data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Extend the definition of the binomial coefficient to $C_x^m = \\frac{x(x-1)\\dots(x-m+1)}{m!}$ where $x\\in\\mathbb{R}$ and $m$ is a positive integer, with $C_x^0=1$. This is a generalization of the binomial coefficient $C_n^m$ (where $n$ and $m$ are positive integers and $m\\l... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 18849,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ ABC$ be a triangle with $ AB \\equal{} AC$ . The angle bisectors of $ \\angle C AB$ and $ \\angle AB C$ meet the sides $ B C$ and $ C A$ at $ D$ and $ E$ , respectively. Let $ K$ be the incentre of triangle $ ADC$. Suppose that $ \\angle B E K \\equal{} 45^\\circ$ . Find ... | math | {
"style": "rule",
"ground_truth": "60^\\circ \\text{ and } 90^\\circ"
} | {
"split": "train",
"index": 18850,
"solution": "\nGiven a triangle \\( ABC \\) with \\( AB = AC \\) (isosceles triangle), we are tasked with finding all possible values of \\( \\angle CAB \\) given the specific geometric conditions.\n\n### Problem Setup\n\n1. **Notation and Known Values:**\n - Let \\( ABC \\) ... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\alpha$ and $\\beta$ are acute angles, and $\\cos(\\alpha + \\beta) = \\frac{3}{5}$, $\\sin\\alpha = \\frac{5}{13}$, find the value of $\\cos\\beta$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{56}{65}"
} | {
"split": "train",
"index": 18851,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a $\\text{good~word}$ as a sequence of letters that consists only of the letters $A$, $B$, and $C$ - some of these letters may not appear in the sequence - and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, and $C$ is never i... | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 18852,
"solution": "There are three choices for the first letter and two choices for each subsequent letter, so there are $3\\cdot2^{n-1}\\ n$-letter good words. Substitute $n=7$ to find there are $3\\cdot2^6=\\boxed{192}$ seven-letter good words. ~ aopsav (Credit to AoPS Alcumus)"
} |
deepscaler | [
{
"role": "user",
"content": "A point $P$ is chosen uniformly at random inside a square of side length 2. If $P_{1}, P_{2}, P_{3}$, and $P_{4}$ are the reflections of $P$ over each of the four sides of the square, find the expected value of the area of quadrilateral $P_{1} P_{2} P_{3} P_{4}$. Let's think st... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 18853,
"solution": "Let $A B C D$ denote the square defined in the problem. We see that if $P_{1}$ is the reflection of $P$ over $\\overline{A B}$, then the area of $P_{1} A B$ is the same as the area of $P A B$. Furthermore, if $P_{4}$ is the reflection of $P$ over $\\overline{D A}, ... |
deepscaler | [
{
"role": "user",
"content": "Convex quadrilateral $A B C D$ has right angles $\\angle A$ and $\\angle C$ and is such that $A B=B C$ and $A D=C D$. The diagonals $A C$ and $B D$ intersect at point $M$. Points $P$ and $Q$ lie on the circumcircle of triangle $A M B$ and segment $C D$, respectively, such that ... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 18854,
"solution": "Note that $m \\angle Q P B=m \\angle M P B=m \\angle M A B=m \\angle C A B=\\angle B C A=\\angle C D B$. Thus, $M P \\cdot M Q=M B \\cdot M D$. On the other hand, segment $C M$ is an altitude of right triangle $B C D$, so $M B \\cdot M D=M C^{2}=36$."
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x) = 2$ for all real numbers $x$, what is the value of $f(x + 2)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 18855,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be positive real numbers such that $3a^2 + 2b^2 = 3a + 2b$. Find the minimum value of $A =\\sqrt{\\frac{a}{b(3a+2)}} + \\sqrt{\\frac{b}{a(2b+3)}} $ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{\\sqrt{5}}"
} | {
"split": "train",
"index": 18856,
"solution": "\nLet \\( a \\) and \\( b \\) be positive real numbers such that:\n\n\\[\n3a^2 + 2b^2 = 3a + 2b.\n\\]\n\nWe aim to find the minimum value of:\n\n\\[\nA = \\sqrt{\\frac{a}{b(3a+2)}} + \\sqrt{\\frac{b}{a(2b+3)}}.\n\\]\n\nFirst, observe the given equality:\n\n\\[\n3a^... |
deepscaler | [
{
"role": "user",
"content": "The sum of two numbers is $S$. Suppose $3$ is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2S + 12"
} | {
"split": "train",
"index": 18857,
"solution": "1. Let the two numbers be $a$ and $b$. According to the problem, the sum of these two numbers is $S$. Therefore, we have:\n \\[\n a + b = S\n \\]\n\n2. According to the problem, $3$ is added to each number. Thus, the new numbers become $a+3$ and $b+3$.\n\n3. ... |
deepscaler | [
{
"role": "user",
"content": "A right triangle has perimeter $2008$ , and the area of a circle inscribed in the triangle is $100\\pi^3$ . Let $A$ be the area of the triangle. Compute $\\lfloor A\\rfloor$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31541"
} | {
"split": "train",
"index": 18858,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A shopper buys a $100$ dollar coat on sale for $20\\%$ off. An additional $5$ dollars are taken off the sale price by using a discount coupon. A sales tax of $8\\%$ is paid on the final selling price. The total amount the shopper pays for the coat is Let's think step by step... | math | {
"style": "rule",
"ground_truth": "81.00"
} | {
"split": "train",
"index": 18859,
"solution": "1. **Calculate the price after the initial discount**: \n The original price of the coat is $100$ dollars. The discount offered is $20\\%$. \n The price after the discount is calculated as follows:\n \\[\n 100 \\times (1 - 0.20) = 100 \\times 0.80 = 80 \\... |
deepscaler | [
{
"role": "user",
"content": "Find the arithmetic mean of the prime numbers in this list: 21, 23, 25, 27, 29 Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 18860,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the greatest common divisor of $8!$ and $(6!)^3.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11520"
} | {
"split": "train",
"index": 18861,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the quadratic function $f(x)=ax^{2}+(2b+1)x-a-2 (a,b \\in R, a \\neq 0)$ has at least one root in the interval $[3,4]$, calculate the minimum value of $a^{2}+b^{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{100}"
} | {
"split": "train",
"index": 18862,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Eight positive integers are written on the faces of a cube. Each vertex is labeled with the product of the three numbers on the faces adjacent to that vertex. If the sum of the numbers on the vertices is equal to $2107$, what is the sum of the numbers written on the faces? Let'... | math | {
"style": "rule",
"ground_truth": "57"
} | {
"split": "train",
"index": 18863,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer \\(n\\) such that \\(\\frac{n}{n+75}\\) is equal to a terminating decimal? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 18864,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A chocolate bar consists of \\(5 \\times 8\\) square pieces. The bar is broken along lines dividing the pieces until there are 40 individual pieces. How many times must the bar be broken? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 18865,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In square \\(ABCD\\) with a side length of 10, points \\(P\\) and \\(Q\\) lie on the segment joining the midpoints of sides \\(AD\\) and \\(BC\\). Connecting \\(PA\\), \\(PC\\), \\(QA\\), and \\(QC\\) divides the square into three regions of equal area. Find the length of segme... | math | {
"style": "rule",
"ground_truth": "20/3"
} | {
"split": "train",
"index": 18866,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $11^5\\equiv n\\pmod 9$, where $0\\le n<9$.\n\nWhat is the value of $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 18867,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a cubic planet, cubic mice live only on the faces of the cube and not on the edges or vertices. The number of mice on different faces is different and on any two neighboring faces, this number differs by at least 2. What is the minimum number of cubic mice that can live on t... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 18868,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Numbers $m$ and $n$ are on the number line. What is the value of $n-m$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "55"
} | {
"split": "train",
"index": 18869,
"solution": "On a number line, the markings are evenly spaced. Since there are 6 spaces between 0 and 30, each space represents a change of $\\frac{30}{6}=5$. Since $n$ is 2 spaces to the right of 60, then $n=60+2 \\times 5=70$. Since $m$ is 3 spaces to the left of 30, then $m=... |
deepscaler | [
{
"role": "user",
"content": "The distance between Luga and Volkhov is 194 km, between Volkhov and Lodeynoe Pole is 116 km, between Lodeynoe Pole and Pskov is 451 km, and between Pskov and Luga is 141 km. What is the distance between Pskov and Volkhov? Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "335"
} | {
"split": "train",
"index": 18870,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two boys start moving from the same point A on a circular track but in opposite directions. Their speeds are 5 ft. per second and 9 ft. per second. If they start at the same time and finish when they first meet at the point A again, then the number of times they meet, excluding... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 18871,
"solution": "1. **Define the problem setup**: Two boys start at the same point A on a circular track and move in opposite directions with speeds of 5 ft/s and 9 ft/s respectively. We need to find the number of times they meet excluding the start and finish.\n\n2. **Calculate re... |
deepscaler | [
{
"role": "user",
"content": "Let $C = (3, 5)$ be the midpoint of $AB,$ where $A = (1, 8)$ and $B = (x, y).$ What is the value of $xy$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 18872,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers less than $201$ are multiples of either $6$ or $8$, but not both at once? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 18873,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$r$ and $s$ are integers such that $3 r \\geq 2 s-3 \\text { and } 4 s \\geq r+12$. What is the smallest possible value of $r / s$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 18874,
"solution": "We simply plot the two inequalities in the $s r$-plane and find the lattice point satisfying both inequalities such that the slope from it to the origin is as low as possible. We find that this point is $(2,4)$ (or $(3,6))$, as circled in the figure, so the answer ... |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of the first $n$ terms of the sequence $\\{a_n\\}$ is $S_n=\\ln (1+ \\frac {1}{n})$, find the value of $e^{a_7+a_8+a_9}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {20}{21}"
} | {
"split": "train",
"index": 18875,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $AB$ be a diameter of a circle centered at $O$. Let $E$ be a point on the circle, and let the tangent at $B$ intersect the tangent at $E$ and $AE$ at $C$ and $D$, respectively. If $\\angle BAE = 43^\\circ$, find $\\angle CED$, in degrees.\n\n[asy]\nimport graph;\n\nunitsi... | math | {
"style": "rule",
"ground_truth": "47^\\circ"
} | {
"split": "train",
"index": 18876,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The cube of $a$ and the fourth root of $b$ vary inversely. If $a=3$ when $b=256$, then find $b$ when $ab=81$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 18877,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers from 1 to 1000 are written in a circle. Starting from the first one, every 15th number (i.e., the numbers 1, 16, 31, etc.) is crossed out, and during subsequent rounds, already crossed-out numbers are also taken into account. The crossing out continues until it turn... | math | {
"style": "rule",
"ground_truth": "800"
} | {
"split": "train",
"index": 18878,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "120 schools each send 20 people to form 20 teams, with each team having exactly 1 person from each school. Find the smallest positive integer \\( k \\) such that when \\( k \\) people are selected from each team, there will be at least 20 people from the same school among all t... | math | {
"style": "rule",
"ground_truth": "115"
} | {
"split": "train",
"index": 18879,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive divisors do 8400 and 7560 have in common? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 18880,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the two-digit integer $MM$, where both digits are equal, when multiplied by the one-digit integer $K$ (different from $M$ and only $1\\leq K \\leq 9$), it results in a three-digit number $NPK$. Identify the digit pairs $(M, K)$ that yield the highest value of $NPK$. Let's... | math | {
"style": "rule",
"ground_truth": "891"
} | {
"split": "train",
"index": 18881,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The following bar graph represents the length (in letters) of the names of 19 people. What is the median length of these names? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 18882,
"solution": "\nTo find the median length of the names represented in the bar graph, we need to determine the position of the median in a sorted list of the name lengths. Since there are 19 names, the median will be the length of the 10th name when the names are arranged in incr... |
deepscaler | [
{
"role": "user",
"content": "Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2009,0),(2009,2010),$ and $(0,2010)$. What is the probability that $x > 7y$? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "\\frac{287}{4020}"
} | {
"split": "train",
"index": 18883,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two of the altitudes of an acute triangle divide the sides into segments of lengths $5,3,2$ and $x$ units, as shown. What is the value of $x$? [asy]\ndefaultpen(linewidth(0.7)); size(75);\npair A = (0,0);\npair B = (1,0);\npair C = (74/136,119/136);\npair D = foot(B, A, C);\npa... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 18884,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given positive integers $x$ and $y$ such that $x\\neq y$ and $\\frac{1}{x} + \\frac{1}{y} = \\frac{1}{15}$, what is the smallest possible value for $x + y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 18885,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The $10\\times15$ rectangle $EFGH$ is cut into two congruent pentagons, which are repositioned to form a square. Determine the length $z$ of one side of the pentagons that aligns with one side of the square.\nA) $5\\sqrt{2}$ \nB) $5\\sqrt{3}$ \nC) $10\\sqrt{2}$ \nD) $10\\s... | math | {
"style": "rule",
"ground_truth": "5\\sqrt{3}"
} | {
"split": "train",
"index": 18886,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "China has become the world's largest electric vehicle market. Electric vehicles have significant advantages over traditional vehicles in ensuring energy security and improving air quality. After comparing a certain electric vehicle with a certain fuel vehicle, it was found that... | math | {
"style": "rule",
"ground_truth": "5000"
} | {
"split": "train",
"index": 18887,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f(x)=(m^{2}-m-1)x^{m^{2}+m-1}$ is a power function, and it is decreasing on $(0,+\\infty)$. Find the real number $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 18888,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( x, y, z \\) be positive integers such that\n\\[ (x+y)(y+z) = 2016 \\]\n\\[ (x+y)(z+x) = 1080 \\]\nDetermine the smallest possible value for \\( x+y+z \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "61"
} | {
"split": "train",
"index": 18889,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The polynomial \\( f(x) \\) satisfies the equation \\( f(x) - f(x-2) = (2x-1)^{2} \\) for all \\( x \\). Find the sum of the coefficients of \\( x^{2} \\) and \\( x \\) in \\( f(x) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"split": "train",
"index": 18890,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The value of \\(2 \\frac{1}{10} + 3 \\frac{11}{100} + 4 \\frac{111}{1000}\\) is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9.321"
} | {
"split": "train",
"index": 18891,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A number is considered a visible factor number if it is divisible by each of its non-zero digits. For example, 204 is divisible by 2 and 4 and is therefore a visible factor number. Determine how many visible factor numbers exist from 200 to 250, inclusive. Let's think step by s... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 18892,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the sequence of non-negative integers $\\left\\{a_{n}\\right\\}$ satisfy:\n$$\na_{n} \\leqslant n \\quad (n \\geqslant 1), \\quad \\text{and} \\quad \\sum_{k=1}^{n-1} \\cos \\frac{\\pi a_{k}}{n} = 0 \\quad (n \\geqslant 2).\n$$\n\nFind all possible values of $a_{2021}$. Let... | math | {
"style": "rule",
"ground_truth": "2021"
} | {
"split": "train",
"index": 18893,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\(\\frac{\\left(\\frac{a}{c}+\\frac{a}{b}+1\\right)}{\\left(\\frac{b}{a}+\\frac{b}{c}+1\\right)}=11\\), where \\(a, b\\), and \\(c\\) are positive integers, find the number of different ordered triples \\((a, b, c)\\) such that \\(a+2b+c \\leq 40\\). Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 18894,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In isosceles $\\triangle A B C, A B=A C$ and $P$ is a point on side $B C$. If $\\angle B A P=2 \\angle C A P, B P=\\sqrt{3}$, and $C P=1$, compute $A P$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 18895,
"solution": "Let $\\angle C A P=\\alpha$, By the Law of Sines, $\\frac{\\sqrt{3}}{\\sin 2 \\alpha}=\\frac{1}{\\sin \\alpha}$ which rearranges to $\\cos \\alpha=\\frac{\\sqrt{3}}{2} \\Rightarrow \\alpha=\\frac{\\pi}{6}$. This implies that $\\angle B A C=\\frac{\\pi}{2}$. By the ... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a > b$, $a=5$, $c=6$, and $\\sin B=\\dfrac{3}{5}$.\n\n1. Find the values of $b$ and $\\sin A$.\n2. Find the value of $\\sin (2A+\\dfrac{\\pi}{4})$. Let's think st... | math | {
"style": "rule",
"ground_truth": "\\dfrac {7 \\sqrt {2}}{26}"
} | {
"split": "train",
"index": 18896,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given triangle $ABC$ where $AB=6$, $\\angle A=30^\\circ$, and $\\angle B=120^\\circ$, find the area of $\\triangle ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9\\sqrt{3}"
} | {
"split": "train",
"index": 18897,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $w_1$ and $w_2$ denote the circles $x^2+y^2+10x-24y-87=0$ and $x^2 +y^2-10x-24y+153=0,$ respectively. Let $m$ be the smallest positive value of $a$ for which the line $y=ax$ contains the center of a circle that is externally tangent to $w_2$ and internally tangent to $w_1.$... | math | {
"style": "rule",
"ground_truth": "169"
} | {
"split": "train",
"index": 18898,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( f(x) = x^{9} + x^{8} + x^{7} + x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1 \\). When \\( f(x^{10}) \\) is divided by \\( f(x) \\), the remainder is \\( b \\). Find the value of \\( b \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 18899,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cara is sitting at a circular table with her six friends. How many different sets of two friends can Cara be directly sitting between? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 18900,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Grandma told her grandchildren: \"Today I am 60 years and 50 months and 40 weeks and 30 days old.\"\n\nHow old was Grandma on her last birthday? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 18901,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\tan (\\alpha-\\beta)= \\frac {1}{2}$, $\\tan \\beta=- \\frac {1}{7}$, and $\\alpha$, $\\beta\\in(0,\\pi)$, find the value of $2\\alpha-\\beta$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac {3\\pi}{4}"
} | {
"split": "train",
"index": 18902,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many times in a day is the angle between the hour and minute hands exactly $19^{\\circ}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 18903,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different positive three-digit integers can be formed using only the digits in the set $\\{1, 3, 4, 4, 7, 7, 7\\}$ if no digit may be used more times than it appears in the given set of available digits? Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "43"
} | {
"split": "train",
"index": 18904,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A root of unity is a complex number that is a solution to $z^n = 1$ for some positive integer $n$. Determine the number of roots of unity that are also roots of $z^2 + az + b = 0$ for some integers $a$ and $b$. Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 18905,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In \\(\\triangle PMO\\), \\(PM = 6\\sqrt{3}\\), \\(PO = 12\\sqrt{3}\\), and \\(S\\) is a point on \\(MO\\) such that \\(PS\\) is the angle bisector of \\(\\angle MPO\\). Let \\(T\\) be the reflection of \\(S\\) across \\(PM\\). If \\(PO\\) is parallel to \\(MT\\), find the leng... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{183}"
} | {
"split": "train",
"index": 18906,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)= \\sqrt {3}\\sin x\\cdot\\cos x- \\frac {1}{2}\\cos 2x$ $(x\\in\\mathbb{R})$. \n$(1)$ Find the minimum value and the smallest positive period of the function $f(x)$. \n$(2)$ Let $\\triangle ABC$ have internal angles $A$, $B$, $C$ opposite to sides $a$... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt {3}"
} | {
"split": "train",
"index": 18907,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that 30 balls are put into four boxes A, B, C, and D, such that the sum of the number of balls in A and B is greater than the sum of the number of balls in C and D, find the total number of possible ways. Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "2600"
} | {
"split": "train",
"index": 18908,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $$x^{5}=a_{0}+a_{1}(2-x)+a_{2}(2-x)^{2}+…+a_{5}(2-x)^{5}$$, find the value of $$\\frac {a_{0}+a_{2}+a_{4}}{a_{1}+a_{3}}$$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac {61}{60}"
} | {
"split": "train",
"index": 18909,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a polynomial $(a+b+c+d+e+1)^N$, when expanded and like terms are combined, the expression contains exactly 2002 terms that include all five variables $a, b, c, d, e$, each to some positive power, determine the value of $N$. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 18910,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a sequence of coin tosses, one can keep a record of instances in which a tail is immediately followed by a head, a head is immediately followed by a head, and etc. We denote these by TH, HH, and etc. For example, in the sequence TTTHHTHTTTHHTTH of 15 coin tosses we observe t... | math | {
"style": "rule",
"ground_truth": "560"
} | {
"split": "train",
"index": 18911,
"solution": "Let's consider each of the sequences of two coin tosses as an operation instead; this operation takes a string and adds the next coin toss on (eg, THHTH + HT = THHTHT). We examine what happens to the last coin toss. Adding HH or TT is simply an identity for the las... |
deepscaler | [
{
"role": "user",
"content": "Given a sequence ${a_n}$ defined by $|a_{n+1}|+|a_n|=3$ for any positive integer $n$ and $a_1=2$, find the minimum value of the sum $S_{2019}$ of its first 2019 terms. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3025"
} | {
"split": "train",
"index": 18912,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify\n\\[\\frac{\\tan^3 75^\\circ + \\cot^3 75^\\circ}{\\tan 75^\\circ + \\cot 75^\\circ}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 18913,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define an operation between sets A and B: $A*B = \\{x | x = x_1 + x_2, \\text{ where } x_1 \\in A, x_2 \\in B\\}$. If $A = \\{1, 2, 3\\}$ and $B = \\{1, 2\\}$, then the sum of all elements in $A*B$ is ____. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 18914,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a parabola $C$: $y^{2}=2px (p > 0)$ with focus $F$, directrix $x=-1$, and a point $M$ on the directrix below the $x$-axis. A line $l$ passing through $M$ and $F$ intersects with $C$ at point $N$, and $F$ is the midpoint of segment $MN$.\n\n1. Find the equations of parabol... | math | {
"style": "rule",
"ground_truth": "\\frac {16}{3}"
} | {
"split": "train",
"index": 18915,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Forty slips of paper numbered $1$ to $40$ are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, \"I can't tell who has the larger number.\" Then Bob says, \"I know who has the larger n... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 18916,
"solution": "1. **Alice's Statement Analysis**:\n - Alice says she cannot determine who has the larger number. This implies that Alice's number, $A$, must be such that it is not the smallest number (1) because if Alice had 1, she would know for sure that Bob has a larger numb... |
deepscaler | [
{
"role": "user",
"content": "Six six-sided dice are rolled. We are told there are no four-of-a-kind, but there are two different pairs of dice showing the same numbers. These four dice (two pairs) are set aside, and the other two dice are re-rolled. What is the probability that after re-rolling these two d... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 18917,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a trapezoid with labeled points as shown in the diagram, the point of intersection of the extended non-parallel sides is labeled as \\(E\\), and the point of intersection of the diagonals is labeled as \\(F\\). Similar right triangles \\(BFC\\) and \\(DFA\\) have correspo... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{\\sqrt{3}}"
} | {
"split": "train",
"index": 18918,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest integer divisible by all positive integers less than its cube root? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "420"
} | {
"split": "train",
"index": 18919,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A group of 12 friends decides to form a committee of 5. Calculate the number of different committees that can be formed. Additionally, if there are 4 friends who refuse to work together, how many committees can be formed without any of these 4 friends? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 18920,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When evaluated, the sum of the digits of the integer equal to \\(10^{2021} - 2021\\) is calculated. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18185"
} | {
"split": "train",
"index": 18921,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that an ellipse and a hyperbola $(x^{2}-y^{2}=1)$ share the same foci and the eccentricity is $\\frac{\\sqrt{2}}{2}$.\n(I) Find the standard equation of the ellipse;\n(II) A line passing through point $P(0,1)$ intersects the ellipse at points $A$ and $B$. $O$ is the origi... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{126}}{8}"
} | {
"split": "train",
"index": 18922,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the quadrilateral $ABCD$ , $AB = BC = CD$ and $\\angle BMC = 90^\\circ$ , where $M$ is the midpoint of $AD$ . Determine the acute angle between the lines $AC$ and $BD$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 18923,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two moving points \\( A(x_1, y_1) \\) and \\( B(x_2, y_2) \\) on the parabola \\( y^2 = 6x \\), where \\( x_1 \\neq x_2 \\) and \\( x_1 + x_2 = 4 \\). The perpendicular bisector of segment \\( AB \\) intersects the \\( x \\)-axis at point \\( C \\). Find the maximum area ... | math | {
"style": "rule",
"ground_truth": "\\frac{14}{3} \\sqrt{7}"
} | {
"split": "train",
"index": 18924,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\alpha$ and $\\beta$ be reals. Find the least possible value of $(2 \\cos \\alpha+5 \\sin \\beta-8)^{2}+(2 \\sin \\alpha+5 \\cos \\beta-15)^{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 18925,
"solution": "Let the vector $\\vec{v}=(2 \\cos \\alpha, 2 \\sin \\alpha)$ and $\\vec{w}=(5 \\sin \\beta, 5 \\cos \\beta)$. The locus of ends of vectors expressible in the form $\\vec{v}+\\vec{w}$ are the points which are five units away from a point on the circle of radius two ... |
deepscaler | [
{
"role": "user",
"content": "The numbers $1, 2, 3, 4, 5, 6, 7,$ and $8$ are randomly written on the faces of a regular octahedron so that each face contains a different number. The probability that no two consecutive numbers, where $8$ and $1$ are considered to be consecutive, are written on faces that sha... | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 18926,
"solution": "[asy] import three; draw((0,0,0)--(0,0,1)--(0,1,1)--(1,1,1)--(1,0,1)--(1,0,0)--(1,1,0)--(0,1,0)--(0,0,0)); draw((1,1,0)--(1,1,1)); draw((0,1,0)--(0,1,1)); draw((0,0,0)--(1,0,0)); draw((0,0,1)--(1,0,1)); for(int i = 0; i < 2; ++i) { for(int j = 0; j < 2; ++j) { for(... |
deepscaler | [
{
"role": "user",
"content": "Jason wishes to purchase some comic books. He has $15 and each comic book costs $1.20, tax included. Additionally, there is a discount of $0.10 on each comic book if he buys more than 10 comic books. What is the maximum number of comic books he can buy? Let's think step by step... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 18927,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integer values of $x$ satisfy $|x|<3\\pi$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 18928,
"solution": "1. **Convert the inequality involving absolute value**: The inequality $|x| < 3\\pi$ can be rewritten as $-3\\pi < x < 3\\pi$.\n\n2. **Approximate $\\pi$ and calculate $3\\pi$**: Since $\\pi \\approx 3.14$, we calculate $3\\pi \\approx 3 \\times 3.14 = 9.42$.\n\n3.... |
deepscaler | [
{
"role": "user",
"content": "A tourist spends half of their money and an additional $100 \\mathrm{Ft}$ each day. By the end of the fifth day, all their money is gone. How much money did they originally have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6200"
} | {
"split": "train",
"index": 18929,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A total of 17 teams play in a single-elimination tournament. (A single-elimination tournament is one where once a team has lost, it is removed from the competition.) How many total games must be played before a winner can be declared, assuming there is no possibility of ties?... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 18930,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 2009 numbers arranged in a circle, each of which is either 1 or -1, and not all numbers are the same. Consider all possible consecutive groups of ten numbers. Compute the product of the numbers in each group of ten and sum these products. What is the maximum possible ... | math | {
"style": "rule",
"ground_truth": "2005"
} | {
"split": "train",
"index": 18931,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a given positive integer $ k$ denote the square of the sum of its digits by $ f_1(k)$ and let $ f_{n\\plus{}1}(k) \\equal{} f_1(f_n(k)).$ Determine the value of $ f_{1991}(2^{1990}).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "256"
} | {
"split": "train",
"index": 18932,
"solution": "Let \\( k \\) be a positive integer, and define the function \\( f_1(k) \\) as the square of the sum of the digits of \\( k \\). We are also given a recursive function \\( f_{n+1}(k) = f_1(f_n(k)) \\). We need to find the value of \\( f_{1991}(2^{1990}) \\).\n\n###... |
deepscaler | [
{
"role": "user",
"content": "Let $f(n)$ denote the largest odd factor of $n$ , including possibly $n$ . Determine the value of\n\\[\\frac{f(1)}{1} + \\frac{f(2)}{2} + \\frac{f(3)}{3} + \\cdots + \\frac{f(2048)}{2048},\\]\nrounded to the nearest integer. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "1365"
} | {
"split": "train",
"index": 18933,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the first eight prime numbers that have a units digit of 3. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "404"
} | {
"split": "train",
"index": 18934,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A wooden model of a square pyramid has a base edge of 12 cm and an altitude of 8 cm. A cut is made parallel to the base of the pyramid that separates it into two pieces: a smaller pyramid and a frustum. Each base edge of the smaller pyramid is 6 cm and its altitude is 4 cm. How... | math | {
"style": "rule",
"ground_truth": "336"
} | {
"split": "train",
"index": 18935,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle has an area of $\\pi$ square units. What is the length of the circle's diameter, in units? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 18936,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $\\dfrac {x^{2}}{a^{2}}-\\dfrac {y^{2}}{16}=1$ with a point $P$ on its right branch. The difference in distances from $P$ to the left and right foci is $6$, and the distance from $P$ to the left directrix is $\\dfrac {34}{5}$. Calculate the distance from $P$... | math | {
"style": "rule",
"ground_truth": "\\dfrac {16}{3}"
} | {
"split": "train",
"index": 18937,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The witch Gingema cast a spell on a wall clock so that the minute hand moves in the correct direction for five minutes, then three minutes in the opposite direction, then five minutes in the correct direction again, and so on. How many minutes will the hand show after 2022 minu... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 18938,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow {m}=(\\sin x，-1)$ and $\\overrightarrow {n}=( \\sqrt {3}\\cos x，- \\frac {1}{2})$, and the function $f(x)= \\overrightarrow {m}^{2}+ \\overrightarrow {m}\\cdot \\overrightarrow {n}-2$.\n(I) Find the maximum value of $f(x)$ and the set of values o... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 18939,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A standard deck of cards has 52 cards divided into 4 suits, each of which has 13 cards. Two of the suits ($\\heartsuit$ and $\\diamondsuit$) are red, and the other two ($\\spadesuit$ and $\\clubsuit$) are black. The cards in the deck are shuffled. What is the probability that t... | math | {
"style": "rule",
"ground_truth": "\\frac{169}{2652}"
} | {
"split": "train",
"index": 18940,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sarah and John run on a circular track. Sarah runs counterclockwise and completes a lap every 120 seconds, while John runs clockwise and completes a lap every 75 seconds. Both start from the same line at the same time. A photographer standing inside the track takes a picture at... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"split": "train",
"index": 18941,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that in triangle $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $2\\sin A + \\sin B = 2\\sin C\\cos B$ and the area of $\\triangle ABC$ is $S = \\frac{\\sqrt{3}}{2}c$, find the minimum value of $ab$. Let's think step ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 18942,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the sides \\( AB \\) and \\( AD \\) of the square \\( ABCD \\), points \\( E \\) and \\( F \\) are marked such that \\( BE : EA = AF : FD = 2022 : 2023 \\). The segments \\( EC \\) and \\( FC \\) intersect the diagonal of the square \\( BD \\) at points \\( G \\) and \\( H \... | math | {
"style": "rule",
"ground_truth": "\\frac{12271519}{36814556}"
} | {
"split": "train",
"index": 18943,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three squares with a perimeter of 28cm each are combined to form a rectangle. What is the area of this rectangle? Additionally, if four such squares are combined to form a larger square, what is the perimeter of this larger square? Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 18944,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all values of \\(a\\) such that the roots \\(x_1, x_2, x_3\\) of the polynomial \\(x^3 - 6x^2 + ax + a\\) satisfy \\((x_1 - 3)^2 + (x_2 - 3)^3 + (x_3 - 3)^3 = 0\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-9"
} | {
"split": "train",
"index": 18945,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the terminal side of $\\alpha$ passes through the point $(a, 2a)$ (where $a < 0$),\n(1) Find the values of $\\cos\\alpha$ and $\\tan\\alpha$.\n(2) Simplify and find the value of $$\\frac {\\sin(\\pi-\\alpha)\\cos(2\\pi-\\alpha)\\sin(-\\alpha+ \\frac {3\\pi}{2})}{\\ta... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{10}"
} | {
"split": "train",
"index": 18946,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A man, standing on a lawn, is wearing a circular sombrero of radius 3 feet. Unfortunately, the hat blocks the sunlight so effectively that the grass directly under it dies instantly. If the man walks in a circle of radius 5 feet, what area of dead grass will result? Let's think... | math | {
"style": "rule",
"ground_truth": "60\\pi"
} | {
"split": "train",
"index": 18947,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a particular day in Salt Lake, UT, the temperature was given by $-t^2 +12t+50$ where $t$ is the time in hours past noon. What is the largest $t$ value at which the temperature was exactly 77 degrees? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 18948,
"solution": ""
} |
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