data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Three congruent isosceles triangles are constructed inside an equilateral triangle with a side length of $\\sqrt{2}$. Each base of the isosceles triangle is placed on one side of the equilateral triangle. If the total area of the isosceles triangles equals $\\frac{1}{2}$ the ar... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 4500,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle has a radius of three inches. The distance from the center of the circle to chord $CD$ is two inches. How many inches long is chord $CD$? 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": "2\\sqrt{5}"
} | {
"split": "train",
"index": 4501,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an opaque bag, there are four small balls labeled with the Chinese characters \"阳\", \"过\", \"阳\", and \"康\" respectively. Apart from the characters, the balls are indistinguishable. Before each draw, the balls are thoroughly mixed.<br/>$(1)$ If one ball is randomly drawn fr... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 4502,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A book with 53 pages numbered 1 to 53 has its pages renumbered in reverse, from 53 to 1. For how many pages do the new page number and old page number share the same units digit? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 4503,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An entry in a grid is called a saddle point if it is the largest number in its row and the smallest number in its column. Suppose that each cell in a $3 \\times 3$ grid is filled with a real number, each chosen independently and uniformly at random from the interval $[0,1]$. Co... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"split": "train",
"index": 4504,
"solution": "With probability 1, all entries of the matrix are unique. If this is the case, we claim there can only be one saddle point. To see this, suppose $A_{i j}$ and $A_{k l}$ are both saddle points. They cannot be in the same row, since they cannot both be the greatest nu... |
deepscaler | [
{
"role": "user",
"content": "Find the common ratio of the infinite geometric series: $$\\frac{7}{8} - \\frac{5}{12} + \\frac{35}{144} - \\dots$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{10}{21}"
} | {
"split": "train",
"index": 4505,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of distinct terms in the simplified expansion of $[(x+5y)^3(x-5y)^3]^{3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 4506,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Cubs are playing the Red Sox in the World Series. To win the world series, a team must win 4 games before the other team does. If the Cubs win each game with probability $\\dfrac{3}{5}$ and there are no ties, what is the probability that the Cubs will win the World Series? ... | math | {
"style": "rule",
"ground_truth": "71"
} | {
"split": "train",
"index": 4507,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $\\tan \\left( -\\frac{3 \\pi}{4} \\right).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 4508,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The expression $x^2 + 15x + 54$ can be written as $(x + a)(x + b),$ and the expression $x^2 - 17x + 72$ written as $(x - b)(x - c)$, where $a$, $b$, and $c$ are integers. What is the value of $a + b + c$? Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 4509,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\{a\\_n\\}$ satisfies $a\\_1=1$, and for any $n∈N^∗$, $a_{n+1}=a\\_n+n+1$, find the value of $$\\frac {1}{a_{1}}+ \\frac {1}{a_{2}}+…+ \\frac {1}{a_{2017}}+ \\frac {1}{a_{2016}}+ \\frac {1}{a_{2019}}$$. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\frac{2019}{1010}"
} | {
"split": "train",
"index": 4510,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $r = \\sqrt{\\frac{\\sqrt{53}}{2} + \\frac{3}{2}}$. There is a unique triple of positive integers $(a, b, c)$ such that $r^{100} = 2r^{98} + 14r^{96} + 11r^{94} - r^{50} + ar^{46} + br^{44} + cr^{40}$. What is the value of $a^{2} + b^{2} + c^{2}$? Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "15339"
} | {
"split": "train",
"index": 4511,
"solution": "Suppose that $r = \\sqrt{\\frac{\\sqrt{53}}{2} + \\frac{3}{2}}$. Thus, $r^{2} = \\frac{\\sqrt{53}}{2} + \\frac{3}{2}$ and so $2r^{2} = \\sqrt{53} + 3$ or $2r^{2} - 3 = \\sqrt{53}$. Squaring both sides again, we obtain $(2r^{2} - 3)^{2} = 53$ or $4r^{4} - 12r^{2} + 9... |
deepscaler | [
{
"role": "user",
"content": "Find the cross product of $\\begin{pmatrix} 2 \\\\ 0 \\\\ 3 \\end{pmatrix}$ and $\\begin{pmatrix} 5 \\\\ -1 \\\\ 7 \\end{pmatrix}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 3 \\\\ 1 \\\\ -2 \\end{pmatrix}"
} | {
"split": "train",
"index": 4512,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all the triples of positive integers $(a,b,c)$ for which the number\n\\[\\frac{(a+b)^4}{c}+\\frac{(b+c)^4}{a}+\\frac{(c+a)^4}{b}\\]\nis an integer and $a+b+c$ is a prime. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(1, 1, 1), (2, 2, 1), (6, 3, 2)"
} | {
"split": "train",
"index": 4513,
"solution": "\nTo solve this problem, we are tasked with finding all triples of positive integers \\((a, b, c)\\) such that the expression \n\n\\[\n\\frac{(a+b)^4}{c} + \\frac{(b+c)^4}{a} + \\frac{(c+a)^4}{b}\n\\]\n\nis an integer and the sum \\(a + b + c\\) is a prime number.\n... |
deepscaler | [
{
"role": "user",
"content": "In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different painti... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 4514,
"solution": "To solve this problem, we will use Burnside's Lemma, which states that the number of distinct colorings, up to symmetry, is the average number of colorings fixed by each group action. The group actions in this case are the symmetries of a hexagon, which include rota... |
deepscaler | [
{
"role": "user",
"content": "Call a positive integer monotonous if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, $3$, $23578$, and $987620$ are monotonous, but $88$, $7434$, and $23557$ are not. How ma... | math | {
"style": "rule",
"ground_truth": "1524"
} | {
"split": "train",
"index": 4515,
"solution": "1. **Understanding Monotonous Numbers**: A monotonous number is defined as a number whose digits are either strictly increasing or strictly decreasing when read from left to right. This includes all one-digit numbers.\n\n2. **Counting One-Digit Monotonous Numbers**:... |
deepscaler | [
{
"role": "user",
"content": "Given that the point $(1, \\frac{1}{3})$ lies on the graph of the function $f(x)=a^{x}$ ($a > 0$ and $a \\neq 1$), and the sum of the first $n$ terms of the geometric sequence $\\{a_n\\}$ is $f(n)-c$, the first term and the sum $S_n$ of the sequence $\\{b_n\\}$ ($b_n > 0$) sati... | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 4516,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three chiefs of Indian tribes are sitting by a fire with three identical pipes. They are holding a war council and smoking. The first chief can finish a whole pipe in ten minutes, the second in thirty minutes, and the third in an hour. How should the chiefs exchange the pipes a... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 4517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest possible median for the five number set $\\{x, 2x, 3, 2, 5\\}$ if $x$ can be any integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 4518,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Knowing that segment \\( CD \\) has a length of 6 and its midpoint is \\( M \\), two triangles \\( \\triangle ACD \\) and \\( \\triangle BCD \\) are constructed on the same side with \\( CD \\) as a common side and both having a perimeter of 16, such that \\( \\angle AMB = 90^{... | math | {
"style": "rule",
"ground_truth": "400/41"
} | {
"split": "train",
"index": 4519,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Create a cube $C_{1}$ with edge length 1. Take the centers of the faces and connect them to form an octahedron $O_{1}$. Take the centers of the octahedron's faces and connect them to form a new cube $C_{2}$. Continue this process infinitely. Find the sum of all the surface area... | math | {
"style": "rule",
"ground_truth": "\\frac{54+9 \\sqrt{3}}{8}"
} | {
"split": "train",
"index": 4520,
"solution": "The lengths of the second cube are one-third of the lengths of the first cube, so the surface area decreases by a factor of one-ninth. Since the first cube has surface area 6 and the first octahedron has surface area $\\sqrt{3}$, the total area is $(6+\\sqrt{3}) \\c... |
deepscaler | [
{
"role": "user",
"content": "Given $f(z) = z^2-19z$, there are complex numbers $z$ with the property that $z$, $f(z)$, and $f(f(z))$ are the vertices of a right triangle in the complex plane with a right angle at $f(z)$. There are positive integers $m$ and $n$ such that one such value of $z$ is $m+\\sqrt{n... | math | {
"style": "rule",
"ground_truth": "230"
} | {
"split": "train",
"index": 4521,
"solution": "Notice that we must have \\[\\frac{f(f(z))-f(z)}{f(z)-z}=-\\frac{f(f(z))-f(z)}{z-f(z)}\\in i\\mathbb R .\\]However, $f(t)-t=t(t-20)$, so \\begin{align*} \\frac{f(f(z))-f(z)}{f(z)-z}&=\\frac{(z^2-19z)(z^2-19z-20)}{z(z-20)}\\\\ &=\\frac{z(z-19)(z-20)(z+1)}{z(z-20)}\\\... |
deepscaler | [
{
"role": "user",
"content": "A line with a slope of $2$ passing through the right focus of the ellipse $\\frac{x^2}{5} + \\frac{y^2}{4} = 1$ intersects the ellipse at points $A$ and $B$. If $O$ is the origin, then the area of $\\triangle OAB$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"split": "train",
"index": 4522,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rise of $600$ feet is required to get a railroad line over a mountain. The grade can be kept down by lengthening the track and curving it around the mountain peak. The additional length of track required to reduce the grade from $3\\%$ to $2\\%$ is approximately: Let's think ... | math | {
"style": "rule",
"ground_truth": "10000"
} | {
"split": "train",
"index": 4523,
"solution": "1. **Understanding the problem**: The problem states that a railroad needs to rise 600 feet to cross a mountain. The grade of the railroad, which is the ratio of the rise to the horizontal length, can be adjusted by changing the length of the track.\n\n2. **Calculat... |
deepscaler | [
{
"role": "user",
"content": "Carl and Bob can demolish a building in 6 days, Anne and Bob can do it in 3, Anne and Carl in 5. How many days does it take all of them working together if Carl gets injured at the end of the first day and can't come back? Express your answer as a fraction in lowest terms. Let'... | math | {
"style": "rule",
"ground_truth": "\\frac{59}{20}"
} | {
"split": "train",
"index": 4524,
"solution": "Let $a$ be the portion of the work that Anne does in one day, similarly $b$ for Bob and $c$ for Carl. Then what we are given is the system of equations $b+c=1 / 6, a+b=1 / 3$, and $a+c=1 / 5$. Thus in the first day they complete $a+b+c=\\frac{1}{2}(1 / 6+1 / 3+1 / 5... |
deepscaler | [
{
"role": "user",
"content": "The two banners below:\n中华少年 杯赛联谊 切磋勾股\n炎黄子孙 惠州弘志 振兴中华\n\nEach character represents a non-zero natural number less than 25. Different characters represent different numbers, and the same characters represent the same number. It is known that the average value of these 34 charac... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 4525,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p,$ $q,$ $r,$ $s$ be real numbers such that $p + q + r + s = 10$ and\n\\[ pq + pr + ps + qr + qs + rs = 20. \\]\nFind the largest possible value of $s$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5 + \\sqrt{105}}{2}"
} | {
"split": "train",
"index": 4526,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the product of the numerator and the denominator when $0.\\overline{018}$ is expressed as a fraction in lowest terms? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "222"
} | {
"split": "train",
"index": 4527,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On side \\( AB \\) of parallelogram \\( ABCD \\), point \\( F \\) is selected, and on the extension of side \\( BC \\) beyond vertex \\( B \\), point \\( H \\) is chosen such that \\( \\frac{AB}{BF} = \\frac{BC}{BH} = 5 \\). Point \\( G \\) is selected so that \\( BFGH \\) is a... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 4528,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Diagonal $DB$ of rectangle $ABCD$ is divided into three segments of length $1$ by parallel lines $L$ and $L'$ that pass through $A$ and $C$ and are perpendicular to $DB$. The area of $ABCD$, rounded to the one decimal place, is Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "4.2"
} | {
"split": "train",
"index": 4529,
"solution": "1. **Identify the Geometry and Key Points**: \n Let $ABCD$ be a rectangle with diagonal $DB$. Lines $L$ and $L'$ are parallel to each other, pass through points $A$ and $C$ respectively, and are perpendicular to diagonal $DB$. Let $E$ and $F$ be the points where ... |
deepscaler | [
{
"role": "user",
"content": "Calculate the volume of the tetrahedron with vertices at points \\( A_{1}, A_{2}, A_{3}, A_{4} \\) and its height, dropped from vertex \\( A_{4} \\) onto the face \\( A_{1} A_{2} A_{3} \\).\n\n\\( A_{1}(7, 2, 4) \\) \n\\( A_{2}(7, -1, -2) \\) \n\\( A_{3}(3, 3, 1) \\) \n\\( A... | math | {
"style": "rule",
"ground_truth": "21.5"
} | {
"split": "train",
"index": 4530,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle with radius $\\frac{\\sqrt{2}}{2}$ and a regular hexagon with side length 1 share the same center. Calculate the area inside the circle, but outside the hexagon. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2} - \\frac{3\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 4531,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. It is given that $\\frac{-b + \\sqrt{2}c}{\\cos B} = \\frac{a}{\\cos A}$, \n$(I)$ Find the size of angle $A$; \n$(II)$ If $a=2$, find the maximum value of the area $S$. Let... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} + 1"
} | {
"split": "train",
"index": 4532,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The edges of a regular tetrahedron with vertices $A$, $B$, $C$, and $D$ each have length one. Find the least possible distance between a pair of points $P$ and $Q$, where $P$ is on edge $AB$ and $Q$ is on edge $CD$. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 4533,
"solution": "1. **Identify the Key Points and Their Coordinates:**\n - Let $P$ be a point on edge $AB$ and $Q$ be a point on edge $CD$ of a regular tetrahedron with vertices $A, B, C, D$ and each edge of length 1.\n - Assume $P$ and $Q$ are the midpoints of $AB$ and $CD$, re... |
deepscaler | [
{
"role": "user",
"content": "Kelvin and 15 other frogs are in a meeting, for a total of 16 frogs. During the meeting, each pair of distinct frogs becomes friends with probability $\\frac{1}{2}$. Kelvin thinks the situation after the meeting is cool if for each of the 16 frogs, the number of friends they ma... | math | {
"style": "rule",
"ground_truth": "1167"
} | {
"split": "train",
"index": 4534,
"solution": "Consider the multivariate polynomial $$\\prod_{1 \\leq i<j \\leq 16}\\left(1+x_{i} x_{j}\\right)$$ We're going to filter this by summing over all $4^{16} 16$-tuples $\\left(x_{1}, x_{2}, \\ldots, x_{16}\\right)$ such that $x_{j}= \\pm 1, \\pm i$. Most of these evalu... |
deepscaler | [
{
"role": "user",
"content": "There are 4 representatives from each of 4 companies at a convention. At the start of the convention, every person shakes hands once with every person except the other representatives from their company. How many handshakes are there? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 4535,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that on a parabola with vertex $V$ and a focus $F$ there exists a point $A$ such that $AF=20$ and $AV=21$. What is the sum of all possible values of the length $FV?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{40}{3}"
} | {
"split": "train",
"index": 4536,
"solution": "1. **Understanding the Parabola and its Properties**: \n - The parabola $\\mathcal{P}$ is defined such that for any point $T$ on $\\mathcal{P}$, the distance from $T$ to the directrix $\\ell$ is equal to the distance from $T$ to the focus $F$.\n - Let $V$ be the... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides corresponding to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $b = a \\sin C + c \\cos A$,\n\n(1) Find the value of $A + B$;\n\n(2) If $c = \\sqrt{2}$, find the maximum area of $\\triangle ABC$. Let's think... | math | {
"style": "rule",
"ground_truth": "\\frac{1 + \\sqrt{2}}{2}"
} | {
"split": "train",
"index": 4537,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Bethany has 11 pound coins and some 20 pence coins and some 50 pence coins in her purse. The mean value of the coins is 52 pence. Which could not be the number of coins in the purse? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 4538,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Giraldo wrote five distinct natural numbers on the vertices of a pentagon. And next he wrote on each side of the pentagon the least common multiple of the numbers written of the two vertices who were on that side and noticed that the five numbers written on the sides were equal... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 4539,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $10$ by $4$ rectangle has the same center as a circle of radius $5$. Calculate the area of the region common to both the rectangle and the circle.\nA) $40 + 2\\pi$\nB) $36 + 4\\pi$\nC) $40 + 4\\pi$\nD) $44 + 4\\pi$\nE) $48 + 2\\pi$ Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "40 + 4\\pi"
} | {
"split": "train",
"index": 4540,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A collection of seven positive integers has a mean of 6, a unique mode of 4, and a median of 6. If a 12 is added to this collection, what is the new median? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6.5"
} | {
"split": "train",
"index": 4541,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A large square region is paved with $n^2$ square black tiles, where each tile measures $t$ inches on each side. Surrounding each tile is a white border that is $w$ inches wide. When $n=30$, it's given that the black tiles cover $81\\%$ of the area of the large square region. Fi... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"split": "train",
"index": 4542,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, medians $AD$ and $CE$ intersect at $P$, $PE=2$, $PD=6$, and $DE=2\\sqrt{10}$. Determine the area of quadrilateral $AEDC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 4543,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( x_{i} \\geq 0 \\) for \\( i = 1, 2, \\cdots, n \\) and \\( \\sum_{i=1}^{n} x_{i} = 1 \\) with \\( n \\geq 2 \\), find the maximum value of \\( \\sum_{1 \\leq i \\leq j \\leq n} x_{i} x_{j} (x_{i} + x_{j}) \\). Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 4544,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$3-i(i)$ is a root of the equation $x^{2}+px+10=0(p∈R)$ with respect to $x$. Find the value of $p$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 4545,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \\ldots$ for $n \\geq 0$ as\n$$a_{n+1} = \n\\begin{cases}\n\\sqrt{a_n} & \\text{if } \\sqrt{a_n} \\text{ is an integer,} \\\\\na_n + 3 & \\text{otherwise.}\n\\end{cases}\n$$\nDetermine all values of $a_0$ such that... | math | {
"style": "rule",
"ground_truth": "3 \\mid a_0"
} | {
"split": "train",
"index": 4546,
"solution": "\nWe are given a sequence defined by \\( a_0, a_1, a_2, \\ldots \\) where the recurrence relation for \\( n \\geq 0 \\) is:\n\\[\na_{n+1} = \n\\begin{cases}\n\\sqrt{a_n} & \\text{if } \\sqrt{a_n} \\text{ is an integer}, \\\\\na_n + 3 & \\text{otherwise}.\n\\end{case... |
deepscaler | [
{
"role": "user",
"content": "Given the coordinates of the three vertices of $\\triangle ABC$ are $A(0,1)$, $B(1,0)$, $C(0,-2)$, and $O$ is the origin, if a moving point $M$ satisfies $|\\overrightarrow{CM}|=1$, calculate the maximum value of $|\\overrightarrow{OA}+ \\overrightarrow{OB}+ \\overrightarrow{OM... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}+1"
} | {
"split": "train",
"index": 4547,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $F_{1}$ and $F_{2}$ are two foci of ellipse $C$, $P$ is a point on $C$, and $\\angle F_{1}PF_{2}=60^{\\circ}$, $|PF_{1}|=3|PF_{2}|$, calculate the eccentricity of $C$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{7}}{4}"
} | {
"split": "train",
"index": 4548,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of\n\\[2 \\sqrt{x} + \\frac{1}{x}\\]for $x > 0.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 4549,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Seven students are standing in a row for a graduation photo. Among them, student A must stand in the middle, and students B and C must stand together. How many different arrangements are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 4550,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many whole numbers are between $\\sqrt[3]{50}$ and $\\sqrt{200}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 4551,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( f : \\mathbb{C} \\to \\mathbb{C} \\) be defined by \\( f(z) = z^2 - 2iz + 2 \\). Determine how many complex numbers \\( z \\) exist such that \\( \\text{Im}(z) > 0 \\) and both the real and the imaginary parts of \\( f(z) \\) are integers within \\( |a|, |b| \\leq 5 \\)... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"split": "train",
"index": 4552,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the minimum possible value of the sum\n\\[\n\\frac{a}{3b} + \\frac{b}{6c} + \\frac{c}{9a},\n\\]\nwhere \\(a\\), \\(b\\), and \\(c\\) are positive real numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{\\sqrt[3]{162}}"
} | {
"split": "train",
"index": 4553,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the $2 \\times 2$ matrix $\\mathbf{M}$ such that $\\mathbf{M} \\begin{pmatrix} 3 \\\\ 0 \\end{pmatrix} = \\begin{pmatrix} 6 \\\\ 21 \\end{pmatrix}$ and $\\mathbf{M} \\begin{pmatrix} -1 \\\\ 5 \\end{pmatrix} = \\begin{pmatrix} 3 \\\\ -17 \\end{pmatrix}.$ Let's think step by... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 2 & 1 \\\\ 7 & -2 \\end{pmatrix}"
} | {
"split": "train",
"index": 4554,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(a\\) and \\(b\\) be positive integers such that \\(90 < a + b < 99\\) and \\(0.9 < \\frac{a}{b} < 0.91\\). Find \\(ab\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2346"
} | {
"split": "train",
"index": 4555,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sasha and Masha each picked a natural number and communicated them to Vasya. Vasya wrote the sum of these numbers on one piece of paper and their product on another piece, then hid one of the pieces and showed the other (on which the number 2002 was written) to Sasha and Masha.... | math | {
"style": "rule",
"ground_truth": "1001"
} | {
"split": "train",
"index": 4556,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all real solutions to the equation $(x+1)(2x+1)(3x+1)(4x+1)=17x^{4}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{25+5\\sqrt{17}}{8}"
} | {
"split": "train",
"index": 4557,
"solution": "First, note that $(x+1)(2x+1)(3x+1)(4x+1)=((x+1)(4x+1))((2x+1)(3x+$ $1))=\\left(4x^{2}+5x+1\\right)\\left(6x^{2}+5x+1\\right)=\\left(5x^{2}+5x+1-x^{2}\\right)\\left(5x^{2}+5x+1+x^{2}\\right)=\\left(5x^{2}+5x+1\\right)^{2}-x^{4}$. Therefore, the equation is equivalen... |
deepscaler | [
{
"role": "user",
"content": "Given a positive integer $n$, find all $n$-tuples of real number $(x_1,x_2,\\ldots,x_n)$ such that\n\\[ f(x_1,x_2,\\cdots,x_n)=\\sum_{k_1=0}^{2} \\sum_{k_2=0}^{2} \\cdots \\sum_{k_n=0}^{2} \\big| k_1x_1+k_2x_2+\\cdots+k_nx_n-1 \\big| \\]\nattains its minimum. Let's think step b... | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{1}{n+1}, \\frac{1}{n+1}, \\ldots, \\frac{1}{n+1} \\right)"
} | {
"split": "train",
"index": 4558,
"solution": "\nGiven a positive integer \\( n \\), we aim to find all \\( n \\)-tuples of real numbers \\( (x_1, x_2, \\ldots, x_n) \\) such that\n\\[\nf(x_1, x_2, \\cdots, x_n) = \\sum_{k_1=0}^{2} \\sum_{k_2=0}^{2} \\cdots \\sum_{k_n=0}^{2} \\left| k_1 x_1 + k_2 x_2 + \\cdots +... |
deepscaler | [
{
"role": "user",
"content": "For a finite sequence $B = (b_1, b_2, \\dots, b_{50})$ of numbers, the Cesaro sum of $B$ is defined as\n\\[\\frac{T_1 + \\cdots + T_{50}}{50},\\]\nwhere $T_k = b_1 + \\cdots + b_k$ for $1 \\leq k \\leq 50$.\n\nIf the Cesaro sum of the 50-term sequence $(b_1, \\dots, b_{50})$ is... | math | {
"style": "rule",
"ground_truth": "198.078431372549"
} | {
"split": "train",
"index": 4559,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Except for the first two terms, each term of the sequence $2000, y, 2000 - y,\\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $y$ produces a sequence of m... | math | {
"style": "rule",
"ground_truth": "1333"
} | {
"split": "train",
"index": 4560,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many diagonals are in a convex polygon with 25 sides, if we only consider diagonals that skip exactly one vertex? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 4561,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f : Q \\to Q$ be a function satisfying the equation $f(x + y) = f(x) + f(y) + 2547$ for all rational numbers $x, y$ . If $f(2004) = 2547$ , find $f(2547)$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2547"
} | {
"split": "train",
"index": 4562,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $$h \\cdot a \\cdot r \\cdot v \\cdot a \\cdot r \\cdot d=m \\cdot i \\cdot t=h \\cdot m \\cdot m \\cdot t=100$$ Find $(r \\cdot a \\cdot d) \\cdot(t \\cdot r \\cdot i \\cdot v \\cdot i \\cdot a)$. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "10000"
} | {
"split": "train",
"index": 4563,
"solution": "The answer is $$\\frac{\\text { harvard } \\cdot \\text { mit } \\cdot \\text { mit }}{h m m t}=100^{2}=10000$$"
} |
deepscaler | [
{
"role": "user",
"content": "Roll a die twice in succession, observing the number of points facing up each time, and calculate: \n(1) The probability that the sum of the two numbers is 5; \n(2) The probability that at least one of the two numbers is odd; \n(3) The probability that the point (x, y), with... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"split": "train",
"index": 4564,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find how many integer values of \\( x \\) are there such that \\( \\lceil{\\sqrt{x}}\\rceil=18 \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 4565,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be a triangle with incenter $I$ and circumcenter $O$. Let the circumradius be $R$. What is the least upper bound of all possible values of $I O$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "R"
} | {
"split": "train",
"index": 4566,
"solution": "$I$ always lies inside the convex hull of $A B C$, which in turn always lies in the circumcircle of $A B C$, so $I O<R$. On the other hand, if we first draw the circle $\\Omega$ of radius $R$ about $O$ and then pick $A, B$, and $C$ very close together on it, we can ... |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all values of $k$ such that the equation $2x^2-kx+8=0$ has two distinct integer solutions? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 4567,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Andrea and Lauren are $20$ kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of $1$ kilometer per minute. After $5$ minutes, Andrea stops biking because of a flat tire and waits... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 4568,
"solution": "1. **Define Variables:**\n Let $v_A$ be Andrea's speed and $v_L$ be Lauren's speed, both in kilometers per hour.\n\n2. **Set Up Equations:**\n Given that Andrea travels at three times the speed of Lauren, we have:\n \\[ v_A = 3v_L \\]\n Also, the rate at whi... |
deepscaler | [
{
"role": "user",
"content": "In rectangle ABCD, AB=2, BC=3, and points E, F, and G are midpoints of BC, CD, and AD, respectively. Point H is the midpoint of EF. What is the area of the quadrilateral formed by the points A, E, H, and G? Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"split": "train",
"index": 4569,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A class collects 50 dollars to buy flowers for a classmate who is in the hospital. Roses cost 3 dollars each, and carnations cost 2 dollars each. No other flowers are to be used. How many different bouquets could be purchased for exactly 50 dollars? Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 4570,
"solution": "1. **Formulate the Problem:**\n Let $r$ represent the number of roses, and $c$ represent the number of carnations. The cost of roses is $3$ dollars each and carnations $2$ dollars each. The total amount spent is $50$ dollars. Therefore, we can set up the equation:... |
deepscaler | [
{
"role": "user",
"content": "A circle is tangent to two adjacent sides \\(AB\\) and \\(AD\\) of square \\(ABCD\\) and cuts off a segment of length 8 cm from vertices \\(B\\) and \\(D\\) at the points of tangency. On the other two sides, the circle cuts off segments of 4 cm and 2 cm respectively from the ve... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 4571,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Previously, on an old truck, I traveled from village $A$ through $B$ to village $C$. After five minutes, I asked the driver how far we were from $A$. \"Half as far as from $B,\" was the answer. Expressing my concerns about the slow speed of the truck, the driver assured me that... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 4572,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many multiples of 7 between $10^{6}$ and $10^{9}$ are perfect squares? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4375"
} | {
"split": "train",
"index": 4573,
"solution": "$\\left[\\sqrt{\\frac{10^{9}}{7^{2}}}\\right]-\\left[\\sqrt{\\frac{10^{6}}{7^{2}}}\\right]=4517-142=4375$."
} |
deepscaler | [
{
"role": "user",
"content": "During the 2013 National Day, a city organized a large-scale group calisthenics performance involving 2013 participants, all of whom were students from the third, fourth, and fifth grades. The students wore entirely red, white, or blue sports uniforms. It was known that the fou... | math | {
"style": "rule",
"ground_truth": "213"
} | {
"split": "train",
"index": 4574,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain district's education department wants to send 5 staff members to 3 schools for earthquake safety education. Each school must receive at least 1 person and no more than 2 people. How many different arrangements are possible? (Answer with a number) Let's think step by s... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 4575,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cyclic quadrilateral $ABCD$ satisfies $\\angle ABD = 70^\\circ$ , $\\angle ADB=50^\\circ$ , and $BC=CD$ . Suppose $AB$ intersects $CD$ at point $P$ , while $AD$ intersects $BC$ at point $Q$ . Compute $\\angle APQ-\\angle AQP$ . Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 4576,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A ball bounces back up $\\frac{2}{3}$ of the height from which it falls. If the ball is dropped from a height of $243$ cm, after how many bounces does the ball first rise less than $30$ cm? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 4577,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain state issues license plates consisting of six digits (from 0 through 9). The state requires that any two plates differ in at least two places. (Thus the plates $\\boxed{027592}$ and $\\boxed{020592}$ cannot both be used.) Determine, with proof, the maximum number of d... | math | {
"style": "rule",
"ground_truth": "10^5"
} | {
"split": "train",
"index": 4578,
"solution": "Consider license plates of $n$ digits, for some fixed $n$ , issued with the same criteria.\nWe first note that by the pigeonhole principle, we may have at most $10^{n-1}$ distinct plates. Indeed, if we have more, then there must be two plates which agree on the fir... |
deepscaler | [
{
"role": "user",
"content": "The graph of the rational function $\\frac{p(x)}{q(x)}$ is shown below, with a horizontal asymptote of $y = 0$ and a vertical asymptote of $ x=-1 $. If $q(x)$ is quadratic, $p(2)=1$, and $q(2) = 3$, find $p(x) + q(x).$\n[asy]\nsize(8cm);\nimport graph;\n\nLabel f; \nf.p=fontsiz... | math | {
"style": "rule",
"ground_truth": "x^2 + x - 2"
} | {
"split": "train",
"index": 4579,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For real numbers $t,$ the point\n\\[(x,y) = (\\cos^2 t, \\sin^2 t)\\]is plotted. All the plotted points lie on what kind of curve?\n\n(A) Line\n(B) Circle\n(C) Parabola\n(D) Ellipse\n(E) Hyperbola\n\nEnter the letter of the correct option. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "\\text{(A)}"
} | {
"split": "train",
"index": 4580,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If there exists a line $l$ that is a tangent to the curve $y=x^{2}$ and also a tangent to the curve $y=a\\ln x$, then the maximum value of the real number $a$ is ____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2e"
} | {
"split": "train",
"index": 4581,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A six place number is formed by repeating a three place number; for example, $256256$ or $678678$, etc. Any number of this form is always exactly divisible by: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1001"
} | {
"split": "train",
"index": 4582,
"solution": "1. **Express the number in a general form**: \n Let the three-digit number be represented as $\\overline{abc}$, where $a$, $b$, and $c$ are its digits. The six-digit number formed by repeating $\\overline{abc}$ is $\\overline{abcabc}$.\n\n2. **Convert the six-dig... |
deepscaler | [
{
"role": "user",
"content": "Let $f : N \\to N$ be a strictly increasing function such that $f(f(n))= 3n$ , for all $n \\in N$ . Find $f(2010)$ . \n\nNote: $N = \\{0,1,2,...\\}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3015"
} | {
"split": "train",
"index": 4583,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers less than $50$ have an odd number of positive integer divisors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 4584,
"solution": "To solve this problem, we need to determine how many positive integers less than $50$ have an odd number of positive integer divisors.\n\n1. **Understanding the Divisor Counting**: A number has an odd number of divisors if and only if it is a perfect square. This is... |
deepscaler | [
{
"role": "user",
"content": "Find the radius of the circle with the equation $x^2 - 8x + y^2 - 10y + 34 = 0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{7}"
} | {
"split": "train",
"index": 4585,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $150\\times 324\\times 375$ rectangular solid is made by gluing together $1\\times 1\\times 1$ cubes. An internal diagonal of this solid passes through the interiors of how many of the $1\\times 1\\times 1$ cubes?\n Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "768"
} | {
"split": "train",
"index": 4586,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can one choose distinct numbers a and b from {1, 2, 3, ..., 2005} such that a + b is a multiple of 5? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "401802"
} | {
"split": "train",
"index": 4587,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the smallest positive integer $m$ such that $11m-3$ and $8m + 5$ have a common factor greater than $1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 4588,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $A$, $B$, and $C$ form an isosceles triangle with $AB = AC$. Points $M$, $N$, and $O$ are the midpoints of sides $AB$, $BC$, and $CA$ respectively. Find the number of noncongruent triangles that can be drawn using any three of these six points as vertices. Let's think st... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 4589,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If two sides of a triangle are 8 and 15 units, and the angle between them is 30 degrees, what is the length of the third side? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{289 - 120\\sqrt{3}}"
} | {
"split": "train",
"index": 4590,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the equation whose graph is a parabola with vertex $(2,4)$, vertical axis of symmetry, and contains the point $(1,1)$. Express your answer in the form \"$ax^2+bx+c$\". Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3x^2+12x-8"
} | {
"split": "train",
"index": 4591,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the product of all constants $t$ such that the quadratic $x^2 + tx - 24$ can be factored in the form $(x+a)(x+b)$, where $a$ and $b$ are integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5290000"
} | {
"split": "train",
"index": 4592,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$A, B, C$ and $D$ are distinct positive integers such that the product $AB = 60$, the product $CD = 60$ and $A - B = C + D$ . What is the value of $A$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 4593,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest perfect square such that when the last two digits are removed, it is still a perfect square. (It is assumed that one of the digits removed is not zero.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1681"
} | {
"split": "train",
"index": 4594,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the random variable $X$ follows a normal distribution $N(2, \\sigma^2)$, and $P(X \\leq 4) = 0.84$, determine the value of $P(X < 0)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.16"
} | {
"split": "train",
"index": 4595,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p$ be the largest prime with 2010 digits. What is the smallest positive integer $k$ such that $p^2 - k$ is divisible by 12? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "k = 1"
} | {
"split": "train",
"index": 4596,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( n \\in \\mathbf{Z}_{+} \\), and\n$$\n\\begin{array}{l}\na, b, c \\in \\{x \\mid x \\in \\mathbf{Z} \\text{ and } x \\in [1,9]\\}, \\\\\nA_{n} = \\underbrace{\\overline{a a \\cdots a}}_{n \\text{ digits}}, B_{n} = \\underbrace{b b \\cdots b}_{2n \\text{ digits}}, C_{n} =... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 4597,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pen costs $\\mathrm{Rs.}\\, 13$ and a note book costs $\\mathrm{Rs.}\\, 17$ . A school spends exactly $\\mathrm{Rs.}\\, 10000$ in the year $2017-18$ to buy $x$ pens and $y$ note books such that $x$ and $y$ are as close as possible (i.e., $|x-y|$ is minimum). ... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 4598,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A gardener wants to plant 3 maple trees, 4 oak trees, and 5 birch trees in a row. He will randomly determine the order of these trees. What is the probability that no two birch trees are adjacent? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7/99"
} | {
"split": "train",
"index": 4599,
"solution": ""
} |
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