data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Given $α \\in \\left(0, \\frac{\\pi}{2}\\right)$, $\\cos \\left(α+ \\frac{\\pi}{3}\\right) = -\\frac{2}{3}$, then $\\cos α =$ \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{15}-2}{6}"
} | {
"split": "train",
"index": 23049,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve\n\\[(x^3 + 3x^2 \\sqrt{2} + 6x + 2 \\sqrt{2}) + (x + \\sqrt{2}) = 0.\\]Enter all the solutions, separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\sqrt{2}, -\\sqrt{2} + i, -\\sqrt{2} - i"
} | {
"split": "train",
"index": 23050,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\\frac{n}{180}$ terminate? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 23051,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the equation csa\\sin (+ \\dfrac {\\pi}{6})+siasn(a- \\dfrac {\\pi}{3})=, evaluate the expression. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{2}"
} | {
"split": "train",
"index": 23052,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $n$ is $1$ less than a multiple of $50$, then what is the remainder when $n^2+2n+3$ is divided by $50$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 23053,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $(4^4 \\div 4^3) \\cdot 2^8$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1024"
} | {
"split": "train",
"index": 23054,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $A$, $B$, and $C$ are noncollinear points in the plane with integer coordinates\nsuch that the distances $AB$, $AC$, and $BC$ are integers, what is the smallest possible value of $AB$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 23055,
"solution": "The smallest distance is 3, achieved by $A = (0,0)$, $B = (3,0)$, $C = (0,4)$.\nTo check this, it suffices to check that $AB$ cannot equal 1 or 2. (It cannot equal 0\nbecause if two of the points were to coincide, the three points would be collinear.)\n\nThe triang... |
deepscaler | [
{
"role": "user",
"content": "A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(6,0)$, $(6,2)$, and $(0,2)$. What is the probability that $x + y < 3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 23056,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The difference \\(\\sqrt{|40 \\sqrt{2}-57|}-\\sqrt{40 \\sqrt{2}+57}\\) is an integer. Find this number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-10"
} | {
"split": "train",
"index": 23057,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The set $T = \\{1, 2, 3, \\ldots, 59, 60\\}$ contains the first 60 positive integers. After the multiples of 2, the multiples of 3, and multiples of 5 are removed, how many integers remain in the set $T$? Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 23058,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of three different numbers is 67. The two larger numbers differ by 7 and the two smaller numbers differ by 3. What is the value of the largest number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 23059,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $4^{n}=64^{2}$, what is the value of $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 23060,
"solution": "We note that $64=4 \\times 4 \\times 4$. Thus, $64^{2}=64 \\times 64=4 \\times 4 \\times 4 \\times 4 \\times 4 \\times 4$. Since $4^{n}=64^{2}$, then $4^{n}=4 \\times 4 \\times 4 \\times 4 \\times 4 \\times 4$ and so $n=6$."
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ for which $315^2-n^2$ evenly divides $315^3-n^3$ .\n\n*Proposed by Kyle Lee* Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 23061,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many kilometers will a traveler cover in 17 days, spending 10 hours a day on this, if he has already covered 112 kilometers in 29 days, traveling 7 hours each day? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "93.79"
} | {
"split": "train",
"index": 23062,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct sequences of five letters can be made from the letters in FREQUENCY if each sequence must begin with F, end with Y, and no letter can appear in a sequence more than once? Further, the second letter must be a vowel. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 23063,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x = -3$, what is the value of $(x-3)^{2}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 23064,
"solution": "Evaluating, $(x-3)^{2}=(-3-3)^{2}=(-6)^{2}=36$."
} |
deepscaler | [
{
"role": "user",
"content": "In a regular tetrahedron \\(ABCD\\), points \\(E\\) and \\(F\\) are on edges \\(AB\\) and \\(AC\\) respectively, satisfying \\(BE=3\\) and \\(EF=4\\), and \\(EF \\parallel\\) plane \\(BCD\\). Find the area of \\(\\triangle DEF\\). Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{33}"
} | {
"split": "train",
"index": 23065,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a small pond there are eleven lily pads in a row labeled 0 through 10. A frog is sitting on pad 1. When the frog is on pad $N$, $0<N<10$, it will jump to pad $N-1$ with probability $\\frac{N}{10}$ and to pad $N+1$ with probability $1-\\frac{N}{10}$. Each jump is independent ... | math | {
"style": "rule",
"ground_truth": "\\frac{63}{146}"
} | {
"split": "train",
"index": 23066,
"solution": "Define \\( P(N) \\) as the probability that the frog survives starting from pad \\( N \\). We need to find \\( P(1) \\), the probability that the frog escapes starting from pad 1.\n\nThe recursive relationship given in the problem is:\n\\[ P(N) = \\frac{N}{10} P(N-... |
deepscaler | [
{
"role": "user",
"content": "Rounded to 3 decimal places, what is $\\frac{8}{11}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.727"
} | {
"split": "train",
"index": 23067,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Floyd looked at a standard $12$ hour analogue clock at $2\\!:\\!36$ . When Floyd next looked at the clock, the angles through which the hour hand and minute hand of the clock had moved added to $247$ degrees. How many minutes after $3\\!:\\!00$ was that? Let's think step... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 23068,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "John has 15 marbles of different colors, including one red, one green, one blue, and three yellow marbles. In how many ways can he choose 5 marbles, if he must choose exactly one marble that is red, green, blue, or yellow? Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "756"
} | {
"split": "train",
"index": 23069,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T$ be the set of numbers of the form $2^{a} 3^{b}$ where $a$ and $b$ are integers satisfying $0 \\leq a, b \\leq 5$. How many subsets $S$ of $T$ have the property that if $n$ is in $S$ then all positive integer divisors of $n$ are in $S$ ? Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "924"
} | {
"split": "train",
"index": 23070,
"solution": "Consider the correspondence $(a, b) \\leftrightarrow 2^{a} 3^{b}$ for non-negative integers $a$ and $b$. So we can view $T$ as the square of lattice points $(a, b)$ where $0 \\leq a, b \\leq 5$, and subsets of $T$ as subsets of this square. Notice then that the int... |
deepscaler | [
{
"role": "user",
"content": "A binary string of length $n$ is a sequence of $n$ digits, each of which is 0 or 1 . The distance between two binary strings of the same length is the number of positions in which they disagree; for example, the distance between the strings 01101011 and 00101110 is 3 since they... | math | {
"style": "rule",
"ground_truth": "\\begin{tabular}{ll} 00000000 & 00110101 \\ 11001010 & 10011110 \\ 11100001 & 01101011 \\ 11010100 & 01100110 \\ 10111001 & 10010011 \\ 01111100 & 11001101 \\ 00111010 & 10101100 \\ 01010111 & 11110010 \\ 00001111 & 01011001 \\ 10100111 & 11111111 \\ \\end{tabular}"
} | {
"split": "train",
"index": 23071,
"solution": "The maximum possible number of such strings is 20 . An example of a set attaining this bound is \\begin{tabular}{ll} 00000000 & 00110101 \\\\ 11001010 & 10011110 \\\\ 11100001 & 01101011 \\\\ 11010100 & 01100110 \\\\ 10111001 & 10010011 \\\\ 01111100 & 11001101 \\\... |
deepscaler | [
{
"role": "user",
"content": "Tim is organizing a week-long series of pranks. On Monday, he gets his friend Joe to help. On Tuesday, he can choose between two friends, either Ambie or John. For Wednesday, there are four new people willing to help. However, on Thursday, none of these previous individuals can... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 23072,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=2\\sin (\\\\omega x)$, where $\\\\omega > 0$.\n\n(1) When $ \\\\omega =1$, determine the even-odd property of the function $F(x)=f(x)+f(x+\\\\dfrac{\\\\mathrm{ }\\\\!\\\\!\\\\pi\\\\!\\\\!{ }}{2})$ and explain the reason.\n\n(2) When $ \\\\omega =2$, the... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 23073,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Knot is on an epic quest to save the land of Hyruler from the evil Gammadorf. To do this, he must collect the two pieces of the Lineforce, then go to the Temple of Lime. As shown on the figure, Knot starts on point $K$, and must travel to point $T$, where $O K=2$ and $O T=4$. H... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{5}"
} | {
"split": "train",
"index": 23074,
"solution": "Let $l_{1}$ and $l_{2}$ be the lines as labeled in the above diagram. First, suppose Knot visits $l_{1}$ first, at point $P_{1}$, then $l_{2}$, at point $P_{2}$. Let $K^{\\prime}$ be the reflection of $K$ over $l_{1}$, and let $T^{\\prime}$ be the reflection of $T$... |
deepscaler | [
{
"role": "user",
"content": "Calculate: $\\frac{145}{273} \\times 2 \\frac{173}{245} \\div 21 \\frac{13}{15}=$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7395}{112504}"
} | {
"split": "train",
"index": 23075,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system $(xOy)$, the slope angle of line $l$ passing through point $M(2,1)$ is $\\frac{\\pi}{4}$. Establish a polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, using the same unit length for both coordi... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{30}}{7}"
} | {
"split": "train",
"index": 23076,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the sum of the series $\\sum_{n=1}^{\\infty} \\frac{(-1)^{n+1}}{n^{5}}$ with an accuracy of $10^{-3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.973"
} | {
"split": "train",
"index": 23077,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the natural numbers from 1 to 800, inclusive, contain the digit 7 at least once? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "152"
} | {
"split": "train",
"index": 23078,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square with sides of 10 inches is shown. If $P$ is a point such that the segments $\\overline{PA}$, $\\overline{PB}$, and $\\overline{PC}$ are equal in length, and segment $\\overline{PC}$ is perpendicular to segment $\\overline{GD}$, what is the area, in square inches, of tr... | math | {
"style": "rule",
"ground_truth": "\\frac{75}{4}"
} | {
"split": "train",
"index": 23079,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $P$ is on the circle $C_{1}: x^{2}+y^{2}-8x-4y+11=0$, and point $Q$ is on the circle $C_{2}: x^{2}+y^{2}+4x+2y+1=0$. What is the minimum value of $|PQ|$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{5} - 5"
} | {
"split": "train",
"index": 23080,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the figure, the visible gray area within the larger circle is equal to three times the area of the white circular region. What is the ratio of the radius of the small circle to the radius of the large circle? Express your answer as a common fraction.\n\n[asy]size(101);\nfill... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 23081,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On Monday, 5 students in the class received A's in math, on Tuesday 8 students received A's, on Wednesday 6 students, on Thursday 4 students, and on Friday 9 students. None of the students received A's on two consecutive days. What is the minimum number of students that could h... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 23082,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $20\\left(\\frac{256}{4} + \\frac{64}{16} + \\frac{16}{64} + 2\\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1405"
} | {
"split": "train",
"index": 23083,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the number of square units in the area of the hexagon below?\n\n[asy]\nunitsize(0.5cm);\ndefaultpen(linewidth(0.7)+fontsize(10));\ndotfactor = 4;\nint i,j;\nfor(i=0;i<=4;++i)\n\n{\n\nfor(j=-3;j<=3;++j)\n\n{\n\ndot((i,j));\n\n}\n\n}\n\nfor(i=1;i<=4;++i)\n\n{\n\ndraw((i,-... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 23084,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f\\left(x\\right)=ax^{2}-bx-1$, sets $P=\\{1,2,3,4\\}$, $Q=\\{2,4,6,8\\}$, if a number $a$ and a number $b$ are randomly selected from sets $P$ and $Q$ respectively to form a pair $\\left(a,b\\right)$.<br/>$(1)$ Let event $A$ be \"the monotonically increasin... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{16}"
} | {
"split": "train",
"index": 23085,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The altitudes of a triangle are $12$, $15$, and $20$. What is the area of this triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 23086,
"solution": "\nTo find the area of a triangle with altitudes \\( 12 \\), \\( 15 \\), and \\( 20 \\), we begin by letting the sides of the triangle be \\( a \\), \\( b \\), and \\( c \\) opposite to the altitudes \\( h_a = 12 \\), \\( h_b = 15 \\), and \\( h_c = 20 \\), respecti... |
deepscaler | [
{
"role": "user",
"content": "A speaker talked for sixty minutes to a full auditorium. Twenty percent of the audience heard the entire talk and ten percent slept through the entire talk. Half of the remainder heard one third of the talk and the other half heard two thirds of the talk. What was the average n... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 23087,
"solution": "1. **Assumptions and Initial Setup**:\n Let's assume there are $100$ people in the audience for simplicity of calculation. The total duration of the talk is $60$ minutes.\n\n2. **Distribution of Audience**:\n - **20% heard the entire talk**: This is $20\\%$ of ... |
deepscaler | [
{
"role": "user",
"content": "Given that the ceiling is 3 meters above the floor, the light bulb is 15 centimeters below the ceiling, Alice is 1.6 meters tall and can reach 50 centimeters above the top of her head, and a 5 centimeter thick book is placed on top of a stool to reach the light bulb, find the h... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 23088,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the total number of digits used when the first 3003 positive even integers are written? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11460"
} | {
"split": "train",
"index": 23089,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The population size (in number of animals) of a certain animal species is given by the equation $y=a\\log_{2}(x+1)$. Suppose that the population size of this animal species in the first year was 100 animals. What will be the population size in the 15th year. Let's think step by... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 23090,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest number of consecutive integers whose sum is $36$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 23091,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A number contains only two kinds of digits: 3 or 4, and both 3 and 4 appear at least once. The number is a multiple of both 3 and 4. What is the smallest such number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3444"
} | {
"split": "train",
"index": 23092,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product $\\left(1-\\frac{1}{2^{2}}\\right)\\left(1-\\frac{1}{3^{2}}\\right)\\ldots\\left(1-\\frac{1}{9^{2}}\\right)\\left(1-\\frac{1}{10^{2}}\\right)$ equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{20}"
} | {
"split": "train",
"index": 23093,
"solution": "1. **Factor each term as a difference of squares**: \n Each term in the product can be written as $\\left(1-\\frac{1}{n^2}\\right)$, which factors as $\\left(1-\\frac{1}{n}\\right)\\left(1+\\frac{1}{n}\\right)$.\n\n2. **Write the product using the factored terms*... |
deepscaler | [
{
"role": "user",
"content": "Part of the graph of $f(x) = ax^3 + bx^2 + cx + d$ is shown. What is $b$?\n\n[asy]\nunitsize(1.5 cm);\n\nreal func(real x) {\n return((x + 1)*(x - 1)*(x - 2));\n}\n\ndraw(graph(func,-1.1,1.5));\ndraw((-1.5,0)--(1.5,0),Arrows(6));\ndraw((0,-1)--(0,2.5),Arrows(6));\n\nlabel(\"$x... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 23094,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation $y=-16t^2+22t+45$ describes the height (in feet) of a ball thrown upwards at $22$ feet per second from $45$ feet above the ground. Find the time (in seconds) when the ball will hit the ground. Express your answer as a common fraction. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"split": "train",
"index": 23095,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area of a square inscribed in a semicircle of radius 1, with one of its sides flush with the diameter of the semicircle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"split": "train",
"index": 23096,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "**Q8.** Given a triangle $ABC$ and $2$ point $K \\in AB, \\; N \\in BC$ such that $BK=2AK, \\; CN=2BN$ and $Q$ is the common point of $AN$ and $CK$ . Compute $\\dfrac{ S_{ \\triangle ABC}}{S_{\\triangle BCQ}}.$ Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "7/4"
} | {
"split": "train",
"index": 23097,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nDecide whether there is an integer \\( n > 1 \\) with the following properties:\n\n(a) \\( n \\) is not a prime number.\n\n(b) For all integers \\( a \\), \\( a^{n} - a \\) is divisible by \\( n \\). Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "561"
} | {
"split": "train",
"index": 23098,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the greatest $a$ such that $\\frac{7\\sqrt{(2a)^2+(1)^2}-4a^2-1}{\\sqrt{1+4a^2}+3}=2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 23099,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the system of equations:\n\\begin{align*}\n8x - 5y &= a, \\\\\n10y - 15x &= b.\n\\end{align*}\nIf this system has a solution \\((x, y)\\) where both \\(x\\) and \\(y\\) are nonzero, calculate \\(\\frac{a}{b}\\), assuming \\(b\\) is nonzero. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{15}"
} | {
"split": "train",
"index": 23100,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( m \\) and \\( n \\) are two distinct positive integers and the last four digits of \\( 2019^{m} \\) and \\( 2019^{n} \\) are the same, find the minimum value of \\( m+n \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "502"
} | {
"split": "train",
"index": 23101,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Igor Gorshkov has all seven books about Harry Potter. In how many ways can Igor arrange these seven volumes on three different shelves, such that each shelf has at least one book? (Arrangements that differ in the order of books on a shelf are considered different). Let's think ... | math | {
"style": "rule",
"ground_truth": "75600"
} | {
"split": "train",
"index": 23102,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider $9$ points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of $\\,n\\,$ such that whenever exactly $\\,n\\,$ edges are co... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 23103,
"solution": "\nConsider a configuration where you have 9 points in space, with each pair of points joined by an edge, for a total of \\(\\binom{9}{2} = 36\\) edges. We want to find the smallest \\( n \\) such that if exactly \\( n \\) edges are colored (either blue or red), the... |
deepscaler | [
{
"role": "user",
"content": "Suppose point $P$ is on the curve represented by the equation $\\sqrt{(x-5)^2+y^2} - \\sqrt{(x+5)^2+y^2} = 6$, and $P$ is also on the line $y=4$. Determine the x-coordinate of point $P$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3\\sqrt{2}"
} | {
"split": "train",
"index": 23104,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function f(x) = 2x^3 - ax^2 + 1, where a ∈ R.\n(I) When a = 6, the line y = -6x + m is tangent to f(x). Find the value of m.\n(II) If the function f(x) has exactly one zero in the interval (0, +∞), find the monotonic intervals of the function.\n(III) When a > 0, if th... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 23105,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $k$ such that $z^{10}+z^{9}+z^{6}+z^{5}+z^{4}+z+1$ divides $z^{k}-1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 23106,
"solution": "Let $Q(z)$ denote the polynomial divisor. We need that the roots of $Q$ are $k$-th roots of unity. With this in mind, we might observe that solutions to $z^{7}=1$ and $z \\neq 1$ are roots of $Q$, which leads to its factorization. Alternatively, we note that $$(z-1... |
deepscaler | [
{
"role": "user",
"content": "A game involves jumping to the right on the real number line. If $a$ and $b$ are real numbers\nand $b > a$, the cost of jumping from $a$ to $b$ is $b^3-ab^2$. For what real numbers\n$c$ can one travel from $0$ to $1$ in a finite number of jumps with total cost exactly $c$? Let'... | math | {
"style": "rule",
"ground_truth": "1/3 < c \\leq 1"
} | {
"split": "train",
"index": 23107,
"solution": "The desired real numbers $c$ are precisely those for which $1/3 < c \\leq 1$.\nFor any positive integer $m$ and any\nsequence $0 = x_0 < x_1 < \\cdots < x_m = 1$,\nthe cost of jumping along this sequence is\n$\\sum_{i=1}^m (x_i - x_{i-1})x_i^2$. Since\n\\begin{alig... |
deepscaler | [
{
"role": "user",
"content": "Given \\(\\alpha \\in (0, \\pi)\\), if \\(\\sin \\alpha + \\cos \\alpha = \\frac{\\sqrt{3}}{3}\\), then \\(\\cos^2 \\alpha - \\sin^2 \\alpha = \\) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{5}}{3}"
} | {
"split": "train",
"index": 23108,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p$ and $q$ be constants. Suppose that the equation \n\\[\\frac{(x+p)(x+q)(x-15)}{(x-5)^2} = 0\\]\nhas exactly $3$ distinct roots, while the equation \n\\[\\frac{(x-2p)(x-5)(x+10)}{(x+q)(x-15)} = 0\\]\nhas exactly $2$ distinct roots. Compute $100p + q.$ Let's think step by ... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 23109,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of the function \\( f(x) = x^{2} + 2x + \\frac{6}{x} + \\frac{9}{x^{2}} + 4 \\) for \\( x > 0 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10 + 4 \\sqrt{3}"
} | {
"split": "train",
"index": 23110,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $C: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$ with the length of the minor axis being $2$ and the eccentricity being $\\frac{\\sqrt{2}}{2}$, the line $l: y = kx + m$ intersects the ellipse $C$ at points $A$ and $B$, and the perpendicular bisector of... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 23111,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point P is located on side AB of triangle ABC. What is the probability that the area of triangle PBC is less than or equal to 1/3 of the area of triangle ABC. 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": 23112,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a $3 \\times 3$ board the numbers from $1$ to $9$ are written in some order and without repeating. We say that the arrangement obtained is *Isthmian* if the numbers in any two adjacent squares have different parity. Determine the number of different Isthmian arrangemen... | math | {
"style": "rule",
"ground_truth": "720"
} | {
"split": "train",
"index": 23113,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mike rides his bicycle at a constant speed of $30 \\mathrm{~km} / \\mathrm{h}$. How many kilometres does Mike travel in 20 minutes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 23114,
"solution": "Since 1 hour equals 60 minutes, then 20 minutes equals $\\frac{1}{3}$ of an hour. Since Mike rides at $30 \\mathrm{~km} / \\mathrm{h}$, then in $\\frac{1}{3}$ of an hour, he travels $\\frac{1}{3} \\times 30 \\mathrm{~km}=10 \\mathrm{~km}$."
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate plane $(xOy)$, a moving point $P$ passes through the chords $PA$ and $PB$ of the circles $C_{1}$: $x^{2}+y^{2}+2x+2y+1=0$ and $C_{2}$: $x^{2}+y^{2}-4x-6y+9=0$ respectively, where $A$ and $B$ are the points of intersection. If $|PA|=|PB|$, find the mi... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"split": "train",
"index": 23115,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the smallest positive real number $r$ such that there exist differentiable functions $f\\colon \\mathbb{R} \\to \\mathbb{R}$ and $g\\colon \\mathbb{R} \\to \\mathbb{R}$ satisfying \\begin{enumerate} \\item[(a)] $f(0) > 0$, \\item[(b)] $g(0) = 0$, \\item[(c)] $|f'(x)| ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"split": "train",
"index": 23116,
"solution": "The answer is $r=\\frac{\\pi}{2}$, which manifestly is achieved by setting $f(x)=\\cos x$ and $g(x)=\\sin x$. \\n\\n\\textbf{First solution.} Suppose by way of contradiction that there exist some $f,g$ satisfying the stated conditions for some $0 < r<\\frac{\\pi}{2... |
deepscaler | [
{
"role": "user",
"content": "Mario is once again on a quest to save Princess Peach. Mario enters Peach's castle and finds himself in a room with 4 doors. This room is the first in a sequence of 6 indistinguishable rooms. In each room, 1 door leads to the next room in the sequence (or, for the last room, Bo... | math | {
"style": "rule",
"ground_truth": "5460"
} | {
"split": "train",
"index": 23117,
"solution": "This problem works in the same general way as the last problem, but it can be more succinctly solved using the general formula, which is provided below in the solution to the next problem."
} |
deepscaler | [
{
"role": "user",
"content": "For positive integers $n,$ let $\\tau (n)$ denote the number of positive integer divisors of $n,$ including 1 and $n.$ Define $S(n)$ by $S(n)=\\tau(1)+ \\tau(2) + \\cdots + \\tau(n).$ Let $a$ denote the number of positive integers $n \\leq 3000$ with $S(n)$ odd, and let $b$ den... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 23118,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three different numbers from the list $2, 3, 4, 6$ have a sum of 11. What is the product of these numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 23119,
"solution": "The sum of 2, 3 and 6 is $2 + 3 + 6 = 11$. Their product is $2 \\cdot 3 \\cdot 6 = 36$."
} |
deepscaler | [
{
"role": "user",
"content": "Simplify this expression to a common fraction: $\\frac{1}{\\frac{1}{(\\frac{1}{2})^{1}}+\\frac{1}{(\\frac{1}{2})^{2}}+\\frac{1}{(\\frac{1}{2})^{3}}}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{14}"
} | {
"split": "train",
"index": 23120,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the $2013\\cdot Jining$ test, if the sum of the first $n$ terms of the sequence $\\{a_n\\}$ is $S_n = n^2 - 4n + 2$, then $|a_1| + |a_2| + \\ldots + |a_{10}| = \\_\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "66"
} | {
"split": "train",
"index": 23121,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given triangle $\\triangle ABC$ with angles $A$, $B$, $C$ and their respective opposite sides $a$, $b$, $c$, such that $\\frac{\\sqrt{3}c}{\\cos C} = \\frac{a}{\\cos(\\frac{3\\pi}{2} + A)}$.\n(I) Find the value of $C$;\n(II) If $\\frac{c}{a} = 2$, $b = 4\\sqrt{3}$, find the are... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{15} - 2\\sqrt{3}"
} | {
"split": "train",
"index": 23122,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of the $20$ balls is tossed independently and at random into one of the $5$ bins. Let $p$ be the probability that some bin ends up with $3$ balls, another with $5$ balls, and the other three with $4$ balls each. Let $q$ be the probability that every bin ends up with $4$ ba... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 23123,
"solution": "1. **Define the sets and probabilities**: Let $A$ be the set of all configurations where the balls are distributed as $3{-}5{-}4{-}4{-}4$ among the bins, and let $B$ be the set of all configurations where the balls are distributed as $4{-}4{-}4{-}4{-}4$. Define $p ... |
deepscaler | [
{
"role": "user",
"content": " $p(n) $ is a product of all digits of n.Calculate: $ p(1001) + p(1002) + ... + p(2011) $ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "91125"
} | {
"split": "train",
"index": 23124,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Walter gets up at 6:30 a.m., catches the school bus at 7:30 a.m., has 6 classes that last 50 minutes each, has 30 minutes for lunch, and has 2 hours additional time at school. He takes the bus home and arrives at 4:00 p.m. Calculate the total number of minutes Walter spent on t... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 23125,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $n$ that satisfies $\\frac{1}{n+1} + \\frac{2}{n+1} + \\frac{n}{n+1} = 3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 23126,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the equations:\n\n(1) 2x^2 - 5x + 1 = 0;\n\n(2) 3x(x - 2) = 2(2 - x). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{3}"
} | {
"split": "train",
"index": 23127,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sector $OAB$ is a quarter of a circle with radius 5 cm. Inside this sector, a circle is inscribed, tangent at three points. Find the radius of the inscribed circle in simplest radical form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5\\sqrt{2} - 5"
} | {
"split": "train",
"index": 23128,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a, b, c$ are the sides opposite to angles $A, B, C$ respectively in $\\triangle ABC$, and $\\cos A \\sin B \\sin C + \\cos B \\sin A \\sin C = 2 \\cos C \\sin A \\sin B$, find the maximum value of $C$. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 23129,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Natural numbers \\( a, b, c \\) are chosen such that \\( a < b < c \\). It is also known that the system of equations \\( 2x + y = 2021 \\) and \\( y = |x - a| + |x - b| + |x - c| \\) has exactly one solution. Find the minimum possible value of \\( c \\). Let's think step by st... | math | {
"style": "rule",
"ground_truth": "1011"
} | {
"split": "train",
"index": 23130,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The opposite of the arithmetic square root of $\\sqrt{81}$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-9"
} | {
"split": "train",
"index": 23131,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given sets $A=\\{x|x^{2}+2x-3=0,x\\in R\\}$ and $B=\\{x|x^{2}-\\left(a+1\\right)x+a=0,x\\in R\\}$.<br/>$(1)$ When $a=2$, find $A\\cap C_{R}B$;<br/>$(2)$ If $A\\cup B=A$, find the set of real numbers for $a$. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "\\{1\\}"
} | {
"split": "train",
"index": 23132,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A group of $12$ pirates agree to divide a treasure chest of gold coins among themselves as follows. The $k^{\\text{th}}$ pirate to take a share takes $\\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for whi... | math | {
"style": "rule",
"ground_truth": "1925"
} | {
"split": "train",
"index": 23133,
"solution": "\nTo solve this problem, we need to determine the smallest number of coins in the chest initially such that each pirate receives a whole number of coins when they take their share according to the given rule.\n\n1. **Initial Setup and Recursive Formula**:\n Let $... |
deepscaler | [
{
"role": "user",
"content": "There are $n \\geq 2$ coins, each with a different positive integer value. Call an integer $m$ sticky if some subset of these $n$ coins have total value $m$. We call the entire set of coins a stick if all the sticky numbers form a consecutive range of integers. Compute the mini... | math | {
"style": "rule",
"ground_truth": "199"
} | {
"split": "train",
"index": 23134,
"solution": "Sort a stick by increasing value. Note that all sticks must contain 1 by necessity, or the largest and second largest sticky values would not be consecutive. So, let's say a stick's highest coin value is $a$, and all the other terms have a value of $S$. If $a \\geq... |
deepscaler | [
{
"role": "user",
"content": "The number $1025$ can be written as $23q + r$ where $q$ and $r$ are positive integers. What is the greatest possible value of $q - r$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 23135,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Place 6 balls, labeled from 1 to 6, into 3 different boxes with each box containing 2 balls. If the balls labeled 1 and 2 cannot be placed in the same box, the total number of different ways to do this is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 23136,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a square ABCD with side length 8 units. On side AB, semicircles are constructed inside the square both with diameter AB. Inside the square and tangent to AB at its midpoint, another quarter circle with its center at the midpoint of AB is also constructed pointing inwar... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 23137,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Bernardo chooses a three-digit positive integer $N$ and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer $S$. For example, if... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 23138,
"solution": "1. **Understanding the Problem**: Bernardo writes the base-5 and base-6 representations of a three-digit number $N$ on a blackboard. LeRoy, treating these numbers as base-10, adds them to get $S$. We need to find how many such $N$ exist such that the last two digit... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest solution to the equation \\[\\lfloor x^2 \\rfloor - \\lfloor x \\rfloor^2 = 24.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6\\sqrt{2}"
} | {
"split": "train",
"index": 23139,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "1. The focal distance of the parabola $4x^{2}=y$ is \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\n2. The equation of the hyperbola that has the same asymptotes as the hyperbola $\\frac{x^{2}}{2} -y^{2}=1$ and passes through $(2,0)$ is \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\n3. In the pla... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 23140,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many real values of $c$ do we have $\\left|\\frac12-ci\\right| = \\frac34$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 23141,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At 30 palm trees on different parts of an uninhabited island, a sign is attached.\n\n- On 15 of them it says: \"Exactly under 15 signs a treasure is buried.\"\n- On 8 of them it says: \"Exactly under 8 signs a treasure is buried.\"\n- On 4 of them it says: \"Exactly under 4 sig... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 23142,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From Moscow to city \\( N \\), a passenger can travel by train, taking 20 hours. If the passenger waits for a flight (waiting will take more than 5 hours after the train departs), they will reach city \\( N \\) in 10 hours, including the waiting time. By how many times is the p... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 23143,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\log_2\\frac{1}{16}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 23144,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\tan \\left(α- \\frac {π}{4}\\right)=2$, find the value of $\\sin \\left(2α- \\frac {π}{4}\\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {\\sqrt {2}}{10}"
} | {
"split": "train",
"index": 23145,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence $(a_n)$ is defined as follows: $a_1 = 1,$ $a_2 = \\frac{1}{2},$ and\n\\[a_n = \\frac{1 - a_{n - 1}}{2a_{n - 2}}\\]for all $n \\ge 3.$ Find $a_{120}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{20}{41}"
} | {
"split": "train",
"index": 23146,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A large equilateral triangle with side length $12$ is to be filled completely with non-overlapping equilateral triangles, each of side length $2$. Calculate the number of small triangles needed to fill the large triangle. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 23147,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Factorize the expression $27x^6 - 512y^6$ and find the sum of all integer coefficients in its factorized form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "92"
} | {
"split": "train",
"index": 23148,
"solution": ""
} |
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