data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "If $2x - y = 5$ and $x + 2y = 5$, what is the value of $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 22449,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a plane, equilateral triangle $A B C$, square $B C D E$, and regular dodecagon $D E F G H I J K L M N O$ each have side length 1 and do not overlap. Find the area of the circumcircle of $\\triangle A F N$. Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "(2+\\sqrt{3}) \\pi"
} | {
"split": "train",
"index": 22450,
"solution": "Note that $\\angle A C D=\\angle A C B+\\angle B C D=60^{\\circ}+90^{\\circ}=150^{\\circ}$. In a dodecagon, each interior angle is $180^{\\circ} \\cdot \\frac{12-2}{12}=150^{\\circ}$ meaning that $\\angle F E D=\\angle D O N=150^{\\circ}$. since $E F=F D=1$ and $D ... |
deepscaler | [
{
"role": "user",
"content": "Given a right circular cone $(P-ABC)$ with lateral edges $(PA)$, $(PB)$, $(PC)$ being pairwise perpendicular, and base edge $AB = \\sqrt{2}$, find the surface area of the circumscribed sphere of the right circular cone $(P-ABC)$. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "3\\pi"
} | {
"split": "train",
"index": 22451,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a certain product requires $6$ processing steps, where $2$ of these steps must be consecutive and another $2$ steps cannot be consecutive, calculate the number of possible processing sequences. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 22452,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The mayor of a city wishes to establish a transport system with at least one bus line, in which:\n- each line passes exactly three stops,\n- every two different lines have exactly one stop in common,\n- for each two different bus stops there is exactly one line that passes thro... | math | {
"style": "rule",
"ground_truth": "3,7"
} | {
"split": "train",
"index": 22453,
"solution": "\nTo solve this problem, we need to establish the conditions that lead to the correct number of bus stops in the city given the requirements for the bus lines. Let's break down the problem and find a systematic way to achieve the solution.\n\n### Conditions:\n\n1. ... |
deepscaler | [
{
"role": "user",
"content": "All vertices of a regular 2016-gon are initially white. What is the least number of them that can be painted black so that:\n(a) There is no right triangle\n(b) There is no acute triangle\nhaving all vertices in the vertices of the 2016-gon that are still white? Let's think ste... | math | {
"style": "rule",
"ground_truth": "1008"
} | {
"split": "train",
"index": 22454,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the point $\\left(m,n\\right)$ in the first quadrant is symmetric with respect to the line $x+y-2=0$ and lies on the line $2x+y+3=0$, calculate the minimum value of $\\frac{1}{m}+\\frac{8}{n}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{25}{9}"
} | {
"split": "train",
"index": 22455,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On each spin of the spinner shown, the arrow is equally likely to stop on any one of the four numbers. Deanna spins the arrow on the spinner twice. She multiplies together the two numbers on which the arrow stops. Which product is most likely to occur? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 22456,
"solution": "We make a chart that lists the possible results for the first spin down the left side, the possible results for the second spin across the top, and the product of the two results in the corresponding cells:\n\\begin{tabular}{c|cccc} \n& 1 & 2 & 3 & 4 \\\\\n\\hline ... |
deepscaler | [
{
"role": "user",
"content": "Box $A$ contains 1 red ball and 5 white balls, and box $B$ contains 3 white balls. Three balls are randomly taken from box $A$ and placed into box $B$. After mixing thoroughly, three balls are then randomly taken from box $B$ and placed back into box $A$. What is the probabilit... | math | {
"style": "rule",
"ground_truth": "1/4"
} | {
"split": "train",
"index": 22457,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Screamers are coached by Coach Yellsalot. The Screamers have 12 players, but two of them, Bob and Yogi, refuse to play together. How many starting lineups (of 5 players) can Coach Yellsalot make, if the starting lineup can't contain both Bob and Yogi? (The order of the 5... | math | {
"style": "rule",
"ground_truth": "672"
} | {
"split": "train",
"index": 22458,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest possible value of $x$ in the simplified form $x=\\frac{a+b\\sqrt{c}}{d}$ if $\\frac{7x}{5}-2=\\frac{4}{x}$, where $a, b, c,$ and $d$ are integers. What is $\\frac{acd}{b}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5775"
} | {
"split": "train",
"index": 22459,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The digits 1, 3, 4, and 5 are arranged randomly to form a four-digit number. What is the probability that the number is odd? 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": 22460,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow {a} = (\\sin\\theta, \\cos\\theta - 2\\sin\\theta)$ and $\\overrightarrow {b} = (1, 2)$. \n(1) If $\\overrightarrow {a} \\parallel \\overrightarrow {b}$, find the value of $\\tan\\theta$; \n(2) If $|\\overrightarrow {a}| = |\\overrightarrow {b}... | math | {
"style": "rule",
"ground_truth": "\\frac {3\\pi}{4}"
} | {
"split": "train",
"index": 22461,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The right triangles $\\triangle M D C$ and $\\triangle A D K$ share a common right angle $\\angle D$. Point $K$ is on $C D$ and divides it in the ratio $2: 3$ counting from point $C$. Point $M$ is the midpoint of side $A D$. Find the sum of $\\angle A K D$ and $\\angle M C D$, ... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 22462,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given point $P(2,-1)$,\n(1) Find the general equation of the line that passes through point $P$ and has a distance of 2 units from the origin.\n(2) Find the general equation of the line that passes through point $P$ and has the maximum distance from the origin. Calculate the ma... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 22463,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the triangle \\( \\triangle ABC \\), if \\( \\frac{\\overrightarrow{AB} \\cdot \\overrightarrow{BC}}{3} = \\frac{\\overrightarrow{BC} \\cdot \\overrightarrow{CA}}{2} = \\frac{\\overrightarrow{CA} \\cdot \\overrightarrow{AB}}{1} \\), find \\( \\tan A \\). Let's think step by ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{11}"
} | {
"split": "train",
"index": 22464,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 8 keys numbered 1 to 8 and 8 boxes numbered 1 to 8. Each key can only open the box with the same number. All keys are placed in these boxes and locked up so that each box contains one key. How many different ways are there to place the keys in the boxes such that at l... | math | {
"style": "rule",
"ground_truth": "35280"
} | {
"split": "train",
"index": 22465,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a rectangular coordinate system, a point whose coordinates are both integers is called a lattice point. How many lattice points \\((x, y)\\) satisfy the inequality \\((|x|-1)^{2} + (|y|-1)^{2} < 2\\)? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 22466,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $(-1)^1+(-1)^2+\\cdots+(-1)^{2006}$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 22467,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the degree measure of the least positive angle $\\theta$ for which\n\\[\\cos 5^\\circ = \\sin 25^\\circ + \\sin \\theta.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35^\\circ"
} | {
"split": "train",
"index": 22468,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The value of the expression \\[(2^{1004}+5^{1005})^2-(2^{1004}-5^{1005})^2\\]is $k\\cdot10^{1004}$ for some positive integer $k$. What is $k$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 22469,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the positive integer $n$ such that\n\\[\\arctan\\frac {1}{3} + \\arctan\\frac {1}{4} + \\arctan\\frac {1}{5} + \\arctan\\frac {1}{n} = \\frac {\\pi}{4}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 22470,
"solution": "Adding a series of angles is the same as multiplying the complex numbers whose arguments they are. In general, $\\arctan\\frac{1}{n}$, is the argument of $n+i$. The sum of these angles is then just the argument of the product\n\\[(3+i)(4+i)(5+i)(n+i)\\]\n and expan... |
deepscaler | [
{
"role": "user",
"content": "Two real numbers are selected independently at random from the interval $[-15, 15]$. The product of those numbers is considered only if both numbers are outside the interval $[-5, 5]$. What is the probability that the product of those numbers, when considered, is greater than z... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"split": "train",
"index": 22471,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The skeletal structure of coronene, a hydrocarbon with the chemical formula $\\mathrm{C}_{24} \\mathrm{H}_{12}$, is shown below. Each line segment between two atoms is at least a single bond. However, since each carbon (C) requires exactly four bonds connected to it and each hy... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 22472,
"solution": "Note that each carbon needs exactly one double bond. Label the six carbons in the center $1,2,3,4,5,6$ clockwise. We consider how these six carbons are double-bonded. If a carbon in the center is not double-bonded to another carbon in the center, it must double-bon... |
deepscaler | [
{
"role": "user",
"content": "A cube with an edge length of 1 and its circumscribed sphere intersect with a plane to form a cross section that is a circle and an inscribed equilateral triangle. What is the distance from the center of the sphere to the plane of the cross section? Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{6}"
} | {
"split": "train",
"index": 22473,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Inside a convex 7-sided polygon, we mark 10 points so that in the set $H$ consisting of the 7 vertices of the polygon and the marked points, no 3 points are collinear. Then we triangulate the heptagon in such a way that the set $H$ is exactly the set of the vertices of the tria... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 22474,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three primes $p, q$, and $r$ satisfy $p + q = r$ and $1 < p < q$. Then $p$ equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 22475,
"solution": "1. **Identify the nature of the primes**: We know $p, q, r$ are primes and $p + q = r$. Since $r$ is a prime and primes (except $2$) are odd, the sum $p + q$ must be odd.\n\n2. **Analyze the parity of $p$ and $q$**: \n - If both $p$ and $q$ were odd, their sum $p... |
deepscaler | [
{
"role": "user",
"content": "In a class of $40$ students, $18$ said they liked apple pie, $15$ said they liked chocolate cake, and $12$ said they did not like either. How many students in the class liked both? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 22476,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{A} = \\begin{pmatrix} 2 & 3 \\\\ 0 & 1 \\end{pmatrix}.$ Find $\\mathbf{A}^{20} - 2 \\mathbf{A}^{19}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 0 & 3 \\\\ 0 & -1 \\end{pmatrix}"
} | {
"split": "train",
"index": 22477,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$, the sum of its first $n$ terms is $S_n$. It is known that $S_8 \\leq 6$ and $S_{11} \\geq 27$. Determine the minimum value of $S_{19}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "133"
} | {
"split": "train",
"index": 22478,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The minimum and maximum values of the function $f(x)=\\cos 2x+\\sin x+1$ are $\\_\\_\\_\\_\\_\\_$ and $\\_\\_\\_\\_\\_\\_$, respectively. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{8}"
} | {
"split": "train",
"index": 22479,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a science project, Sammy observed a chipmunk and squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer ho... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 22480,
"solution": "1. **Define Variables:**\n Let $x$ be the number of holes the chipmunk dug. Each hole contains 3 acorns, so the total number of acorns the chipmunk hid is $3x$.\n Let $y$ be the number of holes the squirrel dug. Each hole contains 4 acorns, so the total number ... |
deepscaler | [
{
"role": "user",
"content": "Given a cubic polynomial $ 60 x^3 - 80x^2 + 24x - 2 $, which has three roots $\\alpha, \\beta, \\gamma$, all distinct and lying in the interval $(0, 1)$. Find the sum \n\\[\n\\frac{1}{1-\\alpha} + \\frac{1}{1-\\beta} + \\frac{1}{1-\\gamma}.\n\\] Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 22481,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Chandra has four bowls. Each one is a different color (red, blue, yellow, green). She also has exactly one glass the same color as each bowl. If she chooses a bowl and a glass from the cupboard, how many pairings are possible? One such pairing is a blue bowl and a yellow gl... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 22482,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x$ is a real number and $\\lceil x \\rceil = 9,$ how many possible values are there for $\\lceil x^2 \\rceil$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 22483,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(A B C\\) be an acute triangle with circumcenter \\(O\\) such that \\(A B=4, A C=5\\), and \\(B C=6\\). Let \\(D\\) be the foot of the altitude from \\(A\\) to \\(B C\\), and \\(E\\) be the intersection of \\(A O\\) with \\(B C\\). Suppose that \\(X\\) is on \\(B C\\) bet... | math | {
"style": "rule",
"ground_truth": "\\frac{96}{41}"
} | {
"split": "train",
"index": 22484,
"solution": "Let \\(A X\\) intersect the circumcircle of \\(\\triangle A B C\\) again at \\(K\\). Let \\(O Y\\) intersect \\(A K\\) and \\(B C\\) at \\(T\\) and \\(L\\), respectively. We have \\(\\angle L O A=\\angle O Y X=\\angle T D X=\\angle L A K\\), so \\(A L\\) is tangent... |
deepscaler | [
{
"role": "user",
"content": "In how many ways can the natural numbers from 1 to 10 (each used exactly once) be arranged in a $2 \\times 5$ table so that the sum of the numbers in each of the five columns is odd? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "460800"
} | {
"split": "train",
"index": 22485,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A table can seat 6 people. Two tables joined together can seat 10 people. Three tables joined together can seat 14 people. Following this pattern, if 10 tables are arranged in two rows with 5 tables in each row, how many people can sit? Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 22486,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In February 1983, $789$ millimeters of rain fell in Jorhat, India. What was the average rainfall in millimeters per hour during that particular month?\nA) $\\frac{789}{672}$\nB) $\\frac{789 \\times 28}{24}$\nC) $\\frac{789 \\times 24}{28}$\nD) $\\frac{28 \\times 24}{789}$\nE) $... | math | {
"style": "rule",
"ground_truth": "\\frac{789}{672}"
} | {
"split": "train",
"index": 22487,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any positive integer $a$ , define $M(a)$ to be the number of positive integers $b$ for which $a+b$ divides $ab$ . Find all integer(s) $a$ with $1\\le a\\le 2013$ such that $M(a)$ attains the largest possible value in the range of $a$ . Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "1680"
} | {
"split": "train",
"index": 22488,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the absolute values of the terms of a finite arithmetic progression is 100. If all its terms are increased by 1 or all its terms are increased by 2, the sum of the absolute values of the terms of the resulting progression will also be 100. What values can the quantit... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 22489,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum value of the positive real number \\( A \\) such that for any real numbers \\( x, y, z \\), the inequality\n$$\nx^{4} + y^{4} + z^{4} + x^{2} y z + x y^{2} z + x y z^{2} - A(x y + y z + z x)^{2} \\geq 0\n$$\nholds. Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 22490,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three equally spaced parallel lines intersect a circle, creating three chords of lengths 38, 38, and 34. What is the distance between two adjacent parallel lines? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 22491,
"solution": "1. **Identify the setup and apply Stewart's Theorem**: We are given three equally spaced parallel lines intersecting a circle, creating three chords of lengths 38, 38, and 34. Let's denote the center of the circle as $O$, and the points where the chords intersect t... |
deepscaler | [
{
"role": "user",
"content": "Let $\\varphi(n)$ denote the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Let $S$ be the set of positive integers $n$ such that $\\frac{2 n}{\\varphi(n)}$ is an integer. Compute the sum $\\sum_{n \\in S} \\frac{1}{n}$. Let's think ste... | math | {
"style": "rule",
"ground_truth": "\\frac{10}{3}"
} | {
"split": "train",
"index": 22492,
"solution": "Let $T_{n}$ be the set of prime factors of $n$. Then $\\frac{2 n}{\\phi(n)}=2 \\prod_{p \\in T} \\frac{p}{p-1}$. We can check that this is an integer for the following possible sets: $\\varnothing,\\{2\\},\\{3\\},\\{2,3\\},\\{2,5\\},\\{2,3,7\\}$. For each set $T$, ... |
deepscaler | [
{
"role": "user",
"content": "The center of the circle with equation $x^2+y^2=-2x-10y-16$ is the point $(x,y)$. What is $x+y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 22493,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\left\\{a_{n}\\right\\}$ where $a_{1}=1$, $a_{2}=2$, and $a_{n} a_{n+1} a_{n+2}=a_{n}+a_{n+1}+a_{n+2}$ for all $n$, and $a_{n+1} a_{n+2} \\neq 1$, find the sum $S_{1999}=\\sum_{n=1}^{1999} a_{n}$. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "3997"
} | {
"split": "train",
"index": 22494,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $b^{2}=ac$, and $a+c=3$, $\\cos B= \\frac{3}{4}$, then $\\overset{→}{AB} \\cdot \\overset{→}{BC} =$______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"split": "train",
"index": 22495,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A broken line consists of $31$ segments. It has no self intersections, and its start and end points are distinct. All segments are extended to become straight lines. Find the least possible number of straight lines. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 22496,
"solution": "\nLet us consider a broken line made up of 31 segments with no self-intersections, where the start and end points are distinct. Each segment of the broken line can be extended indefinitely to form a straight line. The problem asks us to find the least possible numb... |
deepscaler | [
{
"role": "user",
"content": "When the greatest common divisor and least common multiple of two integers are multiplied, the product is 180. How many different values could be the greatest common divisor of the two integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 22497,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ with vertices $A(-2, 0)$, $B(1, 4)$ and $C(-3, 2)$ is reflected over the $y$-axis to form triangle $A'B'C'$. What is the length of a segment drawn from $C$ to $C'$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 22498,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $m$ and $n$ are two non-coincident lines, and $\\alpha$, $\\beta$, $\\gamma$ are three pairwise non-coincident planes, consider the following four propositions: \n$(1)$ If $m \\perp \\alpha$ and $m \\perp \\beta$, then $\\alpha \\parallel \\beta$; \n$(2)$ If $\\alp... | math | {
"style": "rule",
"ground_truth": "(2)(3)(4)"
} | {
"split": "train",
"index": 22499,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different routes are there from point $A$ to point $B$ in a 3x3 grid (where you can only move to the right or down along the drawn segments)?\n\n[asy]\nunitsize(0.09inch);\ndraw((0,0)--(15,0)--(15,15)--(0,15)--cycle);\ndraw((5,0)--(5,15));\ndraw((10,0)--(10,15));\ndraw... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 22500,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The greatest common divisor of 15 and some number between 75 and 90 is 5. What is the number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 22501,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Twelve congruent disks are placed on a circle $C^{}_{}$ of radius 1 in such a way that the twelve disks cover $C^{}_{}$, no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure ... | math | {
"style": "rule",
"ground_truth": "135"
} | {
"split": "train",
"index": 22502,
"solution": "We wish to find the radius of one circle, so that we can find the total area.\n Notice that for them to contain the entire circle, each pair of circles must be tangent on the larger circle. Now consider two adjacent smaller circles. This means that the line connect... |
deepscaler | [
{
"role": "user",
"content": "Given that a set of $n$ people participate in an online video soccer tournament, the statistics from the tournament reveal: The average number of complete teams wholly contained within randomly chosen subsets of $10$ members equals twice the average number of complete teams fou... | math | {
"style": "rule",
"ground_truth": "450"
} | {
"split": "train",
"index": 22503,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of solutions to:\n\\[\\sin x = \\left( \\frac{1}{3} \\right)^x\\]\non the interval $(0,50 \\pi)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 22504,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of $245_8$, $174_8$, and $354_8$ in base 8. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1015_8"
} | {
"split": "train",
"index": 22505,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ that satisfies the following two properties:\n\n1. $n$ has exactly 144 distinct positive divisors.\n2. Among the positive divisors of $n$, there are ten consecutive integers. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "110880"
} | {
"split": "train",
"index": 22506,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The expression $3y^2-y-24$ can be written as $(3y + a)(y + b),$ where $a$ and $b$ are integers. What is $a - b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 22507,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The average of two 2-digit positive integers is equal to the decimal number obtained by writing one of the two-digit integers before the decimal point and the other two-digit integer after the decimal point. What is the smaller of the two integers? Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 22508,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any real number \\( x \\), let \\( \\lfloor x \\rfloor \\) denote the greatest integer less than or equal to \\( x \\), and let \\( \\{ x \\} \\) denote the fractional part of \\( x \\). Then\n$$\n\\left\\{\\frac{2014}{2015}\\right\\}+\\left\\{\\frac{2014^{2}}{2015}\\right\... | math | {
"style": "rule",
"ground_truth": "1007"
} | {
"split": "train",
"index": 22509,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We plotted the graph of the function \\( f(x) = \\frac{1}{x} \\) in the coordinate system. How should we choose the new, still equal units on the axes, if we want the curve to become the graph of the function \\( g(x) = \\frac{2}{x} \\)? Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 22510,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow {m}$=(cosx, sinx) and $\\overrightarrow {n}$=(cosx, $\\sqrt {3}$cosx), where x∈R, define the function f(x) = $\\overrightarrow {m}$$\\cdot \\overrightarrow {n}$+ $\\frac {1}{2}$.\n(1) Find the analytical expression and the interval where the func... | math | {
"style": "rule",
"ground_truth": "\\sqrt {3}-1"
} | {
"split": "train",
"index": 22511,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the smallest positive integer $n$ such that $5^n\\equiv n^5\\pmod 3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 22512,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The American Mathematics College is holding its orientation for incoming freshmen. The incoming freshman class contains fewer than $500$ people. When the freshmen are told to line up in columns of $23$, $22$ people are in the last column. When the freshmen are told to line up i... | math | {
"style": "rule",
"ground_truth": "413"
} | {
"split": "train",
"index": 22513,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers $1000^{2}, 1001^{2}, 1002^{2}, \\ldots$ have their last two digits discarded. How many of the first terms of the resulting sequence form an arithmetic progression? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 22514,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiao Ming set a six-digit passcode for his phone using the numbers $0-9$, but he forgot the last digit. The probability that Xiao Ming can unlock his phone with just one try is ____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{10}"
} | {
"split": "train",
"index": 22515,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define $ a \\circledast b = a + b-2ab $ . Calculate the value of $$ A=\\left( ...\\left(\\left(\\frac{1}{2014}\\circledast \\frac{2}{2014}\\right)\\circledast\\frac{3}{2014}\\right)...\\right)\\circledast\\frac{2013}{2014} $$ Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 22516,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The length of the curve given by the parametric equations \\(\\left\\{\\begin{array}{l}x=2 \\cos ^{2} \\theta \\\\ y=3 \\sin ^{2} \\theta\\end{array}\\right.\\) (where \\(\\theta\\) is the parameter) is? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\sqrt{13}"
} | {
"split": "train",
"index": 22517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a right-angled triangle, the sum of the squares of the three side lengths is 1800. What is the length of the hypotenuse of this triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 22518,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many natural numbers between 200 and 400 are divisible by 8? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 22519,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\arccos(\\sin 3)$, where all functions are in radians. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 - \\frac{\\pi}{2}"
} | {
"split": "train",
"index": 22520,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the non-decreasing sequence of odd integers $\\{a_1,a_2,a_3,\\ldots \\}=\\{1,3,3,3,5,5,5,5,5,\\ldots \\}$ each odd positive integer $k$ appears $k$ times. It is a fact that there are integers $b, c$, and $d$ such that for all positive integers $n$, $a_n=b\\lfloor \\sqrt{n+c}... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 22521,
"solution": "1. **Understanding the sequence**: The sequence $\\{a_1, a_2, a_3, \\ldots\\}$ is defined such that each odd integer $k$ appears exactly $k$ times. For example, $1$ appears once, $3$ appears three times, $5$ appears five times, and so on.\n\n2. **Form of the sequen... |
deepscaler | [
{
"role": "user",
"content": "Given the function $y=ax^2+bx+c$, where $a, b, c \\in \\{0, 1, 2, 3, 4\\}$, calculate the total number of different quadratic functions. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 22522,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The fictional country of Isoland uses a 6-letter license plate system using the same 12-letter alphabet as the Rotokas of Papua New Guinea (A, E, G, I, K, O, P, R, T, U, V). Design a license plate that starts with a vowel (A, E, I, O, U), ends with a consonant (G, K, P, R, T, V... | math | {
"style": "rule",
"ground_truth": "151200"
} | {
"split": "train",
"index": 22523,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On an indoor circular track of circumference 50 feet, Joneal starts at point $S$, runs in a counterclockwise direction, and then stops when he has run exactly one mile (5280 feet). On which quarter of the circle, $A$, $B$, $C$ or $D$, did Joneal stop?\n\n[asy]\nimport graph;\nd... | math | {
"style": "rule",
"ground_truth": "C"
} | {
"split": "train",
"index": 22524,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The integer $N$ is the smallest positive integer that is a multiple of 2024, has more than 100 positive divisors (including 1 and $N$), and has fewer than 110 positive divisors (including 1 and $N$). What is the sum of the digits of $N$? Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 22525,
"solution": "Throughout this solution, we use the fact that if $N$ is a positive integer with $N>1$ and $N$ has prime factorization $p_{1}^{a_{1}} p_{2}^{a_{2}} \\cdots p_{m}^{a_{m}}$ for some distinct prime numbers $p_{1}, p_{2}, \\ldots, p_{m}$ and positive integers $a_{1}, a... |
deepscaler | [
{
"role": "user",
"content": "Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the liquid surfaces are $3$ cm and $6$ cm. Into each cone is dropped a spherical marble of radius $1$ cm, which sinks to the bottom and is... | math | {
"style": "rule",
"ground_truth": "4:1"
} | {
"split": "train",
"index": 22526,
"solution": "\n#### Initial Scenario\nLet the heights of the narrow cone and the wide cone be \\( h_1 \\) and \\( h_2 \\), respectively. The volumes of the cones before the marbles are dropped are given by:\n- Narrow Cone: \\( V_1 = \\frac{1}{3} \\pi r_1^2 h_1 = \\frac{1}{3} \\... |
deepscaler | [
{
"role": "user",
"content": "Given that $f(x) = x^{2}-2x+5$ and $g(x) =x+3$, what is the value of $f(g(5)) -g(f(5))$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 22527,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point is randomly thrown onto the interval \\( [11; 18] \\) and let \\( k \\) be the resulting value. Find the probability that the roots of the equation \\((k^2 + 2k - 99)x^2 + (3k - 7)x + 2 = 0\\) satisfy the condition \\( x_1 \\leq 2x_2 \\). Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 22528,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the positive difference between the $2000^{\\mathrm{th}}$ term and the $2005^{\\mathrm{th}}$ term of the arithmetic sequence $-8,$ $-2,$ $4,$ $10,$ $\\ldots$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 22529,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Inside a right circular cone with base radius $5$ and height $12$ are three congruent spheres with radius $r$. Each sphere is tangent to the other two spheres and also tangent to the base and side of the cone. What is $r$? Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "\\frac{90-40\\sqrt{3}}{11}"
} | {
"split": "train",
"index": 22530,
"solution": "1. **Identify the Geometry of the Problem:**\n The problem involves three congruent spheres inside a right circular cone, each sphere tangent to the others and to the base and side of the cone. The cone has a base radius of $5$ and a height of $12$.\n\n2. **Spher... |
deepscaler | [
{
"role": "user",
"content": "Given that the terminal side of angle $θ$ is symmetric to the terminal side of a $480^\\circ$ angle with respect to the $x$-axis, and point $P(x,y)$ is on the terminal side of angle $θ$ (not the origin), then the value of $\\frac{xy}{{x}^2+{y}^2}$ is equal to __. Let's think st... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 22531,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 203 students in the third grade, which is 125 fewer than the fourth grade. How many students are there in total in the third and fourth grades? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "531"
} | {
"split": "train",
"index": 22532,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $325_{8}-237_{8}$. Express your answer in base $8$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "66_8"
} | {
"split": "train",
"index": 22533,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5 minutes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 22534,
"solution": "1. **Determine individual rates**: \n - Cagney's rate of frosting cupcakes is 1 cupcake per 20 seconds.\n - Lacey's rate of frosting cupcakes is 1 cupcake per 30 seconds.\n\n2. **Calculate combined rate**:\n - The combined rate of frosting cupcakes when Cagne... |
deepscaler | [
{
"role": "user",
"content": "I have written a strictly increasing sequence of six positive integers, such that each number (besides the first) is a multiple of the one before it, and the sum of all six numbers is 79 . What is the largest number in my sequence? Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 22535,
"solution": "If the fourth number is \\geq 12, then the last three numbers must sum to at least $12+$ $2 \\cdot 12+2^{2} \\cdot 12=84>79$. This is impossible, so the fourth number must be less than 12. Then the only way we can have the required divisibilities among the first fo... |
deepscaler | [
{
"role": "user",
"content": "It is given that one root of $2x^2 + rx + s = 0$, with $r$ and $s$ real numbers, is $3+2i (i = \\sqrt{-1})$. The value of $s$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 22536,
"solution": "1. **Identify the nature of the roots:** Given that one root of the quadratic equation $2x^2 + rx + s = 0$ is $3+2i$, and knowing that the coefficients $r$ and $s$ are real numbers, the other root must be the complex conjugate of $3+2i$. Therefore, the other root i... |
deepscaler | [
{
"role": "user",
"content": "Find all integers $n$ satisfying $n \\geq 2$ and $\\dfrac{\\sigma(n)}{p(n)-1} = n$, in which $\\sigma(n)$ denotes the sum of all positive divisors of $n$, and $p(n)$ denotes the largest prime divisor of $n$. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 22537,
"solution": "\nGiven the problem, we need to find all integers \\( n \\) such that \\( n \\geq 2 \\) and \n\n\\[\n\\frac{\\sigma(n)}{p(n) - 1} = n,\n\\]\n\nwhere \\(\\sigma(n)\\) denotes the sum of all positive divisors of \\(n\\), and \\(p(n)\\) denotes the largest prime divis... |
deepscaler | [
{
"role": "user",
"content": "In $\\Delta ABC$, it is known that $\\overrightarrow{AB} \\cdot \\overrightarrow{AC} + 2\\overrightarrow{BA} \\cdot \\overrightarrow{BC} = 3\\overrightarrow{CA} \\cdot \\overrightarrow{CB}$. The minimum value of $\\cos C$ is ______. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "\\dfrac{ \\sqrt{2}}{3}"
} | {
"split": "train",
"index": 22538,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A quadrilateral has vertices $P(a,b)$, $Q(b,a)$, $R(-a, -b)$, and $S(-b, -a)$, where $a$ and $b$ are integers with $a>b>0$. The area of $PQRS$ is $16$. What is $a+b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 22539,
"solution": "1. **Identify the slopes of the sides to determine the shape of the quadrilateral**:\n - The slope of line segment $PQ$ is calculated as:\n \\[\n \\text{slope of } PQ = \\frac{Q_y - P_y}{Q_x - P_x} = \\frac{a - b}{b - a} = -1\n \\]\n - The slope of ... |
deepscaler | [
{
"role": "user",
"content": "A regular pentagon has an area numerically equal to its perimeter, and a regular hexagon also has its area numerically equal to its perimeter. Compare the apothem of the pentagon with the apothem of the hexagon. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "1.06"
} | {
"split": "train",
"index": 22540,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The degree measure of the sum of the interior angles of a convex polygon with $n$ sides is 1800. What is the degree measure of the sum of the interior angles of a convex polygon with $n + 2$ sides? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2160"
} | {
"split": "train",
"index": 22541,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five persons wearing badges with numbers $1, 2, 3, 4, 5$ are seated on $5$ chairs around a circular table. In how many ways can they be seated so that no two persons whose badges have consecutive numbers are seated next to each other? (Two arrangements obtained by rotation ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 22542,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be the centroid of triangle $ABC$. Let $G_1$, $G_2$, and $G_3$ be the centroids of triangles $PBC$, $PCA$, and $PAB$, respectively. The area of triangle $ABC$ is 24. Find the area of triangle $G_1 G_2 G_3$. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}"
} | {
"split": "train",
"index": 22543,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, the side \\(AB\\) of \\(\\triangle ABC\\) is divided into \\(n\\) equal parts (\\(n > 1990\\)). Through the \\(n-1\\) division points, lines parallel to \\(BC\\) are drawn intersecting \\(AC\\) at points \\(B_i, C_i\\) respectively for \\(i=1, 2, 3, \\cdots, n-1... | math | {
"style": "rule",
"ground_truth": "1: 3979"
} | {
"split": "train",
"index": 22544,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ and the parabola $y^{2}=8x$ share a common focus $F$, and one of the intersection points of the two curves is $P$. If $|PF|=4$, calculate the eccentricity of the hyperbola. Let's think step by step... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}+1"
} | {
"split": "train",
"index": 22545,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The greatest prime number that is a divisor of $16{,}384$ is $2$ because $16{,}384 = 2^{14}$. What is the sum of the digits of the greatest prime number that is a divisor of $16{,}383$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 22546,
"solution": "1. **Identify the number to factorize**: We start with the number $16{,}383$. We know that $16{,}384 = 2^{14}$, so $16{,}383 = 2^{14} - 1$.\n\n2. **Factorize $16{,}383$ using difference of squares**:\n \\[\n 16{,}383 = 2^{14} - 1 = (2^7)^2 - 1^2 = (2^7 + 1)(2^7... |
deepscaler | [
{
"role": "user",
"content": "A class of 10 students took a math test. Each problem was solved by exactly 7 of the students. If the first nine students each solved 4 problems, how many problems did the tenth student solve? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 22547,
"solution": "Suppose the last student solved $n$ problems, and the total number of problems on the test was $p$. Then the total number of correct solutions written was $7 p$ (seven per problem), and also equal to $36+n$ (the sum of the students' scores), so $p=(36+n) / 7$. The ... |
deepscaler | [
{
"role": "user",
"content": "Find all three-digit numbers $\\overline{\\Pi B \\Gamma}$, consisting of distinct digits $\\Pi, B$, and $\\Gamma$, for which the following equality holds: $\\overline{\\Pi B \\Gamma} = (\\Pi + B + \\Gamma) \\times (\\Pi + B + \\Gamma + 1)$. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "156"
} | {
"split": "train",
"index": 22548,
"solution": ""
} |
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