data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "What is the least possible positive four-digit palindrome that is divisible by 4? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1881"
} | {
"split": "train",
"index": 26049,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence is defined as \\( a_{0}=134, a_{1}=150, a_{k+1}=a_{k-1}-\\frac{k}{a_{k}} \\) for \\( k=1,2, \\cdots, n-1 \\). Determine the value of \\( n \\) for which \\( a_{n}=0 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "201"
} | {
"split": "train",
"index": 26050,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the last two digits of $\\tbinom{200}{100}$ . Express the answer as an integer between $0$ and $99$ . (e.g. if the last two digits are $05$ , just write $5$ .) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 26051,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The projection of $\\begin{pmatrix} 0 \\\\ 3 \\\\ z \\end{pmatrix}$ onto $\\begin{pmatrix} -3 \\\\ 5 \\\\ -1 \\end{pmatrix}$ is\n\\[\\frac{12}{35} \\begin{pmatrix} -3 \\\\ 5 \\\\ -1 \\end{pmatrix}.\\]Find $z.$ Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 26052,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many pairs of parallel edges, such as $\\overline{AB}$ and $\\overline{GH}$ or $\\overline{EH}$ and $\\overline{FG}$, does a cube have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 26053,
"solution": "\nTo find the number of pairs of parallel edges in a cube, we can consider the cube's structure and symmetry. A cube has 12 edges, and each edge has exactly one parallel counterpart in each of the three dimensions (length, width, height).\n\n1. **Identify Parallel ... |
deepscaler | [
{
"role": "user",
"content": "Find $AB$ in the triangle below.\n\n[asy]\nunitsize(1inch);\npair A,B,C;\nA = (0,0);\nB = (1,0);\nC = (0.5,sqrt(3)/2);\ndraw (A--B--C--A,linewidth(0.9));\ndraw(rightanglemark(B,A,C,3));\nlabel(\"$A$\",A,S);\nlabel(\"$B$\",B,S);\nlabel(\"$C$\",C,N);\nlabel(\"$18$\", (A+C)/2,W);\... | math | {
"style": "rule",
"ground_truth": "18\\sqrt{3}"
} | {
"split": "train",
"index": 26054,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the last two digits of this portion of the Fibonacci Factorial Series: $1!+1!+2!+3!+5!+8!+13!+21!+34!+55!+89!$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 26055,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive divisors of 50! are either prime or the product of exactly two distinct primes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 26056,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a function \\( f(x) \\) on \\( \\mathbb{R} \\) that satisfies \\( f(-x) = -f(x) \\), \\( f(x-2) = f(x+2) \\), and when \\( x \\in (-1,0) \\), \\( f(x) = 2^x + \\frac{1}{5} \\). Then, evaluate \\( f(\\log_{2} 20) \\). Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 26057,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ordered quadruples \\((a, b, c, d)\\) of positive odd integers are there that satisfy the equation \\(a + b + c + 2d = 15?\\) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 26058,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the absolute values of the roots of $x^4-4x^3-4x^2+16x-8=0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2+2\\sqrt{2}+2\\sqrt{3}"
} | {
"split": "train",
"index": 26059,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiaoming's father departs from home to go shopping at the supermarket. If he first rides a bicycle for 12 minutes and then walks for 20 minutes, he can reach the supermarket; if he first rides a bicycle for 8 minutes and then walks for 36 minutes, he can also reach the supermar... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 26060,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the parametric equations of curve $C_{1}$ are $\\left\\{{\\begin{array}{l}{x=2\\cos\\varphi}\\\\{y=\\sqrt{2}\\sin\\varphi}\\end{array}}\\right.$ (where $\\varphi$ is the parameter). Taking point $O$ as the pole and the positive half-axi... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{7}}{3}"
} | {
"split": "train",
"index": 26061,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $ABCD$ is a rectangle whose diagonals meet at $E$ . The perimeter of triangle $ABE$ is $10\\pi$ and the perimeter of triangle $ADE$ is $n$ . Compute the number of possible integer values of $n$ . Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 26062,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The volume of the solid formed by rotating an isosceles right triangle with legs of length 1 around its hypotenuse is __________. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{6}\\pi"
} | {
"split": "train",
"index": 26063,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from $(c,0)$ to $(3,3)$, divides the entire region into two regions of equal area. What is $c$? Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 26064,
"solution": "\nWe are given a configuration of five unit squares in the coordinate plane, and a line extending from $(c,0)$ to $(3,3)$ that divides the entire region into two regions of equal area. We need to find the value of $c$.\n\n#### Step-by-step Analysis:\n\n1. **Total A... |
deepscaler | [
{
"role": "user",
"content": "There is a unique angle $\\theta$ between $0^{\\circ}$ and $90^{\\circ}$ such that for nonnegative integers $n$, the value of $\\tan{\\left(2^{n}\\theta\\right)}$ is positive when $n$ is a multiple of $3$, and negative otherwise. The degree measure of $\\theta$ is $\\tfrac{p}{q... | math | {
"style": "rule",
"ground_truth": "547"
} | {
"split": "train",
"index": 26065,
"solution": "Note that if $\\tan \\theta$ is positive, then $\\theta$ is in the first or third quadrant, so $0^{\\circ} < \\theta < 90^{\\circ} \\pmod{180^{\\circ}}$. Also notice that the only way $\\tan{\\left(2^{n}\\theta\\right)}$ can be positive for all $n$ that are multipl... |
deepscaler | [
{
"role": "user",
"content": "The diagram shows a square, its two diagonals, and two line segments, each of which connects two midpoints of the sides of the square. What fraction of the area of the square is shaded?\n\nA) $\\frac{1}{8}$ \nB) $\\frac{1}{10}$ \nC) $\\frac{1}{12}$ \nD) $\\frac{1}{16}$ \nE)... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"split": "train",
"index": 26066,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Spinners $A$ and $B$ are spun. On each spinner, the arrow is equally likely to land on each number. What is the probability that the product of the two spinners' numbers is even? Express your answer as a common fraction. [asy]\n/* AMC8 2004 #21 Problem */\ndraw(circle((0,0), 2... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 26067,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of terms in an A.P. (Arithmetic Progression) is even. The sum of the odd and even-numbered terms are 24 and 30, respectively. If the last term exceeds the first by 10.5, the number of terms in the A.P. is Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 26068,
"solution": "1. **Define the variables:**\n Let $a$ be the first term, $n$ be the number of terms, and $d$ be the common difference in the arithmetic progression (A.P.). The last term of the A.P. can be expressed as $a + (n-1)d$.\n\n2. **Set up the equation for the difference... |
deepscaler | [
{
"role": "user",
"content": "The numbers from 1 to 8 are arranged at the vertices of a cube in such a way that the sum of the numbers at any three vertices on the same face is at least 10. What is the minimum possible sum of the numbers on the vertices of one face? Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 26069,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a sequence defined as $500, x, 500 - x, \\ldots$ where each term of the sequence after the second one is obtained by subtracting the previous term from the term before it. The sequence terminates as soon as a negative term appears. Determine the positive integer $x$ th... | math | {
"style": "rule",
"ground_truth": "309"
} | {
"split": "train",
"index": 26070,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the value of the following expression:\n$$\n\\left\\lfloor\\frac{11}{2010}\\right\\rfloor+\\left\\lfloor\\frac{11 \\times 2}{2010}\\right\\rfloor+\\left\\lfloor\\frac{11 \\times 3}{2010}\\right\\rfloor+\\\\left\\lfloor\\frac{11 \\times 4}{2010}\\right\\rfloor+\\cdots+... | math | {
"style": "rule",
"ground_truth": "10045"
} | {
"split": "train",
"index": 26071,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is 30% of 200? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 26072,
"solution": "$30\\%$ of 200 equals $\\frac{30}{100} \\times 200=60$. Alternatively, we could note that $30\\%$ of 100 is 30 and $200=2 \\times 100$, so $30\\%$ of 200 is $30 \\times 2$ which equals 60."
} |
deepscaler | [
{
"role": "user",
"content": "The Dunbar family consists of a mother, a father, and some children. The average age of the members of the family is $20$, the father is $48$ years old, and the average age of the mother and children is $16$. How many children are in the family? Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 26073,
"solution": "Let $m$ be the age of the mother.\nLet $x$ be the number of children and $y$ be the average age of the children. The total age of the children is then $xy$. \n\nWe are given two key pieces of information:\n1. The average age of the family (mother, father, and child... |
deepscaler | [
{
"role": "user",
"content": "My friend and I both have the same math homework one day. I work at a rate of $p$ problems per hour and it takes me $t$ hours to finish my homework. My friend works at a rate of $2p-4$ problems per hour and it only takes him $t-2$ hours to finish his homework. Given that $p$ an... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 26074,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the maximum possible value of \\[\\frac{\\left(x^2+5x+12\\right)\\left(x^2+5x-12\\right)\\left(x^2-5x+12\\right)\\left(-x^2+5x+12\\right)}{x^4}\\] over all non-zero real numbers $x$ .\n\n*2019 CCA Math Bonanza Lightning Round #3.4* Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "576"
} | {
"split": "train",
"index": 26075,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from $1$ to $15$ in clockwise order. Committee rules state that a M... | math | {
"style": "rule",
"ground_truth": "346"
} | {
"split": "train",
"index": 26076,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(n)$ be a function that, given an integer $n$, returns an integer $k$, where $k$ is the smallest possible integer such that $k!$ is divisible by $n$. Given that $n$ is a multiple of 15, what is the smallest value of $n$ such that $f(n) > 15$? Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "n = 255"
} | {
"split": "train",
"index": 26077,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the left focus $F$ of the ellipse $C$: $$\\frac {x^{2}}{a^{2}}$$+ $$\\frac {y^{2}}{b^{2}}$$\\=1 ($a>b>0$), a line is drawn through the upper endpoint $B$ of $C$ and intersects the ellipse at another point $A$. If $|BF|=3|AF|$, find the eccentricity of $C$. Let's think ste... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 26078,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle inscribed in a square has two chords as shown in a pair. It has radius 2, and $P$ bisects $T U$. The chords' intersection is where? Answer the question by giving the distance of the point of intersection from the center of the circle. Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2} - 2"
} | {
"split": "train",
"index": 26079,
"solution": "The point lies between $X$ and $Q$. Then $M N X Q$ is a parallelogram. For, $O B \\| N M$ by homothety at $C$ and $P M \\| N X$ because $M N X P$ is an isoceles trapezoid. It follows that $Q X=M N$. Considering that the center of the circle together with points $M,... |
deepscaler | [
{
"role": "user",
"content": "The polynomial $f(z)=az^{2018}+bz^{2017}+cz^{2016}$ has real coefficients not exceeding $2019$, and $f\\left(\\tfrac{1+\\sqrt{3}i}{2}\\right)=2015+2019\\sqrt{3}i$. Find the remainder when $f(1)$ is divided by $1000$. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "53"
} | {
"split": "train",
"index": 26080,
"solution": "We have $\\frac{1+\\sqrt{3}i}{2} = \\omega$ where $\\omega = e^{\\frac{i\\pi}{3}}$ is a primitive 6th root of unity. Then we have\n\\begin{align*} f(\\omega) &= a\\omega^{2018} + b\\omega^{2017} + c\\omega^{2016}\\\\ &= a\\omega^2 + b\\omega + c \\end{align*}\n We ... |
deepscaler | [
{
"role": "user",
"content": "Find the number of ordered triples $(x,y,z)$ of real numbers such that $x + y = 2$ and $xy - z^2 = 1.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 26081,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the radius of the circle inscribed in triangle $ABC$ if $AB = 5, AC=6, BC=7$? 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": "\\frac{2\\sqrt{6}}{3}"
} | {
"split": "train",
"index": 26082,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $10$ points in the space such that each $4$ points are not lie on a plane. Connect some points with some segments such that there are no triangles or quadrangles. Find the maximum number of the segments. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 26083,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest integer that must divide the product of any $5$ consecutive integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 26084,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of ways to arrange the letters of the word COMBINATION. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4,\\!989,\\!600"
} | {
"split": "train",
"index": 26085,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\( a, b, c \\) are real numbers such that \\( |a-b|=1 \\), \\( |b-c|=1 \\), \\( |c-a|=2 \\) and \\( abc=60 \\), calculate the value of \\( \\frac{a}{bc} + \\frac{b}{ca} + \\frac{c}{ab} - \\frac{1}{a} - \\frac{1}{b} - \\frac{1}{c} \\). Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{10}"
} | {
"split": "train",
"index": 26086,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Factor $36-9x^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9(2-x)(2+x)"
} | {
"split": "train",
"index": 26087,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the random variable $X$ follows a normal distribution $N(-1, \\sigma^2)$, and $P(-3 \\leq X \\leq -1) = 0.4$, calculate the probability of $X$ being greater than or equal to $1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.1"
} | {
"split": "train",
"index": 26088,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Count the number of functions $f: \\mathbb{Z} \\rightarrow\\{$ 'green', 'blue' $\\}$ such that $f(x)=f(x+22)$ for all integers $x$ and there does not exist an integer $y$ with $f(y)=f(y+2)=$ 'green'. Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "39601"
} | {
"split": "train",
"index": 26089,
"solution": "It is clear that $f$ is determined by $f(0), \\ldots, f(21)$. The colors of the 11 even integers are independent of those of the odd integers because evens and odds are never exactly 2 apart. First, we count the number of ways to 'color' the even integers. $f(0)$ c... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\{a\\_n\\}$ is a geometric sequence, $a\\_2=2$, $a\\_6=162$, find $a\\_{10}$ = $\\_\\_\\_\\_\\_\\_$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13122"
} | {
"split": "train",
"index": 26090,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The distances from a certain point inside a regular hexagon to three of its consecutive vertices are 1, 1, and 2, respectively. What is the side length of this hexagon? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 26091,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ten digits 0, 1, 2, 3, …, 9 and the imaginary unit i, determine the total number of distinct imaginary numbers that can be formed. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 26092,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In Idaho, 472 people were asked what they call soft drinks. The results of the survey are shown in the pie chart. The central angle of the ``Pop'' sector of the graph is $251^\\circ$, to the nearest whole degree. How many of the people surveyed chose ``Pop''? Express your answe... | math | {
"style": "rule",
"ground_truth": "329"
} | {
"split": "train",
"index": 26093,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mark has 75% more pencils than John, and Luke has 50% more pencils than John. Find the percentage relationship between the number of pencils that Mark and Luke have. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16.67\\%"
} | {
"split": "train",
"index": 26094,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$, respectively.\n(1) If $A$, $B$, $C$ form an arithmetic sequence, find the range of $\\cos A + \\cos C$.\n(2) If $a$, $b$, $c$ form a geometric sequence, and $\\cos B = \\frac{4}{5}$, find the val... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"split": "train",
"index": 26095,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an arithmetic sequence $\\{a_n\\}$, $a_{10} < 0$, $a_{11} > 0$, and $a_{11} > |a_{10}|$. The maximum negative value of the partial sum $S_n$ of the first $n$ terms of the sequence $\\{a_n\\}$ is the sum of the first ______ terms. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 26096,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the function $f(x)$ is an even function with a period of $2$, and when $x \\in (0,1)$, $f(x) = 2^x - 1$, find the value of $f(\\log_{2}{12})$. 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": 26097,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( ABC \\) be a right triangle with the hypotenuse \\( BC \\) measuring \\( 4 \\) cm. The tangent at \\( A \\) to the circumcircle of \\( ABC \\) meets the line \\( BC \\) at point \\( D \\). Suppose \\( BA = BD \\). Let \\( S \\) be the area of triangle \\( ACD \\), expre... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 26098,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the maximum number of colours that can be used to paint an $8 \\times 8$ chessboard so that every square is painted in a single colour, and is adjacent , horizontally, vertically but not diagonally, to at least two other squares of its own colour?\n\n(A Shapovalov) Let'... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 26099,
"solution": "\nTo find the maximum number of colors that can be used to paint an \\(8 \\times 8\\) chessboard such that each square is adjacent (horizontally or vertically) to at least two other squares of its own color, we need to carefully analyze and construct a feasible col... |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ and $c$ be complex numbers such that $|a| = |b| = |c| = 1$ and\n\\[\\frac{a^2}{bc} + \\frac{b^2}{ac} + \\frac{c^2}{ab} = -1.\\]Find all possible values of $|a + b + c|.$\n\nEnter all the possible values, separated by commas. Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "1,2"
} | {
"split": "train",
"index": 26100,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bored student walks down a hall that contains a row of closed lockers, numbered $1$ to $1024$. He opens the locker numbered 1, and then alternates between skipping and opening each locker thereafter. When he reaches the end of the hall, the student turns around and starts bac... | math | {
"style": "rule",
"ground_truth": "342"
} | {
"split": "train",
"index": 26101,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a sports league, each team uses a set of at most $t$ signature colors. A set $S$ of teams is[i] color-identifiable[/i] if one can assign each team in $S$ one of their signature colors, such that no team in $S$ is assigned any signature color of a different team in $S$.\n\nFo... | math | {
"style": "rule",
"ground_truth": "\\lceil \\frac{n}{t} \\rceil"
} | {
"split": "train",
"index": 26102,
"solution": "\nIn a sports league, each team uses a set of at most \\( t \\) signature colors. A set \\( S \\) of teams is color-identifiable if one can assign each team in \\( S \\) one of their signature colors, such that no team in \\( S \\) is assigned any signature color o... |
deepscaler | [
{
"role": "user",
"content": "You have a whole cake in your pantry. On your first trip to the pantry, you eat one-third of the cake. On each successive trip, you eat one-third of the remaining cake. After four trips to the pantry, what fractional part of the cake have you eaten? Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "\\frac{40}{81}"
} | {
"split": "train",
"index": 26103,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Fill in the blanks with appropriate numbers. \n6.8 + 4.1 + __ = 12 __ + 6.2 + 7.6 = 20 19.9 - __ - 5.6 = 10 Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4.3"
} | {
"split": "train",
"index": 26104,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Patrícia wrote, in ascending order, the positive integers formed only by odd digits: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 31, 33, ... What was the 157th number she wrote? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1113"
} | {
"split": "train",
"index": 26105,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two right triangles share a side as follows: [asy]\npair pA, pB, pC, pD, pE;\npA = (0, 0);\npB = pA + 6 * dir(0);\npC = pA + 10 * dir(90);\npD = pB + 6 * dir(90);\npE = (6 * pA + 10 * pD) / 16;\ndraw(pA--pB--pC--pA);\ndraw(pA--pB--pD--pA);\nlabel(\"$A$\", pA, SW);\nlabel(\"$B$\... | math | {
"style": "rule",
"ground_truth": "\\frac{75}{4}"
} | {
"split": "train",
"index": 26106,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Marisa has a collection of $2^{8}-1=255$ distinct nonempty subsets of $\\{1,2,3,4,5,6,7,8\\}$. For each step she takes two subsets chosen uniformly at random from the collection, and replaces them with either their union or their intersection, chosen randomly with equal probabi... | math | {
"style": "rule",
"ground_truth": "\\frac{1024}{255}"
} | {
"split": "train",
"index": 26107,
"solution": "It suffices to compute the probability of each number appearing in the final subset. For any given integer $n \\in[1,8]$, there are $2^{7}=128$ subsets with $n$ and $2^{7}-1=127$ without. When we focus on only this element, each operation is equivalent to taking tw... |
deepscaler | [
{
"role": "user",
"content": "In rectangle $ABCD$, $AB = 10$ cm, $BC = 14$ cm, and $DE = DF$. The area of triangle $DEF$ is one-fifth the area of rectangle $ABCD$. What is the length in centimeters of segment $EF$? Express your answer in simplest radical form. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{7}"
} | {
"split": "train",
"index": 26108,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An electric company installed a total of 402 poles along both sides of the road, with a distance of 20 meters between each adjacent pole. Later, all poles were replaced, and only 202 poles were installed. The distance between each adjacent pole after the replacement is $\\qquad... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 26109,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $5^n$ inclusive. Find the smallest positive integer $n$ for which $T_n$ is an integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "504"
} | {
"split": "train",
"index": 26110,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has right angle at $B$, and contains a point $P$ for which $PA = 10$, $PB = 6$, and $\\angle APB = \\angle BPC = \\angle CPA$. Find $PC$.\n[asy] unitsize(0.2 cm); pair A, B, C, P; A = (0,14); B = (0,0); C = (21*sqrt(3),0); P = intersectionpoint(arc(B,6,0,180),arc... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 26111,
"solution": "Let $PC = x$. Since $\\angle APB = \\angle BPC = \\angle CPA$, each of them is equal to $120^\\circ$. By the Law of Cosines applied to triangles $\\triangle APB$, $\\triangle BPC$ and $\\triangle CPA$ at their respective angles $P$, remembering that $\\cos 120^\\ci... |
deepscaler | [
{
"role": "user",
"content": "For the power function $y = (m^2 - m - 1)x^{2m+1}$, if it is a decreasing function for $x \\in (0, +\\infty)$, then the value of the real number $m$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 26112,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $a_1,$ $a_2,$ $a_3,$ $\\dots$ satisfies $a_1 = 19,$ $a_9 = 99,$ and for all $n \\ge 3,$ $a_n$ is the arithmetic mean of the first $n - 1$ terms. Find $a_2.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "179"
} | {
"split": "train",
"index": 26113,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\frac{x}{3} = y^2$ and $\\frac{x}{6} = 3y$, solve for the value of $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 26114,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a convex quadrilateral such that $AB + BC = 2021$ and $AD = CD$ . We are also given that $\\angle ABC = \\angle CDA = 90^o$ . Determine the length of the diagonal $BD$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2021 \\sqrt{2}}{2}"
} | {
"split": "train",
"index": 26115,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among the following functions, identify which pairs represent the same function.\n\n1. $f(x) = |x|, g(x) = \\sqrt{x^2}$;\n2. $f(x) = \\sqrt{x^2}, g(x) = (\\sqrt{x})^2$;\n3. $f(x) = \\frac{x^2 - 1}{x - 1}, g(x) = x + 1$;\n4. $f(x) = \\sqrt{x + 1} \\cdot \\sqrt{x - 1}, g(x) = \\s... | math | {
"style": "rule",
"ground_truth": "(1)"
} | {
"split": "train",
"index": 26116,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Carl has 5 cubes each having side length 1, and Kate has 5 cubes each having side length 2. What is the total volume of these 10 cubes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 26117,
"solution": "1. **Calculate the volume of one of Carl's cubes**: \n The volume $V$ of a cube with side length $s$ is given by the formula:\n \\[\n V = s^3\n \\]\n For Carl's cubes, each has a side length of $1$. Therefore, the volume of one cube is:\n \\[\n V = 1^... |
deepscaler | [
{
"role": "user",
"content": "Given the functions $f(x)=x^{2}+px+q$ and $g(x)=x+\\frac{1}{x^{2}}$ on the interval $[1,2]$, determine the maximum value of $f(x)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4 - \\frac{5}{2} \\sqrt[3]{2} + \\sqrt[3]{4}"
} | {
"split": "train",
"index": 26118,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a rectangular grid comprising 5 rows and 4 columns of squares, how many different rectangles can be traced using the lines in the grid? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 26119,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A running competition on an unpredictable distance is conducted as follows. On a circular track with a length of 1 kilometer, two points \\( A \\) and \\( B \\) are randomly selected (using a spinning arrow). The athletes then run from \\( A \\) to \\( B \\) along the shorter a... | math | {
"style": "rule",
"ground_truth": "0.25"
} | {
"split": "train",
"index": 26120,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the Riemann function defined on the interval $\\left[0,1\\right]$ is: $R\\left(x\\right)=\\left\\{\\begin{array}{l}{\\frac{1}{q}, \\text{when } x=\\frac{p}{q} \\text{(p, q are positive integers, } \\frac{p}{q} \\text{ is a reduced proper fraction)}}\\\\{0, \\text{whe... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"split": "train",
"index": 26121,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a$ is a positive real number and $b$ is an integer between $1$ and $201$, inclusive, find the number of ordered pairs $(a,b)$ such that $(\\log_b a)^{2023}=\\log_b(a^{2023})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "603"
} | {
"split": "train",
"index": 26122,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the first 1000 positive integers can be expressed in the form\n$\\lfloor 2x \\rfloor + \\lfloor 4x \\rfloor + \\lfloor 6x \\rfloor + \\lfloor 8x \\rfloor$,\n where $x$ is a real number, and $\\lfloor z \\rfloor$ denotes the greatest integer less than or equal to $z$... | math | {
"style": "rule",
"ground_truth": "600"
} | {
"split": "train",
"index": 26123,
"solution": "Because $2,4,6,8$ are all multiples of $2$, we can speed things up. We only need to check up to $\\frac{12}{24}$, and the rest should repeat. As shown before, we hit 6 integers ($1,2,4,5,6,10$) from $\\frac{1}{24}$ to $\\frac{12}{24}$. Similarly, this should repeat... |
deepscaler | [
{
"role": "user",
"content": "Compute $\\sin(-30^\\circ)$ and verify by finding $\\cos(-30^\\circ)$, noticing the relationship, and confirming with the unit circle properties. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 26124,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_1,$ $a_2,$ $\\dots,$ $a_{12}$ be positive real numbers such that $a_1 + a_2 + \\dots + a_{12} = 1.$ Find the minimum value of\n\\[\\frac{1}{a_1} + \\frac{1}{a_2} + \\dots + \\frac{1}{a_{12}}.\\] Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 26125,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a three-digit number \\( A \\). By placing a decimal point in front of one of its digits, we get a number \\( B \\). If \\( A - B = 478.8 \\), find \\( A \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "532"
} | {
"split": "train",
"index": 26126,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\Delta A_{1} B_{1} C$ be a triangle with $\\angle A_{1} B_{1} C=90^{\\circ}$ and $\\frac{C A_{1}}{C B_{1}}=\\sqrt{5}+2$. For any $i \\geq 2$, define $A_{i}$ to be the point on the line $A_{1} C$ such that $A_{i} B_{i-1} \\perp A_{1} C$ and define $B_{i}$ to be the point o... | math | {
"style": "rule",
"ground_truth": "4030"
} | {
"split": "train",
"index": 26127,
"solution": "We claim that $\\Gamma_{2}$ is the incircle of $\\triangle B_{1} A_{2} C$. This is because $\\triangle B_{1} A_{2} C$ is similar to $A_{1} B_{1} C$ with dilation factor $\\sqrt{5}-2$, and by simple trigonometry, one can prove that $\\Gamma_{2}$ is similar to $\\Gam... |
deepscaler | [
{
"role": "user",
"content": "What is the least common multiple of 3, 4, 6 and 15? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 26128,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Every evening, Laura gathers three socks randomly to pack for her gym session next morning. She has 12 black socks, 10 white socks, and 6 striped socks in her drawer. What is the probability that all three socks she picks are of the same type? Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "\\frac{60}{546}"
} | {
"split": "train",
"index": 26129,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find at the point \\( M_{0}(1,1,1) \\) the direction of the greatest change of the scalar field \\( u = xy + yz + xz \\) and the magnitude of this greatest change at this point. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 26130,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a hyperbola with left and right foci at $F_1$ and $F_2$ respectively, a chord $AB$ on the left branch passing through $F_1$ with a length of 5. If $2a=8$, determine the perimeter of $\\triangle ABF_2$. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 26131,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=kx+b$, whose graph intersects the $x$ and $y$ axes at points A and B respectively, with $\\overrightarrow{AB}=2\\overrightarrow{i}+2\\overrightarrow{j}$ ($\\overrightarrow{i}$, $\\overrightarrow{j}$ are unit vectors in the positive direction of the $x$ ... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 26132,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 26133,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $x$ that satisfies $\\frac{\\sqrt{5x}}{\\sqrt{3(x-1)}}=2$. Express your answer in simplest fractional form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{12}{7}"
} | {
"split": "train",
"index": 26134,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two numbers are independently selected from the set of positive integers less than or equal to 6. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 26135,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest integer value of $x$ such that $\\frac{x^2 + 2x + 5}{x-3}$ is an integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 26136,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Initially, there are 33 ones written on the board. Every minute, Karlson erases any two numbers and writes their sum on the board, then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he can eat in 33 minutes? Let's... | math | {
"style": "rule",
"ground_truth": "528"
} | {
"split": "train",
"index": 26137,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\sin \\alpha + \\cos \\alpha = \\frac{1}{5}$, and $- \\frac{\\pi}{2} \\leqslant \\alpha \\leqslant \\frac{\\pi}{2}$, find the value of $\\tan \\alpha$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{3}{4}"
} | {
"split": "train",
"index": 26138,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In tetrahedron \\(ABCD\\), it is known that \\(\\angle ADB = \\angle BDC = \\angle CDA = 60^\\circ\\), \\(AD = BD = 3\\), and \\(CD = 2\\). Find the radius of the circumscribed sphere of tetrahedron \\(ABCD\\). Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 26139,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The integer \\( n \\) has a total of 10 divisors. These divisors are arranged in ascending order, and the 8th divisor is \\( \\frac{n}{3} \\). Find the maximum value of the integer \\( n \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "162"
} | {
"split": "train",
"index": 26140,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The real numbers $a$, $b$, and $c$ satisfy the equation $({a}^{2}+\\frac{{b}^{2}}{4}+\\frac{{c}^{2}}{9}=1)$. Find the maximum value of $a+b+c$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{14}"
} | {
"split": "train",
"index": 26141,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Select 5 elements from the set $\\{x|1\\leq x \\leq 11, \\text{ and } x \\in \\mathbb{N}^*\\}$ to form a subset of this set, and any two elements in this subset do not sum up to 12. How many different subsets like this are there? (Answer with a number). Let's think step by step... | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 26142,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I have two 10-sided dice where each die has 3 gold sides, 4 silver sides, 2 diamond sides, and 1 rainbow side. If I roll both dice, what is the probability that they come up showing the same color or pattern? Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"split": "train",
"index": 26143,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x>y>0$ , then $\\frac{x^y y^x}{y^y x^x}=$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "{\\left(\\frac{x}{y}\\right)}^{y-x}"
} | {
"split": "train",
"index": 26144,
"solution": "1. Start by rewriting the given expression:\n \\[\n \\frac{x^y y^x}{y^y x^x}\n \\]\n This can be factored into two separate fractions:\n \\[\n \\frac{x^y}{y^y} \\times \\frac{y^x}{x^x}\n \\]\n\n2. Simplify each fraction:\n - For the first fraction, ... |
deepscaler | [
{
"role": "user",
"content": "Suppose a point $P$ has coordinates $(m, n)$, where $m$ and $n$ are the numbers obtained by rolling two dice consecutively. The probability that point $P$ lies outside the circle $x^{2}+y^{2}=16$ is $\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{9}"
} | {
"split": "train",
"index": 26145,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$1000 \\times 1993 \\times 0.1993 \\times 10 =$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(1993)^2"
} | {
"split": "train",
"index": 26146,
"solution": "1. **Break down the expression**: We start by simplifying the expression $1000 \\times 1993 \\times 0.1993 \\times 10$. We can rearrange the terms for easier computation:\n \\[\n 1000 \\times 10 \\times 1993 \\times 0.1993\n \\]\n\n2. **Simplify the powers of... |
deepscaler | [
{
"role": "user",
"content": "The constant term in the expansion of the binomial $\\left(\\frac{1}{\\sqrt{x}} - x^2\\right)^{10}$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 26147,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the number of zeros in the integer equal to $(10^{100}) \times (100^{10})$ is sought, what is this number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 26148,
"solution": "Since $100=10^{2}$, then $100^{10}=(10^{2})^{10}=10^{20}$. Therefore, $(10^{100}) \times (100^{10})=(10^{100}) \times (10^{20})=10^{120}$. When written out, this integer consists of a 1 followed by 120 zeros."
} |
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