data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "What percent of the five-by-five square is shaded?\n\n[asy]\nsize(5cm,5cm);\nfill((0,0)--(10,0)--(10,10)--(0,10)--cycle,gray(0.7));\nfill((0,20)--(10,20)--(10,30)--(0,30)--cycle,gray(0.7));\nfill((0,40)--(10,40)--(10,50)--(0,50)--cycle,gray(0.7));\nfill((10,10)--(20,10)--(20,20... | math | {
"style": "rule",
"ground_truth": "52\\%"
} | {
"split": "train",
"index": 19549,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of $N$ consecutive four-digit positive integers is divisible by $2010^{2}$. What is the least possible value of $N$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 19550,
"solution": "First, we note that $2010=10(201)=2(5)(3)(67)$ and so $2010^{2}=2^{2} 3^{2} 5^{2} 67^{2}$. Consider $N$ consecutive four-digit positive integers. For the product of these $N$ integers to be divisible by $2010^{2}$, it must be the case that two different integers ar... |
deepscaler | [
{
"role": "user",
"content": "Find the greatest positive integer $x$ such that $23^{6+x}$ divides $2000!$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "83"
} | {
"split": "train",
"index": 19551,
"solution": "\nTo find the greatest positive integer \\( x \\) such that \\( 23^{6+x} \\) divides \\( 2000! \\), we need to determine how many times the prime factor 23 appears in the prime factorization of \\( 2000! \\).\n\nThe exponent of a prime \\( p \\) in \\( n! \\) is gi... |
deepscaler | [
{
"role": "user",
"content": "Six students are to be arranged into two classes, with two students in each class, and there are six classes in total. Calculate the number of different arrangement plans. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 19552,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It takes 42 seconds for a clock to strike 7 times. How many seconds does it take for it to strike 10 times? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 19553,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bowl contained 320 grams of pure white sugar. Mixture \\( Y \\) was formed by taking \\( x \\) grams of the white sugar out of the bowl, adding \\( x \\) grams of brown sugar to the bowl, and then mixing uniformly. In Mixture \\( Y \\), the ratio of the mass of the white suga... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 19554,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cara is sitting at a circular table with her seven friends. How many different possible pairs of people could Cara be sitting between? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 19555,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the form of $n$ such that $2^n + 2$ is divisible by $n$ where $n$ is less than 100. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "n=6, 66, 946"
} | {
"split": "train",
"index": 19556,
"solution": "Note that $2^n+2=2(2^{n-1}+1)$ so that $n$ is of the form $2r$ with $r$ odd. We will consider two cases. i) $n=2p$ with $p$ prime. $2p \\mid 2^{2p}+2$, implies that $p \\mid 2^{2n-1}+1$ and hence $p \\mid 2^{4p-2}-1$. On the other hand Fermat's little theorem guara... |
deepscaler | [
{
"role": "user",
"content": "What is the volume of the region in three-dimensional space defined by the inequalities $|x|+|y|+|z|\\le2$ and $|x|+|y|+|z-2|\\le2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}"
} | {
"split": "train",
"index": 19557,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers 1, 2, ..., 2002 are written in order on a blackboard. Then the 1st, 4th, 7th, ..., 3k+1th, ... numbers in the list are erased. The process is repeated on the remaining list (i.e., erase the 1st, 4th, 7th, ... 3k+1th numbers in the new list). This continues until no ... | math | {
"style": "rule",
"ground_truth": "1598"
} | {
"split": "train",
"index": 19558,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The real numbers \\(x_{1}, x_{2}, \\cdots, x_{2001}\\) satisfy \\(\\sum_{k=1}^{2000}\\left|x_{k}-x_{k+1}\\right|=2001\\). Let \\(y_{k}=\\frac{1}{k}\\left(x_{1}+ x_{2} + \\cdots + x_{k}\\right)\\) for \\(k=1, 2, \\cdots, 2001\\). Find the maximum possible value of \\(\\sum_{k=1}... | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"split": "train",
"index": 19559,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In unit square $ABCD,$ the inscribed circle $\\omega$ intersects $\\overline{CD}$ at $M,$ and $\\overline{AM}$ intersects $\\omega$ at a point $P$ different from $M.$ What is $AP?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{10}"
} | {
"split": "train",
"index": 19560,
"solution": "1. **Positioning and Equation of Circle $\\omega$:**\n The circle $\\omega$ is inscribed in the unit square $ABCD$. We place the center of $\\omega$ at the origin of the Cartesian plane. Since the side length of the square is 1, the radius of the circle is half t... |
deepscaler | [
{
"role": "user",
"content": "In the figure shown, arc $ADB$ and arc $BEC$ are semicircles, each with a radius of one unit. Point $D$, point $E$ and point $F$ are the midpoints of arc $ADB$, arc $BEC$ and arc $DFE$, respectively. If arc $DFE$ is also a semicircle, what is the area of the shaded region?\n\n[... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 19561,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If real numbers \\(a\\), \\(b\\), and \\(c\\) satisfy \\(a^{2} + b^{2} + c^{2} = 9\\), then what is the maximum value of the algebraic expression \\((a - b)^{2} + (b - c)^{2} + (c - a)^{2}\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 19562,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a chemistry quiz, there were $7y$ questions. Tim missed $2y$ questions. What percent of the questions did Tim answer correctly? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "71.43\\%"
} | {
"split": "train",
"index": 19563,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You and five friends need to raise $1500$ dollars in donations for a charity, dividing the fundraising equally. How many dollars will each of you need to raise? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "250"
} | {
"split": "train",
"index": 19564,
"solution": "1. **Identify the total amount to be raised and the number of people involved:**\n - Total amount to be raised: $1500$ dollars.\n - Number of people involved: $6$ (you and five friends).\n\n2. **Calculate the amount each person needs to raise:**\n - To find o... |
deepscaler | [
{
"role": "user",
"content": "How many unordered pairs of coprime numbers are there among the integers 2, 3, ..., 30? Recall that two integers are called coprime if they do not have any common natural divisors other than one. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "248"
} | {
"split": "train",
"index": 19565,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $z_{1}, z_{2}, z_{3}, z_{4}$ be the solutions to the equation $x^{4}+3 x^{3}+3 x^{2}+3 x+1=0$. Then $\\left|z_{1}\\right|+\\left|z_{2}\\right|+\\left|z_{3}\\right|+\\left|z_{4}\\right|$ can be written as $\\frac{a+b \\sqrt{c}}{d}$, where $c$ is a square-free positive intege... | math | {
"style": "rule",
"ground_truth": "7152"
} | {
"split": "train",
"index": 19566,
"solution": "Note that $x=0$ is clearly not a solution, so we can divide the equation by $x^{2}$ to get $\\left(x^{2}+2+\\frac{1}{x^{2}}\\right)+3\\left(x+\\frac{1}{x}\\right)+1=0$. Letting $y=x+\\frac{1}{x}$, we get that $y^{2}+3 y+1=0$, so $y=x+\\frac{1}{x}=\\frac{-3 \\pm \\s... |
deepscaler | [
{
"role": "user",
"content": "Anya, Vanya, Dania, Sanya, and Tanya were collecting apples. It turned out that each of them collected an integer percentage of the total number of apples, and all these percentages are different and greater than zero. What is the minimum number of apples that could have been c... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 19567,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive divisors does $24$ have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 19568,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Eight chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least four adjacent chairs. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "288"
} | {
"split": "train",
"index": 19569,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A \\(3 \\times 3 \\times 3\\) cube composed of 27 unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes (order is irrelevant) from a \\(3 \\times 3 \\times 1\\) block with the property that the line joining the centers of the ... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 19570,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is 50% more than the regular. After both consume $\\frac{3}{4}$ of their drinks, Ann gives Ed a third of what she has left, and 2 additional ounces. When they finish t... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 19571,
"solution": "1. **Define the variables:**\n Let the size of Ed's drink be $x$ ounces. Since Ann's drink is 50% more than Ed's, her drink size is $\\frac{3}{2}x$ ounces.\n\n2. **Calculate the amount consumed and remaining:**\n Both Ed and Ann consume $\\frac{3}{4}$ of their ... |
deepscaler | [
{
"role": "user",
"content": "In the figure, $\\overline{AB}$ and $\\overline{CD}$ are diameters of the circle with center $O$, $\\overline{AB} \\perp \\overline{CD}$, and chord $\\overline{DF}$ intersects $\\overline{AB}$ at $E$. If $DE = 6$ and $EF = 2$, then the area of the circle is Let's think step by ... | math | {
"style": "rule",
"ground_truth": "24 \\pi"
} | {
"split": "train",
"index": 19572,
"solution": "\n#### Step 1: Establish Known Values and Relationships\nGiven that $\\overline{AB}$ and $\\overline{CD}$ are diameters of the circle with center $O$ and $\\overline{AB} \\perp \\overline{CD}$, we know that $O$ is the midpoint of both $\\overline{AB}$ and $\\overli... |
deepscaler | [
{
"role": "user",
"content": "The equation\n\\[(x - \\sqrt[3]{17})(x - \\sqrt[3]{67})(x - \\sqrt[3]{97}) = \\frac{1}{2}\\]\nhas three distinct solutions $u,$ $v,$ and $w.$ Calculate the value of $u^3 + v^3 + w^3.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "181.5"
} | {
"split": "train",
"index": 19573,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number 25 is expressed as the sum of positive integers \\(x_{1}, x_{2}, \\cdots, x_{k}\\), where \\(k \\leq 25\\). What is the maximum value of the product of \\(x_{1}, x_{2}, x_{3}, \\cdots\\), and \\(x_{k}\\)? Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "8748"
} | {
"split": "train",
"index": 19574,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A function $f: \\mathbb{Z} \\rightarrow \\mathbb{Z}$ satisfies: $f(0)=0$ and $$\\left|f\\left((n+1) 2^{k}\\right)-f\\left(n 2^{k}\\right)\\right| \\leq 1$$ for all integers $k \\geq 0$ and $n$. What is the maximum possible value of $f(2019)$? Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 19575,
"solution": "Consider a graph on $\\mathbb{Z}$ with an edge between $(n+1) 2^{k}$ and $n 2^{k}$ for all integers $k \\geq 0$ and $n$. Each vertex $m$ is given the value $f(m)$. The inequality $\\left|f\\left((n+1) 2^{k}\\right)-f\\left(n 2^{k}\\right)\\right| \\leq 1$ means tha... |
deepscaler | [
{
"role": "user",
"content": "The power function $f(x)=(m^{2}+2m-2)x^{m}$ is a decreasing function on $(0,+\\infty)$. Find the value of the real number $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 19576,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Arrange the positive integers in the following number matrix:\n\\begin{tabular}{lllll}\n1 & 2 & 5 & 10 & $\\ldots$ \\\\\n4 & 3 & 6 & 11 & $\\ldots$ \\\\\n9 & 8 & 7 & 12 & $\\ldots$ \\\\\n16 & 15 & 14 & 13 & $\\ldots$ \\\\\n$\\ldots$ & $\\ldots$ & $\\ldots$ & $\\ldots$ & $\\ldot... | math | {
"style": "rule",
"ground_truth": "421"
} | {
"split": "train",
"index": 19577,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A$ , $B$ , and $C$ be distinct points on a line with $AB=AC=1$ . Square $ABDE$ and equilateral triangle $ACF$ are drawn on the same side of line $BC$ . What is the degree measure of the acute angle formed by lines $EC$ and $BF$ ?\n\n*Ray Li* Let's think step b... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 19578,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A box contains 5 white balls and 5 black balls. I draw them out of the box, one at a time. What is the probability that all of my draws alternate colors, starting and ending with the same color? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{126}"
} | {
"split": "train",
"index": 19579,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the inequality $x^{2}+ax+1\\geqslant 0$, if this inequality holds for all $x\\in(0, \\frac {1}{2}]$, find the minimum value of the real number $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac {5}{2}"
} | {
"split": "train",
"index": 19580,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Al's age is $16$ more than the sum of Bob's age and Carl's age, and the square of Al's age is $1632$ more than the square of the sum of Bob's age and Carl's age. What is the sum of the ages of Al, Bob, and Carl? Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "102"
} | {
"split": "train",
"index": 19581,
"solution": "1. **Define Variables:**\n Let $a$ be Al's age, $b$ be Bob's age, and $c$ be Carl's age. \n\n2. **Set Up Equations:**\n From the problem, we have two equations:\n \\[ a = 16 + b + c \\]\n \\[ a^2 = 1632 + (b+c)^2 \\]\n\n3. **Substitute and Expand:**\n Sub... |
deepscaler | [
{
"role": "user",
"content": "Given the sets \\( A=\\{x \\mid 5x - a \\leqslant 0\\} \\) and \\( B=\\{x \\mid 6x - b > 0\\} \\), where \\( a, b \\in \\mathbf{N} \\), and \\( A \\cap B \\cap \\mathbf{N} = \\{2, 3, 4\\} \\), determine the number of integer pairs \\( (a, b) \\). Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 19582,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An influenza outbreak occurred in regions $A$, $B$, and $C$. The percentages of people with influenza in these three regions are $6\\%$, $5\\%$, and $4\\%$ respectively. It is assumed that the populations of these three regions are in the ratio of $3:5:2$. Now, a person is rand... | math | {
"style": "rule",
"ground_truth": "\\frac{18}{51}"
} | {
"split": "train",
"index": 19583,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $$f(x)=(2-a)\\ln x+ \\frac {1}{x}+2ax \\quad (a\\leq0)$$.\n(Ⅰ) When $a=0$, find the extreme value of $f(x)$;\n(Ⅱ) When $a<0$, discuss the monotonicity of $f(x)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2-2\\ln2"
} | {
"split": "train",
"index": 19584,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the integers from 1 to 100, inclusive, have at least one digit equal to 6? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 19585,
"solution": "The integers between 1 and 100 that have a ones digit equal to 6 are \\(6, 16, 26, 36, 46, 56, 66, 76, 86, 96\\), of which there are 10. The additional integers between 1 and 100 that have a tens digit equal to 6 are \\(60, 61, 62, 63, 64, 65, 67, 68, 69\\), of whi... |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of\n\\[\\sqrt{x^2 + (1 + 2x)^2} + \\sqrt{(x - 1)^2 + (x - 1)^2}\\]over all real numbers $x.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} \\times \\sqrt{5}"
} | {
"split": "train",
"index": 19586,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 400 students at Pascal H.S., where the ratio of boys to girls is $3: 2$. There are 600 students at Fermat C.I., where the ratio of boys to girls is $2: 3$. What is the ratio of boys to girls when considering all students from both schools? Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "12:13"
} | {
"split": "train",
"index": 19587,
"solution": "Since the ratio of boys to girls at Pascal H.S. is $3: 2$, then $\frac{3}{3+2}=\frac{3}{5}$ of the students at Pascal H.S. are boys. Thus, there are $\frac{3}{5}(400)=\frac{1200}{5}=240$ boys at Pascal H.S. Since the ratio of boys to girls at Fermat C.I. is $2: 3$,... |
deepscaler | [
{
"role": "user",
"content": "A plane is uniquely determined by three non-collinear points. What is the maximum possible number of planes that can be determined by 12 points in space? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "220"
} | {
"split": "train",
"index": 19588,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S_n$ be the sum of the first $n$ terms of a geometric sequence $\\{a_n\\}$. Given that $a_3 = 8a_6$, find the value of $\\frac{S_4}{S_2}$. 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": 19589,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\tan\\left(\\frac{\\pi}{9}\\right)\\tan\\left(\\frac{2\\pi}{9}\\right)\\tan\\left(\\frac{4\\pi}{9}\\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 19590,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find an ordered pair $(a, b)$ of real numbers for which $x^{2}+a x+b$ has a non-real root whose cube is 343. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(7, 49)"
} | {
"split": "train",
"index": 19591,
"solution": "The cube roots of 343 are the roots of $x^{3}-343$, which is $(x-7)\\left(x^{2}+7 x+49\\right)$. Therefore the ordered pair we want is $(\\mathbf{7, 49})$."
} |
deepscaler | [
{
"role": "user",
"content": "Let $a, b, c,$ and $d$ be positive integers such that $\\gcd(a, b)=24$, $\\gcd(b, c)=36$, $\\gcd(c, d)=54$, and $70<\\gcd(d, a)<100$. Which of the following must be a divisor of $a$?\n$\\textbf{(A)} \\text{ 5} \\qquad \\textbf{(B)} \\text{ 7} \\qquad \\textbf{(C)} \\text{ 11} \... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 19592,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$ Calculate the value of $p + q + r + s.$ Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "2028"
} | {
"split": "train",
"index": 19593,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Petya places \"+\" and \"-\" signs in all possible ways into the expression $1 * 2 * 3 * 4 * 5 * 6$ at the positions of the asterisks. For each arrangement of signs, he calculates the resulting value and writes it on the board. Some numbers may appear on the board multiple time... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 19594,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different primes are in the prime factorization of $87\\cdot89\\cdot91\\cdot93$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 19595,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, each of the three identical circles touch the other two. The circumference of each circle is 36. What is the perimeter of the shaded region? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 19596,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $AB = AC = 17$, $BC = 16$, and point $G$ is the centroid of the triangle. Points $A'$, $B'$, and $C'$ are the images of $A$, $B$, and $C$, respectively, after a $90^\\circ$ clockwise rotation about $G$. Determine the area of the union of the two regions enclo... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 19597,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a large 15 by 20 rectangular region, one quarter area of the rectangle is shaded. If the shaded quarter region itself represents one fourth of its quarter area, calculate the fraction of the total area that is shaded.\nA) $\\frac{1}{16}$\nB) $\\frac{1}{12}$\nC) $\\frac{1}{4}... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"split": "train",
"index": 19598,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If we express $3x^2 - 6x - 2$ in the form $a(x - h)^2 + k$, then what is $a + h + k$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 19599,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four positive integers $p$, $q$, $r$, $s$ satisfy $p \\cdot q \\cdot r \\cdot s = 9!$ and $p < q < r < s$. What is the smallest possible value of $s-p$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 19600,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system, the equation of circle $C$ is $(x-2)^2 + y^2 = 4$, with the center at point $C$. Using the origin as the pole and the non-negative half of the $x$-axis as the initial ray, establish a polar coordinate system. The curve $C_1: \\rho = -4\\sqrt{... | math | {
"style": "rule",
"ground_truth": "1 : 2"
} | {
"split": "train",
"index": 19601,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of natural numbers \\( k \\) not exceeding 353500 such that \\( k^{2} + k \\) is divisible by 505. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2800"
} | {
"split": "train",
"index": 19602,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the geometric sequence $\\{a_{n}\\}$, $a_{2}$ and $a_{18}$ are the two roots of the equation $x^{2}+15x+16=0$, find the value of $a_{3}a_{10}a_{17}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-64"
} | {
"split": "train",
"index": 19603,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_n= \\frac {1}{n}\\sin \\frac {n\\pi}{25}$, and $S_n=a_1+a_2+\\ldots+a_n$. Find the number of positive terms among $S_1, S_2, \\ldots, S_{100}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 19604,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the minimum number of equilateral triangles, each of side length 1 unit, needed to completely cover an equilateral triangle of side length 15 units? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 19605,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of the positive differences of the sum of eight integers (28) is 6. To what power will it always be divisible? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6^7"
} | {
"split": "train",
"index": 19606,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the expected value of the roll of a fair octahedral die? (An octahedral die has 8 faces, numbered from 1 to 8.) 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": "4.5"
} | {
"split": "train",
"index": 19607,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\sum_{k=1}^{2009} k\\left(\\left\\lfloor\\frac{2009}{k}\\right\\rfloor-\\left\\lfloor\\frac{2008}{k}\\right\\rfloor\\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2394"
} | {
"split": "train",
"index": 19608,
"solution": "The summand is equal to $k$ if $k$ divides 2009 and 0 otherwise. Thus the sum is equal to the sum of the divisors of 2009, or 2394."
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest common divisor of $1729$ and $1768$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 19609,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the adjoining figure, two circles with radii $8$ and $6$ are drawn with their centers $12$ units apart. At $P$, one of the points of intersection, a line is drawn in such a way that the chords $QP$ and $PR$ have equal length. Find the square of the length of $QP$.\n[asy]size... | math | {
"style": "rule",
"ground_truth": "130"
} | {
"split": "train",
"index": 19610,
"solution": "Let $QP = PR = x$. Extend the line containing the centers of the two circles to meet $R$, and to meet the other side of the large circle at a point $S$.\n The part of this line from $R$ to the point nearest to $R$ where it intersects the larger circle has length $6... |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a} = (\\cos x, -\\sqrt{3}\\cos x)$ and $\\overrightarrow{b} = (\\cos x, \\sin x)$, and the function $f(x) = \\overrightarrow{a} \\cdot \\overrightarrow{b} + 1$.\n\n(Ⅰ) Find the interval of monotonic increase for the function $f(x)$;\n\n(Ⅱ) If $f(... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3} - \\sqrt{5}}{6}"
} | {
"split": "train",
"index": 19611,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiao Wang plans to finish reading a 200-page book within a certain number of days. After reading for 5 days, he changed his plan and decided to read 5 more pages each day. As a result, he finished reading one day earlier than planned. How many pages did he originally plan to re... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 19612,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $a_{1}, a_{2}, a_{3}, \\ldots$ of real numbers satisfies the recurrence $a_{n+1}=\\frac{a_{n}^{2}-a_{n-1}+2 a_{n}}{a_{n-1}+1}$. Given that $a_{1}=1$ and $a_{9}=7$, find $a_{5}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 19613,
"solution": "Let $b_{n}=a_{n}+1$. Then the recurrence becomes $b_{n+1}-1=\\left(b_{n}^{2}-b_{n-1}\\right) / b_{n-1}=b_{n}^{2} / b_{n-1}-1$, so $b_{n+1}=b_{n}^{2} / b_{n-1}$. It follows that the sequence $\\left(b_{n}\\right)$ is a geometric progression, from which $b_{5}^{2}=b_... |
deepscaler | [
{
"role": "user",
"content": "For how many integers \\( n \\) between 1 and 20 (inclusive) is \\( \\frac{n}{18} \\) a repeating decimal? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 19614,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sqrt{3}\\sin x\\cos x+\\cos^{2}x$.\n\n(1) Find the value of $f(\\frac{\\pi }{24})$;\n\n(2) If the function $f(x)$ is monotonically increasing in the interval $[-m,m]$, find the maximum value of the real number $m$. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"split": "train",
"index": 19615,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\\left(\\sin A+\\sin B\\right)\\left(a-b\\right)=c\\left(\\sin C-\\sin B\\right)$, and $D$ is a point on side $BC$ such that $AD$ bisects angle $BAC$ and $A... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 19616,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{A}$ be the set of finite sequences of positive integers $a_1,a_2,\\dots,a_k$ such that $|a_n-a_{n-1}|=a_{n-2}$ for all $3\\leqslant n\\leqslant k$ . If $a_1=a_2=1$ , and $k=18$ , determine the number of elements of $\\mathcal{A}$ . Let's think step by s... | math | {
"style": "rule",
"ground_truth": "1597"
} | {
"split": "train",
"index": 19617,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Max sold glasses of lemonade for 25 cents each. He sold 41 glasses on Saturday and 53 glasses on Sunday. What were his total sales for these two days? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$23.50"
} | {
"split": "train",
"index": 19618,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $z$ be a complex number satisfying $z^2 + z + 1 = 0.$ Compute\n\\[z^{97} + z^{98} + z^{99} + z^{100} + z^{101}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 19619,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four fair coins are to be flipped. What is the probability that all four will be heads or all four will be tails? 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{1}{8}"
} | {
"split": "train",
"index": 19620,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The minimum value of the quotient of a (base ten) number of three different non-zero digits divided by the sum of its digits is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10.5"
} | {
"split": "train",
"index": 19621,
"solution": "1. **Expression Setup**: We start by expressing the number with three different non-zero digits $a$, $b$, and $c$ as $100a + 10b + c$. The sum of its digits is $a + b + c$. Thus, the quotient of the number by the sum of its digits is:\n \\[\n \\frac{100a + 10b ... |
deepscaler | [
{
"role": "user",
"content": "A projection takes $\\begin{pmatrix} 4 \\\\ 4 \\end{pmatrix}$ to $\\begin{pmatrix} \\frac{60}{13} \\\\ \\frac{12}{13} \\end{pmatrix}.$ Which vector does the projection take $\\begin{pmatrix} -2 \\\\ 2 \\end{pmatrix}$ to? Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -20/13 \\\\ -4/13 \\end{pmatrix}"
} | {
"split": "train",
"index": 19622,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all positive integers $n$ such that $1+2+\\cdots+n$ divides $15\\left[(n+1)^{2}+(n+2)^{2}+\\cdots+(2 n)^{2}\\right]$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 19623,
"solution": "We can compute that $1+2+\\cdots+n=\\frac{n(n+1)}{2}$ and $(n+1)^{2}+(n+2)^{2}+\\cdots+(2 n)^{2}=\\frac{2 n(2 n+1)(4 n+1)}{6}-\\frac{n(n+1)(2 n+1)}{6}=\\frac{n(2 n+1)(7 n+1)}{6}$, so we need $\\frac{15(2 n+1)(7 n+1)}{3(n+1)}=\\frac{5(2 n+1)(7 n+1)}{n+1}$ to be an i... |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all real solutions to the equation \\[\\frac{x-3}{x^2+3x+2} = \\frac{x-7}{x^2-7x+12}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{26}{3}"
} | {
"split": "train",
"index": 19624,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I bought a lottery ticket, the sum of the digits of its five-digit number turned out to be equal to the age of my neighbor. Determine the number of the ticket, given that my neighbor easily solved this problem. Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "99999"
} | {
"split": "train",
"index": 19625,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $(a_n)_{n \\ge 1}$ be a sequence of positive real numbers such that the sequence $(a_{n+1}-a_n)_{n \\ge 1}$ is convergent to a non-zero real number. Evaluate the limit $$ \\lim_{n \\to \\infty} \\left( \\frac{a_{n+1}}{a_n} \\right)^n. $$ Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "e"
} | {
"split": "train",
"index": 19626,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{a}, \\mathbf{b},$ and $\\mathbf{c}$ be nonzero vectors, no two of which are parallel, such that\n\\[(\\mathbf{a} \\times \\mathbf{b}) \\times \\mathbf{c} = \\frac{1}{2} \\|\\mathbf{b}\\| \\|\\mathbf{c}\\| \\mathbf{a}.\\]\nLet $\\theta$ be the angle between $\\math... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 19627,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A church rings its bells every 15 minutes, the school rings its bells every 20 minutes and the day care center rings its bells every 25 minutes. If they all ring their bells at noon on the same day, at what time will they next all ring their bells together? (Answer in the form... | math | {
"style": "rule",
"ground_truth": "05\\!:\\!00"
} | {
"split": "train",
"index": 19628,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the maximum value that the expression \\(\\frac{1}{a+\\frac{2010}{b+\\frac{1}{c}}}\\) can take, where \\(a, b, c\\) are distinct non-zero digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/203"
} | {
"split": "train",
"index": 19629,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The faces of each of $7$ standard dice are labeled with the integers from $1$ to $6$. Let $p$ be the probabilities that when all $7$ dice are rolled, the sum of the numbers on the top faces is $10$. What other sum occurs with the same probability as $p$? Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 19630,
"solution": "To solve this problem, we need to understand the symmetry in the sum of the numbers on the faces of the dice when rolled. The minimum sum occurs when all dice show a $1$, which is $7 \\times 1 = 7$. The maximum sum occurs when all dice show a $6$, which is $7 \\tim... |
deepscaler | [
{
"role": "user",
"content": "A bag contains 2 red balls, 3 white balls, and 4 yellow balls. If 4 balls are randomly selected from the bag, what is the probability that the selection includes balls of all three colors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4/7"
} | {
"split": "train",
"index": 19631,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider two sets of consecutive integers. Let $A$ be the least common multiple (LCM) of the integers from $15$ to $25$ inclusive. Let $B$ be the least common multiple of $A$ and the integers $26$ to $45$. Compute the value of $\\frac{B}{A}$.\nA) 4536\nB) 18426\nC) 3 * 37 * 41 ... | math | {
"style": "rule",
"ground_truth": "3 \\cdot 37 \\cdot 41 \\cdot 43"
} | {
"split": "train",
"index": 19632,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points \\( E, F, M \\) are located on the sides \\( AB, BC, \\) and \\( AC \\) of triangle \\( ABC \\), respectively. The segment \\( AE \\) is one third of side \\( AB \\), the segment \\( BF \\) is one sixth of side \\( BC \\), and the segment \\( AM \\) is two fifths of side... | math | {
"style": "rule",
"ground_truth": "23/90"
} | {
"split": "train",
"index": 19633,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence \\(\\left\\{a_{n}\\right\\}\\) with the general term\n\\[ a_{n} = n^{4} + 6n^{3} + 11n^{2} + 6n, \\]\nfind the sum of the first 12 terms \\( S_{12} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "104832"
} | {
"split": "train",
"index": 19634,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $\\triangle ABC$, $\\triangle BCD$, and $\\triangle CDE$ are right-angled at $B$, $C$, and $D$ respectively, with $\\angle ACB=\\angle BCD = \\angle CDE = 45^\\circ$, and $AB=15$. [asy]\npair A, B, C, D, E;\nA=(0,15);\nB=(0,0);\nC=(10.6066,0);\nD=(15,0);\nE=(21.... | math | {
"style": "rule",
"ground_truth": "\\frac{15\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 19635,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(n)$ and $g(n)$ be functions satisfying \\[f(n) = \\begin{cases}\\sqrt{n} & \\text{ if } \\sqrt{n} \\text{ is an integer}\\\\ 1 + f(n+1) & \\text{ otherwise} \\end{cases}\\]and \\[g(n) = \\begin{cases}\\sqrt{n} & \\text{ if } \\sqrt{n} \\text{ is an integer}\\\\ 2 + g(n+2... | math | {
"style": "rule",
"ground_truth": "258"
} | {
"split": "train",
"index": 19636,
"solution": "Since $n$ isn't a perfect square, let $n=m^2+k$ with $0<k<2m+1$. If $m$ is odd, then $f(n)=g(n)$. If $m$ is even, then \\begin{align*} f(n)&=(m+1)^2-(m^2+k)+(m+1)=3m+2-k, \\\\ g(n)&=(m+2)^2-(m^2+k)+(m+2)=5m+6-k, \\end{align*} from which \\begin{align*} 7(3m+2-k)&=4... |
deepscaler | [
{
"role": "user",
"content": "If $\\log_M{N}=\\log_N{M}$, $M \\ne N$, $MN>0$, $M \\ne 1$, $N \\ne 1$, then $MN$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 19637,
"solution": "1. Given that $\\log_M N = \\log_N M$ and $M \\neq N$, $MN > 0$, $M \\neq 1$, and $N \\neq 1$.\n\n2. We can use the property of logarithms that states $\\log_a b = \\frac{\\log_c b}{\\log_c a}$ to change the base of the logarithms. Let's change the base of $\\log_M... |
deepscaler | [
{
"role": "user",
"content": "A ferry boat begins shuttling tourists to an island every hour starting at 10 AM, with its last trip starting at 4 PM. On the 10 AM trip, there were 100 tourists on the ferry, and on each successive trip, the number of tourists decreased by 2 from the previous trip. Calculate t... | math | {
"style": "rule",
"ground_truth": "658"
} | {
"split": "train",
"index": 19638,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ a, b \\in \\mathbb{N}$ with $ 1 \\leq a \\leq b,$ and $ M \\equal{} \\left[\\frac {a \\plus{} b}{2} \\right].$ Define a function $ f: \\mathbb{Z} \\mapsto \\mathbb{Z}$ by\n\\[ f(n) \\equal{} \\begin{cases} n \\plus{} a, & \\text{if } n \\leq M, \\\\\nn \\minus{} b, & \\t... | math | {
"style": "rule",
"ground_truth": "\\frac {a + b}{\\gcd(a,b)}"
} | {
"split": "train",
"index": 19639,
"solution": "\nLet \\( a, b \\in \\mathbb{N} \\) with \\( 1 \\leq a \\leq b \\), and let \\( M = \\left\\lfloor \\frac{a + b}{2} \\right\\rfloor \\). The function \\( f: \\mathbb{Z} \\to \\mathbb{Z} \\) is defined as:\n\n\\[\nf(n) = \n\\begin{cases} \nn + a, & \\text{if } n \\l... |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$, with sides of length $5$, $6$, and $7$, has one vertex on the positive $x$-axis, one on the positive $y$-axis, and one on the positive $z$-axis. Let $O$ be the origin. What is the volume of tetrahedron $OABC$? Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{95}"
} | {
"split": "train",
"index": 19640,
"solution": "1. **Assigning Coordinates**: Assume without loss of generality that vertex $A$ is on the $x$-axis, vertex $B$ is on the $y$-axis, and vertex $C$ is on the $z$-axis. Let the coordinates of $A$, $B$, and $C$ be $(a,0,0)$, $(0,b,0)$, and $(0,0,c)$ respectively.\n\n2.... |
deepscaler | [
{
"role": "user",
"content": "900 cards are inscribed with all natural numbers from 1 to 900. Cards inscribed with squares of integers are removed, and the remaining cards are renumbered starting from 1.\n\nThen, the operation of removing the squares is repeated. How many times must this operation be repeat... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 19641,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $\\frac{2a^{-1}+\\frac{a^{-1}}{2}}{a}$ when $a= \\frac{1}{2}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 19642,
"solution": "1. **Rewrite the expression with negative exponents:** \n Given the expression $\\frac{2a^{-1}+\\frac{a^{-1}}{2}}{a}$, we first simplify the numerator. Recall that $a^{-1} = \\frac{1}{a}$. Thus, the expression becomes:\n \\[\n \\frac{2\\left(\\frac{1}{a}\\ri... |
deepscaler | [
{
"role": "user",
"content": "Digital clocks display hours and minutes (for example, 16:15). While practicing arithmetic, Buratino finds the sum of the digits on the clock $(1+6+1+5=13)$. Write down such a time of day when the sum of the digits on the clock will be the greatest. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "19:59"
} | {
"split": "train",
"index": 19643,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has side lengths $AB=7, BC=8,$ and $CA=9.$ Circle $\\omega_1$ passes through $B$ and is tangent to line $AC$ at $A.$ Circle $\\omega_2$ passes through $C$ and is tangent to line $AB$ at $A.$ Let $K$ be the intersection of circles $\\omega_1$ and $\\omega_2$ not e... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 19644,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the graph of the linear function $y=kx+b$ ($k \\neq 0$) intersects the positive half-axes of the $x$-axis and $y$-axis at points $A$ and $B$, respectively, and the area of $\\triangle OAB$ is equal to $|OA|+|OB|+3$. \n(1) Express $k$ i... | math | {
"style": "rule",
"ground_truth": "7+2\\sqrt{10}"
} | {
"split": "train",
"index": 19645,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the function \\( f:(0,1) \\rightarrow \\mathbf{R} \\) be defined as\n$$\nf(x)=\\left\\{\\begin{array}{l}\nx, \\text{ if } x \\text{ is irrational; } \\\\\n\\frac{p+1}{q}, \\text{ if } x=\\frac{p}{q}, \\text{ where } (p, q)=1 \\text{ and } 0<p<q.\n\\end{array}\\right.\n$$\n\... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{17}"
} | {
"split": "train",
"index": 19646,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function \\( g(x) \\) satisfies\n\\[ g(x) - 2 g \\left( \\frac{1}{x} \\right) = 3^x + x \\]\nfor all \\( x \\neq 0 \\). Find \\( g(2) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4 - \\frac{2\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 19647,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $M(3,7)$ is the midpoint of $\\overline{AB}$. If point $A$ has coordinates $(9,3)$, what is the sum of the coordinates of point $B$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 19648,
"solution": ""
} |
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