data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "The sum of the lengths of the twelve edges of a rectangular box is $140$, and the distance from one corner of the box to the farthest corner is $21$. What is the total surface area of the box? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "784"
} | {
"split": "train",
"index": 33249,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$Q$ is the point of intersection of the diagonals of one face of a cube whose edges have length 2 units. Calculate the length of $QR$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"split": "train",
"index": 33250,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pyramid with volume 40 cubic inches has a rectangular base. If the length of the base is doubled, the width tripled and the height increased by $50\\%$, what is the volume of the new pyramid, in cubic inches? Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "360"
} | {
"split": "train",
"index": 33251,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a French class, I need to master a list of 600 vocabulary words for an upcoming test. The score on the test is based on the percentage of words I recall correctly. In this class, I have noticed that even when guessing the words I haven't studied, I have about a 10% chance o... | math | {
"style": "rule",
"ground_truth": "534"
} | {
"split": "train",
"index": 33252,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The length of the longer side of rectangle \\(R'\\) is \\(15\\%\\) more than the length of a side of square \\(S'\\). The length of the shorter side of rectangle \\(R'\\) is \\(5\\%\\) less than the length of a side of square \\(S'\\). What is the ratio of the area of rectangle... | math | {
"style": "rule",
"ground_truth": "\\frac{10925}{10000}"
} | {
"split": "train",
"index": 33253,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The axis cross-section $SAB$ of a cone with an equal base triangle side length of 2, $O$ as the center of the base, and $M$ as the midpoint of $SO$. A moving point $P$ is on the base of the cone (including the circumference). If $AM \\perp MP$, then the length of the trajectory... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{7}}{2}"
} | {
"split": "train",
"index": 33254,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the USA, standard letter-size paper is 8.5 inches wide and 11 inches long. What is the largest integer that cannot be written as a sum of a whole number (possibly zero) of 8.5's and a whole number (possibly zero) of 11's? Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "159"
} | {
"split": "train",
"index": 33255,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an isosceles trapezoid with \\(AB = 24\\) units, \\(CD = 10\\) units, and legs \\(AD\\) and \\(BC\\) each measuring \\(13\\) units. Find the length of diagonal \\(AC\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 33256,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that three fair coins are tossed once and for each head, one fair die is rolled, with a second roll occurring if the first roll is 6, determine the probability that the sum of all die rolls is odd. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{16}"
} | {
"split": "train",
"index": 33257,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the product \\[\\frac{9}{3}\\cdot\\frac{15}{9}\\cdot\\frac{21}{15} \\dotsm \\frac{3n+6}{3n} \\dotsm \\frac{3003}{2997}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1001"
} | {
"split": "train",
"index": 33258,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be a list of positive integers--not necessarily distinct--in which the number $68$ appears. The average (arithmetic mean) of the numbers in $S$ is $56$. However, if $68$ is removed, the average of the remaining numbers drops to $55$. What is the largest number that can ... | math | {
"style": "rule",
"ground_truth": "649"
} | {
"split": "train",
"index": 33259,
"solution": "Suppose that $S$ has $n$ numbers other than the $68.$ We have the following table: \\[\\begin{array}{c|c|c|c} & & & \\\\ [-2.5ex] & \\textbf{Count} & \\textbf{Arithmetic Mean} & \\textbf{Sum} \\\\ \\hline & & & \\\\ [-2.5ex] \\textbf{Initial} & n+1 & 56 & 56(n+1) \... |
deepscaler | [
{
"role": "user",
"content": "Find the least positive integer $k$ for which the equation $\\left\\lfloor\\frac{2002}{n}\\right\\rfloor=k$ has no integer solutions for $n$. (The notation $\\lfloor x\\rfloor$ means the greatest integer less than or equal to $x$.) Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 33260,
"solution": "Rewriting the given information and simplifying it a bit, we have \\begin{align*} k \\le \\frac{2002}{n} < k+1 &\\implies \\frac{1}{k} \\ge \\frac{n}{2002} > \\frac{1}{k+1}. \\\\ &\\implies \\frac{2002}{k} \\ge n > \\frac{2002}{k+1}. \\end{align*}\n Now note that i... |
deepscaler | [
{
"role": "user",
"content": "If a worker receives a $20$% cut in wages, he may regain his original pay exactly by obtaining a raise of: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25\\%"
} | {
"split": "train",
"index": 33261,
"solution": "1. **Understanding the wage cut**: Let the original wage of the worker be $W$. A $20\\%$ cut in wages means the worker now earns $80\\%$ of $W$. Mathematically, this can be expressed as:\n \\[\n \\text{New Wage} = 0.8W = \\frac{4}{5}W\n \\]\n\n2. **Calculatin... |
deepscaler | [
{
"role": "user",
"content": "Calculate:<br/>$(1)(\\sqrt{3})^2+|1-\\sqrt{3}|+\\sqrt[3]{-27}$;<br/>$(2)(\\sqrt{12}-\\sqrt{\\frac{1}{3}})×\\sqrt{6}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5\\sqrt{2}"
} | {
"split": "train",
"index": 33262,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles with radius $2$ and radius $4$ have a common center at P. Points $A, B,$ and $C$ on the larger circle are the vertices of an equilateral triangle. Point $D$ is the intersection of the smaller circle and the line segment $PB$ . Find the square of the are... | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 33263,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of $\\log_{2}9 \\cdot \\log_{3}5 \\cdot \\log_{\\sqrt{5}}8 = \\_\\_\\_\\_\\_\\_.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 33264,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain store sells a batch of helmets for $80 each. It can sell 200 helmets per month. During the \"Creating a Civilized City\" period, the store plans to reduce the price of the helmets for sale. After investigation, it was found that for every $1 decrease in price, an addi... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 33265,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cars A and B travel the same distance. Car A travels half that distance at $u$ miles per hour and half at $v$ miles per hour. Car B travels half the time at $u$ miles per hour and half at $v$ miles per hour. The average speed of Car A is $x$ miles per hour and that of Car B is ... | math | {
"style": "rule",
"ground_truth": "x \\leq y"
} | {
"split": "train",
"index": 33266,
"solution": "1. **Define the average speed for Car A**:\n Car A travels half the distance at speed $u$ mph and the other half at $v$ mph. Let the total distance be $D$. Then each half is $\\frac{D}{2}$. The time taken to travel the first half at speed $u$ is $\\frac{\\frac{D}... |
deepscaler | [
{
"role": "user",
"content": "Find the number of subsets of $\\{1, 2, 3, 4, 5, 6, 7, 8\\}$ that are subsets of neither $\\{1, 2, 3, 4, 5\\}$ nor $\\{4, 5, 6, 7, 8\\}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "196"
} | {
"split": "train",
"index": 33267,
"solution": "Note that by Principle of Inclusion and Exclusion, the total number of subsets must be $2^8-2^5-2^5+2^2$ as denoted by above. Thus our answer is $64(3)+4 = \\boxed{196}$"
} |
deepscaler | [
{
"role": "user",
"content": "In the sequence\n\\[..., a, b, c, d, 0, 1, 1, 2, 3, 5, 8,...\\]\neach term is the sum of the two terms to its left. Find $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 33268,
"solution": "To find the value of $a$, we will use the given property of the sequence that each term is the sum of the two terms to its left. We will calculate the values of $d$, $c$, $b$, and finally $a$ by working backwards from the known terms.\n\n1. **Finding $d$:**\n Giv... |
deepscaler | [
{
"role": "user",
"content": "The integer $n$ is the largest positive multiple of $15$ such that every digit of $n$ is either $8$ or $0$. Compute $\\frac{n}{15}$.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "592"
} | {
"split": "train",
"index": 33269,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "[asy] pair A = (0,0), B = (7,4.2), C = (10, 0), D = (3, -5), E = (3, 0), F = (7,0); draw(A--B--C--D--cycle,dot); draw(A--E--F--C,dot); draw(D--E--F--B,dot); markscalefactor = 0.1; draw(rightanglemark(B, A, D)); draw(rightanglemark(D, E, C)); draw(rightanglemark(B, F, A)); draw(... | math | {
"style": "rule",
"ground_truth": "4.2"
} | {
"split": "train",
"index": 33270,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $|\\overrightarrow{a}|=1$, $|\\overrightarrow{b}|=2$ and $|2\\overrightarrow{a}+\\overrightarrow{b}|=2\\sqrt{3}$, find the angle between vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$. Let's think st... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 33271,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the function $f(x)=\\frac{1}{2}(2-m)x^{2}+(n-8)x+1$ $(m > 2)$ is monotonically decreasing in the interval $[-2,-1]$, find the maximum value of $mn$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 33272,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different ways are there to split the number 2004 into natural summands that are approximately equal? There can be one or several summands. Numbers are considered approximately equal if their difference is no more than 1. Ways that differ only by the order of summands ... | math | {
"style": "rule",
"ground_truth": "2004"
} | {
"split": "train",
"index": 33273,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\frac{m}{x} - m + \\ln x$ (where $m$ is a constant).\n1. Find the monotonic intervals of $f(x)$.\n2. For what values of $m$ does $f(x) \\geq 0$ hold true? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "m = 1"
} | {
"split": "train",
"index": 33274,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\( x \\) and \\( y \\) are positive integers with \\( x>y \\) and \\( x+x y=391 \\), what is the value of \\( x+y \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 33275,
"solution": "Since \\( x+x y=391 \\), then \\( x(1+y)=391 \\). We note that \\( 391=17 \\cdot 23 \\). Since 17 and 23 are both prime, then if 391 is written as the product of two positive integers, it must be \\( 1 \\times 391 \\) or \\( 17 \\times 23 \\) or \\( 23 \\times 17 \... |
deepscaler | [
{
"role": "user",
"content": "When $\\sqrt[4]{2^6 \\cdot 3^5 \\cdot 5^2}$ is fully simplified, the result is $x\\sqrt[4]{y}$, where $x$ and $y$ are positive integers. What is $x+y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "306"
} | {
"split": "train",
"index": 33276,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Detached calculation.\n\n327 + 46 - 135\n\n1000 - 582 - 128\n\n(124 - 62) × 6\n\n500 - 400 ÷ 5 Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "420"
} | {
"split": "train",
"index": 33277,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the numbers from the set $\\{1,\\ 2,\\ 3,\\ldots,\\ 50\\}$ have a perfect square factor other than one? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 33278,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of circle $C$ is $$\\rho=4 \\sqrt {2}\\sin\\left( \\frac {3\\pi}{4}-\\theta\\right)$$ \n(1) Convert the polar equation of circle $C$ in... | math | {
"style": "rule",
"ground_truth": "\\frac {1}{2}"
} | {
"split": "train",
"index": 33279,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An ice cream shop offers 8 different flavors of ice cream. What is the greatest number of sundaes that can be made if each sundae can consist of 1, 2, or 3 scoops, with each scoop possibly being a different type of ice cream and no two sundaes having the same combination of fla... | math | {
"style": "rule",
"ground_truth": "92"
} | {
"split": "train",
"index": 33280,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\ell_A$ and $\\ell_B$ be two distinct perpendicular lines. For positive integers $m$ and $n$, distinct points $A_1, A_2, \\allowbreak A_3, \\allowbreak \\ldots, \\allowbreak A_m$ lie on $\\ell_A$, and distinct points $B_1, B_2, B_3, \\ldots, B_n$ lie on $\\ell_B$. Additio... | math | {
"style": "rule",
"ground_truth": "244"
} | {
"split": "train",
"index": 33281,
"solution": "We want to derive a general function $f(m,n)$ that indicates the number of bounded regions. Observing symmetry, we know this is a symmetric function about $m$ and $n$. Now let's focus on $f(m+1, n)-f(m, n)$, which is the difference caused by adding one point to the... |
deepscaler | [
{
"role": "user",
"content": "How many natural numbers greater than 10 but less than 100 are relatively prime to 21? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 33282,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the ellipse $\\frac {x^{2}}{3}+ \\frac {y^{2}}{2}=1$, the distance from a point P to the left focus is $\\frac { \\sqrt {3}}{2}$. Find the distance from P to the right directrix. 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": 33283,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of $1.43$ dollars. Some of the $30$ sixth graders each bought a pencil, and they paid a total of $1.95$ dollars. How many more sixth grader... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 33284,
"solution": "1. **Convert the total payments to cents**: \n - Seventh graders paid $1.43$ dollars, which is $143$ cents.\n - Sixth graders paid $1.95$ dollars, which is $195$ cents.\n\n2. **Determine the cost of one pencil**:\n - The cost of a pencil must be a common fact... |
deepscaler | [
{
"role": "user",
"content": "Let \\( n = 2^{40}5^{15} \\). How many positive integer divisors of \\( n^2 \\) are less than \\( n \\) but do not divide \\( n \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "599"
} | {
"split": "train",
"index": 33285,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "2 + \\sqrt{2}"
} | {
"split": "train",
"index": 33286,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a regular tetrahedron $ABCD$ with edge length of 1 cm, an ant starts at point $A$ and follows these rules: at each vertex, it randomly chooses one of the three edges connected to that vertex and crawls along this edge to the next vertex. What is the probability that, afte... | math | {
"style": "rule",
"ground_truth": "61/243"
} | {
"split": "train",
"index": 33287,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ratio of boys to girls in Mr. Smith's class is 3:4, and there are 42 students in total. What percent of the students are boys. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42.86\\%"
} | {
"split": "train",
"index": 33288,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of an equilateral triangle is numerically equal to the length of one of its sides. What is the perimeter of the triangle, in units? Express your answer in simplest radical form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"split": "train",
"index": 33289,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $x \\gt -1$, $y \\gt 0$, and $x+2y=1$, find the minimum value of $\\frac{1}{x+1}+\\frac{1}{y}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3+2\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 33290,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of 11-digit positive integers such that the digits from left to right are non-decreasing. (For example, 12345678999, 55555555555, 23345557889.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75582"
} | {
"split": "train",
"index": 33291,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the first four terms of an arithmetic progression, as well as the sum of the first nine terms, are natural numbers. Additionally, the first term \\( b_{1} \\) of this progression satisfies the inequality \\( b_{1} \\leq \\frac{3}{4} \\). What is the greatest possible... | math | {
"style": "rule",
"ground_truth": "11/15"
} | {
"split": "train",
"index": 33292,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $7$ fair standard $6$-sided dice are thrown, the probability that the sum of the numbers on the top faces is $10$ can be written as $\\frac{n}{6^{7}}$, where $n$ is a positive integer. What is $n$? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 33293,
"solution": "\nTo find the number of ways to get a sum of $10$ when rolling $7$ fair $6$-sided dice, we can use the stars and bars method, considering the constraints that each die must show at least $1$ and at most $6$.\n\n1. **Initial Setup**: Each die must show at least $1$.... |
deepscaler | [
{
"role": "user",
"content": "Math City has eight streets, all of which are straight. No street is parallel to another street. One police officer is stationed at each intersection. What is the greatest number of police officers needed? Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 33294,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $x, y > 0$ and $\\frac{1}{x} + \\frac{1}{y} = 2$, find the minimum value of $x + 2y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3 + 2\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 33295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all positive integers $ n$ for which the numbers in the set $ S \\equal{} \\{1,2, \\ldots,n \\}$ can be colored red and blue, with the following condition being satisfied: The set $ S \\times S \\times S$ contains exactly $ 2007$ ordered triples $ \\left(x, y, z\\right)$ s... | math | {
"style": "rule",
"ground_truth": "n \\in \\{69, 84\\}"
} | {
"split": "train",
"index": 33296,
"solution": "\nLet us consider a set \\( S = \\{1, 2, \\ldots, n\\} \\) whose elements are to be colored either red or blue. We need to find all positive integers \\( n \\) for which the set \\( S \\times S \\times S \\) contains exactly 2007 ordered triples \\( (x, y, z) \\) s... |
deepscaler | [
{
"role": "user",
"content": "The area of the shaded region is 78 square inches. All angles are right angles and all measurements are given in inches. What is the perimeter of the non-shaded region?\n\n[asy]size(101);\nfilldraw(((0,0)--(0,8)--(10,8)--(10,-2)--(6,-2)--(6,0)--cycle^^(2.5,3)--(2.5,5)--(7.5,5)-... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 33297,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate: \\((56 \\times 0.57 \\times 0.85) \\div(2.8 \\times 19 \\times 1.7) =\\) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.3"
} | {
"split": "train",
"index": 33298,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Read the following material before solving the problem: In mathematics, there are numbers with square roots that contain another square root, which can be simplified by using the complete square formula and the properties of quadratic surds. For example, $\\sqrt{3+2\\sqrt{2}}=\... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}-2"
} | {
"split": "train",
"index": 33299,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all pairs of integers $a,b$ for which there exists a polynomial $P(x) \\in \\mathbb{Z}[X]$ such that product $(x^2+ax+b)\\cdot P(x)$ is a polynomial of a form \\[ x^n+c_{n-1}x^{n-1}+\\cdots+c_1x+c_0 \\] where each of $c_0,c_1,\\ldots,c_{n-1}$ is equal to $1$ or $-1$. Let'... | math | {
"style": "rule",
"ground_truth": "{(a,b)\\in \\{(-2,1), (-1,1), (0,1), (1,1), (2,1), (-1,-1), (0,-1), (1,-1)\\}}"
} | {
"split": "train",
"index": 33300,
"solution": "\nTo solve this problem, we need to determine all integer pairs \\((a, b)\\) such that there exists a polynomial \\( P(x) \\in \\mathbb{Z}[X] \\) with the product \\((x^2 + ax + b) \\cdot P(x)\\) having all coefficients either \\(1\\) or \\(-1\\).\n\nAssume \\( P(x... |
deepscaler | [
{
"role": "user",
"content": "For each positive integer $n$ , let $g(n)$ be the sum of the digits when $n$ is written in binary. For how many positive integers $n$ , where $1\\leq n\\leq 2007$ , is $g(n)\\geq 3$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1941"
} | {
"split": "train",
"index": 33301,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the line $x - 2y + 2k = 0$ encloses a triangle with an area of $1$ together with the two coordinate axes, find the value of the real number $k$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 33302,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( P = 3659893456789325678 \\) and \\( 342973489379256 \\), the product \\( P \\) is calculated. The number of digits in \\( P \\) is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 33303,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $z$ and $w$ be complex numbers such that $|z + 1 + 3i| = 1$ and $|w - 7 - 8i| = 3.$ Find the smallest possible value of $|z - w|.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{185} - 4"
} | {
"split": "train",
"index": 33304,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From 1 to 100, take a pair of integers (repetitions allowed) so that their sum is greater than 100. How many ways are there to pick such pairs? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5050"
} | {
"split": "train",
"index": 33305,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As shown in the diagram, circles \\( \\odot O_{1} \\) and \\( \\odot O_{2} \\) are externally tangent. The line segment \\( O_{1}O_{2} \\) intersects \\( \\odot O_{1} \\) at points \\( A \\) and \\( B \\), and intersects \\( \\odot O_{2} \\) at points \\( C \\) and \\( D \\). C... | math | {
"style": "rule",
"ground_truth": "1.2"
} | {
"split": "train",
"index": 33306,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a cube \\(ABCD-A_1B_1C_1D_1\\) with side length 1, and \\(E\\) as the midpoint of \\(D_1C_1\\), find the following:\n1. The distance between skew lines \\(D_1B\\) and \\(A_1E\\).\n2. The distance from \\(B_1\\) to plane \\(A_1BE\\).\n3. The distance from \\(D_1C\\) to pla... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 33307,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid $PQRS$, the lengths of the bases $PQ$ and $RS$ are 10 and 23, respectively. The legs of the trapezoid are extended beyond $P$ and $Q$ to meet at point $T$. What is the ratio of the area of triangle $TPQ$ to the area of trapezoid $PQRS$? Express your answer as a com... | math | {
"style": "rule",
"ground_truth": "\\frac{100}{429}"
} | {
"split": "train",
"index": 33308,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\cos\\left(\\alpha- \\frac{\\beta}{2}\\right) = -\\frac{1}{9}$ and $\\sin\\left(\\frac{\\alpha}{2} - \\beta\\right) = \\frac{2}{3}$, with $\\frac{\\pi}{2} < \\alpha < \\pi$ and $0 < \\beta < \\frac{\\pi}{2}$, find $\\cos(\\alpha + \\beta)$. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "-\\frac{239}{729}"
} | {
"split": "train",
"index": 33309,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A balloon that inflates into the shape of a perfect cube is being blown up at a rate such that at time \\( t \\) (in fortnights), it has a surface area of \\( 6t \\) square furlongs. At what rate, in cubic furlongs per fortnight, is the air being pumped in when the surface area... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{6}"
} | {
"split": "train",
"index": 33310,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Take a unit sphere \\(S\\), i.e., a sphere with radius 1. Circumscribe a cube \\(C\\) about \\(S\\), and inscribe a cube \\(D\\) in \\(S\\) such that every edge of cube \\(C\\) is parallel to some edge of cube \\(D\\). What is the shortest possible distance from a point on a fa... | math | {
"style": "rule",
"ground_truth": "1 - \\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 33311,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\log_2(\\log_3(\\log_4 x))=\\log_3(\\log_4(\\log_2 y))=\\log_4(\\log_2(\\log_3 z))=0$, then the sum $x+y+z$ is equal to Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "89"
} | {
"split": "train",
"index": 33312,
"solution": "Given the equations:\n1. \\(\\log_2(\\log_3(\\log_4 x)) = 0\\)\n2. \\(\\log_3(\\log_4(\\log_2 y)) = 0\\)\n3. \\(\\log_4(\\log_2(\\log_3 z)) = 0\\)\n\nFrom the property of logarithms, if \\(\\log_b(a) = 0\\), then \\(a = 1\\). Applying this property to each equation... |
deepscaler | [
{
"role": "user",
"content": "Form a three-digit number using the digits 0, 1, 2, 3. Repeating digits is not allowed. \n① How many three-digit numbers can be formed? \n② If the three-digit numbers from ① are sorted in ascending order, what position does 230 occupy? \n③ If repeating digits is allowed, how... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 33313,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be positive real numbers. Determine the minimum possible value of $$\\sqrt{a^{2}+b^{2}}+\\sqrt{(a-1)^{2}+b^{2}}+\\sqrt{a^{2}+(b-1)^{2}}+\\sqrt{(a-1)^{2}+(b-1)^{2}}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{2}"
} | {
"split": "train",
"index": 33314,
"solution": "Let $A B C D$ be a square with $A=(0,0), B=(1,0), C=(1,1), D=(0,1)$, and $P$ be a point in the same plane as $A B C D$. Then the desired expression is equivalent to $A P+B P+C P+D P$. By the triangle inequality, $A P+C P \\geq A C$ and $B P+D P \\geq B D$, so the m... |
deepscaler | [
{
"role": "user",
"content": "How many positive integers divide $5n^{11}-2n^5-3n$ for all positive integers $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 33315,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The union of sets \\( A \\) and \\( B \\) is \\( A \\cup B = \\{a_1, a_2, a_3\\} \\). When \\( A \\neq B \\), \\((A, B)\\) and \\((B, A)\\) are considered different pairs. How many such pairs \\((A, B)\\) exist? Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 33316,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The password consists of four different digits whose sum equals 27. How many possible passwords are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 33317,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be a positive integer. Find all $n \\times n$ real matrices $A$ with only real eigenvalues satisfying $$A+A^{k}=A^{T}$$ for some integer $k \\geq n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "A = 0"
} | {
"split": "train",
"index": 33318,
"solution": "Solution 1. Taking the transpose of the matrix equation and substituting we have $$A^{T}+\\left(A^{T}\\right)^{k}=A \\Longrightarrow A+A^{k}+\\left(A+A^{k}\\right)^{k}=A \\Longrightarrow A^{k}\\left(I+\\left(I+A^{k-1}\\right)^{k}\\right)=0$$ Hence $p(x)=x^{k}\\left... |
deepscaler | [
{
"role": "user",
"content": "Let $(a,b,c)$ be the real solution of the system of equations $x^3 - xyz = 2$, $y^3 - xyz = 6$, $z^3 - xyz = 20$. The greatest possible value of $a^3 + b^3 + c^3$ can be written in the form $\\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.... | math | {
"style": "rule",
"ground_truth": "158"
} | {
"split": "train",
"index": 33319,
"solution": "Add the three equations to get $a^3 + b^3 + c^3 = 28 + 3abc$. Now, let $abc = p$. $a = \\sqrt [3]{p + 2}$, $b = \\sqrt [3]{p + 6}$ and $c = \\sqrt [3]{p + 20}$, so $p = abc = (\\sqrt [3]{p + 2})(\\sqrt [3]{p + 6})(\\sqrt [3]{p + 20})$. Now cube both sides; the $p^3... |
deepscaler | [
{
"role": "user",
"content": "If \\( a + \\log_{3} 2017 \\), \\( a + \\log_{9} 2017 \\), \\( a + \\log_{27} 2017 \\) (\\( a \\in \\mathbb{R} \\)) form a geometric sequence, what is the common ratio? 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": 33320,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, point $E$ lies on line segment $AB$, and triangles $AED$ and $BEC$ are isosceles. Also, $\\angle DEC$ is twice $\\angle ADE$. What is the measure of $\\angle EBC$ in degrees? [asy]\nimport olympiad;\nimport math;\n\nsize(7cm);\n\n// Draw triangles\npair a = (-2 ... | math | {
"style": "rule",
"ground_truth": "75^\\circ"
} | {
"split": "train",
"index": 33321,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In land of Nyemo, the unit of currency is called a *quack*. The citizens use coins that are worth $1$ , $5$ , $25$ , and $125$ quacks. How many ways can someone pay off $125$ quacks using these coins?\n\n*Proposed by Aaron Lin* Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "82"
} | {
"split": "train",
"index": 33322,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "While waiting at the post office, Lena moved 40 feet closer to the counter over a period of 20 minutes. At this rate, how many minutes will it take her to move the remaining 100 meters to the counter? Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "164.042"
} | {
"split": "train",
"index": 33323,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the plane rectangular coordinate system $xOy$, it is known that $MN$ is a chord of the circle $C: (x-2)^{2} + (y-4)^{2} = 2$, and satisfies $CM\\perp CN$. Point $P$ is the midpoint of $MN$. As the chord $MN$ moves on the circle $C$, there exist two points $A$ and $B$ on the ... | math | {
"style": "rule",
"ground_truth": "\\frac{6\\sqrt{5}}{5} + 2"
} | {
"split": "train",
"index": 33324,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Add $4_6 + 14_6.$ Express your answer in base $6.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22_6"
} | {
"split": "train",
"index": 33325,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For some positive integers $p$, there is a quadrilateral $ABCD$ with positive integer side lengths, perimeter $p$, right angles at $B$ and $C$, $AB=2$, and $CD=AD$. How many different values of $p<2015$ are possible?\n$\\textbf{(A) }30\\qquad\\textbf{(B) }31\\qquad\\textbf{(C) ... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 33326,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of two prime numbers is $102$ and one of the prime numbers is greater than $30$, calculate the product of these two prime numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2201"
} | {
"split": "train",
"index": 33327,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the vectors $\\overrightarrow{a} = (1, 1)$, $\\overrightarrow{b} = (2, 0)$, the angle between vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 33328,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Player A and Player B play a number guessing game. First, Player A thinks of a number denoted as $a$, then Player B guesses the number that Player A is thinking of, and denotes this guessed number as $b$. Both $a$ and $b$ belong to the set $\\{1,2,3,4,5,6\\}$. If $|a-b| \\leqsl... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"split": "train",
"index": 33329,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Storage space on a computer is measured in gigabytes (GB) and megabytes (MB), where $1 \\mathrm{~GB} = 1024 \\mathrm{MB}$. Julia has an empty 300 GB hard drive and puts 300000 MB of data onto it. How much storage space on the hard drive remains empty? Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "7200 \\mathrm{MB}"
} | {
"split": "train",
"index": 33330,
"solution": "Since $1 \\mathrm{~GB} = 1024 \\mathrm{MB}$, then Julia's 300 GB hard drive has $300 \\times 1024 = 307200 \\mathrm{MB}$ of storage space. When Julia puts 300000 MB of data on the empty hard drive, the amount of empty space remaining is $307200 - 300000 = 7200 \\ma... |
deepscaler | [
{
"role": "user",
"content": "An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is $20$, and one of the base angles is $\\arcsin(0.6)$. Find the area of the trapezoid given that the height of the trapezoid is $9$. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 33331,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $\\frac {x^{2}}{a^{2}} + \\frac {y^{2}}{b^{2}} = 1$ ($a > b > 0$) and a line $l: y = -\\frac { \\sqrt {3}}{3}x + b$ intersect at two distinct points P and Q. The distance from the origin to line $l$ is $\\frac { \\sqrt {3}}{2}$, and the eccentricity of the elli... | math | {
"style": "rule",
"ground_truth": "-\\frac {7}{6}"
} | {
"split": "train",
"index": 33332,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer divisor of $10!$ is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 33333,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A beam of light is emitted from point $P(1,2,3)$, reflected by the $Oxy$ plane, and then absorbed at point $Q(4,4,4)$. The distance traveled by the light beam is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{62}"
} | {
"split": "train",
"index": 33334,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A magician and their assistant are planning to perform the following trick. A spectator writes a sequence of $N$ digits on a board. The magician's assistant covers two adjacent digits with a black circle. Then the magician enters. Their task is to guess both of the covered digi... | math | {
"style": "rule",
"ground_truth": "101"
} | {
"split": "train",
"index": 33335,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cryptographer designed the following method to encode natural numbers: first, represent the natural number in base 5, then map the digits in the base 5 representation to the elements of the set $\\{V, W, X, Y, Z\\}$ in a one-to-one correspondence. Using this correspondence, h... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 33336,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle XYZ$ be a triangle in the plane, and let $W$ be a point outside the plane of $\\triangle XYZ$, so that $WXYZ$ is a pyramid whose faces are all triangles.\n\nSuppose that the edges of $WXYZ$ have lengths of either $24$ or $49$, and no face of $WXYZ$ is equilatera... | math | {
"style": "rule",
"ground_truth": "48 \\sqrt{2257}"
} | {
"split": "train",
"index": 33337,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $P$ is the product of $n$ quantities in Geometric Progression, $S$ their sum, and $S'$ the sum of their reciprocals, then $P$ in terms of $S, S'$, and $n$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(S/S')^{\\frac{1}{2}n}"
} | {
"split": "train",
"index": 33338,
"solution": "To solve the problem, we first need to understand the properties of a geometric progression (GP). Let the first term of the GP be $a$ and the common ratio be $r$. Then the $n$ terms of the GP are $a, ar, ar^2, \\ldots, ar^{n-1}$.\n\n1. **Calculate the Product $P$:*... |
deepscaler | [
{
"role": "user",
"content": "What is the value of $\frac{8+4}{8-4}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 33339,
"solution": "Simplifying, $\frac{8+4}{8-4}=\frac{12}{4}=3$."
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, the sides opposite angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and it is given that $3a\\cos A = \\sqrt{6}(b\\cos C + c\\cos B)$. \n(1) Calculate the value of $\\tan 2A$. \n(2) If $\\sin\\left(\\frac{\\pi}{2} + B\\right) = \\frac{1}{3}$ and $c = 2\... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{5}\\sqrt{2}"
} | {
"split": "train",
"index": 33340,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest number of digits that can be erased from the 1000-digit number 201820182018....2018 so that the sum of the remaining digits is 2018? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "741"
} | {
"split": "train",
"index": 33341,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\sqrt{2.1}=1.449$ and $\\sqrt{21}=4.573$, find the value of $\\sqrt{21000}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "144.9"
} | {
"split": "train",
"index": 33342,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Crestview's school colors are purple and gold. The students are designing a flag using three solid-colored horizontal stripes, as shown. Using one or both of the school colors, how many different flags are possible if adjacent stripes may be the same color?\n\n[asy]\nsize(75);\... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 33343,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The center of a hyperbola is at the origin, the foci are on the coordinate axes, and the distance from point \\( P(-2,0) \\) to its asymptote is \\( \\frac{2 \\sqrt{6}}{3} \\). If a line with slope \\( \\frac{\\sqrt{2}}{2} \\) passes through point \\( P \\) and intersects the h... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 33344,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\sqrt{x+2}=2$, then $(x+2)^2$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 33345,
"solution": "1. Start with the given equation:\n \\[\n \\sqrt{x+2} = 2\n \\]\n\n2. Square both sides to eliminate the square root:\n \\[\n (\\sqrt{x+2})^2 = 2^2\n \\]\n \\[\n x+2 = 4\n \\]\n\n3. To find $(x+2)^2$, square the expression $x+2$:\n \\[\n (x+2)... |
deepscaler | [
{
"role": "user",
"content": "Given a power function $y = f(x)$ whose graph passes through the point $(4, 2)$, find $f\\left( \\frac{1}{2} \\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 33346,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the four numbers $0,1,2,3$, we want to select $3$ digits to form a three-digit number with no repeating digits. What is the probability that this three-digit number is divisible by $3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{5}{9}"
} | {
"split": "train",
"index": 33347,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P(z)=x^3+ax^2+bx+c$, where $a,$ $b,$ and $c$ are real. There exists a complex number $w$ such that the three roots of $P(z)$ are $w+3i$, $w+9i$, and $2w-4$, where $i^2=-1$. Find $a+b+c$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-136"
} | {
"split": "train",
"index": 33348,
"solution": ""
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.