data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "To meet the shopping needs of customers during the \"May Day\" period, a fruit supermarket purchased cherries and cantaloupes from the fruit production base for $9160$ yuan, totaling $560$ kilograms. The purchase price of cherries is $35$ yuan per kilogram, and the purchase pri... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 2400,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a rectangular coordinate system, what is the number of units in the distance from the origin to the point $(-15, 8)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 2401,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many positive integers $a$ does the polynomial $x^{2}-a x+a$ have an integer root? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 2402,
"solution": "Let $r, s$ be the roots of $x^{2}-a x+a=0$. By Vieta's, we have $r+s=r s=a$. Note that if one root is an integer, then both roots must be integers, as they sum to an integer $a$. Then, $$r s-(r+s)+1=1 \\Longrightarrow(r-1)(s-1)=1$$ Because we require $r, s$ to be bo... |
deepscaler | [
{
"role": "user",
"content": "If $x$ and $y$ are positive integers such that $xy - 8x + 9y = 632$, what is the minimal possible value of $|x - y|$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 2403,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,b,c,d,e,f,g,h,i$ be distinct integers from $1$ to $9.$ The minimum possible positive value of \\[\\dfrac{a \\cdot b \\cdot c - d \\cdot e \\cdot f}{g \\cdot h \\cdot i}\\] can be written as $\\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m... | math | {
"style": "rule",
"ground_truth": "289"
} | {
"split": "train",
"index": 2404,
"solution": "To minimize a positive fraction, we minimize its numerator and maximize its denominator. It is clear that $\\frac{a \\cdot b \\cdot c - d \\cdot e \\cdot f}{g \\cdot h \\cdot i} \\geq \\frac{1}{7\\cdot8\\cdot9}.$\n If we minimize the numerator, then $a \\cdot b \\cd... |
deepscaler | [
{
"role": "user",
"content": "An economist and a statistician play a game on a calculator which does only one\noperation. The calculator displays only positive integers and it is used in the following\nway: Denote by $n$ an integer that is shown on the calculator. A person types an integer,\n$m$, chosen fro... | math | {
"style": "rule",
"ground_truth": "951"
} | {
"split": "train",
"index": 2405,
"solution": "\nTo solve this problem, we need to understand the specific condition under which the current displayed number \\( n \\) on the calculator can be transformed to another integer through the operation described, where \\( m \\) is chosen from the set \\(\\{1, 2, \\ldo... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest integer satisfying the following conditions:\n\n$\\bullet$ I. The sum of the squares of its digits is $85$.\n\n$\\bullet$ II. Each digit is larger than the one on its left.\n\nWhat is the product of the digits of this integer? Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 2406,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Call an integer $n>1$ radical if $2^{n}-1$ is prime. What is the 20th smallest radical number? If $A$ is your answer, and $S$ is the correct answer, you will get $\\max \\left(25\\left(1-\\frac{|A-S|}{S}\\right), 0\\right)$ points, rounded to the nearest integer. Let's think st... | math | {
"style": "rule",
"ground_truth": "4423"
} | {
"split": "train",
"index": 2407,
"solution": "The answer is 4423."
} |
deepscaler | [
{
"role": "user",
"content": "The symbol $\\lfloor x \\rfloor$ denotes the largest integer not exceeding $x$. For example, $\\lfloor 3 \\rfloor = 3,$ and $\\lfloor 9/2 \\rfloor = 4.$ Compute \\[\\lfloor \\sqrt{1} \\rfloor + \\lfloor \\sqrt{2} \\rfloor + \\lfloor \\sqrt{3} \\rfloor + \\cdots + \\lfloor \\sqr... | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 2408,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\{a_n\\}$ is a sequence of positive numbers, and the sum of its first $n$ terms $2S_n = a_n^2 + a_n$ ($n \\in \\mathbb{N}^*$), the sequence $\\{b_n\\}$ satisfies $b_1 = \\frac{3}{2}$, $b_{n+1} = b_n + 3^{a_n}$ ($n \\in \\mathbb{N}^*$).\n\n(I) Find the general formu... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 2409,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a given triangle, for $\\angle P$ to be the largest angle of the triangle, it must be that $a < y < b$. The side lengths are given by $y+6$, $2y+1$, and $5y-10$. What is the least possible value of $b-a$, expressed as a common fraction? Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"split": "train",
"index": 2410,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The \ffirst number in the following sequence is $1$ . It is followed by two $1$ 's and two $2$ 's. This is followed by three $1$ 's, three $2$ 's, and three $3$ 's. The sequence continues in this fashion.\n\\[1,1,1,2,2,1,1,1,2,2,2,3,3,3,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,\\d... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 2411,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a circle with 800 points labeled in sequence clockwise as \\(1, 2, \\ldots, 800\\), dividing the circle into 800 arcs. Initially, one point is painted red, and subsequently, additional points are painted red according to the following rule: if the \\(k\\)-th point is alre... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 2412,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\frac{cos2α}{sin(α-\\frac{π}{4})}=-\\frac{\\sqrt{6}}{2}$, express $cos(α-\\frac{π}{4})$ in terms of radicals. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{4}"
} | {
"split": "train",
"index": 2413,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $\\frac{2}{3}$ of $10$ bananas are worth as much as $8$ oranges. How many oranges are worth as much as $\\frac{1}{2}$ of $5$ bananas? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 2414,
"solution": "1. **Establish the given relationship**: We are given that $\\frac{2}{3}$ of $10$ bananas are worth as much as $8$ oranges. This can be written as:\n \\[\n \\frac{2}{3} \\times 10 \\text{ bananas} = 8 \\text{ oranges}\n \\]\n Simplifying the left side, we ge... |
deepscaler | [
{
"role": "user",
"content": "Given that point $P$ is a moving point on the circle $x^2+y^2=18$, and $PQ \\perp x$-axis at point $Q$, if the moving point $M$ satisfies $\\overrightarrow{OM}=\\frac{1}{3}\\overrightarrow{OP}+\\frac{2}{3}\\overrightarrow{OQ}$.\n\n(Ⅰ) Find the equation of the trajectory $C$ of ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{3}"
} | {
"split": "train",
"index": 2415,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a circle centered at $O$ . Parallel chords $AB$ of length $8$ and $CD$ of length $10$ are of distance $2$ apart such that $AC < AD$ . We can write $\\tan \\angle BOD =\\frac{a}{b}$ , where $a, b$ are positive integers such that gcd $(a, b) = 1$ . Compu... | math | {
"style": "rule",
"ground_truth": "113"
} | {
"split": "train",
"index": 2416,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A trirectangular tetrahedron $M-ABC$ has three pairs of adjacent edges that are perpendicular, and a point $N$ inside the base triangle $ABC$ is at distances of $2\\sqrt{2}$, $4$, and $5$ from the three faces respectively. Find the surface area of the smallest sphere that passe... | math | {
"style": "rule",
"ground_truth": "49\\pi"
} | {
"split": "train",
"index": 2417,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\mathbf{a}$ and $\\mathbf{b}$ such that $\\|\\mathbf{a}\\| = 6,$ $\\|\\mathbf{b}\\| = 8,$ and $\\|\\mathbf{a} + \\mathbf{b}\\| = 11.$ Find $\\cos \\theta,$ where $\\theta$ is the angle between $\\mathbf{a}$ and $\\mathbf{b}.$ Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{32}"
} | {
"split": "train",
"index": 2418,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The average of the numbers $1, 2, 3, \\dots, 50, y$ is $51y$. What is $y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{51}{104}"
} | {
"split": "train",
"index": 2419,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of the binomial coefficients in the expansion of {(5x-1/√x)^n} is 64, determine the constant term in its expansion. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "375"
} | {
"split": "train",
"index": 2420,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence is $\\frac{1}{2}$, $\\frac{1}{3}$, $\\frac{2}{3}$, $\\frac{1}{4}$, $\\frac{2}{4}$, $\\frac{3}{4}$, ... , $\\frac{1}{m+1}$, $\\frac{2}{m+1}$, ... , $\\frac{m}{m+1}$, ... Find the $20^{th}$ term. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "\\frac{6}{7}"
} | {
"split": "train",
"index": 2421,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that for a positive integer \\( n \\), \\( 2^n + 1 \\) is a prime number. What remainder can this prime have when divided by 240? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 2422,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c,$ $d$ be nonzero integers such that\n\\[\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}^2 = \\begin{pmatrix} 7 & 0 \\\\ 0 & 7 \\end{pmatrix}.\\]Find the smallest possible value of $|a| + |b| + |c| + |d|.$ Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 2423,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The figure shows two concentric circles. If the length of chord AB is 80 units and chord AB is tangent to the smaller circle, what is the area of the shaded region? Express your answer in terms of $\\pi$.\n\n[asy]\ndefaultpen(linewidth(.8pt));\ndotfactor=4;\nfilldraw(circle((0,... | math | {
"style": "rule",
"ground_truth": "1600\\pi"
} | {
"split": "train",
"index": 2424,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that points P and Q are on the curve $f(x) = x^2 - \\ln x$ and the line $x-y-2=0$ respectively, find the minimum distance between points P and Q. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 2425,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $\\triangle ABC$ has a right angle at $C$, $\\angle A = 45^\\circ$, and $AC=12$. Find the radius of the incircle of $\\triangle ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6 - 3\\sqrt{2}"
} | {
"split": "train",
"index": 2426,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The solid \\( T \\) consists of all points \\((x,y,z)\\) such that \\( |x| + |y| \\le 2 \\), \\( |x| + |z| \\le 1 \\), and \\( |z| + |y| \\le 1 \\). Find the volume of \\( T \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 2427,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The coefficient of $x^3y^5$ in the expansion of $(x+y)(x-y)^7$ is __________. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 2428,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ ABC$ be an isosceles triangle with $ \\left|AB\\right| \\equal{} \\left|AC\\right| \\equal{} 10$ and $ \\left|BC\\right| \\equal{} 12$ . $ P$ and $ R$ are points on $ \\left[BC\\right]$ such that $ \\left|BP\\right| \\equal{} \\left|RC\\right| \\equal{} 3$ . ... | math | {
"style": "rule",
"ground_truth": "\\frac {10\\sqrt {13} }{13}"
} | {
"split": "train",
"index": 2429,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An ATM password at Fred's Bank is composed of four digits from $0$ to $9$, with repeated digits allowable. If no password may begin with the sequence $9,1,1,$ then how many passwords are possible? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9990"
} | {
"split": "train",
"index": 2430,
"solution": "\nTo find the total number of possible ATM passwords, we need to consider the constraints given in the problem. The password is a four-digit number, where each digit can range from $0$ to $9$. However, the password cannot begin with the sequence $9,1,1$.\n\n**Step 1... |
deepscaler | [
{
"role": "user",
"content": "Let \\( M = 42 \\cdot 43 \\cdot 75 \\cdot 196 \\). Find the ratio of the sum of the odd divisors of \\( M \\) to the sum of the even divisors of \\( M \\). 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": 2431,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest $n$ such that $n!$ ends with 10 zeroes. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 2432,
"solution": "The number of zeroes that $n!$ ends with is the largest power of 10 dividing $n!$. The exponent of 5 dividing $n!$ exceeds the exponent of 2 dividing $n!$, so we simply seek the exponent of 5 dividing $n!$. For a number less than 125, this exponent is just the numbe... |
deepscaler | [
{
"role": "user",
"content": "$\\frac{\\text{华杯赛}}{\\text{少} \\times \\text{俊} + \\text{金坛} + \\text{论} \\times \\text{数}} = 15$\n\nIn the above equation, different Chinese characters represent different digits between $1$ and $9$. When the three-digit number \"华杯赛\" reaches its maximum value, please write ... | math | {
"style": "rule",
"ground_truth": "975"
} | {
"split": "train",
"index": 2433,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Call a $3$-digit number geometric if it has $3$ distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "840"
} | {
"split": "train",
"index": 2434,
"solution": "The smallest geometric number is $124$ because $123$ and any number containing a zero does not work. $964$ is the largest geometric number because the middle digit cannot be 8 or 7. Subtracting the numbers gives $\\boxed{840}.$\n ~IceMatrix\n ~Shreyas S"
} |
deepscaler | [
{
"role": "user",
"content": "Given that a square $S_1$ has an area of $25$, the area of the square $S_3$ constructed by bisecting the sides of $S_2$ is formed by the points of bisection of $S_2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6.25"
} | {
"split": "train",
"index": 2435,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $\\left(\\frac{6}{7}\\right)^2 \\cdot \\left(\\frac{1}{2}\\right)^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{49}"
} | {
"split": "train",
"index": 2436,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the three sides of triangle $\\triangle ABC$ are $a$, $a+3$, and $a+6$, and the largest angle is twice the smallest angle, calculate the cosine value of the smallest angle. 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": 2437,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 10 questions available, among which 6 are Type A questions and 4 are Type B questions. Student Xiao Ming randomly selects 3 questions to solve.\n\n(1) Calculate the probability that Xiao Ming selects at least 1 Type B question.\n\n(2) Given that among the 3 selected q... | math | {
"style": "rule",
"ground_truth": "\\dfrac{93}{125}"
} | {
"split": "train",
"index": 2438,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The centers of two circles are $41$ inches apart. The smaller circle has a radius of $4$ inches and the larger one has a radius of $5$ inches. \nThe length of the common internal tangent is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40 \\text{ inches}"
} | {
"split": "train",
"index": 2439,
"solution": "1. **Identify the centers and radii of the circles**: Let $A$ be the center of the circle with radius $5$ inches, and $B$ be the center of the circle with radius $4$ inches. The distance between the centers $A$ and $B$ is given as $41$ inches.\n\n2. **Understanding ... |
deepscaler | [
{
"role": "user",
"content": "Define a function $f(x)$ on $\\mathbb{R}$ that satisfies $f(x+6)=f(x)$. When $x \\in [-3, -1]$, $f(x) = -(x+2)^2$, and when $x \\in [-1, 3)$, $f(x) = x$. Calculate the value of $f(1) + f(2) + f(3) + \\ldots + f(2015)$. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "336"
} | {
"split": "train",
"index": 2440,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the first \\( n \\) terms of a geometric sequence \\(\\left\\{a_{n}\\right\\}\\) with all real numbers is \\( S_{n} \\). If \\( S_{10} = 10 \\) and \\( S_{30} = 70 \\), then \\( S_{40} \\) is equal to: Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 2441,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be the number to partition 2013 into an ordered tuple of prime numbers? What is $\\log _{2}(P)$? If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\\left\\lfloor\\frac{125}{2}\\left(\\min \\left(\\frac{C}{A}, \\frac{A}{C}\\rig... | math | {
"style": "rule",
"ground_truth": "614.519."
} | {
"split": "train",
"index": 2442,
"solution": "We use the following facts and heuristics. (1) The ordered partitions of $n$ into any positive integers (not just primes) is $2^{n-1}$. This can be guessed by checking small cases and finding a pattern, and is not difficult to prove. (2) The partitions of $\\frac{20... |
deepscaler | [
{
"role": "user",
"content": "Find an approximate value of $0.998^6$ such that the error is less than $0.001$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.988"
} | {
"split": "train",
"index": 2443,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $1 + 7 + 7^2 + \\cdots + 7^{2004}$ is divided by $1000$, a remainder of $N$ is obtained. Determine the value of $N$.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "801"
} | {
"split": "train",
"index": 2444,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the inequality (e-a)e^x + x + b + 1 ≤ 0, where e is the natural constant, find the maximum value of $\\frac{b+1}{a}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{e}"
} | {
"split": "train",
"index": 2445,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many pairs of positive integer solutions \\((x, y)\\) satisfy \\(\\frac{1}{x+1} + \\frac{1}{y} + \\frac{1}{(x+1) y} = \\frac{1}{1991}\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 2446,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The distance from point P(1, -1) to the line $ax+3y+2a-6=0$ is maximized when the line passing through P is perpendicular to the given line. 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": 2447,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In Zuminglish, words consist only of the letters $M, O,$ and $P$. $O$ is a vowel, while $M$ and $P$ are consonants. A valid Zuminglish word must have at least two consonants between any two $O's$. Determine the number of valid 7-letter words in Zuminglish and find the remainder... | math | {
"style": "rule",
"ground_truth": "912"
} | {
"split": "train",
"index": 2448,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two runners start simultaneously and run on a circular track in opposite directions at constant speeds. One runner completes a lap in 5 minutes, and the other runner completes a lap in 8 minutes. Find the number of distinct meeting points of the runners on the track if they run... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 2449,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A group consists of 4 boys and 3 girls.\n(1) If one boy and one girl are each chosen to host a program, how many different selections are possible?\n(2) If two boys and two girls are chosen to form a small choir, and are to be arranged in a row with the requirement that the two... | math | {
"style": "rule",
"ground_truth": "216"
} | {
"split": "train",
"index": 2450,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest $2$-digit prime factor of the integer $n = {300\\choose 150}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "97"
} | {
"split": "train",
"index": 2451,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all integer values $n$ for which $\\binom{20}{n}+\\binom{20}{10}=\\binom{21}{11}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 2452,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the remainder when 369,963 is divided by 6? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 2453,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the equation of line $l$ is $y=x+4$, and the parametric equation of circle $C$ is $\\begin{cases} x=2\\cos \\theta \\\\ y=2+2\\sin \\theta \\end{cases}$ (where $\\theta$ is the parameter), with the origin as the pole and the positive half-axis of $x$ as the polar axis. Es... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}+2"
} | {
"split": "train",
"index": 2454,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A box contains 5 white balls and 6 black balls. A ball is drawn out of the box at random. What is the probability that the ball is white? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{5}{11}"
} | {
"split": "train",
"index": 2455,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To meet market demand, a clothing supermarket purchased a short-sleeved $T$-shirt at the beginning of June. The cost price of each shirt is $80$ yuan, and the supermarket stipulates that the selling price of each shirt must not be less than $90$ yuan. According to a survey, whe... | math | {
"style": "rule",
"ground_truth": "12000"
} | {
"split": "train",
"index": 2456,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $[x]$ is the greatest integer less than or equal to $x$, calculate $\\sum_{N=1}^{1024}\\left[\\log _{2} N\\right]$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8204"
} | {
"split": "train",
"index": 2457,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $y=x+\\frac{1}{x}$, then $x^4+x^3-4x^2+x+1=0$ becomes: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x^2(y^2+y-6)=0"
} | {
"split": "train",
"index": 2458,
"solution": "1. **Express $y$ in terms of $x$:**\n Given $y = x + \\frac{1}{x}$.\n\n2. **Square $y$ to find $y^2$:**\n \\[\n y^2 = \\left(x + \\frac{1}{x}\\right)^2 = x^2 + 2 + \\frac{1}{x^2}\n \\]\n\n3. **Express $x^2 y^2$ in terms of $x$:**\n \\[\n x^2 y^2 = x^2 \\... |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle ABC$ be a triangle with $AB=85$ , $BC=125$ , $CA=140$ , and incircle $\\omega$ . Let $D$ , $E$ , $F$ be the points of tangency of $\\omega$ with $\\overline{BC}$ , $\\overline{CA}$ , $\\overline{AB}$ respectively, and furthermore denote by $X$ ... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 2459,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{F}$ be a family of subsets of $\\{1,2,\\ldots, 2017\\}$ with the following property: if $S_1$ and $S_2$ are two elements of $\\mathcal{F}$ with $S_1\\subsetneq S_2$ , then $|S_2\\setminus S_1|$ is odd. Compute the largest number of subsets $\\mathca... | math | {
"style": "rule",
"ground_truth": "2 \\binom{2017}{1008}"
} | {
"split": "train",
"index": 2460,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Intermediate Maths Challenge has 25 questions with the following scoring rules:\n5 marks are awarded for each correct answer to Questions 1-15;\n6 marks are awarded for each correct answer to Questions 16-25;\nEach incorrect answer to Questions 16-20 loses 1 mark;\nEach inc... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 2461,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four friends went fishing one day and caught a total of 11 fish. Each person caught at least one fish. Which statement must be true: (A) At least one person caught exactly one fish. (B) At least one person caught exactly three fish. (C) At least one person caught more than thre... | math | {
"style": "rule",
"ground_truth": "D"
} | {
"split": "train",
"index": 2462,
"solution": "Choice (A) is not necessarily true, since the four friends could have caught 2,3, 3, and 3 fish. Choice (B) is not necessarily true, since the four friends could have caught 1, 1, 1, and 8 fish. Choice (C) is not necessarily true, since the four friends could have c... |
deepscaler | [
{
"role": "user",
"content": "The bases \\( AB \\) and \\( CD \\) of the trapezoid \\( ABCD \\) are 155 and 13 respectively, and its diagonals are mutually perpendicular. Find the dot product of the vectors \\( \\overrightarrow{AD} \\) and \\( \\overrightarrow{BC} \\). Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "2015"
} | {
"split": "train",
"index": 2463,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A natural number undergoes the following operation: the rightmost digit of its decimal representation is discarded, and then the number obtained after discarding is added to twice the discarded digit. For example, $157 \\mapsto 15 + 2 \\times 7 = 29$, $5 \\mapsto 0 + 2 \\times ... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 2464,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If I have a $4\\times 4$ chess board, in how many ways can I place four distinct pawns on the board such that each column and row of the board contains no more than one pawn? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "576"
} | {
"split": "train",
"index": 2465,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Danielle picks a positive integer $1 \\leq n \\leq 2016$ uniformly at random. What is the probability that \\operatorname{gcd}(n, 2015)=1? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1441}{2016}"
} | {
"split": "train",
"index": 2466,
"solution": "We split the interval $[1,2016]$ into $[1,2015]$ and 2016. The number of integers in $[1,2015]$ that are relatively prime to 2015 is $\\phi(2015)=\\frac{4}{5} \\cdot \\frac{12}{13} \\cdot \\frac{30}{31} \\cdot 2015=1440$. Also, 2016 is relatively prime to 2015, so t... |
deepscaler | [
{
"role": "user",
"content": "A truck delivered 4 bags of cement. They are stacked in the truck. A worker can carry one bag at a time either from the truck to the gate or from the gate to the shed. The worker can carry the bags in any order, each time taking the top bag, carrying it to the respective destin... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 2467,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $C$: $\\dfrac{x^2}{a^2} + \\dfrac{y^2}{b^2} = 1 (a > b > 0)$ passes through the point $(1, \\dfrac{2\\sqrt{3}}{3})$, with the left and right foci being $F_1$ and $F_2$ respectively. The circle $x^2 + y^2 = 2$ intersects with the line $x + y + b = 0$ at a chord... | math | {
"style": "rule",
"ground_truth": "\\dfrac{2\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 2468,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\frac{\\sin \\alpha \\cos \\alpha }{1-\\cos 2\\alpha }=\\frac{1}{4}$, $\\tan (\\alpha-\\beta)=2$, calculate $\\tan (\\beta-2\\alpha)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 2469,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the coefficient of $x^{-4}$ in the expansion of $\\left(1- \\frac {1}{x^{2}}\\right)^{n}$ (where $n\\in\\mathbb{N}_{+}$) be denoted as $a_{n}$. Calculate the value of $$\\frac {1}{a_{2}}+ \\frac {1}{a_{3}}+…+ \\frac {1}{a_{2015}}$$. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "\\frac {4028}{2015}"
} | {
"split": "train",
"index": 2470,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$ABCD$ is a regular tetrahedron (right triangular pyramid). If $M$ is the midpoint of $\\overline{CD}$, then what is $\\cos \\angle AMB$? 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": 2471,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\triangle PQR$ is right-angled at $P$ with $PR=12$, $SQ=11$, and $SR=13$, what is the perimeter of $\\triangle QRS$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 2472,
"solution": "By the Pythagorean Theorem in $\\triangle PRS$, $PS=\\sqrt{RS^{2}-PR^{2}}=\\sqrt{13^{2}-12^{2}}=\\sqrt{169-144}=\\sqrt{25}=5$ since $PS>0$. Thus, $PQ=PS+SQ=5+11=16$. By the Pythagorean Theorem in $\\triangle PRQ$, $RQ=\\sqrt{PR^{2}+PQ^{2}}=\\sqrt{12^{2}+16^{2}}=\\sq... |
deepscaler | [
{
"role": "user",
"content": "Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.295"
} | {
"split": "train",
"index": 2473,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct positive factors does 32 have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 2474,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a circle, ten points \\(A_{1}, A_{2}, A_{3}, \\ldots, A_{10}\\) are equally spaced. If \\(C\\) is the center of the circle, what is the size, in degrees, of the angle \\(A_{1} A_{5} C\\) ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 2475,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A workshop has fewer than $60$ employees. When these employees are grouped in teams of $8$, $5$ employees remain without a team. When arranged in teams of $6$, $3$ are left without a team. How many employees are there in the workshop? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 2476,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "With the rapid development of the \"Internet + transportation\" model, \"shared bicycles\" have appeared successively in many cities. In order to understand the satisfaction of users in a certain area with the services provided, a certain operating company randomly surveyed 10 ... | math | {
"style": "rule",
"ground_truth": "50\\%"
} | {
"split": "train",
"index": 2477,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the squares of three consecutive positive integers is 7805. What is the sum of the cubes of the three original integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "398259"
} | {
"split": "train",
"index": 2478,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three red beads, two white beads, and one blue bead are placed in line in random order. What is the probability that no two neighboring beads are the same color? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 2479,
"solution": "1. **Calculate the total number of orderings for the beads**:\n Given that there are three red beads, two white beads, and one blue bead, the total number of ways to arrange these beads can be calculated using the formula for permutations of multiset:\n \\[\n ... |
deepscaler | [
{
"role": "user",
"content": "If \\(\\cos ^{4} \\theta + \\sin ^{4} \\theta + (\\cos \\theta \\cdot \\sin \\theta)^{4} + \\frac{1}{\\cos ^{4} \\theta + \\sin ^{4} \\theta} = \\frac{41}{16}\\), find the value of \\(\\sin ^{2} \\theta\\). Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 2480,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $A=60^{\\circ}$, $AC=4$, $BC=2\\sqrt{3}$, the area of $\\triangle ABC$ equals \\_\\_\\_\\_\\_\\_. 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": 2481,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$, $Q$, and $R$ be points on a circle of radius $12$. If $\\angle PRQ = 110^\\circ,$ find the circumference of the minor arc $PQ$. Express your answer in terms of $\\pi$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{22}{3}\\pi"
} | {
"split": "train",
"index": 2482,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line parallel to side \\( AB \\) of triangle \\( ABC \\) intersects side \\( BC \\) at point \\( M \\) and side \\( AC \\) at point \\( N \\). The area of triangle \\( MCN \\) is twice the area of trapezoid \\( ABMN \\). Find the ratio \\( CM:MB \\). Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "2 + \\sqrt{6}"
} | {
"split": "train",
"index": 2483,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A number is called *6-composite* if it has exactly 6 composite factors. What is the 6th smallest 6-composite number? (A number is *composite* if it has a factor not equal to 1 or itself. In particular, 1 is not composite.)\n\n*Ray Li.* Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "441"
} | {
"split": "train",
"index": 2484,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given four points $P, A, B, C$ on a sphere, if $PA$, $PB$, $PC$ are mutually perpendicular and $PA=PB=PC=1$, calculate the surface area of this sphere. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\pi"
} | {
"split": "train",
"index": 2485,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an addition problem where the digits were written on cards, two cards were swapped, resulting in an incorrect expression: $37541 + 43839 = 80280$. Find the error and write the correct value of the sum. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "81380"
} | {
"split": "train",
"index": 2486,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Some of the digits in the following correct addition have been replaced by the letters \\( P, Q, R, \\) and \\( S \\), as shown.\nWhat is the value of \\( P+Q+R+S \\)?\n\\[\n\\begin{array}{r}\nP 45 \\\\\n+\\quad Q R \\quad S \\\\\n\\hline\n654\n\\end{array}\n\\] Let's think ste... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 2487,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many solid $1 \times 1 \times 1$ cubes are required to make a solid $2 \times 2 \times 2$ cube? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 2488,
"solution": "The volume of a $1 \times 1 \times 1$ cube is 1 . The volume of a $2 \times 2 \times 2$ cube is 8 . Thus, 8 of the smaller cubes are needed to make the larger cube."
} |
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": 2489,
"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 Ar... |
deepscaler | [
{
"role": "user",
"content": "Find the area of the region of the \\( xy \\)-plane defined by the inequality \\( |x|+|y|+|x+y| \\leq 1 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3/4"
} | {
"split": "train",
"index": 2490,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose there are initially 1001 townspeople and two goons. What is the probability that, when the game ends, there are exactly 1000 people in jail? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{1003}"
} | {
"split": "train",
"index": 2491,
"solution": "By considering the parity of the number of people in jail, we see that this situation arises if and only if the goons win after the 500th night. That means that at this point we must have exactly one townsperson and two goons remaining. In other words, this situatio... |
deepscaler | [
{
"role": "user",
"content": "\nThe side $AB$ of triangle $ABC$ is divided into $n$ equal parts (with division points $B_0 = A, B_1, B_2, \\ldots, B_n = B$), and the side $AC$ of this triangle is divided into $n+1$ equal parts (with division points $C_0 = A, C_1, C_2, \\ldots, C_{n+1} = C$). The triangles $... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 2492,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The price of a pair of shoes at Barry's Boutique was $50. Find the price of the shoes on Thursday after a 15% price increase. Then, calculate the price of the shoes on Friday after a 20% discount is applied to the new price. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 2493,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest positive integer $N$ so that the number of integers in the set $\\{1,2,\\dots,N\\}$ which are divisible by 3 is equal to the number of integers which are divisible by 5 or 7 (or both). Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 2494,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of solutions to the equation\n\\[\\tan (10 \\pi \\cos \\theta) = \\cot (10 \\pi \\sin \\theta)\\]\nwhere $\\theta \\in (0, 2 \\pi).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 2495,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest integer \\( k \\) such that for at least one natural number \\( n > 1000 \\), the number \\( n! = 1 \\cdot 2 \\cdot \\ldots \\cdot n \\) is divisible by \\( 2^{n+k+2} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 2496,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ilya takes a triplet of numbers and transforms it following the rule: at each step, each number is replaced by the sum of the other two. What is the difference between the largest and the smallest numbers in the triplet after the 1989th application of this rule, if the initial ... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 2497,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Giselle will combine blue paint, green paint, and white paint in the ratio $3:2:4$, respectively. If she uses $12$ quarts of white paint, how many quarts of green paint should she use? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 2498,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the absolute values of the roots of $x^{4}-4 x^{3}-4 x^{2}+16 x-8=0$. 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": 2499,
"solution": "$$\\begin{aligned} x^{4}-4 x^{3}-4 x^{2}+16 x-8 & =\\left(x^{4}-4 x^{3}+4 x^{2}\\right)-\\left(8 x^{2}-16 x+8\\right) \\\\ & =x^{2}(x-2)^{2}-8(x-1)^{2} \\\\ & =\\left(x^{2}-2 x\\right)^{2}-(2 \\sqrt{2} x-2 \\sqrt{2})^{2} \\\\ & =\\left(x^{2}-(2+2 \\sqrt{2}) x+2 \\sq... |
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