data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "In a competition with five participants A, B, C, D, and E, determine the probability that neither B nor C appears adjacent to A. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"split": "train",
"index": 29949,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the equation\n\n$$\n\\sin ^{4} x + 5(x - 2 \\pi)^{2} \\cos x + 5 x^{2} + 20 \\pi^{2} = 20 \\pi x\n$$\n\nFind the sum of its roots that belong to the interval $[-\\pi ; 6 \\pi]$, and provide the answer, rounding to two decimal places if necessary. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "31.42"
} | {
"split": "train",
"index": 29950,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer $N$ is a palindrome if the integer obtained by reversing the sequence of digits of $N$ is equal to $N$. The year 1991 is the only year in the current century with the following 2 properties:\n(a) It is a palindrome\n(b) It factors as a product of a 2-digit pr... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 29951,
"solution": "1. **Identify the properties of a four-digit palindrome**: A four-digit palindrome has the form $abba$, where $a$ and $b$ are digits and $a \\neq 0$. For example, $1991$ is a palindrome because reversing the digits gives the same number.\n\n2. **Check divisibility ... |
deepscaler | [
{
"role": "user",
"content": "During the regular season, Washington Redskins achieve a record of 10 wins and 6 losses. Compute the probability that their wins came in three streaks of consecutive wins, assuming that all possible arrangements of wins and losses are equally likely. (For example, the record LL... | math | {
"style": "rule",
"ground_truth": "\\frac{315}{2002}"
} | {
"split": "train",
"index": 29952,
"solution": "Suppse the winning streaks consist of $w_{1}, w_{2}$, and $w_{3}$ wins, in chronological order, where the first winning streak is preceded by $l_{0}$ consecutive losses and the $i$ winning streak is immediately succeeded by $l_{i}$ losses. Then $w_{1}, w_{2}, w_{3}... |
deepscaler | [
{
"role": "user",
"content": "A train, an hour after starting, meets with an accident which detains it a half hour, after which it proceeds at $\\frac{3}{4}$ of its former rate and arrives $3\\tfrac{1}{2}$ hours late. Had the accident happened $90$ miles farther along the line, it would have arrived only $3... | math | {
"style": "rule",
"ground_truth": "600"
} | {
"split": "train",
"index": 29953,
"solution": "1. **Define Variables:**\n Let the speed of the train be $x$ miles per hour, and let $D$ miles be the total distance of the trip.\n\n2. **Analyze the First Scenario:**\n - The train travels for 1 hour before the accident, covering $x$ miles.\n - After the acc... |
deepscaler | [
{
"role": "user",
"content": "If person A has either a height or weight greater than person B, then person A is considered not inferior to person B. Among 100 young boys, if a person is not inferior to the other 99, he is called an outstanding boy. What is the maximum number of outstanding boys among the 10... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 29954,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $0 < \\beta < \\alpha < \\frac{\\pi}{2}$, point $P(1,4 \\sqrt{3})$ is a point on the terminal side of angle $\\alpha$, and $\\sin \\alpha \\sin \\left(\\frac{\\pi}{2}-\\beta \\right)+\\cos \\alpha \\cos \\left(\\frac{\\pi}{2}+\\beta \\right)= \\frac{3 \\sqrt{3}}{14}$, cal... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 29955,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Paul Erdős was one of the most prolific mathematicians of all time and was renowned for his many collaborations. The Erdős number of a mathematician is defined as follows. Erdős has an Erdős number of 0, a mathematician who has coauthored a paper with Erdős has an Erdős number ... | math | {
"style": "rule",
"ground_truth": "4.65"
} | {
"split": "train",
"index": 29956,
"solution": "We'll suppose that each mathematician collaborates with approximately 20 people (except for Erdős himself, of course). Furthermore, if a mathematician has Erdős number $k$, then we'd expect him to be the cause of approximately $\\frac{1}{2^{k}}$ of his collaborator... |
deepscaler | [
{
"role": "user",
"content": "Let \\( n \\) be a positive integer. When \\( n > 100 \\), what are the first two decimal places of the fractional part of \\( \\sqrt{n^2 + 3n + 1} \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 29957,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $b$ is a positive integer greater than or equal to $2.$ When $197$ is converted to base $b$, the resulting representation has $4$ digits. What is the number of possible values for $b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 29958,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the smallest positive value of $x,$ in degrees, for which the function\n\\[f(x) = \\sin \\frac{x}{3} + \\sin \\frac{x}{11}\\]achieves its maximum value. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8910^\\circ"
} | {
"split": "train",
"index": 29959,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a circle with center $O$, $AD$ is a diameter, $ABC$ is a chord, $BO = 5$, and $\\angle ABO = \\text{arc } CD = 60^\\circ$. Find the length of $BC$.\n\n[asy]\nimport graph;\n\nunitsize(2 cm);\n\npair O, A, B, C, D;\n\nO = (0,0);\nA = dir(30);\nC = dir(160);\nB = (2*C + A)/3;... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 29960,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sandy and Sam each selected a positive integer less than 250. Sandy's number is a multiple of 15, and Sam's number is a multiple of 20. What is the probability that they selected the same number? Express your answer as a common fraction. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{48}"
} | {
"split": "train",
"index": 29961,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be a triangle with $A B=13, B C=14, C A=15$. Let $O$ be the circumcenter of $A B C$. Find the distance between the circumcenters of triangles $A O B$ and $A O C$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{91}{6}"
} | {
"split": "train",
"index": 29962,
"solution": "Let $S, T$ be the intersections of the tangents to the circumcircle of $A B C$ at $A, C$ and at $A, B$ respectively. Note that $A S C O$ is cyclic with diameter $S O$, so the circumcenter of $A O C$ is the midpoint of $O S$, and similarly for the other side. So the... |
deepscaler | [
{
"role": "user",
"content": "Parallelogram $ABCD$ has area $1,\\!000,\\!000$. Vertex $A$ is at $(0,0)$ and all other vertices are in the first quadrant. Vertices $B$ and $D$ are lattice points on the lines $y = x$ and $y = kx$ for some integer $k > 1$, respectively. How many such parallelograms are there? ... | math | {
"style": "rule",
"ground_truth": "784"
} | {
"split": "train",
"index": 29963,
"solution": "1. **Identify the coordinates of points $B$ and $D$:**\n - Since $A = (0,0)$ and $B$ lies on the line $y = x$, let $B = (b, b)$.\n - Since $D$ lies on the line $y = kx$ where $k$ is an integer greater than 1, let $D = (d, kd)$.\n\n2. **Determine the coordinates... |
deepscaler | [
{
"role": "user",
"content": "What is the minimum value of $5x^2-20x+1357$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1337"
} | {
"split": "train",
"index": 29964,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the sum:\n\\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2)))))))) \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1022"
} | {
"split": "train",
"index": 29965,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\omega = -\\frac{1}{2} + \\frac{i\\sqrt{3}}{2}$ and $\\omega^2 = -\\frac{1}{2} - \\frac{i\\sqrt{3}}{2}$. Define $T$ as the set of all points in the complex plane of the form $a + b\\omega + c\\omega^2 + d$, where $0 \\leq a, b, c \\leq 1$ and $d \\in \\{0, 1\\}$. Find the... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3}"
} | {
"split": "train",
"index": 29966,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the digits of the greatest prime number that is a divisor of 8,191? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 29967,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 10 different natural numbers, their sum is 604, and these 10 numbers have the same sum of digits. What is the largest number among these 10 numbers? $\\qquad Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "109"
} | {
"split": "train",
"index": 29968,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the digits of a natural number can be divided into two groups such that the sum of the digits in each group is equal, the number is called a \"balanced number\". For example, 25254 is a \"balanced number\" because $5+2+2=4+5$. If two adjacent natural numbers are both \"balan... | math | {
"style": "rule",
"ground_truth": "1099"
} | {
"split": "train",
"index": 29969,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of ${\\left[\\log_{10}\\left(5\\log_{10}100\\right)\\right]}^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 29970,
"solution": "1. **Identify the expression to simplify:** \n We start with the expression $[\\log_{10}(5\\log_{10}100)]^2$.\n\n2. **Simplify $\\log_{10}100$:** \n We know that $100 = 10^2$, so:\n \\[\n \\log_{10}100 = \\log_{10}(10^2) = 2 \\cdot \\log_{10}10 = 2\n \\... |
deepscaler | [
{
"role": "user",
"content": "The diagram shows part of a scale of a measuring device. The arrow indicates an approximate reading of Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10.3"
} | {
"split": "train",
"index": 29971,
"solution": "1. **Identify the Range**: The arrow on the scale is positioned between the marks labeled $10.25$ and $10.5$. This observation immediately eliminates options $\\text{(A)}\\ 10.05$, $\\text{(B)}\\ 10.15$, and $\\text{(E)}\\ 10.6$ as they are outside this range.\n\n2... |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $\\dfrac {x^{2}}{4}+ \\dfrac {y^{2}}{3}=1$ with its left and right foci denoted as $F_{1}$ and $F_{2}$ respectively, and a point $P$ on the ellipse. If $\\overrightarrow{PF_{1}}\\cdot \\overrightarrow{PF_{2}}= \\dfrac {5}{2}$, calculate $| \\overrightarrow{PF_{... | math | {
"style": "rule",
"ground_truth": "\\dfrac{7}{2}"
} | {
"split": "train",
"index": 29972,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An urn initially contains two red balls and one blue ball. George undertakes the operation of randomly drawing a ball and then adding two more balls of the same color from a box into the urn. This operation is done three times. After these operations, the urn has a total of nin... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"split": "train",
"index": 29973,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest value that the ratio of the areas of two isosceles right triangles can have, given that three vertices of one of the triangles lie on three different sides of the other triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "1/5"
} | {
"split": "train",
"index": 29974,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence of three real numbers forms an arithmetic progression with a first term of 9. If 2 is added to the second term and 20 is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term of the g... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 29975,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four girls — Mary, Alina, Tina, and Hanna — sang songs in a concert as trios, with one girl sitting out each time. Hanna sang $7$ songs, which was more than any other girl, and Mary sang $4$ songs, which was fewer than any other girl. How many songs did these trios sing? Let's ... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 29976,
"solution": "1. **Identify the number of songs each girl sang**: \n - Hanna sang 7 songs.\n - Mary sang 4 songs.\n - Alina and Tina sang more than Mary but less than Hanna, so they sang either 5 or 6 songs each.\n\n2. **Set up the equation for total songs**:\n - Let $a$... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\overrightarrow{OA}=(1,0)$, $\\overrightarrow{OB}=(1,1)$, and $(x,y)=λ \\overrightarrow{OA}+μ \\overrightarrow{OB}$, if $0\\leqslant λ\\leqslant 1\\leqslant μ\\leqslant 2$, then the maximum value of $z= \\frac {x}{m}+ \\frac{y}{n}(m > 0,n > 0)$ is $2$. Find the min... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}+ \\sqrt{6}"
} | {
"split": "train",
"index": 29977,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider positive integers $n$ where $D(n)$ denotes the number of pairs of different adjacent digits in the binary (base two) representation of $n$. Determine the number of positive integers less than or equal to $50$ for which $D(n) = 3$. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 29978,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point $P$ is a moving point on the parabola $y^2=2x$, find the minimum value of the sum of the distance from point $P$ to point $A(0,2)$ and the distance from $P$ to the directrix of the parabola. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{17}}{2}"
} | {
"split": "train",
"index": 29979,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $\\frac{3}{1250}$ is written as a decimal, how many zeros are there between the decimal point and the first non-zero digit? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 29980,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\sin^2 6^\\circ + \\sin^2 12^\\circ + \\sin^2 18^\\circ + \\dots + \\sin^2 174^\\circ.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15.5"
} | {
"split": "train",
"index": 29981,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of the expression $\\frac{1 \\cdot 2 \\cdot 3 \\cdot 4 \\cdot 5 \\cdot 6 \\cdot 7 \\cdot 8}{1+2+3+4+5+6+7+8}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1120"
} | {
"split": "train",
"index": 29982,
"solution": "1. **Calculate the numerator**: The numerator of the given expression is the product of the first 8 positive integers:\n \\[\n 1 \\cdot 2 \\cdot 3 \\cdot 4 \\cdot 5 \\cdot 6 \\cdot 7 \\cdot 8 = 8!\n \\]\n Using the factorial of 8, we find:\n \\[\n 8! = ... |
deepscaler | [
{
"role": "user",
"content": "A line with a slope of 2 and a line with a slope of -4 each have a $y$-intercept of 6. What is the distance between the $x$-intercepts of these lines? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"split": "train",
"index": 29983,
"solution": "The line with a slope of 2 and $y$-intercept 6 has equation $y=2x+6$. To find its $x$-intercept, we set $y=0$ to obtain $0=2x+6$ or $2x=-6$, which gives $x=-3$. The line with a slope of -4 and $y$-intercept 6 has equation $y=-4x+6$. To find its $x$-intercept, we se... |
deepscaler | [
{
"role": "user",
"content": "Let \\( O \\) be a point inside \\( \\triangle ABC \\), such that \\(\\overrightarrow{AO} = \\frac{1}{3}\\overrightarrow{AB} + \\frac{1}{4}\\overrightarrow{AC}\\). Find the ratio of the area of \\( \\triangle OAB \\) to the area of \\( \\triangle OBC \\). Let's think step by st... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 29984,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Corners are sliced off from a cube of side length 2 so that all its six faces each become regular octagons. Find the total volume of the removed tetrahedra.\nA) $\\frac{80 - 56\\sqrt{2}}{3}$\nB) $\\frac{80 - 48\\sqrt{2}}{3}$\nC) $\\frac{72 - 48\\sqrt{2}}{3}$\nD) $\\frac{60 - 42... | math | {
"style": "rule",
"ground_truth": "\\frac{80 - 56\\sqrt{2}}{3}"
} | {
"split": "train",
"index": 29985,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cowboy is initially 6 miles south and 2 miles west of a stream that flows due northeast. His cabin is located 12 miles east and 9 miles south of his initial position. He wants to water his horse at the stream and then return to his cabin. What is the shortest distance he can ... | math | {
"style": "rule",
"ground_truth": "8 + \\sqrt{545}"
} | {
"split": "train",
"index": 29986,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be an integer greater than or equal to $1$. Find, as a function of $n$, the smallest integer $k\\ge 2$ such that, among any $k$ real numbers, there are necessarily two of which the difference, in absolute value, is either strictly less than $1 / n$, either strictly gre... | math | {
"style": "rule",
"ground_truth": "n^2 + 2"
} | {
"split": "train",
"index": 29987,
"solution": "Let \\( n \\) be an integer such that \\( n \\geq 1 \\). We need to find the smallest integer \\( k \\geq 2 \\) such that for any set of \\( k \\) real numbers, there exist at least two numbers, say \\( x \\) and \\( y \\), where either \\( |x - y| < \\frac{1}{n} \... |
deepscaler | [
{
"role": "user",
"content": "If the height on the base of an isosceles triangle is $18 \\mathrm{~cm}$ and the median on the leg is $15 \\mathrm{~cm}$, what is the area of this isosceles triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 29988,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a quadrilateral with an inscribed circle $\\omega$ that has center $I$ . If $IA = 5, IB = 7, IC = 4, ID = 9$ , find the value of $\\frac{AB}{CD}$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35/36"
} | {
"split": "train",
"index": 29989,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the following infinite geometric series: $$\\frac{7}{8}-\\frac{14}{27}+\\frac{28}{81}-\\dots$$ Find the common ratio of this series. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{3}"
} | {
"split": "train",
"index": 29990,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point is chosen at random on the number line between 0 and 1, and the point is colored green. Then, another point is chosen at random on the number line between 0 and 1, and this point is colored purple. What is the probability that the number of the purple point is greater t... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 29991,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse C: $$\\frac{x^2}{a^2}+ \\frac{y^2}{b^2}=1 \\quad (a>b>0)$$ which passes through the point $(1, \\frac{2\\sqrt{3}}{3})$, with its foci denoted as $F_1$ and $F_2$. The circle $x^2+y^2=2$ intersects the line $x+y+b=0$ forming a chord of length 2.\n(I) Determine th... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 29992,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five candidates are to be selected to perform four different jobs, where one candidate can only work as a driver and the other four can do all the jobs. Determine the number of different selection schemes. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 29993,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\{a\\_n\\}$, where $a\\_n= \\sqrt {5n-1}$, $n\\in\\mathbb{N}^*$, arrange the integer terms of the sequence $\\{a\\_n\\}$ in their original order to form a new sequence $\\{b\\_n\\}$. Find the value of $b_{2015}$. Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "5037"
} | {
"split": "train",
"index": 29994,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p(n)$ denote the product of decimal digits of a positive integer $n$ . Computer the sum $p(1)+p(2)+\\ldots+p(2001)$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "184320"
} | {
"split": "train",
"index": 29995,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle has width $x$ and length $y$. The rectangle is cut along the horizontal and vertical dotted lines to produce four smaller rectangles. The sum of the perimeters of these four rectangles is 24. What is the value of $x+y$? Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 29996,
"solution": "The sum of the lengths of the horizontal line segments is $4x$, because the tops of the four small rectangles contribute a total of $2x$ to their combined perimeter and the bottoms contribute a total of $2x$. Similarly, the sum of the lengths of the vertical line s... |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system, the coordinates of point P(a,b) satisfy $a \\neq b$, and both $a$ and $b$ are elements of the set $\\{1,2,3,4,5,6\\}$. Additionally, the distance from point P to the origin, $|OP|$, is greater than or equal to 5. The number of such points P i... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 29997,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $2\\sin ^{2} \\frac{A}{2}= \\sqrt{3}\\sin A$, $\\sin (B-C)=2\\cos B\\sin C$, find the value of $\\frac{AC}{AB}$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1+\\sqrt{13}}{2}"
} | {
"split": "train",
"index": 29998,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alex is thinking of a number that is divisible by all of the positive integers 1 through 200 inclusive except for two consecutive numbers. What is the smaller of these numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 29999,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If we express $2x^2 + 6x + 11$ in the form $a(x - h)^2 + k$, then what is $h$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"split": "train",
"index": 30000,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five packages are delivered to five different houses, with each house receiving one package. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to their correct houses? Express your answer as a common fraction. Let's think... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"split": "train",
"index": 30001,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( ABC \\), angle \\( B \\) is \\( 80^\\circ \\). On side \\( BC \\), point \\( D \\) is marked such that \\( AB = AD = CD \\). On side \\( AB \\), point \\( F \\) is marked such that \\( AF = BD \\). On segment \\( AC \\), point \\( E \\) is marked such that \\( A... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 30002,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point $P$ is a moving point on the parabola $y=\\frac{1}{4}x^2$, determine the minimum value of the sum of the distance from point $P$ to the line $x+2y+4=0$ and the $x$-axis. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6\\sqrt{5}}{5}-1"
} | {
"split": "train",
"index": 30003,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider \\(A \\in \\mathcal{M}_{2020}(\\mathbb{C})\\) such that\n\\[ A + A^{\\times} = I_{2020} \\]\n\\[ A \\cdot A^{\\times} = I_{2020} \\]\nwhere \\(A^{\\times}\\) is the adjugate matrix of \\(A\\), i.e., the matrix whose elements are \\(a_{ij} = (-1)^{i+j} d_{ji}\\), where ... | math | {
"style": "rule",
"ground_truth": "673"
} | {
"split": "train",
"index": 30004,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The domain of the function $y=\\sin x$ is $[a,b]$, and its range is $\\left[-1, \\frac{1}{2}\\right]$. Calculate the maximum value of $b-a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4\\pi}{3}"
} | {
"split": "train",
"index": 30005,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be a region in the plane with area 10. When we apply the matrix\n\\[\\begin{pmatrix} 2 & 1 \\\\ 7 & -3 \\end{pmatrix}\\]to $S,$ we obtain the region $S'.$ Find the area of $S'.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "130"
} | {
"split": "train",
"index": 30006,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a school fundraising campaign, $25\\%$ of the money donated came from parents. The rest of the money was donated by teachers and students. The ratio of the amount of money donated by teachers to the amount donated by students was $2:3$. What is the ratio of the amount of mon... | math | {
"style": "rule",
"ground_truth": "5:9"
} | {
"split": "train",
"index": 30007,
"solution": "Since $25\\%$ of the money donated came from parents, then the remaining $100\\%-25\\%=75\\%$ came from the teachers and students. Since the ratio of the amount donated by teachers to the amount donated by students is $2:3$, then the students donated $\\frac{3}{2+3... |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all the integer solutions of $x^4 - 25x^2 + 144 = 0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 30008,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Oscar buys $13$ pencils and $3$ erasers for $1.00$. A pencil costs more than an eraser, and both items cost a whole number of cents. What is the total cost, in cents, of one pencil and one eraser? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 30009,
"solution": "1. **Define Variables:**\n Let $p$ represent the cost of one pencil and $e$ represent the cost of one eraser in cents. We are given the equation:\n \\[\n 13p + 3e = 100\n \\]\n where $p > e > 0$ and both $p$ and $e$ are whole numbers.\n\n2. **Constraints ... |
deepscaler | [
{
"role": "user",
"content": "Compute the value of the following expression:\n\\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2))))))))). \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2046"
} | {
"split": "train",
"index": 30010,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each corner cube is removed from this $3\\text{ cm}\\times 3\\text{ cm}\\times 3\\text{ cm}$ cube. The surface area of the remaining figure is\n[asy]\ndraw((2.7,3.99)--(0,3)--(0,0));\ndraw((3.7,3.99)--(1,3)--(1,0));\ndraw((4.7,3.99)--(2,3)--(2,0));\ndraw((5.7,3.99)--(3,3)--(3,0... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 30011,
"solution": "1. **Calculate the original surface area of the cube**: \n The original cube has dimensions $3\\text{ cm} \\times 3\\text{ cm} \\times 3\\text{ cm}$. Each face of the cube is a square with an area of $3^2 = 9\\text{ cm}^2$. Since a cube has 6 faces, the total sur... |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, the sides opposite to angles A, B, and C are denoted by $a$, $b$, and $c$ respectively, with $A+C=\\frac{2\\pi}{3}$ and $b=1$. \n(1) If we let angle A be $x$ and define $f(x)=a+c$, find the range of $f(x)$ when triangle $ABC$ is an acute triangle; \n(2) Det... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 30012,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a similar setup, square $PQRS$ is constructed along diameter $PQ$ of a semicircle. The semicircle and square $PQRS$ are coplanar. Line segment $PQ$ has a length of 8 centimeters. If point $N$ is the midpoint of arc $PQ$, what is the length of segment $NS$? Let's think step b... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{10}"
} | {
"split": "train",
"index": 30013,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cylindrical tank with radius $4$ feet and height $9$ feet is lying on its side. The tank is filled with water to a depth of $2$ feet. What is the volume of water, in cubic feet? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48\\pi - 36 \\sqrt {3}"
} | {
"split": "train",
"index": 30014,
"solution": "1. **Understanding the Geometry**: The cylindrical tank is lying on its side, and we are given that the radius $r = 4$ feet and the height (length of the cylinder) $h = 9$ feet. The water fills up to a depth of $2$ feet from the bottom of the cylinder.\n\n2. **Cros... |
deepscaler | [
{
"role": "user",
"content": "Given positive numbers $a$, $b$, $c$ satisfying: $a^2+ab+ac+bc=6+2\\sqrt{5}$, find the minimum value of $3a+b+2c$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{10}+2\\sqrt{2}"
} | {
"split": "train",
"index": 30015,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\{b_k\\}$ be a sequence of integers such that $b_1 = 2$ and $b_{m+n} = b_m + b_n + mn + 1$, for all positive integers $m$ and $n$. Find $b_{12}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "101"
} | {
"split": "train",
"index": 30016,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In Pascal's Triangle, we know each number is the combination of two numbers just above it. What is the sum of the middle three numbers in each of Rows 5, 6, and 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "157"
} | {
"split": "train",
"index": 30017,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At what value of $b$ do the graphs of $y=bx^2+5x+3$ and $y=-2x-3$ intersect at exactly one point? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{49}{24}"
} | {
"split": "train",
"index": 30018,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a plane Cartesian coordinate system, points where both the vertical and horizontal coordinates are integers are called lattice points. The number of lattice points $(x, y)$ satisfying the inequality $(|x|-1)^{2}+(|y|-1)^{2}<2$ is: Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 30019,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Letters $A, B, C,$ and $D$ represent four different digits selected from $0, 1, 2, \\ldots ,9.$ If $(A+B)/(C+D)$ is an integer that is as large as possible, what is the value of $A+B$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 30020,
"solution": "1. **Objective**: We need to maximize the fraction $\\frac{A+B}{C+D}$ where $A, B, C, D$ are distinct digits from $0$ to $9$. This involves maximizing $A+B$ and minimizing $C+D$.\n\n2. **Maximizing $A+B$**: \n - The maximum value for any digit is $9$. Thus, to ma... |
deepscaler | [
{
"role": "user",
"content": "Some students are required to stand in lines: On June 1st, each column has 15 people; on June 2nd, everyone stands in one column; on June 3rd, each column has one person; on June 4th, each column has 6 people; and so on until June 12th, with a different number of people per col... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 30021,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the curve $C$ is given by $\\frac{x^2}{4} + \\frac{y^2}{3} = 1$. Taking the origin $O$ of the Cartesian coordinate system $xOy$ as the pole and the positive half-axis of $x$ as the polar axis, and using the same unit length, a polar coo... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"split": "train",
"index": 30022,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The lengths of the three sides of $\\triangle ABC$ are 5, 7, and 8, respectively. The radius of its circumcircle is ______, and the radius of its incircle is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 30023,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an experimental field, the number of fruits grown on a single plant of a certain crop, denoted as $x$, follows a normal distribution $N(90, \\sigma ^{2})$, and $P(x < 70) = 0.2$. Ten plants are randomly selected from the field, and the number of plants with fruit numbers in ... | math | {
"style": "rule",
"ground_truth": "2.1"
} | {
"split": "train",
"index": 30024,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be a given positive integer. Solve the system \n\\[x_1 + x_2^2 + x_3^3 + \\cdots + x_n^n = n,\\]\n\\[x_1 + 2x_2 + 3x_3 + \\cdots + nx_n = \\frac{n(n+1)}{2}\\]\nin the set of nonnegative real numbers. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "(x_1, x_2, \\ldots, x_n) = (1, 1, \\ldots, 1)"
} | {
"split": "train",
"index": 30025,
"solution": "\nTo solve the given system of equations:\n\n\\[\nx_1 + x_2^2 + x_3^3 + \\cdots + x_n^n = n,\n\\]\n\n\\[\nx_1 + 2x_2 + 3x_3 + \\cdots + nx_n = \\frac{n(n+1)}{2},\n\\]\n\nwith \\( x_i \\geq 0 \\) for all \\( i \\) and \\( x_i \\) being nonnegative real numbers, we w... |
deepscaler | [
{
"role": "user",
"content": "A triline is a line with the property that three times its slope is equal to the sum of its \\(x\\)-intercept and its \\(y\\)-intercept. For how many integers \\(q\\) with \\(1 \\leq q \\leq 10000\\) is there at least one positive integer \\(p\\) so that there is exactly one tr... | math | {
"style": "rule",
"ground_truth": "57"
} | {
"split": "train",
"index": 30026,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the first $n$ terms of an arithmetic sequence $\\{a_n\\}$ is $S_n$, and the sequence $\\{b_n\\}$ is a geometric sequence, satisfying $a_1=3$, $b_1=1$, $b_2+S_2=10$, and $a_5-2b_2=a_3$. The sum of the first $n$ terms of the sequence $\\left\\{ \\frac{a_n}{b_n} \\right... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 30027,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least integer greater than $\\sqrt{300}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 30028,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Factor $t^2-121$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(t-11)(t+11)"
} | {
"split": "train",
"index": 30029,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths $13,$ $19,$ $20,$ $25$ and $31,$ in some order. Find the area of the pentagon. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "745"
} | {
"split": "train",
"index": 30030,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the function $y=\\sin(\\omega x+ \\frac {5\\pi}{6})$ where $0<\\omega<\\pi$ intersects with the coordinate axes at points closest to the origin, which are $(0, \\frac {1}{2})$ and $( \\frac {1}{2}, 0)$. Determine the axis of symmetry of this graph closest to the y-... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 30031,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A mini soccer team has 12 members. We want to choose a starting lineup of 5 players, which includes one goalkeeper and four outfield players (order of outfield players does not matter). In how many ways can we choose this starting lineup? Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "3960"
} | {
"split": "train",
"index": 30032,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parabola $C$: $y^{2}=4x$ with focus $F$, two lines $l_{1}$ and $l_{2}$ are drawn through point $F$. Line $l_{1}$ intersects the parabola $C$ at points $A$ and $B$, and line $l_{2}$ intersects the parabola $C$ at points $D$ and $E$. If the sum of the squares of the slo... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 30033,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the smallest positive integer \\( a \\) for which \\( 47^n + a \\cdot 15^n \\) is divisible by 1984 for all odd \\( n \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1055"
} | {
"split": "train",
"index": 30034,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Li Shuang rides a bike from location $A$ to location $B$ at a speed of 320 meters per minute. On the way, due to a bike malfunction, he pushes the bike and continues walking for 5 minutes to a location 1800 meters from $B$ to repair the bike. Fifteen minutes later, he resumes r... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 30035,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $z^2 - 8z + 37 = 0$, how many possible values are there for $|z|$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 30036,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function \\( f(n) \\) is an integer-valued function defined on the integers which satisfies \\( f(m + f(f(n))) = -f(f(m+1)) - n \\) for all integers \\( m \\) and \\( n \\). The polynomial \\( g(n) \\) has integer coefficients and satisfies \\( g(n) = g(f(n)) \\) for all \\... | math | {
"style": "rule",
"ground_truth": "-1992"
} | {
"split": "train",
"index": 30037,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\(A, B, C,\\) and \\(D\\) are points on a circle with radius 1, \\(\\overrightarrow{AB} + 2 \\overrightarrow{AC} = \\overrightarrow{AD}\\), and \\(|AC| = 1\\). Find the area of the quadrilateral \\(ABDC\\). Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{3}}{4}"
} | {
"split": "train",
"index": 30038,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p,$ $q,$ $r,$ $s,$ $t,$ $u$ be positive real numbers such that $p + q + r + s + t + u = 8.$ Find the minimum value of\n\\[\\frac{1}{p} + \\frac{9}{q} + \\frac{16}{r} + \\frac{25}{s} + \\frac{36}{t} + \\frac{49}{u}.\\] Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "84.5"
} | {
"split": "train",
"index": 30039,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain township responded to the call of \"green waters and green mountains are mountains of gold and silver\" and tailored the town to be a \"ecological fruit characteristic town\" according to local conditions. After investigation, it was found that the yield $W$ (unit: kg... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 30040,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( PROBLEMZ \\) be a regular octagon inscribed in a circle of unit radius. Diagonals \\( MR \\) and \\( OZ \\) meet at \\( I \\). Compute \\( LI \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 30041,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence {a_n} with the sum of its first n terms denoted as S_n, and given that a_1008 > 0 and a_1007 + a_1008 < 0, find the positive integer value(s) of n that satisfy S_nS_{n+1} < 0. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "2014"
} | {
"split": "train",
"index": 30042,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a=\\sqrt{17}$ and $b=i \\sqrt{19}$, where $i=\\sqrt{-1}$. Find the maximum possible value of the ratio $|a-z| /|b-z|$ over all complex numbers $z$ of magnitude 1 (i.e. over the unit circle $|z|=1$ ). Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 30043,
"solution": "Let $|a-z| /|b-z|=k$. We wish to determine the minimum and maximum value of $k$. Squaring and expansion give: $|a-z|^{2} =|b-z|^{2} \\cdot k^{2} |a|^{2}-2 a \\cdot z+1 =\\left(|b|^{2}-2 b \\cdot z+1\\right) k^{2} |a|^{2}+1-\\left(|b|^{2}+1\\right) k^{2} =2\\left(a-... |
deepscaler | [
{
"role": "user",
"content": "Alice Czarina is bored and is playing a game with a pile of rocks. The pile initially contains 2015 rocks. At each round, if the pile has $N$ rocks, she removes $k$ of them, where $1 \\leq k \\leq N$, with each possible $k$ having equal probability. Alice Czarina continues unti... | math | {
"style": "rule",
"ground_truth": "-501"
} | {
"split": "train",
"index": 30044,
"solution": "We claim that $p=\\frac{1}{5} \\frac{6}{10} \\frac{11}{15} \\frac{16}{20} \\cdots \\frac{2006}{2010} \\frac{2011}{2015}$. Let $p_{n}$ be the probability that, starting with $n$ rocks, the number of rocks left after each round is a multiple of 5. Indeed, using recur... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\tan \\alpha = -\\frac{1}{3}$ and $\\cos \\beta = \\frac{\\sqrt{5}}{5}$, with $\\alpha, \\beta \\in (0, \\pi)$, find:\n1. The value of $\\tan(\\alpha + \\beta)$;\n2. The maximum value of the function $f(x) = \\sqrt{2} \\sin(x - \\alpha) + \\cos(x + \\beta)$. Let's ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 30045,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The highest power of 2 that is a factor of \\(15.13^{4} - 11^{4}\\) needs to be determined. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 30046,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes a lap every 80 seconds. Both start from the same line at the same time. At some random time between 10 minutes and 11 minutes af... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{16}"
} | {
"split": "train",
"index": 30047,
"solution": "1. **Calculate Rachel's running details:**\n - Rachel completes a lap every 90 seconds.\n - In 10 minutes (600 seconds), Rachel completes $\\frac{600}{90} = 6\\frac{2}{3}$ laps. This means she completes 6 full laps and is $\\frac{2}{3}$ of a lap into her sevent... |
deepscaler | [
{
"role": "user",
"content": "How many two-digit primes have a ones digit of 1? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 30048,
"solution": ""
} |
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