data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be an equilateral triangle. Let $P$ and $S$ be points on $AB$ and $AC$ , respectively, and let $Q$ and $R$ be points on $BC$ such that $PQRS$ is a rectangle. If $PQ = \\sqrt3 PS$ and the area of $PQRS$ is $28\\sqrt3$ , what is the length of $PC$... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{7}"
} | {
"split": "train",
"index": 6300,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of rearrangements of the letters in the word MATHMEET that begin and end with the same letter such as TAMEMHET. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "540"
} | {
"split": "train",
"index": 6301,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$908 \\times 501 - [731 \\times 1389 - (547 \\times 236 + 842 \\times 731 - 495 \\times 361)] =$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5448"
} | {
"split": "train",
"index": 6302,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If an integer of two digits is $k$ times the sum of its digits, the number formed by interchanging the digits is the sum of the digits multiplied by Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11-k"
} | {
"split": "train",
"index": 6303,
"solution": "1. **Define the number and its properties**: Let the two-digit integer be represented as $n = 10a + b$, where $a$ and $b$ are the tens and units digits respectively. According to the problem, this number $n$ is $k$ times the sum of its digits. Therefore, we have the... |
deepscaler | [
{
"role": "user",
"content": "At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women an... | math | {
"style": "rule",
"ground_truth": "88"
} | {
"split": "train",
"index": 6304,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "30 students from 5 grades participated in answering 40 questions. Each student answered at least 1 question. Every two students from the same grade answered the same number of questions, and students from different grades answered a different number of questions. How many stude... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 6305,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x = 202$ and $x^3y - 4x^2y + 2xy = 808080$, what is the value of $y$? 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": 6306,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The twelve-sided figure shown has been drawn on $1 \\text{ cm}\\times 1 \\text{ cm}$ graph paper. What is the area of the figure in $\\text{cm}^2$? \n[asy] unitsize(8mm); for (int i=0; i<7; ++i) { draw((i,0)--(i,7),gray); draw((0,i+1)--(7,i+1),gray); } draw((1,3)--(2,4)--(2... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 6307,
"solution": "To find the area of the twelve-sided figure, we can break it down into simpler shapes whose areas we can easily calculate. The figure is composed of several unit squares and triangles on a $1 \\text{ cm} \\times 1 \\text{ cm}$ grid.\n\n1. **Count the full unit squar... |
deepscaler | [
{
"role": "user",
"content": "Given a 4-inch cube constructed from 64 smaller 1-inch cubes, with 50 red and 14 white cubes, arrange these cubes such that the white surface area exposed on the larger cube is minimized, and calculate the fraction of the total surface area of the 4-inch cube that is white. Let... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"split": "train",
"index": 6308,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using the digits 0 to 9, how many three-digit even numbers can be formed without repeating any digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "360"
} | {
"split": "train",
"index": 6309,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One of Euler's conjectures was disproved in the 1960s by three American mathematicians when they showed there was a positive integer such that \\[133^5+110^5+84^5+27^5=n^{5}.\\] Find the value of $n$. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 6310,
"solution": "Note that $n$ is even, since the LHS consists of two odd and two even numbers. By Fermat's Little Theorem, we know $n^5\\equiv n\\pmod{5}.$ Hence, \\[n\\equiv3+0+4+2\\equiv4\\pmod{5}.\\] Continuing, we examine the equation modulo $3,$ \\[n\\equiv1-1+0+0\\equiv0\\pmo... |
deepscaler | [
{
"role": "user",
"content": "There are 4 male students and 2 female students taking the photo. The male student named \"甲\" cannot stand at either end, and the female students must stand next to each other. Find the number of ways to arrange the students. Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 6311,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When simplified, $\\log_{16}{32} \\cdot \\log_{16}{\\frac{1}{2}}$ becomes:\n**A)** $-\\frac{1}{4}$\n**B)** $-\\frac{5}{16}$\n**C)** $\\frac{5}{16}$\n**D)** $-\\frac{1}{16}$\n**E)** $0$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{5}{16}"
} | {
"split": "train",
"index": 6312,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given positive integers \\( a, b, \\) and \\( c \\) such that \\( a < b < c \\). If the product of any two numbers minus 1 is divisible by the third number, what is \\( a^{2} + b^{2} + c^{2} \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 6313,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given in $\\bigtriangleup ABC$, $AB = 75$, and $AC = 120$. A circle with center $A$ and radius $AB$ intersects $\\overline{BC}$ at points $B$ and $X$. Moreover, $\\overline{BX}$ and $\\overline{CX}$ have integer lengths. Find the length of $BC$. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "117"
} | {
"split": "train",
"index": 6314,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Which one of the following is not equivalent to $0.000000375$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8} \\times 10^{-7}"
} | {
"split": "train",
"index": 6315,
"solution": "To find which option is not equivalent to $0.000000375$, we first convert $0.000000375$ into scientific notation:\n\n1. **Convert to Scientific Notation:**\n \\[\n 0.000000375 = 3.75 \\times 10^{-7}\n \\]\n\n2. **Evaluate Each Option:**\n - **Option (A) $3.7... |
deepscaler | [
{
"role": "user",
"content": "An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 6316,
"solution": "Let $R$ denote the action where George selects a red ball and $B$ denote the action where he selects a blue one. To achieve a final state of three red balls and three blue balls, George must select two additional red balls and two additional blue balls during the fo... |
deepscaler | [
{
"role": "user",
"content": "For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 6317,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Billy Bones has two coins - a gold one and a silver one. One of them is symmetric, and the other is not. It is not known which coin is not symmetric, but it is given that the non-symmetric coin lands heads with a probability of $p = 0.6$.\n\nBilly Bones flipped the gold coin, a... | math | {
"style": "rule",
"ground_truth": "0.6"
} | {
"split": "train",
"index": 6318,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $(xOy)$, the sum of the distances from point $P$ to two points $(0,-\\sqrt{3})$ and $(0,\\sqrt{3})$ is equal to $4$. Let the trajectory of point $P$ be $C$.\n(I) Write the equation of $C$;\n(II) Given that the line $y=kx+1$ intersects $C$ at p... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{65}}{17}"
} | {
"split": "train",
"index": 6319,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the sum \\[\\lfloor \\sqrt{1} \\rfloor + \\lfloor \\sqrt{2} \\rfloor + \\lfloor \\sqrt{3} \\rfloor + \\cdots + \\lfloor \\sqrt{25} \\rfloor.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 6320,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Eric plans to compete in a triathlon. He can average $2$ miles per hour in the $\\frac{1}{4}$-mile swim and $6$ miles per hour in the $3$-mile run. His goal is to finish the triathlon in $2$ hours. To accomplish his goal what must his average speed in miles per hour, be for the... | math | {
"style": "rule",
"ground_truth": "\\frac{120}{11}"
} | {
"split": "train",
"index": 6321,
"solution": "1. **Calculate the time taken for the swim**: \n Eric's swimming speed is $2$ miles per hour and he needs to swim $\\frac{1}{4}$ mile. Using the formula for time, $t = \\frac{d}{r}$, where $d$ is distance and $r$ is rate, we find:\n \\[\n t_{\\text{swim}} = \\... |
deepscaler | [
{
"role": "user",
"content": "The five books \"Poetry,\" \"Documents,\" \"Rites,\" \"Changes,\" and \"Spring and Autumn\" all have different numbers of pages. The differences in the number of pages between the books are as follows:\n1. \"Poetry\" and \"Documents\" differ by 24 pages.\n2. \"Documents\" and \... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 6322,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers 2, 3, 5, 7, 11, 13, 17, 19 are arranged in a multiplication table, with four along the top and the other four down the left. The multiplication table is completed and the sum of the sixteen entries is tabulated. What is the largest possible sum of the sixteen entrie... | math | {
"style": "rule",
"ground_truth": "1482"
} | {
"split": "train",
"index": 6323,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of the lunar crescent shape bounded by the portion of the circle of radius 5 and center (0,0), the portion of the circle with radius 2 and center (0,2), and the line segment from (0,0) to (5,0). Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{21\\pi}{4}"
} | {
"split": "train",
"index": 6324,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify\n\\[\\frac{\\sin 10^\\circ + \\sin 20^\\circ + \\sin 30^\\circ + \\sin 40^\\circ + \\sin 50^\\circ + \\sin 60^\\circ + \\sin 70^\\circ + \\sin 80^\\circ}{\\cos 5^\\circ \\cos 10^\\circ \\cos 20^\\circ}.\\] Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "4 \\sqrt{2}"
} | {
"split": "train",
"index": 6325,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC,$ $D$ is on $\\overline{AB}$ such that $AD:DB = 3:2,$ and $E$ is on $\\overline{BC}$ such that $BE:EC = 3:2.$ If lines $DE$ and $AC$ intersect at $F,$ then find $\\frac{DE}{EF}.$ 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": 6326,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two symmetrical coins are flipped. What is the probability that both coins show numbers on their upper sides? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.25"
} | {
"split": "train",
"index": 6327,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers \\(1, 2, 3, \\ldots, 10\\) are written in some order around a circle. Peter computed the sums of all 10 triples of neighboring numbers and wrote the smallest of these sums on the board. What is the maximum possible number that could have been written on the board? L... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 6328,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many numbers should there be in a lottery for the probability of getting an ambo to be $\\frac{5}{473}$, when drawing five numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 6329,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the point $(3,6)$ is on the graph of $y=g(x)$, and $h(x)=(g(x))^2$ for all $x$, then there is one point that must be on the graph of $y=h(x)$. What is the sum of the coordinates of that point? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 6330,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Players A and B have a Go game match, agreeing that the first to win 3 games wins the match. After the match ends, assuming in a single game, the probability of A winning is 0.6, and the probability of B winning is 0.4, with the results of each game being independent. It is kno... | math | {
"style": "rule",
"ground_truth": "2.48"
} | {
"split": "train",
"index": 6331,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a row of 10 chairs, Mary and James each choose their seats at random but are not allowed to sit in the first or the last chair (chairs #1 and #10). What is the probability that they do not sit next to each other? Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 6332,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each page number of a 488-page book is printed one time in the book. The first page is page 1 and the last page is page 488. When printing all of the page numbers, how many more 4's are printed than 8's? Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 6333,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a function \\( g \\) that maps nonnegative integers to real numbers, with \\( g(1) = 1 \\), and for all nonnegative integers \\( m \\ge n \\),\n\\[ g(m + n) + g(m - n) = \\frac{g(3m) + g(3n)}{3} \\]\nFind the sum of all possible values of \\( g(10) \\). Let's think ste... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 6334,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the value of $\\frac{(2210-2137)^2 + (2137-2028)^2}{64}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "268.90625"
} | {
"split": "train",
"index": 6335,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system, with the origin as the pole and the positive x-axis as the polar axis, the polar equation of line $l$ is $$ρ\\cos(θ+ \\frac {π}{4})= \\frac { \\sqrt {2}}{2}$$, and the parametric equation of curve $C$ is $$\\begin{cases} x=5+\\cos\\theta \\\\... | math | {
"style": "rule",
"ground_truth": "2+ \\sqrt {34}"
} | {
"split": "train",
"index": 6336,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $(xOy)$, the equation of circle $C$ is $((x-4)^{2}+y^{2}=1)$. If there exists at least one point on the line $y=kx-3$ such that a circle with this point as the center and $2$ as the radius intersects with circle $C$, then the maximum value of ... | math | {
"style": "rule",
"ground_truth": "\\frac{24}{7}"
} | {
"split": "train",
"index": 6337,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the polynomial\n\\[P(x)=x^3+3x^2+6x+10.\\]\nLet its three roots be $a$ , $b$ , $c$ . Define $Q(x)$ to be the monic cubic polynomial with roots $ab$ , $bc$ , $ca$ . Compute $|Q(1)|$ .\n\n*Proposed by Nathan Xiong* Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 6338,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a cyclic quadrilateral with sides $AB$, $BC$, $CD$, and $DA$. The side lengths are distinct integers less than $10$ and satisfy $BC + CD = AB + DA$. Find the largest possible value of the diagonal $BD$.\nA) $\\sqrt{93}$ \nB) $\\sqrt{\\frac{187}{2}}$\nC) $\\sqrt{\\... | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\frac{191}{2}}"
} | {
"split": "train",
"index": 6339,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a bag of marbles, $\\frac{3}{5}$ of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red? Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{7}"
} | {
"split": "train",
"index": 6340,
"solution": "1. **Identify the fraction of blue and red marbles initially:**\n Given that $\\frac{3}{5}$ of the marbles are blue, the fraction of red marbles is the remainder when subtracted from 1 (since the total fraction must sum up to 1). Thus, the fraction of red marbles ... |
deepscaler | [
{
"role": "user",
"content": "Solve for $y$: $4+2.3y = 1.7y - 20$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-40"
} | {
"split": "train",
"index": 6341,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The perimeter of the triangles that make up rectangle \\(ABCD\\) is 180 cm. \\(BK = KC = AE = ED\\), \\(AK = KD = 17 \\) cm. Find the perimeter of a rectangle, one of whose sides is twice as long as \\(AB\\), and the other side is equal to \\(BC\\). Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 6342,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a sale, a store owner reduces the price of a $\\$10$ scarf by $30\\%$. Later the price is lowered again, this time by $50\\%$ of the reduced price. What is the current price, in dollars? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\$3.50"
} | {
"split": "train",
"index": 6343,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circles with centers $A$, $B$, and $C$ each have radius $r$, where $1 < r < 2$. The distance between each pair of centers is $2$. If $B'$ is the point of intersection of circle $A$ and circle $C$ which is outside circle $B$, and if $C'$ is the point of intersection of circle $A... | math | {
"style": "rule",
"ground_truth": "1+\\sqrt{3(r^2-1)}"
} | {
"split": "train",
"index": 6344,
"solution": "1. **Setting up the problem**: We are given three circles with centers at $A$, $B$, and $C$, each having radius $r$ where $1 < r < 2$. The distance between each pair of centers is $2$. We need to find the length of the line segment $B'C'$, where $B'$ and $C'$ are po... |
deepscaler | [
{
"role": "user",
"content": "Let $x,$ $y,$ $z$ be real numbers such that $0 \\leq x, y, z \\leq 1$. Find the maximum value of\n\\[\n\\frac{1}{(2 - x)(2 - y)(2 - z)} + \\frac{1}{(2 + x)(2 + y)(2 + z)}.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{12}{27}"
} | {
"split": "train",
"index": 6345,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jeff decides to play with a Magic 8 Ball. Each time he asks it a question, it has a 1/3 chance of giving him a positive answer. If he asks it 7 questions, what is the probability that it gives him exactly 3 positive answers? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "\\frac{560}{2187}"
} | {
"split": "train",
"index": 6346,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a circular keychain, I need to arrange six keys. Two specific keys, my house key and my car key, must always be together. How many different ways can I arrange these keys considering that arrangements can be rotated or reflected? Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 6347,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The coefficient of \\\\(x^4\\\\) in the expansion of \\\\((1+x+x^2)(1-x)^{10}\\\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "135"
} | {
"split": "train",
"index": 6348,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers \\(a_{1}, a_{2}, \\ldots, a_{n}\\) are such that the sum of any seven consecutive numbers is negative, and the sum of any eleven consecutive numbers is positive. What is the largest possible \\(n\\) for which this is true? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 6349,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$, it is known that $\\frac{a_{11}}{a_{10}} + 1 < 0$. Determine the maximum value of $n$ for which $S_n > 0$ holds. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 6350,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Alice's car averages 30 miles per gallon of gasoline, and Bob's car averages 20 miles per gallon of gasoline, and Alice drives 120 miles and Bob drives 180 miles, calculate the combined rate of miles per gallon of gasoline for both cars. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "\\frac{300}{13}"
} | {
"split": "train",
"index": 6351,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $3, n$, and $n+1$ cents, $115$ cents is the greatest postage that cannot be formed. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 6352,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the smallest positive integer that does not appear in any problem statement on any round at HMMT November 2023. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 6353,
"solution": "The number 22 does not appear on any round. On the other hand, the numbers 1 through 21 appear as follows. \\begin{tabular}{c|c|c} Number & Round & Problem \\\\ \\hline 1 & Guts & 21 \\\\ 2 & Guts & 13 \\\\ 3 & Guts & 17 \\\\ 4 & Guts & 13 \\\\ 5 & Guts & 14 \\\\ 6 ... |
deepscaler | [
{
"role": "user",
"content": "Three-digit powers of $2$ and $5$ are used in this \"cross-number\" puzzle. What is the only possible digit for the outlined square?\n\\[\\begin{array}{lcl} \\textbf{ACROSS} & & \\textbf{DOWN} \\\\ \\textbf{2}.~ 2^m & & \\textbf{1}.~ 5^n \\end{array}\\] Let's think step by step... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 6354,
"solution": "To solve this problem, we need to identify the three-digit powers of $2$ and $5$ and determine the possible digits that can occupy the outlined square in the cross-number puzzle.\n\n1. **Identify three-digit powers of $5$:**\n - $5^3 = 125$\n - $5^4 = 625$\n -... |
deepscaler | [
{
"role": "user",
"content": "Pirate Bob shares his treasure with Pirate Sam in a peculiar manner. Bob first declares, ``One for me, one for you,'' keeping one coin for himself and starting Sam's pile with one coin. Then Bob says, ``Two for me, and two for you,'' adding two more coins to his pile but updati... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 6355,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The roots of the equation $x^2+kx+5 = 0$ differ by $\\sqrt{61}$. Find the greatest possible value of $k$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 6356,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One student has 6 mathematics books, and another has 8. In how many ways can they exchange three books? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1120"
} | {
"split": "train",
"index": 6357,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the scores of a math exam follow a normal distribution N(102, 4²), the percentage of scores 114 and above is _______ (Note: P(μ-σ<X≤μ+σ)=0.6826, P(μ-2σ<X≤μ+2σ)=0.9544, P(μ-3σ<X≤μ+3σ)=0.9974). Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "0.13\\%"
} | {
"split": "train",
"index": 6358,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of three consecutive even numbers has the form $87XXXXX8$. Provide the 5 missing digits. ($X$ does not necessarily represent the same digits.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "52660"
} | {
"split": "train",
"index": 6359,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Line $l_1$ has equation $4x - 3y = 2$ and passes through point $D = (-2, -3)$. Line $l_2$ has equation $y = 2$ and intersects line $l_1$ at point $E$. Line $l_3$ has a positive slope, passes through point $D$, and meets $l_2$ at point $F$. The area of $\\triangle DEF$ is $6$. W... | math | {
"style": "rule",
"ground_truth": "\\frac{25}{32}"
} | {
"split": "train",
"index": 6360,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We colored the $n^2$ unit squares of an $n\\times n$ square lattice such that in each $2\\times 2$ square, at least two of the four unit squares have the same color. What is the largest number of colors we could have used? Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "[\\frac{n^2+2n-1}{2}]"
} | {
"split": "train",
"index": 6361,
"solution": "\nTo solve the problem, we must determine the largest number of distinct colors that can be used to color an \\( n \\times n \\) square lattice, under the condition that within every \\( 2 \\times 2 \\) sub-square, at least two of the four unit squares share the sam... |
deepscaler | [
{
"role": "user",
"content": "Evaluate $(-27)^{5/3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-243"
} | {
"split": "train",
"index": 6362,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the equation: $4x^2 - (x^2 - 2x + 1) = 0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 6363,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function \\( y = \\sqrt{2x^2 + 2} \\) with its graph represented as curve \\( G \\), and the focus of curve \\( G \\) denoted as \\( F \\), two lines \\( l_1 \\) and \\( l_2 \\) pass through \\( F \\) and intersect curve \\( G \\) at points \\( A, C \\) and \\( B, D \... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 6364,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $\\angle C = 90^\\circ$, $AC = 4$, and $AB = \\sqrt{41}$. What is $\\tan B$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"split": "train",
"index": 6365,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The binomial expansion of $((x-1)^{n}(n∈N_{+}))$ has only the coefficient of the fourth term the largest, find the constant term in the binomial expansion of $(2 \\sqrt {x}- \\frac{1}{ \\sqrt {x}})^{n}$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "-160"
} | {
"split": "train",
"index": 6366,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "**p1.** Triangle $ABC$ has side lengths $AB = 3^2$ and $BC = 4^2$ . Given that $\\angle ABC$ is a right angle, determine the length of $AC$ .**p2.** Suppose $m$ and $n$ are integers such that $m^2+n^2 = 65$ . Find the largest possible value of $m-n$ .**p3.** Six m... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 6367,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are \\(n\\) girls \\(G_{1}, \\ldots, G_{n}\\) and \\(n\\) boys \\(B_{1}, \\ldots, B_{n}\\). A pair \\((G_{i}, B_{j})\\) is called suitable if and only if girl \\(G_{i}\\) is willing to marry boy \\(B_{j}\\). Given that there is exactly one way to pair each girl with a dis... | math | {
"style": "rule",
"ground_truth": "\\frac{n(n+1)}{2}"
} | {
"split": "train",
"index": 6368,
"solution": "We represent the problem as a graph with vertices \\(G_{1}, \\ldots, G_{n}, B_{1}, \\ldots, B_{n}\\) such that there is an edge between vertices \\(G_{i}\\) and \\(B_{j}\\) if and only if \\((G_{i}, B_{j})\\) is suitable, so we want to maximize the number of edges w... |
deepscaler | [
{
"role": "user",
"content": "Let $x_1=97$, and for $n>1$, let $x_n=\\frac{n}{x_{n-1}}$. Calculate the product $x_1x_2x_3x_4x_5x_6x_7x_8$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "384"
} | {
"split": "train",
"index": 6369,
"solution": "Since $x_n=\\frac{n}{x_{n-1}}$, $x_n \\cdot x_{n - 1} = n$. Setting $n = 2, 4, 6$ and $8$ in this equation gives us respectively $x_1x_2 = 2$, $x_3x_4 = 4$, $x_5x_6 = 6$ and $x_7x_8 = 8$ so \\[x_1x_2x_3x_4x_5x_6x_7x_8 = 2\\cdot4\\cdot6\\cdot8 = \\boxed{384}.\\] Noti... |
deepscaler | [
{
"role": "user",
"content": "There are $2^{10} = 1024$ possible 10-letter strings in which each letter is either an A or a B. Find the number of such strings that do not have more than 3 adjacent letters that are identical. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "548"
} | {
"split": "train",
"index": 6370,
"solution": "Let $a_{n}$ be the number of ways to form $n$-letter strings made up of As and Bs such that no more than $3$ adjacent letters are identical.\n Note that, at the end of each $n$-letter string, there are $3$ possibilities for the last letter chain: it must be either $... |
deepscaler | [
{
"role": "user",
"content": "Find the number by which the three numbers 480608, 508811, and 723217 when divided will give the same remainder. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "79"
} | {
"split": "train",
"index": 6371,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The unit's digit (one's digit) of the product of any six consecutive positive whole numbers is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 6372,
"solution": "1. **Identify the nature of the problem**: We need to find the unit's digit of the product of any six consecutive positive whole numbers.\n\n2. **Consider the factors of the product**: Let's denote the smallest of these six consecutive numbers as $n$. Therefore, the... |
deepscaler | [
{
"role": "user",
"content": "A particular triangle has sides of length 14 cm, 8 cm and 9 cm. In centimeters, what is the perimeter of the triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 6373,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest value of $x$ that satisfies the equation $\\sqrt{3x} = 6x^2$? Express your answer in simplest fractional form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt[3]{12}}"
} | {
"split": "train",
"index": 6374,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the integrals:\n1) \\(\\int_{0}^{\\frac{\\pi}{2}} \\sin ^{3} x \\, dx\\);\n2) \\(\\int_{0}^{\\ln 2} \\sqrt{e^{x}-1} \\, dx\\);\n3) \\(\\int_{-a}^{a} x^{2} \\sqrt{a^{2}-x^{2}} \\, dx\\);\n4) \\(\\int_{1}^{2} \\frac{\\sqrt{x^{2}-1}}{x} \\, dx\\). Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3} - \\frac{\\pi}{3}"
} | {
"split": "train",
"index": 6375,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $8000^{50}$ is expanded out, the result is $1$ followed by how many zeros? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 6376,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Gavin has a collection of 50 songs that are each 3 minutes in length and 50 songs that are each 5 minutes in length. What is the maximum number of songs from his collection that he can play in 3 hours? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 6377,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagram shows a smaller rectangle made from three squares, each of area \\(25 \\ \\mathrm{cm}^{2}\\), inside a larger rectangle. Two of the vertices of the smaller rectangle lie on the mid-points of the shorter sides of the larger rectangle. The other two vertices of the sm... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 6378,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a 4 by 4 grid, each of the 16 small squares measures 3 cm by 3 cm and is shaded. Four unshaded circles are then placed on top of the grid as shown. The area of the visible shaded region can be written in the form $A-B\\pi$ square cm. What is the value $A+B$? Let's think step... | math | {
"style": "rule",
"ground_truth": "180"
} | {
"split": "train",
"index": 6379,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $3^8 \\cdot 5^2 = a^b,$ where both $a$ and $b$ are positive integers, find the smallest possible value for $a+b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "407"
} | {
"split": "train",
"index": 6380,
"solution": "1. **Express the given equation in terms of $a$ and $b$:**\n Given the equation $3^8 \\cdot 5^2 = a^b$, we need to find $a$ and $b$ such that $a^b$ equals $3^8 \\cdot 5^2$.\n\n2. **Simplify the expression:**\n \\[\n 3^8 \\cdot 5^2 = (3^4)^2 \\cdot 5^2 = (81)^2... |
deepscaler | [
{
"role": "user",
"content": "(Convert the following binary number to decimal: 101111011<sub>(2)</sub>) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "379"
} | {
"split": "train",
"index": 6381,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) Calculate $\\dfrac{2A_{8}^{5}+7A_{8}^{4}}{A_{8}^{8}-A_{9}^{5}}$,\n\n(2) Calculate $C_{200}^{198}+C_{200}^{196}+2C_{200}^{197}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "67331650"
} | {
"split": "train",
"index": 6382,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $$f(x)=\\cos 2x + 2\\sqrt{3}\\sin x\\cos x$$ \n(1) Find the range of the function $f(x)$ and write down the intervals where $f(x)$ is monotonically increasing; \n(2) If $$0 < \\theta < \\frac{\\pi}{6}$$ and $$f(\\theta) = \\frac{4}{3}$$, calculate the value... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{15} + 2}{6}"
} | {
"split": "train",
"index": 6383,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Fill in the blanks with numbers $1 \\sim 3$, so that each row and each column contains exactly one number appearing twice. The numbers outside the table indicate how many numbers are visible from that direction. A number can block equal or smaller numbers. What is the four-digi... | math | {
"style": "rule",
"ground_truth": "2213"
} | {
"split": "train",
"index": 6384,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine all pairs $(a, b)$ of integers with the property that the numbers $a^{2}+4 b$ and $b^{2}+4 a$ are both perfect squares. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-4,-4),(-5,-6),(-6,-5),(0, k^{2}),(k^{2}, 0),(k, 1-k)"
} | {
"split": "train",
"index": 6385,
"solution": "Without loss of generality, assume that $|b| \\leq|a|$. If $b=0$, then $a$ must be a perfect square. So $(a=k^{2}, b=0)$ for each $k \\in \\mathbb{Z}$ is a solution. Now we consider the case $b \\neq 0$. Because $a^{2}+4 b$ is a perfect square, the quadratic equatio... |
deepscaler | [
{
"role": "user",
"content": "Filling the gas tank of a small car cost, in updated values, $\\mathrm{R} \\$ 29.90$ in 1972 and $\\mathrm{R} \\$ 149.70$ in 1992. Which of the following values best approximates the percentage increase in the price of gasoline during this 20-year period?\n(a) $20 \\%$ \n(b) $... | math | {
"style": "rule",
"ground_truth": "400\\%"
} | {
"split": "train",
"index": 6386,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The expansion of (ax- \\frac {3}{4x}+ \\frac {2}{3})(x- \\frac {3}{x})^{6} is given, and the sum of its coefficients is 16. Determine the coefficient of the x^{3} term in this expansion. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{117}{2}"
} | {
"split": "train",
"index": 6387,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $2^n$ divides $5^{256} - 1$ , what is the largest possible value of $n$ ? $ \n\\textbf{a)}\\ 8\n\\qquad\\textbf{b)}\\ 10\n\\qquad\\textbf{c)}\\ 11\n\\qquad\\textbf{d)}\\ 12\n\\qquad\\textbf{e)}\\ \\text{None of above}\n$ Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 6388,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find $n$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "324"
} | {
"split": "train",
"index": 6389,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the geometric sequence with a first term of $6$ and a second term of $-6$, what is the $205^{th}$ term? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 6390,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Anna and Aaron walk along paths formed by the edges of a region of squares. How far did they walk in total? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "640 \\text{ m}"
} | {
"split": "train",
"index": 6391,
"solution": "Each square has an area of \\( 400 \\text{ m}^2 \\) and side length 20 m. Anna's path is 400 m and Aaron's path is 240 m. Therefore, the total distance walked is \\( 400 + 240 = 640 \\text{ m} \\)."
} |
deepscaler | [
{
"role": "user",
"content": "Find one fourth of 12.8, expressed as a simplified improper fraction and also as a mixed number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 \\frac{1}{5}"
} | {
"split": "train",
"index": 6392,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a new diagram, triangle $A'B'C'$ has an area of 36 square units. The points $A', B', C', D'$ are aligned such that $A'C' = 12$ units and $C'D' = 30$ units. What is the area of triangle $B'C'D'$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 6393,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Months of the year are usually labeled numerically by '01' for January, '02' for February, and so on, through to '12' for December. Lydia notices that during January, the number of letters in the name of the month is greater than the month's numerical label (i.e., $7>1$). For h... | math | {
"style": "rule",
"ground_truth": "121"
} | {
"split": "train",
"index": 6394,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $0.\\overline{36}$ is expressed as a common fraction in lowest terms, what is the sum of the numerator and denominator? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 6395,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A relatively prime date is defined as a date where the day and the month number are coprime. Determine how many relatively prime dates are in the month with 31 days and the highest number of non-relatively prime dates? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 6396,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Moe, Loki, Nick, and Ott are friends. Thor initially had no money, while the other friends did have money. Moe gave Thor one-sixth of his money, Loki gave Thor one-fifth of his money, Nick gave Thor one-fourth of his money, and Ott gave Thor one-third of his money. Each friend ... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"split": "train",
"index": 6397,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many pairs of positive integers $(x, y)$ have the property that the ratio $x: 4$ equals the ratio $9: y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 6398,
"solution": "The equality of the ratios $x: 4$ and $9: y$ is equivalent to the equation $\\frac{x}{4}=\\frac{9}{y}$. This equation is equivalent to the equation $xy=4(9)=36$. The positive divisors of 36 are $1,2,3,4,6,9,12,18,36$, so the desired pairs are $(x, y)=(1,36),(2,18),(... |
deepscaler | [
{
"role": "user",
"content": "Find a three-digit number equal to the sum of the tens digit, the square of the hundreds digit, and the cube of the units digit.\n\nFind the number \\(\\overline{abcd}\\) that is a perfect square, if \\(\\overline{ab}\\) and \\(\\overline{cd}\\) are consecutive numbers, with \\... | math | {
"style": "rule",
"ground_truth": "357"
} | {
"split": "train",
"index": 6399,
"solution": ""
} |
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