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 29622,
"solution": "The number of zeroes in base 7 is the total number of factors of 7 in $1 \\cdot 2 \\cdots n$, which is $$ \\lfloor n / 7\\rfloor+\\left\\lfloor n / 7^{2}\\right\\rfloor+\\left\\lfloor n / 7^{3}\\right\\rfloor+\\cdots $$ The number of zeroes in base 8 is $\\lfloor a... |
deepscaler | [
{
"role": "user",
"content": "Given $a=(2,4,x)$ and $b=(2,y,2)$, if $|a|=6$ and $a \\perp b$, then the value of $x+y$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 29623,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bug travels in the coordinate plane, moving only along the lines that are parallel to the $x$-axis or $y$-axis. Let $A = (-3, 2)$ and $B = (3, -2)$. Consider all possible paths of the bug from $A$ to $B$ of length at most $20$. How many points with integer coordinates lie on ... | math | {
"style": "rule",
"ground_truth": "195"
} | {
"split": "train",
"index": 29624,
"solution": "To solve this problem, we need to find all integer-coordinate points $(x, y)$ that a bug can reach on its path from $A = (-3, 2)$ to $B = (3, -2)$, given that the maximum allowable path length is 20 units. The bug moves only along lines parallel to the $x$-axis or ... |
deepscaler | [
{
"role": "user",
"content": "Four consecutive even integers have a product of 6720. What is the largest of these four integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 29625,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a basketball player has a probability of $a$ for scoring 3 points in a shot, $b$ for scoring 2 points, and $c$ for not scoring any points, where $a, b, c \\in (0, 1)$, and the mathematical expectation for scoring points in one shot is 2, determine the minimum value of $\\... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{3}"
} | {
"split": "train",
"index": 29626,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Several young men and women are seated around a round table. It is known that to the left of exactly 7 women, there are women, and to the left of 12 women, there are men. It is also known that for 75% of the young men, there are women to their right. How many people are seated ... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 29627,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The intercept of the line $5x - 2y - 10 = 0$ on the x-axis is $a$, and on the y-axis is $b$. Find the values of $a$ and $b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 29628,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the limit of the function:\n\n$$\\lim _{x \\rightarrow \\pi}\\left(\\operatorname{ctg}\\left(\\frac{x}{4}\\right)\\right)^{1 / \\cos \\left(\\frac{x}{2}\\right)}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "e"
} | {
"split": "train",
"index": 29629,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sphere is inscribed in a right cone with base radius \\(15\\) cm and height \\(30\\) cm. Determine the radius of the sphere if it can be expressed as \\(b\\sqrt{d} - b\\) cm. What is the value of \\(b + d\\)? Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "12.5"
} | {
"split": "train",
"index": 29630,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For what value of $x$ does $10^{x} \\cdot 100^{2x}=1000^{5}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 29631,
"solution": "1. **Rewrite the equation using properties of exponents**: \n The given equation is $10^x \\cdot 100^{2x} = 1000^5$. We know that $100 = 10^2$ and $1000 = 10^3$. Substituting these values, we get:\n \\[\n 10^x \\cdot (10^2)^{2x} = (10^3)^5\n \\]\n Simplif... |
deepscaler | [
{
"role": "user",
"content": "Given a population of $100$ individuals randomly numbered from $0$ to $99$, and a sample of size $10$ is drawn, with the units digit of the number drawn from the $k$-th group being the same as the units digit of $m + k$, where $m = 6$, find the number drawn from the 7-th group.... | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 29632,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle is tangent to the sides of an angle at points $A$ and $B$. The distance from a point $C$ on the circle to the line $AB$ is 6. Find the sum of the distances from point $C$ to the sides of the angle, given that one of these distances is nine times smaller than the other.... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 29633,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum constant $k$ such that $\\frac{k a b c}{a+b+c} \\leqslant (a+b)^{2} + (a+b+4c)^{2}$ holds for all positive real numbers $a, b, c$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 29634,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a positive integer $n,$ let\n\\[H_n = 1 + \\frac{1}{2} + \\frac{1}{3} + \\dots + \\frac{1}{n}.\\]Compute\n\\[\\sum_{n = 1}^\\infty \\frac{1}{(n + 1) H_n H_{n + 1}}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 29635,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute \\[\\dfrac{2^3-1}{2^3+1}\\cdot\\dfrac{3^3-1}{3^3+1}\\cdot\\dfrac{4^3-1}{4^3+1}\\cdot\\dfrac{5^3-1}{5^3+1}\\cdot\\dfrac{6^3-1}{6^3+1}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{43}{63}"
} | {
"split": "train",
"index": 29636,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, point $D$ is the midpoint of side $BC$. Point $E$ is on $AC$ such that $AE:EC = 1:2$. Point $F$ is on $AD$ such that $AF:FD = 2:1$. If the area of $\\triangle DEF$ is 24, determine the area of $\\triangle ABC$. Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "432"
} | {
"split": "train",
"index": 29637,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain circle's area is $x$ square units, and its circumference is $y$ units. The value of $x + y$ is $80\\pi$. What is the radius of the circle, in units? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 29638,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the arithmetic sequence $1$, $4$, $7$, $10$, $13$, $\\ldots$. Find the $15^{\\text{th}}$ term in the sequence. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"split": "train",
"index": 29639,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cone has a volume of $2592\\pi$ cubic inches and the vertex angle of the vertical cross section is 90 degrees. What is the height of the cone? Express your answer as a decimal to the nearest tenth. Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "20.0"
} | {
"split": "train",
"index": 29640,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The punch machines from before the flood punch some or even all of the nine numbered fields of a ticket. The inspectors request from the machine setter that the machine should not punch the same fields if someone places their ticket in reverse, instead of the prescribed orienta... | math | {
"style": "rule",
"ground_truth": "448"
} | {
"split": "train",
"index": 29641,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\{a_n\\}$ be a geometric sequence with a common ratio not equal to 1, and $a_4=16$. The sum of the first $n$ terms is denoted as $S_n$, and $5S_1$, $2S_2$, $S_3$ form an arithmetic sequence. \n(1) Find the general formula for the sequence $\\{a_n\\}$. \n(2) Let $b_n = \... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 29642,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An academy has $200$ students and $8$ teachers. The class sizes are as follows: $80, 40, 40, 20, 10, 5, 3, 2$. Calculate the average number of students per class as seen by a randomly picked teacher, represented by $t$, and the average number of students per class from the pers... | math | {
"style": "rule",
"ground_truth": "-25.69"
} | {
"split": "train",
"index": 29643,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the seven digits 1, 1, 3, 5, 5, 5, and 9 are arranged to form a seven-digit positive integer, what is the probability that the integer is divisible by 25? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{14}"
} | {
"split": "train",
"index": 29644,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Hoopers, coached by Coach Loud, have 15 players. George and Alex are the two players who refuse to play together in the same lineup. Additionally, if George plays, another player named Sam refuses to play. How many starting lineups of 6 players can Coach Loud create, provid... | math | {
"style": "rule",
"ground_truth": "3795"
} | {
"split": "train",
"index": 29645,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The 30 edges of a regular icosahedron are distinguished by labeling them $1,2,\\dots,30$. How many different ways are there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a ... | math | {
"style": "rule",
"ground_truth": "61917364224"
} | {
"split": "train",
"index": 29646,
"solution": "The number of such colorings is $2^{20} 3^{10} = 61917364224$. Identify the three colors red, white, and blue with (in some order) the elements of the field \\mathbb{F}_3 of three elements (i.e., the ring of integers mod 3). The set of colorings may then be identif... |
deepscaler | [
{
"role": "user",
"content": "Find the angle of inclination of the tangent line to the curve $y=\\frac{1}{3}x^3-5$ at the point $(1,-\\frac{3}{2})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 29647,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A piece of graph paper is folded once so that (0,2) is matched with (4,0), and (7,3) is matched with $(m,n)$. Find $m+n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6.8"
} | {
"split": "train",
"index": 29648,
"solution": "\n1. **Identify the line of fold**: The line of fold is the perpendicular bisector of the segment connecting $(0,2)$ and $(4,0)$. The midpoint of this segment is calculated as follows:\n \\[\n \\left(\\frac{0+4}{2}, \\frac{2+0}{2}\\right) = (2,1)\n \\]\n Th... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.