data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "A light flashes in one of three different colors: red, green, and blue. Every $3$ seconds, it flashes green. Every $5$ seconds, it flashes red. Every $7$ seconds, it flashes blue. If it is supposed to flash in two colors at once, it flashes the more infrequent color. How ... | math | {
"style": "rule",
"ground_truth": "154"
} | {
"split": "train",
"index": 32349,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A set \\( \\mathcal{T} \\) of distinct positive integers has the property that for every integer \\( y \\) in \\( \\mathcal{T}, \\) the arithmetic mean of the set of values obtained by deleting \\( y \\) from \\( \\mathcal{T} \\) is an integer. Given that 2 belongs to \\( \\mat... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 32350,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Which of the following numbers is closest to 1: $\frac{11}{10}$, $\frac{111}{100}$, 1.101, $\frac{1111}{1000}$, 1.011? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.011"
} | {
"split": "train",
"index": 32351,
"solution": "When we convert each of the possible answers to a decimal, we obtain 1.1, 1.11, 1.101, 1.111, and 1.011. Since the last of these is the only one greater than 1 and less than 1.1, it is closest to 1."
} |
deepscaler | [
{
"role": "user",
"content": "The expression $(5 \\times 5)+(5 \\times 5)+(5 \\times 5)+(5 \\times 5)+(5 \\times 5)$ is equal to what? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 32352,
"solution": "The given sum includes 5 terms each equal to $(5 \\times 5)$. Thus, the given sum is equal to $5 \\times(5 \\times 5)$ which equals $5 \\times 25$ or 125."
} |
deepscaler | [
{
"role": "user",
"content": "Harry, who is incredibly intellectual, needs to eat carrots $C_1, C_2, C_3$ and solve *Daily Challenge* problems $D_1, D_2, D_3$ . However, he insists that carrot $C_i$ must be eaten only after solving *Daily Challenge* problem $D_i$ . In how many satisfactory orders can ... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 32353,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five s... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 32354,
"solution": "1. **Understanding the Problem:**\n Cozy the Cat jumps two steps at a time, and Dash the Dog jumps five steps at a time. Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. We need to find the sum of all possible numbers of steps in the stairca... |
deepscaler | [
{
"role": "user",
"content": "In the trapezoid \\(ABCD\\) (\\(AD \\parallel BC\\)), a perpendicular \\(EF\\) is drawn from point \\(E\\) (the midpoint of \\(CD\\)) to line \\(AB\\). Find the area of the trapezoid if \\(AB = 5\\) and \\(EF = 4\\). Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 32355,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the probability of randomly drawing three different numbers from the set {1, 2, ..., 10} such that their sample variance \\( s^2 \\leqslant 1 \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/15"
} | {
"split": "train",
"index": 32356,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A set of numbers $\\{-3, 1, 5, 8, 10, 14\\}$ needs to be rearranged with new rules:\n\n1. The largest isn't in the last position, but it is in one of the last four places.\n2. The smallest isn’t in the first position, but it is in one of the first four places.\n3. The median is... | math | {
"style": "rule",
"ground_truth": "-24"
} | {
"split": "train",
"index": 32357,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tim and Allen are playing a match of tenus. In a match of tenus, the two players play a series of games, each of which is won by one of the two players. The match ends when one player has won exactly two more games than the other player, at which point the player who has won mo... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{3}"
} | {
"split": "train",
"index": 32358,
"solution": "Let the answer be $E$. If Tim wins the first game and Allen wins the second game or vice versa, which occurs with probability $(3 / 4)^{2}+(1 / 4)^{2}=5 / 8$, the expected number of additional games is just $E$, so the expected total number of games is $E+2$. If, o... |
deepscaler | [
{
"role": "user",
"content": "If four fair six-sided dice are rolled, what is the probability that the lowest number appearing on any die is exactly 3? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{175}{1296}"
} | {
"split": "train",
"index": 32359,
"solution": "The probability that all the die rolls are at least 3 is $\\frac{4^{4}}{6}$. The probability they are all at least 4 is $\\frac{3^{4}}{6}$. The probability of being in the former category but not the latter is thus $\\frac{4}{6}^{4}-\\frac{3}{6}^{4}=\\frac{256-81}{... |
deepscaler | [
{
"role": "user",
"content": "Farm A has a total of 625 chickens and ducks, and Farm B has a total of 748 chickens and ducks. The number of chickens in Farm B is 24% more than in Farm A, and the number of ducks in Farm A is 15% less than in Farm B. How many chickens does Farm B have? Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "248"
} | {
"split": "train",
"index": 32360,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers less than $201$ are multiples of either $4$ or $9$, but not both at once? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "62"
} | {
"split": "train",
"index": 32361,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You are given 10 numbers - one one and nine zeros. You are allowed to select two numbers and replace each of them with their arithmetic mean.\n\nWhat is the smallest number that can end up in the place of the one? Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{512}"
} | {
"split": "train",
"index": 32362,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least possible sum of two positive integers $a$ and $b$ where $a \\cdot b = 10! ?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3960"
} | {
"split": "train",
"index": 32363,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the integer $k > 2$ for which\n\\[\\log_{10} (k - 2)! + \\log_{10} (k - 1)! + 2 = 2 \\log_{10} k!.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 32364,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The operation $*$ is defined for non-zero integers as follows: $a * b = \\frac{1}{a} + \\frac{1}{b}$. If $a+b = 9$ and $ a \\times b = 20$, what is the value of $a*b$? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{20}"
} | {
"split": "train",
"index": 32365,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The mean of the set of numbers $\\{91, 89, 85, 88, 90, 87, y\\}$ is 88. What is the median of the set of seven numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "88"
} | {
"split": "train",
"index": 32366,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Eastbound traffic flows at 80 miles per hour and westbound traffic flows at 60 miles per hour. An eastbound driver observes 30 westbound vehicles in a 10-minute period. Calculate the number of westbound vehicles in a 150-mile section of the highway. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "193"
} | {
"split": "train",
"index": 32367,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $(1+\\sin t)(1+\\cos t)=5/4$ and\n$(1-\\sin t)(1-\\cos t)=\\frac mn-\\sqrt{k},$\n where $k, m,$ and $n_{}$ are positive integers with $m_{}$ and $n_{}$ relatively prime, find $k+m+n.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 32368,
"solution": "We want $1+\\sin t \\cos t-\\sin t-\\cos t$. However, note that we only need to find $\\sin t+\\cos t$.\n Let $y = \\sin t+\\cos t \\rightarrow y^2 = \\sin^2 t + \\cos^2 t + 2\\sin t \\cos t = 1 + 2\\sin t \\cos x$\n From this we have $\\sin t \\cos t = \\frac{y^2-... |
deepscaler | [
{
"role": "user",
"content": "Two angles of an isosceles triangle measure $70^\\circ$ and $x^\\circ$. What is the sum of the three possible values of $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "165"
} | {
"split": "train",
"index": 32369,
"solution": "We are given an isosceles triangle with two angles measuring $70^\\circ$ and $x^\\circ$. We need to find the sum of all possible values of $x$.\n\nAn isosceles triangle has at least two sides of equal length, and the angles opposite these sides are equal. The sum o... |
deepscaler | [
{
"role": "user",
"content": "In Middle-Earth, nine cities form a 3 by 3 grid. The top left city is the capital of Gondor and the bottom right city is the capital of Mordor. How many ways can the remaining cities be divided among the two nations such that all cities in a country can be reached from its capi... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 32370,
"solution": "For convenience, we will center the grid on the origin of the coordinate plane and align the outer corners of the grid with the points $( \\pm 1, \\pm 1)$, so that $(-1,1)$ is the capital of Gondor and $(1,-1)$ is the capital of Mordor. We will use casework on whic... |
deepscaler | [
{
"role": "user",
"content": "Two long cylindrical tubes of the same length but different diameters lie parallel to each other on a flat surface. The larger tube has radius $72$ and rolls along the surface toward the smaller tube, which has radius $24$. It rolls over the smaller tube and continues rolling a... | math | {
"style": "rule",
"ground_truth": "312"
} | {
"split": "train",
"index": 32371,
"solution": "1. **Understanding the Problem:**\n Two cylindrical tubes, one with radius 72 and the other with radius 24, are lying parallel on a flat surface. The larger tube rolls over the smaller tube and completes one full revolution, returning to the same point on its cir... |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)$ be an odd function. Is $f(f(f(x)))$ even, odd, or neither?\n\nEnter \"odd\", \"even\", or \"neither\". Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\text{odd}"
} | {
"split": "train",
"index": 32372,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the sum of the base $10$ logarithms of all the proper divisors (all divisors of a number excluding itself) of $1000000$. What is the integer nearest to $S$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "141"
} | {
"split": "train",
"index": 32373,
"solution": "Note that we can just pair terms up such that the product is $10^{6}.$ Now, however, note that $10^{3}$ is not included. Therefore we first exclude. We have $\\displaystyle\\frac{49-1}{2} = 24$ pairs that all multiply to $10^{6}.$ Now we include $10^{3}$ so our cur... |
deepscaler | [
{
"role": "user",
"content": "How many prime numbers are between 30 and 40? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 32374,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that points $P$ and $Q$ are on the circle $x^2 + (y-6)^2 = 2$ and the ellipse $\\frac{x^2}{10} + y^2 = 1$, respectively, what is the maximum distance between $P$ and $Q$?\n\nA) $5\\sqrt{2}$\nB) $\\sqrt{46} + \\sqrt{2}$\nC) $7 + \\sqrt{2}$\nD) $6\\sqrt{2}$ Let's think step... | math | {
"style": "rule",
"ground_truth": "6\\sqrt{2}"
} | {
"split": "train",
"index": 32375,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A list of integers has mode 32 and mean 22. The smallest number in the list is 10. The median m of the list is a member of the list. If the list member m were replaced by m+10, the mean and median of the new list would be 24 and m+10, respectively. If m were instead replac... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 32376,
"solution": "1. **Understanding the problem and setting up equations:**\n Let the number of integers in the list be $n$. The list has a mean of $22$, so the sum of all integers in the list is $22n$.\n\n2. **Analyzing the effect of replacing $m$ with $m+10$:**\n When $m$ is ... |
deepscaler | [
{
"role": "user",
"content": "The number of students in Jakob's graduating class is more than 100 and fewer than 200 and is 2 less than a multiple of 4, 3 less than a multiple of 5, and 4 less than a multiple of 6. How many students are in Jakob's graduating class? Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "122"
} | {
"split": "train",
"index": 32377,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We have 21 pieces of type $\\Gamma$ (each formed by three small squares). We are allowed to place them on an $8 \\times 8$ chessboard (without overlapping, so that each piece covers exactly three squares). An arrangement is said to be maximal if no additional piece can be added... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 32378,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a geometric progression \\( b_1, b_2, \\ldots, b_{3000} \\) with all positive terms and a total sum \\( S \\). It is known that if every term with an index that is a multiple of 3 (i.e., \\( b_3, b_6, \\ldots, b_{3000} \\)) is increased by 50 times, the sum \\( S \\) incr... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{8}"
} | {
"split": "train",
"index": 32379,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A car and a truck start traveling towards each other simultaneously from points $A$ and $B$, respectively. It is known that the car's speed is twice the speed of the truck. The car arrives at point $C$ at 8:30, and the truck arrives at point $C$ at 15:00 on the same day. Both v... | math | {
"style": "rule",
"ground_truth": "10:40"
} | {
"split": "train",
"index": 32380,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $A, B, C$ are the three interior angles, and $a, b, c$ are the sides opposite to angles $A, B, C$ respectively. It is given that $2 \\sqrt{2}\\left(\\sin^2 A - \\sin^2 C\\right) = (a - b) \\sin B$, and the radius of the circumcircle of $\\triangle ABC$ is $... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 32381,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle XYZ$, the ratio $XZ:ZY$ is $5:3$. The bisector of the exterior angle at $Z$ intersects $YX$ extended at $Q$ ($Y$ is between $Q$ and $X$). Find the ratio $QY:YX$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 32382,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Angela has deposited $\\$8,\\!000$ into an account that pays $6\\%$ interest compounded annually.\n\nBob has deposited $\\$10,\\!000$ into an account that pays $7\\%$ simple annual interest.\n\nIn $20$ years Angela and Bob compare their respective balances. To the nearest dolla... | math | {
"style": "rule",
"ground_truth": "\\$1,\\!657"
} | {
"split": "train",
"index": 32383,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest number which when successively divided by \\( 45,454,4545 \\) and 45454 leaves remainders of \\( 4, 45,454 \\) and 4545 respectively. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35641667749"
} | {
"split": "train",
"index": 32384,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three balls labeled 1, 2, and 3 are placed in a jar. A ball is drawn from the jar, its number is recorded, and it is then returned to the jar. This process is repeated three times, with each ball having an equal chance of being drawn in each trial. If the sum of the recorded nu... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{7}"
} | {
"split": "train",
"index": 32385,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular polygon has perimeter 108 cm and each side has length 12 cm. How many sides does this polygon have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 32386,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Quadrilateral $ABCD$ is a square, and segment $AE$ is perpendicular to segment $ED$. If $AE = 8$ units and $DE = 6$ units, what is the area of pentagon $AEDCB$, in square units? [asy]\nsize(150);\npair A, B, C, D, E;\nA=(0,10);\nB=(0,0);\nC=(10,0);\nD=(10,10);\nE=(6.4,5.2);\ndr... | math | {
"style": "rule",
"ground_truth": "76"
} | {
"split": "train",
"index": 32387,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the $x$-intercept point of the line $3x+5y=20$. Provide your answer as an ordered pair. Express the $x$ and $y$ coordinates as common fractions as needed. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left(\\frac{20}{3},0\\right)"
} | {
"split": "train",
"index": 32388,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $x:\\ \\log_2 x+\\log_4 x= 6.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 32389,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a positive integer \\( k \\), find the greatest common divisor (GCD) \\( d \\) of all positive even numbers \\( x \\) that satisfy the following conditions:\n1. Both \\( \\frac{x+2}{k} \\) and \\( \\frac{x}{k} \\) are integers, and the difference in the number of digits of ... | math | {
"style": "rule",
"ground_truth": "1998"
} | {
"split": "train",
"index": 32390,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The midsegment of a trapezoid divides it into two quadrilaterals. The difference in the perimeters of these two quadrilaterals is 24, and the ratio of their areas is $\\frac{20}{17}$. Given that the height of the trapezoid is 2, what is the area of this trapezoid? Let's think s... | math | {
"style": "rule",
"ground_truth": "148"
} | {
"split": "train",
"index": 32391,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the rational number that is the value of the expression\n\n$$\n\\cos ^{6}(3 \\pi / 16)+\\cos ^{6}(11 \\pi / 16)+3 \\sqrt{2} / 16\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5/8"
} | {
"split": "train",
"index": 32392,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\arcsin (-1).$ Express your answer in radians. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\pi}{2}"
} | {
"split": "train",
"index": 32393,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain school is actively preparing for the \"Sunshine Sports\" activity and has decided to purchase a batch of basketballs and soccer balls totaling $30$ items. At a sports equipment store, each basketball costs $80$ yuan, and each soccer ball costs $60$ yuan. During the pu... | math | {
"style": "rule",
"ground_truth": "2080"
} | {
"split": "train",
"index": 32394,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given points $A(-2,0)$ and $B(2,0)$, the slope of line $PA$ is $k_1$, and the slope of line $PB$ is $k_2$, with the product $k_1k_2=-\\frac{3}{4}$.\n$(1)$ Find the equation of the locus $C$ for point $P$.\n$(2)$ Let $F_1(-1,0)$ and $F_2(1,0)$. Extend line segment $PF_1$ to meet... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 32395,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Real numbers \\( x_{1}, x_{2}, \\cdots, x_{2001} \\) satisfy \\( \\sum_{k=1}^{2005}\\left|x_{k}-x_{k+1}\\right|=2007 \\). Define \\( y_{k}=\\frac{1}{k}\\left(x_{1}+x_{2}+\\cdots+x_{k}\\right) \\) for \\( k=1, 2, \\cdots, 2007 \\). Find the maximum possible value of \\( \\sum_{k... | math | {
"style": "rule",
"ground_truth": "2006"
} | {
"split": "train",
"index": 32396,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One integer is selected at random from the following list of 15 integers: $1,2,2,3,3,3,4,4,4,4,5,5,5,5,5$. The probability that the selected integer is equal to $n$ is $\\frac{1}{3}$. What is the value of $n$? Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 32397,
"solution": "Since the list includes 15 integers, then an integer has a probability of $\\frac{1}{3}$ of being selected if it occurs $\\frac{1}{3} \\cdot 15=5$ times in the list. The integer 5 occurs 5 times in the list and no other integer occurs 5 times, so $n=5$."
} |
deepscaler | [
{
"role": "user",
"content": "If the two foci of a hyperbola are respectively $F_1(-3,0)$ and $F_2(3,0)$, and one asymptotic line is $y=\\sqrt{2}x$, calculate the length of the chord that passes through the foci and is perpendicular to the $x$-axis. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"split": "train",
"index": 32398,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The average of the numbers 23 and $x$ is 27. What is the positive difference between 23 and $x$? (Recall that the average of two numbers is their sum divided by 2). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 32399,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The lengths of the three altitudes of a triangle are in the ratio \\(3 : 4 : 6\\). If the radius of the inscribed circle of the triangle is 1, find the radius of the circumcircle of the triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16/5"
} | {
"split": "train",
"index": 32400,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an opaque bag, there are four identical balls labeled with numbers $3$, $4$, $5$, and $6$ respectively. Outside the bag, there are two balls labeled with numbers $3$ and $6$. Determine the probability that a triangle with the drawn ball and the numbers on the two balls outsi... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 32401,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sequence $\\left\\{\\frac{1}{b_{n}}\\right\\}$ is a \"dream sequence\" defined by $\\frac{1}{a_{n+1}}- \\frac{2}{a_{n}}=0$, and that $b_1+b_2+b_3=2$, find the value of $b_6+b_7+b_8$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 32402,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, if $AB=2$, $AC=\\sqrt{2}BC$, find the maximum value of $S_{\\triangle ABC}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 32403,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $\\frac{x^2}{25} + \\frac{y^2}{9} = 1$, the distance from a point $M$ on the ellipse to the left focus $F_1$ is 8. Find the distance from $M$ to the right directrix. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"split": "train",
"index": 32404,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kim earned scores of 87, 83 and 88 on her first three mathematics examinations. If Kim receives a score of 90 on the fourth exam, then by how much will her average increase? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 32405,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \\cdots, a_n \\in S$, all integer roots of the polynomial $a_0+a_1x+\\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form ... | math | {
"style": "rule",
"ground_truth": "\\mathbb{Z}"
} | {
"split": "train",
"index": 32406,
"solution": "\nTo find all rootiful sets of integers \\( S \\) that contain all numbers of the form \\( 2^a - 2^b \\) for positive integers \\( a \\) and \\( b \\), we need to analyze the properties of such sets.\n\n### Step 1: Understand the Definition\n\nA set \\( S \\) is ro... |
deepscaler | [
{
"role": "user",
"content": "Consider the function \\( g(x) = \\left\\{ \\begin{aligned} x-3 & \\quad \\text{ if } x < 5 \\\\ \\sqrt{x-1} & \\quad \\text{ if } x \\ge 5 \\end{aligned} \\right. \\). Find the value of \\( g^{-1}(-6) + g^{-1}(-5) + \\dots + g^{-1}(4) + g^{-1}(5) \\). Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "58"
} | {
"split": "train",
"index": 32407,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of the first $n$ terms of the sequence $\\{a_n\\}$ is $S_n = -a_n - \\left(\\frac{1}{2}\\right)^{n-1} + 2$, and $(1) b_n = 2^n a_n$, find the general term formula for $\\{b_n\\}$. Also, $(2)$ find the maximum term of $\\{a_n\\}$. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 32408,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $(a_1,a_2,a_3,\\ldots,a_{12})$ be a permutation of $(1,2,3,\\ldots,12)$ for which\n$a_1>a_2>a_3>a_4>a_5>a_6 \\mathrm{\\ and \\ } a_6<a_7<a_8<a_9<a_{10}<a_{11}<a_{12}.$\n An example of such a permutation is $(6,5,4,3,2,1,7,8,9,10,11,12).$ Find the number of such permutations... | math | {
"style": "rule",
"ground_truth": "462"
} | {
"split": "train",
"index": 32409,
"solution": "Clearly, $a_6=1$. Now, consider selecting $5$ of the remaining $11$ values. Sort these values in descending order, and sort the other $6$ values in ascending order. Now, let the $5$ selected values be $a_1$ through $a_5$, and let the remaining $6$ be $a_7$ through ... |
deepscaler | [
{
"role": "user",
"content": "What is the largest value of $n$ less than 100,000 for which the expression $9(n-3)^5 - 2n^3 + 17n - 33$ is a multiple of 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "99999"
} | {
"split": "train",
"index": 32410,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive real number $\\lambda$ such that for every numbers $a_1,a_2,a_3 \\in \\left[0, \\frac{1}{2} \\right]$ and $b_1,b_2,b_3 \\in (0, \\infty)$ with $\\sum\\limits_{i=1}^3a_i=\\sum\\limits_{i=1}^3b_i=1,$ we have $$ b_1b_2b_3 \\le \\lambda (a_1b_1+... | math | {
"style": "rule",
"ground_truth": "1/8"
} | {
"split": "train",
"index": 32411,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a division equation, the dividend is 2016 greater than the divisor, the quotient is 15, and the remainder is 0. What is the dividend? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2160"
} | {
"split": "train",
"index": 32412,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $A$ lies at $(0,4)$ and point $B$ lies at $(3,8)$. Find the $x$-coordinate of the point $X$ on the $x$-axis maximizing $\\angle AXB$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5 \\sqrt{2}-3"
} | {
"split": "train",
"index": 32413,
"solution": "Let $X$ be a point on the $x$-axis and let $\\theta=\\angle AXB$. We can easily see that the circle with diameter $AB$ does not meet the $x$-axis, so $\\theta \\leq \\pi$. Thus, maximizing $\\theta$ is equivalent to maximizing $\\sin \\theta$. By the Law of Sines, ... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=a\\cos (x+\\frac{\\pi }{6})$, its graph passes through the point $(\\frac{\\pi }{2}, -\\frac{1}{2})$.\n\n(1) Find the value of $a$;\n\n(2) If $\\sin \\theta =\\frac{1}{3}, 0 < \\theta < \\frac{\\pi }{2}$, find $f(\\theta ).$ Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{6}-1}{6}"
} | {
"split": "train",
"index": 32414,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nCalculate the volume of the tetrahedron with vertices at points $A_{1}, A_{2}, A_{3}, A_{4}$ and its height, dropped from vertex $A_{4}$ onto the face $A_{1} A_{2} A_{3}$.\n\n$A_{1}(2, -1, -2)$\n\n$A_{2}(1, 2, 1)$\n\n$A_{3}(5, 0, -6)$\n\n$A_{4}(-10, 9, -7)$ Let's think step b... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{14}"
} | {
"split": "train",
"index": 32415,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Von Neumann's Poker: The first step in Von Neumann's game is selecting a random number on $[0,1]$. To generate this number, Chebby uses the factorial base: the number $0 . A_{1} A_{2} A_{3} A_{4} \\ldots$ stands for $\\sum_{n=0}^{\\infty} \\frac{A_{n}}{(n+1)!}$, where each $A_{... | math | {
"style": "rule",
"ground_truth": "0.57196"
} | {
"split": "train",
"index": 32416,
"solution": "The expected value of a number generated in the factorial base system is given by the sum of the expected values of each digit divided by its factorial weight. For this specific setup, the expected value is approximately 0.57196."
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $D$ is on segment $BC$ such that $BD:DC = 3:2$. Point $E$ bisects $\\angle BAC$. Given $BD = 45$ units and $DC = 30$ units, find the length of $BE$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 32417,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest power of 2 by which the number \\(10^{10} - 2^{10}\\) is divisible? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 32418,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $2ax^3 - 3bx + 8 = 18$ when $x = -1$, determine the value of $9b - 6a + 2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 32419,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers between 1000 and 2000 have all three of the numbers 15, 20 and 25 as factors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 32420,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square $ABCD$ is circumscribed around a circle. Another square $IJKL$ is inscribed inside a smaller concentric circle. The side length of square $ABCD$ is $4$ units, and the radius of the smaller circle is half the radius of the larger circle. Find the ratio of the area of squa... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 32421,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": " $A$ and $B$ plays a game, with $A$ choosing a positive integer $n \\in \\{1, 2, \\dots, 1001\\} = S$ . $B$ must guess the value of $n$ by choosing several subsets of $S$ , then $A$ will tell $B$ how many subsets $n$ is in. $B$ will do this three times select... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 32422,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $S = i^n + i^{-n}$, where $i = \\sqrt{-1}$ and $n$ is an integer, then the total number of possible distinct values for $S$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 32423,
"solution": "1. **Understanding the Powers of $i$:** \n Recall that $i = \\sqrt{-1}$, and the powers of $i$ cycle every four terms:\n - $i^1 = i$\n - $i^2 = -1$\n - $i^3 = -i$\n - $i^4 = 1$\n - $i^5 = i$, and so on.\n\n2. **Expression for $i^{-n}$:** \n We know ... |
deepscaler | [
{
"role": "user",
"content": "Seven points are marked on the circumference of a circle. How many different chords can be drawn by connecting two of these seven points? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 32424,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is $\\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "187"
} | {
"split": "train",
"index": 32425,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We have a cube with 4 blue faces and 2 red faces. What's the probability that when it is rolled, a blue face will be facing up? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 32426,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a finite graph $G$, let $f(G)$ be the number of triangles and $g(G)$ the number of tetrahedra formed by edges of $G$. Find the least constant $c$ such that \\[g(G)^3\\le c\\cdot f(G)^4\\] for every graph $G$.\n\n[i] Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{32}"
} | {
"split": "train",
"index": 32427,
"solution": "\nLet \\( G \\) be a finite graph. We denote by \\( f(G) \\) the number of triangles and by \\( g(G) \\) the number of tetrahedra in \\( G \\). We seek to establish the smallest constant \\( c \\) such that\n\n\\[\ng(G)^3 \\le c \\cdot f(G)^4\n\\]\n\nfor every grap... |
deepscaler | [
{
"role": "user",
"content": "The graph of $y = f(x)$ is shown below.\n\n[asy]\nunitsize(0.5 cm);\n\nreal func(real x) {\n real y;\n if (x >= -3 && x <= 0) {y = -2 - x;}\n if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}\n if (x >= 2 && x <= 3) {y = 2*(x - 2);}\n return(y);\n}\n\nint i, n;\n\nfor (... | math | {
"style": "rule",
"ground_truth": "\\text{E}"
} | {
"split": "train",
"index": 32428,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square $A B C D$ has side length 1. A dilation is performed about point $A$, creating square $A B^{\\prime} C^{\\prime} D^{\\prime}$. If $B C^{\\prime}=29$, determine the area of triangle $B D C^{\\prime}$. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "420"
} | {
"split": "train",
"index": 32429,
"solution": "$29^{2}-2 \\cdot \\frac{1}{2}(29)\\left(\\frac{29}{2}\\right)-\\frac{1}{2}=420$."
} |
deepscaler | [
{
"role": "user",
"content": "Sam is hired for a 20-day period. On days that he works, he earns $\\$$60. For each day that he does not work, $\\$$30 is subtracted from his earnings. At the end of the 20-day period, he received $\\$$660. How many days did he not work? Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 32430,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $15^{-1} \\equiv 31 \\pmod{53}$, find $38^{-1} \\pmod{53}$, as a residue modulo 53. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 32431,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the infinite sum of \\(\\frac{1^{3}}{3^{1}}+\\frac{2^{3}}{3^{2}}+\\frac{3^{3}}{3^{3}}+\\frac{4^{3}}{3^{4}}+\\cdots\\).\n\n求 \\(\\frac{1^{3}}{3^{1}}+\\frac{2^{3}}{3^{2}}+\\frac{3^{3}}{3^{3}}+\\frac{4^{3}}{3^{4}}+\\cdots\\) 無限項之和。 Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "\\frac{33}{8}"
} | {
"split": "train",
"index": 32432,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A function $f(x) = a \\cos ωx + b \\sin ωx (ω > 0)$ has a minimum positive period of $\\frac{π}{2}$. The function reaches its maximum value of $4$ at $x = \\frac{π}{6}$.\n\n1. Find the values of $a$, $b$, and $ω$.\n2. If $\\frac{π}{4} < x < \\frac{3π}{4}$ and $f(x + \\frac{π}{6... | math | {
"style": "rule",
"ground_truth": "-\\frac{4\\sqrt{6}}{3}"
} | {
"split": "train",
"index": 32433,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\omega$ be a nonreal root of $z^3 = 1.$ Find the number of ordered pairs $(a,b)$ of integers such that $|a \\omega + b| = 1.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 32434,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x, 2x+2, 3x+3, \\dots$ are in geometric progression, the fourth term is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-13\\frac{1}{2}"
} | {
"split": "train",
"index": 32435,
"solution": "1. **Identify the nature of the sequence**: Given that $x, 2x+2, 3x+3, \\dots$ are in geometric progression, the ratio between consecutive terms must be constant. Let's denote this common ratio by $r$.\n\n2. **Set up the equation for the common ratio**:\n \\[\n ... |
deepscaler | [
{
"role": "user",
"content": "A square is inscribed in a circle. A smaller square has one side coinciding with a side of the larger square and has two vertices on the circle, as shown. What percent of the area of the larger square is the area of the smaller square?\n\n[asy]\ndraw(Circle((0,0),1.4142));\nd... | math | {
"style": "rule",
"ground_truth": "4\\%"
} | {
"split": "train",
"index": 32436,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alexio has 100 cards numbered 1-100, inclusive, and places them in a box. Alexio then chooses a card from the box at random. What is the probability that the number on the card he chooses is a multiple of 2, 3 or 5? Express your answer as a common fraction. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{37}{50}"
} | {
"split": "train",
"index": 32437,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the operation $\\#$ be defined as $\\#(a, b, c) = b^2 - 4ac$, for all real numbers $a, b$ and $c$. What is the value of $\\#(1, 2, 3)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"split": "train",
"index": 32438,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of the two $102$-digit numbers $404,040,404,...,040,404$ and $707,070,707,...,070,707$ has thousands digit $A$ and units digit $B$. Calculate the sum of $A$ and $B$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 32439,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $P$ and $Q$ lie in a plane with $PQ=8$. How many locations for point $R$ in this plane are there such that the triangle with vertices $P$, $Q$, and $R$ is a right triangle with area $12$ square units? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 32440,
"solution": "1. **Given Information and Formula for Area**: We know that the area of a triangle can be expressed as \\([PQR] = \\frac{1}{2} \\cdot PQ \\cdot h_R\\), where \\(h_R\\) is the perpendicular distance from point \\(R\\) to line \\(PQ\\). Given that \\(PQ = 8\\) and th... |
deepscaler | [
{
"role": "user",
"content": "What is the base ten equivalent of the base three number $102012_3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "302"
} | {
"split": "train",
"index": 32441,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fenced, rectangular field measures $24$ meters by $52$ meters. An agricultural researcher has 1994 meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the square... | math | {
"style": "rule",
"ground_truth": "702"
} | {
"split": "train",
"index": 32442,
"solution": "Suppose there are $n$ squares in every column of the grid, so there are $\\frac{52}{24}n = \\frac {13}6n$ squares in every row. Then $6|n$, and our goal is to maximize the value of $n$.\n Each vertical fence has length $24$, and there are $\\frac{13}{6}n - 1$ verti... |
deepscaler | [
{
"role": "user",
"content": "A fair coin is tossed six times and the sequence of heads and tails is recorded. What is the probability that the sequence contains exactly two heads? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15}{64}"
} | {
"split": "train",
"index": 32443,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Once, Carlson and Winnie the Pooh competed in the speed of eating honey and jam. Carlson, an expert in jam, eats a jar of jam in 2 minutes, while Winnie the Pooh takes a full 7 minutes to finish a jar of jam. Meanwhile, Winnie the Pooh can finish a pot of honey in 3 minutes, bu... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 32444,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two dice are rolled consecutively, and the numbers obtained are denoted as $a$ and $b$. \n(Ⅰ) Find the probability that the point $(a, b)$ lies on the graph of the function $y=2^x$. \n(Ⅱ) Using the values of $a$, $b$, and $4$ as the lengths of three line segments, find the pr... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{18}"
} | {
"split": "train",
"index": 32445,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(b_n = 7^n + 9^n\\). Determine the remainder when \\(b_{86}\\) is divided by \\(50\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 32446,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $PQR,$ $PQ = 24,$ $QR = 25,$ $PR = 7,$ and point $H$ is the intersection of the altitudes (the orthocenter). Points $P',$ $Q',$ and $R',$ are the images of $P,$ $Q,$ and $R,$ respectively, after a $180^\\circ$ rotation about $H.$ What is the area of the union of the... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 32447,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Captain Zarnin of Planet Hvan has four job openings for his battle station: Assistant Engineer, Weapons Maintenance, Field Technician, and Radio Specialist. Zarnin has received 24 resumes and found himself less than thrilled about half of them and does not hire them. The rest, ... | math | {
"style": "rule",
"ground_truth": "11,\\!880"
} | {
"split": "train",
"index": 32448,
"solution": ""
} |
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