data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Josanna's test scores to date are $90, 80, 70, 60,$ and $85.$ Her goal is to raise her test average at least $3$ points with her next test. What is the minimum test score she would need to accomplish this goal? Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "95"
} | {
"split": "train",
"index": 24649,
"solution": "1. **Calculate the current average score**: Josanna's current test scores are $90, 80, 70, 60,$ and $85$. To find the average of these scores, sum them up and divide by the number of scores:\n \\[\n \\text{Current Average} = \\frac{90 + 80 + 70 + 60 + 85}{5} = ... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive real number $x$ such that\n\\[\\lfloor x^2 \\rfloor - x \\lfloor x \\rfloor = 8.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{89}{9}"
} | {
"split": "train",
"index": 24650,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For positive integers $n,$ let $\\tau (n)$ denote the number of positive integer divisors of $n,$ including 1 and $n.$ For example, $\\tau (1)=1$ and $\\tau(6) =4.$ Define $S(n)$ by $S(n)=\\tau(1)+ \\tau(2) + \\cdots + \\tau(n).$ Let $a$ denote the number of positive integers $... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 24651,
"solution": "Let $\\Delta n$ denote the sum $1+2+3+ \\dots +n-1+n$. We can easily see from the fact \"It is well-known that $\\tau(n)$ is odd if and only if $n$ is a perfect square.\", that\n$a = (2^2-1^2) + (4^2-3^2) \\dots (44^2 - 43^2) = (2+1)(2-1)+(4+3)(4-3) \\dots (44+43)(... |
deepscaler | [
{
"role": "user",
"content": "Complex numbers \\( a \\), \\( b \\), and \\( c \\) form an equilateral triangle with side length 24 in the complex plane. If \\( |a + b + c| = 48 \\), find \\( |ab + ac + bc| \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "768"
} | {
"split": "train",
"index": 24652,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number halfway between $\\frac{1}{8}$ and $\\frac{1}{10}$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"split": "train",
"index": 24653,
"solution": "\n#### Step-by-step Calculation:\n\n1. **Identify the numbers between which the midpoint is to be found:**\n Given numbers are $\\frac{1}{8}$ and $\\frac{1}{10}$.\n\n2. **Convert fractions to a common denominator:**\n \\[\n \\frac{1}{8} = \\frac{5}{40}, \\qua... |
deepscaler | [
{
"role": "user",
"content": "Let\n\\[f(x) = x^3 + 6x^2 + 16x + 28.\\]The graphs of $y = f(x)$ and $y = f^{-1}(x)$ intersect at exactly one point $(a,b).$ Enter the ordered pair $(a,b).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-4,-4)"
} | {
"split": "train",
"index": 24654,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express this sum as a common fraction: $0.\\overline{7} + 0.\\overline{13}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10}{11}"
} | {
"split": "train",
"index": 24655,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Juan wants to calculate the area of a large circular garden, whose actual diameter is 50 meters. Unfortunately, his measurement device has an accuracy error of up to 30%. Compute the largest possible percent error, in percent, in Juan’s computed area of the circle in square met... | math | {
"style": "rule",
"ground_truth": "69\\%"
} | {
"split": "train",
"index": 24656,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to put 4 indistinguishable balls into 2 indistinguishable boxes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 24657,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At the \"Economics and Law\" congress, a \"Best of the Best\" tournament was held, in which more than 220 but fewer than 254 delegates—economists and lawyers—participated. During one match, participants had to ask each other questions within a limited time and record correct an... | math | {
"style": "rule",
"ground_truth": "105"
} | {
"split": "train",
"index": 24658,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f\\left(x\\right)=4^{x}+m\\cdot 2^{x}$, where $m\\in R$. \n$(1)$ If $m=-3$, solve the inequality $f\\left(x\\right) \\gt 4$ with respect to $x$. \n$(2)$ If the minimum value of the function $y=f\\left(x\\right)+f\\left(-x\\right)$ is $-4$, find the value o... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 24659,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of three positive integers $a$, $b$, and $c$ equals 1176. What is the minimum possible value of the sum $a + b + c$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 24660,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The parabola $y^2 = 8x$ and the circle $x^2 + y^2 - 2x - 4y = 0$ intersect at two points $A$ and $B.$ Find the distance $AB.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{5}"
} | {
"split": "train",
"index": 24661,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Charlie folds an $\\frac{17}{2}$-inch by 11-inch piece of paper in half twice, each time along a straight line parallel to one of the paper's edges. What is the smallest possible perimeter of the piece after two such folds? Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\frac{39}{2}"
} | {
"split": "train",
"index": 24662,
"solution": "Note that a piece of paper is folded in half, one pair of opposite sides is preserved and the other pair is halved. Hence, the net effect on the perimeter is to decrease it by one of the side lengths. Hence, the original perimeter is $2\\left(\\frac{17}{2}\\right)+... |
deepscaler | [
{
"role": "user",
"content": "The sides of a triangle have lengths of $15$, $20$, and $25$. Find the length of the shortest altitude. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 24663,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $\\sqrt[3]{2700}$ is simplified, the result is $a\\sqrt[3]{b}$, where $a$ and $b$ are positive integers and $b$ is as small as possible. What is $a+b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "103"
} | {
"split": "train",
"index": 24664,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the remainder when $3^{3^{3^3}}$ is divided by 1000.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "387"
} | {
"split": "train",
"index": 24665,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose \\( x_{1}, x_{2}, \\ldots, x_{2011} \\) are positive integers satisfying\n\\[ x_{1} + x_{2} + \\cdots + x_{2011} = x_{1} x_{2} \\cdots x_{2011} \\]\n\nFind the maximum value of \\( x_{1} + x_{2} + \\cdots + x_{2011} \\). Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "4022"
} | {
"split": "train",
"index": 24666,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The famous Italian mathematician Fibonacci, while studying the problem of rabbit population growth, discovered a sequence of numbers: 1, 1, 2, 3, 5, 8, 13, ..., where starting from the third number, each number is equal to the sum of the two numbers preceding it. This sequence ... | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"split": "train",
"index": 24667,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The first four terms of a sequence are $1,4,2$, and 3. Beginning with the fifth term in the sequence, each term is the sum of the previous four terms. What is the eighth term? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "66"
} | {
"split": "train",
"index": 24668,
"solution": "The first four terms of the sequence are $1,4,2,3$. Since each term starting with the fifth is the sum of the previous four terms, then the fifth term is $1+4+2+3=10$. Also, the sixth term is $4+2+3+10=19$, the seventh term is $2+3+10+19=34$, and the eighth term is... |
deepscaler | [
{
"role": "user",
"content": "Let $(b_1, b_2, b_3, \\ldots, b_{10})$ be a permutation of $(1, 2, 3, \\ldots, 10)$ such that $b_1 > b_2 > b_3 > b_4 > b_5$ and $b_5 < b_6 < b_7 < b_8 < b_9 < b_{10}$. An example of such a permutation is $(5, 4, 3, 2, 1, 6, 7, 8, 9, 10)$. Find the number of such permutations. L... | math | {
"style": "rule",
"ground_truth": "126"
} | {
"split": "train",
"index": 24669,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the pattern of numbers shown, every row begins with a 1 and ends with a 2. Each of the numbers, not on the end of a row, is the sum of the two numbers located immediately above and to the right, and immediately above and to the left. For example, in the fourth row the 9 is t... | math | {
"style": "rule",
"ground_truth": "12288"
} | {
"split": "train",
"index": 24670,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A heptagonal prism has ____ faces and ____ vertices. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 24671,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the tens digit of the smallest six-digit positive integer that is divisible by each of $10,11,12,13,14$, and 15? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 24672,
"solution": "Among the list $10,11,12,13,14,15$, the integers 11 and 13 are prime. Also, $10=2 \\times 5$ and $12=2 \\times 2 \\times 3$ and $14=2 \\times 7$ and $15=3 \\times 5$. For an integer $N$ to be divisible by each of these six integers, $N$ must include at least two fa... |
deepscaler | [
{
"role": "user",
"content": "On the Saturday of a weekend softball tournament, Team A plays Team D, Team B plays Team E, and Team C gets a bye (no match). The winner of Team A vs. Team D plays against Team C in the afternoon, while the winner of Team B vs. Team E has no further matches on Saturday. On Sund... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 24673,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $C:\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with foci $F_{1}$ and $F_{2}$, point $A$ lies on $C$, point $B$ lies on the $y$-axis, and satisfies $\\overrightarrow{AF_{1}}⊥\\overrightarrow{BF_{1}}$, $\\overrightarrow{AF_{2}}=\\frac{2}{3}\\overr... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{5}"
} | {
"split": "train",
"index": 24674,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $r,$ $s,$ and $t$ be the roots of the equation $x^3 - 20x^2 + 18x - 7 = 0.$ Find the value of $\\frac{r}{\\frac{1}{r}+st} + \\frac{s}{\\frac{1}{s}+tr} + \\frac{t}{\\frac{1}{t}+rs}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{91}{2}"
} | {
"split": "train",
"index": 24675,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ages of Jonie's four cousins are distinct single-digit positive integers. Two of the cousins' ages multiplied together give $24$, while the other two multiply to $30$. What is the sum of the ages of Jonie's four cousins? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 24676,
"solution": "1. **Identify possible pairs for the product 24**: The ages of Jonie's cousins are distinct single-digit positive integers. The possible pairs of ages that multiply to 24, considering only single-digit integers, are:\n - $3 \\times 8 = 24$\n - $4 \\times 6 = 24... |
deepscaler | [
{
"role": "user",
"content": "From 6 athletes, 4 are to be chosen to participate in a 4×100 meters relay race, given that athletes A and B both cannot run the first leg, calculate the number of different methods to select the participants. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 24677,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, \\(AB\\) is a diameter of a circle with center \\(O\\). \\(C\\) and \\(D\\) are points on the circle. \\(OD\\) intersects \\(AC\\) at \\(P\\), \\(OC\\) intersects \\(BD\\) at \\(Q\\), and \\(AC\\) intersects \\(BD\\) at \\(R\\). If \\(\\angle BOQ = 60^{\\circ}\\... | math | {
"style": "rule",
"ground_truth": "95"
} | {
"split": "train",
"index": 24678,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $4$ male athletes and $3$ female athletes.<br/>$(1)$ Now $7$ athletes are lined up. If all female athletes are together, how many ways are there to arrange them?<br/>$(2)$ Now the male athletes are sent to two different venues for training, with at least one athlete i... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 24679,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find constants $A,$ $B,$ and $C$ so that\n\\[\\frac{4x}{(x - 5)(x - 3)^2} = \\frac{A}{x - 5} + \\frac{B}{x - 3} + \\frac{C}{(x - 3)^2}.\\]Enter the ordered triple $(A,B,C).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(5,-5,-6)"
} | {
"split": "train",
"index": 24680,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point $E$ on side $CD$ of a rectangle $ABCD$ is such that $\\triangle DBE$ is isosceles and $\\triangle ABE$ is right-angled. Find the ratio between the side lengths of the rectangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\sqrt{5} - 2}"
} | {
"split": "train",
"index": 24681,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways can a 5-person executive board be chosen from a club of 12 people? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "792"
} | {
"split": "train",
"index": 24682,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x)$ is defined for all integers $x \\ge 0,$ $f(1) = 1,$ and\n\\[f(a + b) = f(a) + f(b) - 2f(ab)\\]for all integers $a,$ $b \\ge 0,$ compute $f(1986).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 24683,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest integer $n$ that satisfies $(100^2-99^2)(99^2-98^2)\\dots(3^2-2^2)(2^2-1^2)$ is divisible by $3^n$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 24684,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the digits 0, 1, 2, 3, 4, 5, calculate the number of six-digit numbers that can be formed without repetition and alternating between odd and even digits. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 24685,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alex is at the candy store buying jellybeans. He wants to buy at least 100 jellybeans. He wants to buy the least amount such that he would have exactly $11$ leftover after dividing the jellybeans evenly among $13$ people. How many jellybeans should Alex buy? Let's think step by... | math | {
"style": "rule",
"ground_truth": "102"
} | {
"split": "train",
"index": 24686,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the line $y=x+1$ intersects with the ellipse $mx^2+my^2=1(m > n > 0)$ at points $A$ and $B$, where the x-coordinate of the midpoint of the chord $AB$ is equal to $-\\frac{1}{3}$, find the eccentricity of the hyperbola $\\frac{y^2}{m^2}-\\frac{x^2}{n^2}=1$. Let's think ste... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 24687,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the integer solutions to the system of inequalities\n\\[\n\\begin{cases}\n9x - a \\geq 0, \\\\\n8x - b < 0\n\\end{cases}\n\\]\nare only 1, 2, and 3, how many ordered pairs \\((a, b)\\) of integers satisfy this system? Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 24688,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kevin has four red marbles and eight blue marbles. He arranges these twelve marbles randomly, in a ring. Determine the probability that no two red marbles are adjacent. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{33}"
} | {
"split": "train",
"index": 24689,
"solution": "Select any blue marble and consider the remaining eleven marbles, arranged in a line. The proportion of arrangement for which no two red marbles are adjacent will be the same as for the original twelve marbles, arranged in a ring. The total number of ways of arrang... |
deepscaler | [
{
"role": "user",
"content": "The variables $a$ and $b$ are inversely proportional. When the sum of $a$ and $b$ is 24, their difference is 6. What is $b$ when $a$ equals 5? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 24690,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of an infinite geometric series is $64$ times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio? 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": 24691,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Is there a real number $a$ such that the function $y=\\sin^2x+a\\cos x+ \\frac{5}{8}a- \\frac{3}{2}$ has a maximum value of $1$ on the closed interval $\\left[0,\\frac{\\pi}{2}\\right]$? If it exists, find the corresponding value of $a$. If not, explain why. Let's think step by... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 24692,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A numerical sequence is defined by the conditions: \\( a_{1}=1 \\), \\( a_{n+1}=a_{n}+\\left \\lfloor \\sqrt{a_{n}} \\right \\rfloor \\).\n\nHow many perfect squares are among the first terms of this sequence that do not exceed 1,000,000? Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 24693,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $A=(8,15)$ and $B=(14,9)$ lie on circle $\\omega$ in the plane. Suppose the tangent lines to $\\omega$ at $A$ and $B$ intersect at a point on the x-axis. Find the area of $\\omega$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "306\\pi"
} | {
"split": "train",
"index": 24694,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\triangle ABC$ with its three interior angles $A$, $B$, and $C$, and $2\\sin^{2}(B+C)= \\sqrt{3}\\sin 2A$.\n\n(Ⅰ) Find the degree of $A$; \n(Ⅱ) If $BC=7$ and $AC=5$, find the area $S$ of $\\triangle ABC$. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "10\\sqrt{3}"
} | {
"split": "train",
"index": 24695,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What percent of square $PQRS$ is shaded? All angles in the diagram are right angles. [asy]\nimport graph;\ndefaultpen(linewidth(0.8));\n\nxaxis(0,7,Ticks(1.0,NoZero));\nyaxis(0,7,Ticks(1.0,NoZero));\n\nfill((0,0)--(2,0)--(2,2)--(0,2)--cycle);\nfill((3,0)--(5,0)--(5,5)--(0,5)--(... | math | {
"style": "rule",
"ground_truth": "67.35\\%"
} | {
"split": "train",
"index": 24696,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the geometric sequence {a_n}, a_6 and a_{10} are the two roots of the equation x^2+6x+2=0. Determine the value of a_8. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\sqrt{2}"
} | {
"split": "train",
"index": 24697,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among 50 school teams participating in the HKMO, no team answered all four questions correctly. The first question was solved by 45 teams, the second by 40 teams, the third by 35 teams, and the fourth by 30 teams. How many teams solved both the third and the fourth questions? L... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 24698,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a $2 \\times 2$ grid of squares. David writes a positive integer in each of the squares. Next to each row, he writes the product of the numbers in the row, and next to each column, he writes the product of the numbers in each column. If the sum of the eight numbers he ... | math | {
"style": "rule",
"ground_truth": "88"
} | {
"split": "train",
"index": 24699,
"solution": "Let the four numbers be $a, b, c, d$, so that the other four numbers are $a b, a d, b c, b d$. The sum of these eight numbers is $a+b+c+d+a b+a d+b c+b d=(a+c)+(b+d)+(a+c)(b+d)=2015$, and so $(a+c+1)(b+d+1)=2016$. Since we seek to minimize $a+b+c+d$, we need to fin... |
deepscaler | [
{
"role": "user",
"content": "When simplified, $\\log_{16}{32} \\div \\log_{16}{\\frac{1}{2}}$, calculate the result. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 24700,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $f(-1)$ if $f(x)=x^{2}-2x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 24701,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest value of $c$ such that $-2$ is in the range of $f(x)=x^2+3x+c$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 24702,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ and $c$ be positive real numbers. Find the minimum value of\n\\[\\frac{a}{b} + \\frac{b}{c} + \\frac{c}{a}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 24703,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Construct a square \\( A B C D \\) with a side length of \\( 6 \\text{ cm} \\) and label the intersection of its diagonals as \\( S \\). Construct a point \\( K \\) such that together with points \\( S, B, \\) and \\( C \\), they form a square \\( B K C S \\). Construct a point... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 24704,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many positive integral values of $a$ is it true that $x = 2$ is the only positive integer solution of the system of inequalities $$\n\\begin{cases}\n2x>3x-3\\\\\n3x-a>-6\n\\end{cases}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 24705,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When the radius $r$ of a circle is increased by $5$, the area is quadrupled. What was the original radius $r$? Additionally, find the new perimeter of the circle after this radius increase.\nA) Original radius: 4, New perimeter: $18\\pi$\nB) Original radius: 5, New perimeter: $... | math | {
"style": "rule",
"ground_truth": "20\\pi"
} | {
"split": "train",
"index": 24706,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Anne-Marie has a deck of 16 cards, each with a distinct positive factor of 2002 written on it. She shuffles the deck and begins to draw cards from the deck without replacement. She stops when there exists a nonempty subset of the cards in her hand whose numbers multiply to a pe... | math | {
"style": "rule",
"ground_truth": "\\frac{837}{208}"
} | {
"split": "train",
"index": 24707,
"solution": "Note that $2002=2 \\cdot 7 \\cdot 11 \\cdot 13$, so that each positive factor of 2002 is included on exactly one card. Each card can identified simply by whether or not it is divisible by each of the 4 primes, and we can uniquely achieve all of the $2^{4}$ possibil... |
deepscaler | [
{
"role": "user",
"content": "Jimmy notices $7$ oranges weigh the same as $5$ apples. If Jimmy has $28$ oranges, how many apples would Jimmy need to equal the weight of his $28$ oranges? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 24708,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $DEF$ has side lengths $DE = 15$, $EF = 39$, and $FD = 36$. Rectangle $WXYZ$ has vertex $W$ on $\\overline{DE}$, vertex $X$ on $\\overline{DF}$, and vertices $Y$ and $Z$ on $\\overline{EF}$. In terms of the side length $WX = \\theta$, the area of $WXYZ$ can be expresse... | math | {
"style": "rule",
"ground_truth": "229"
} | {
"split": "train",
"index": 24709,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $g: \\mathbb{R} \\to \\mathbb{R}$ be a function such that\n\\[g((x + y)^2) = g(x)^2 - 2xg(y) + 2y^2\\]\nfor all real numbers $x$ and $y.$ Find the number of possible values of $g(1)$ and the sum of all possible values of $g(1)$. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 24710,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a pyramid P-ABCD whose base ABCD is a rectangle with side lengths AB = 2 and BC = 1, the vertex P is equidistant from all the vertices A, B, C, and D, and ∠APB = 90°. Calculate the volume of the pyramid. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{3}"
} | {
"split": "train",
"index": 24711,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Regular hexagon $PQRSTU$ has vertices $P$ and $R$ at $(0,0)$ and $(8,2)$, respectively. What is its area? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "102\\sqrt{3}"
} | {
"split": "train",
"index": 24712,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of the shaded region BEDC in parallelogram ABCD is to be found, where BC = 15, ED = 9, and the total area of ABCD is 150. If BE is the height of parallelogram ABCD from base BC and is shared with ABE, both of which overlap over BE, calculate the area of the shaded regi... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 24713,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Marsha has two numbers, $a$ and $b$. When she divides $a$ by 70 she gets a remainder of 64. When she divides $b$ by 105 she gets a remainder of 99. What remainder does she get when she divides $a+b$ by 35? Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 24714,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An ellipse has a focus at coordinates $\\left(0,-\\sqrt {2}\\right)$ and is represented by the equation $2x^{2}-my^{2}=1$. Find the value of the real number $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\dfrac{2}{5}"
} | {
"split": "train",
"index": 24715,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three identical square sheets of paper each with side length \\(8\\) are stacked on top of each other. The middle sheet is rotated clockwise \\(20^\\circ\\) about its center and the top sheet is rotated clockwise \\(50^\\circ\\) about its center. Determine the area of the resul... | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 24716,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify first, then evaluate: $4(3a^{2}b-ab^{2})-(2ab^{2}+3a^{2}b)$, where $a=\\frac{1}{3}$ and $b=-1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 24717,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that 10 spots for the finals of the 2009 National High School Mathematics Competition are to be distributed to four different schools in a certain district, with the requirement that one school gets 1 spot, another gets 2 spots, a third gets 3 spots, and the last one gets... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 24718,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Zachary paid for a $\\$1$ burger with 32 coins and received no change. Each coin was either a penny or a nickel. What was the number of nickels Zachary used? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 24719,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many positive integers $n$ less than or equal to 500 is $$(\\cos t - i\\sin t)^n = \\cos nt - i\\sin nt$$ true for all real $t$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "500"
} | {
"split": "train",
"index": 24720,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given 5 distinct real numbers, any two of which are summed to yield 10 sums. Among these sums, the smallest three are 32, 36, and 37, and the largest two are 48 and 51. What is the largest of these 5 numbers? Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "27.5"
} | {
"split": "train",
"index": 24721,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We flip a fair coin 12 times. What is the probability that we get exactly 9 heads and all heads occur consecutively? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{1024}"
} | {
"split": "train",
"index": 24722,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers are greater than $\frac{5}{7}$ and less than $\frac{28}{3}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 24723,
"solution": "The fraction $\frac{5}{7}$ is between 0 and 1. The fraction $\frac{28}{3}$ is equivalent to $9 \frac{1}{3}$ and so is between 9 and 10. Therefore, the integers between these two fractions are $1, 2, 3, 4, 5, 6, 7, 8, 9$, of which there are 9."
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest solution to the equation \\[\\frac{2x}{x-2} + \\frac{2x^2-24}{x} = 11.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1-\\sqrt{65}}{4}"
} | {
"split": "train",
"index": 24724,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bug moves in the coordinate plane, starting at $(0,0)$. On the first turn, the bug moves one unit up, down, left, or right, each with equal probability. On subsequent turns the bug moves one unit up, down, left, or right, choosing with equal probability among the three direct... | math | {
"style": "rule",
"ground_truth": "1/54"
} | {
"split": "train",
"index": 24725,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a positive number $x$ has two square roots, which are $2a-3$ and $5-a$, find the values of $a$ and $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 24726,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence of real numbers $ x_0, x_1, x_2, \\ldots$ is defined as follows: $ x_0 \\equal{} 1989$ and for each $ n \\geq 1$ \r\n\r\n\\[ x_n \\equal{} \\minus{} \\frac{1989}{n} \\sum^{n\\minus{}1}_{k\\equal{}0} x_k.\\]\r\n\r\nCalculate the value of $ \\sum^{1989}_{n\\equal... | math | {
"style": "rule",
"ground_truth": "-1989"
} | {
"split": "train",
"index": 24727,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kolya's parents give him pocket money once a month based on the following criteria: for each A in math, he gets 100 rubles; for each B, he gets 50 rubles; for each C, they subtract 50 rubles; for each D, they subtract 200 rubles. If the total amount is negative, Kolya gets noth... | math | {
"style": "rule",
"ground_truth": "250"
} | {
"split": "train",
"index": 24728,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ratio $\\frac{2^{2001} \\cdot 3^{2003}}{6^{2002}}$ is: 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": 24729,
"solution": "1. **Rewrite the given expression using the properties of exponents:**\n \\[\n \\frac{2^{2001} \\cdot 3^{2003}}{6^{2002}}\n \\]\n\n2. **Express $6^{2002}$ in terms of $2$ and $3$:**\n \\[\n 6^{2002} = (2 \\cdot 3)^{2002} = 2^{2002} \\cdot 3^{2002}\n \\]... |
deepscaler | [
{
"role": "user",
"content": "Define a sequence of integers by $T_1 = 2$ and for $n\\ge2$ , $T_n = 2^{T_{n-1}}$ . Find the remainder when $T_1 + T_2 + \\cdots + T_{256}$ is divided by 255.\n\n*Ray Li.* Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 24730,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\sqrt{5+n}=7$, then what is the value of $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 24731,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An influenza outbreak occurred in three areas, $A$, $B$, and $C$, where $6\\%$, $5\\%$, and $4\\%$ of the population in each area have the flu, respectively. Assuming the population ratios in these three areas are $6:5:4$, if a person is randomly selected from these three areas... | math | {
"style": "rule",
"ground_truth": "\\frac{77}{1500}"
} | {
"split": "train",
"index": 24732,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $X \\sim N(\\mu, \\sigma^2)$, $P(\\mu-\\sigma < X \\leq \\mu+\\sigma) = 0.68$, $P(\\mu-2\\sigma < X \\leq \\mu+2\\sigma) = 0.95$. In a city-wide exam with 20,000 participants, the math scores approximately follow a normal distribution $N(100, 100)$. How many students scor... | math | {
"style": "rule",
"ground_truth": "500"
} | {
"split": "train",
"index": 24733,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\lfloor -4 -.5 \\rfloor \\cdot \\lceil 4 +.5 \\rceil \\cdot \\lfloor -3 -.5 \\rfloor \\cdot \\lceil 3 +.5 \\rceil \\cdot \\dots \\cdot \\lfloor -.5 \\rfloor \\cdot \\lceil .5 \\rceil$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-14400"
} | {
"split": "train",
"index": 24734,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Scientists found a fragment of an ancient mechanics manuscript. It was a piece of a book where the first page was numbered 435, and the last page was numbered with the same digits, but in some different order. How many sheets did this fragment contain? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 24735,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Both $a$ and $b$ are positive integers and $b > 1$. When $a^b$ is the greatest possible value less than 399, what is 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": "21"
} | {
"split": "train",
"index": 24736,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given positive integers \\( N \\) and \\( k \\), we counted how many different ways the number \\( N \\) can be written in the form \\( a + b + c \\), where \\( 1 \\leq a, b, c \\leq k \\), and the order of the summands matters. Could the result be 2007? Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "2007"
} | {
"split": "train",
"index": 24737,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "500"
} | {
"split": "train",
"index": 24738,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": " $\\triangle KWU$ is an equilateral triangle with side length $12$ . Point $P$ lies on minor arc $\\overarc{WU}$ of the circumcircle of $\\triangle KWU$ . If $\\overline{KP} = 13$ , find the length of the altitude from $P$ onto $\\overline{WU}$ .\n\n*Proposed by Brad... | math | {
"style": "rule",
"ground_truth": "\\frac{25\\sqrt{3}}{24}"
} | {
"split": "train",
"index": 24739,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many times does the digit 9 appear in the list of all integers from 1 to 700? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "140"
} | {
"split": "train",
"index": 24740,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point $P$ is chosen at random in the interior of equilateral triangle $ABC$. What is the probability that $\\triangle ABP$ has a greater area than each of $\\triangle ACP$ and $\\triangle BCP$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 24741,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $a$ and $b$ are positive integers such that $a-b=6$ and $\\text{gcd}\\left(\\frac{a^3+b^3}{a+b}, ab\\right) = 9$. Find the smallest possible value of $b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 24742,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sixty cards are placed into a box, each bearing a number 1 through 15, with each number represented on four cards. Four cards are drawn from the box at random without replacement. Let \\(p\\) be the probability that all four cards bear the same number. Let \\(q\\) be the probab... | math | {
"style": "rule",
"ground_truth": "224"
} | {
"split": "train",
"index": 24743,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As shown in the diagram, \\( D \\), \\( E \\), and \\( F \\) are points on the sides \\( BC \\), \\( CA \\), and \\( AB \\) of \\(\\triangle ABC\\), respectively, and \\( AD \\), \\( BE \\), \\( CF \\) intersect at point \\( G \\). Given that the areas of \\(\\triangle BDG\\), ... | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 24744,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(The full score of this question is 12 points) In a box, there are three cards labeled 1, 2, and 3, respectively. Now, two cards are drawn from this box with replacement in succession, and their labels are denoted as $x$ and $y$, respectively. Let $\\xi = |x-2| + |y-x|$. \n(1)... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"split": "train",
"index": 24745,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the numbers 2, 3, 4, 5, 6, 7, 8, 9, two different numbers are selected to be the base and the exponent of a logarithm, respectively. How many different logarithmic values can be formed? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 24746,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\frac{180}{16}\\cdot \\frac{5}{120}\\cdot \\frac{8}{3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"split": "train",
"index": 24747,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A teacher received 10, 6, 8, 5, and 6 letters from Monday to Friday, respectively. The variance $s^2$ of this set of data is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3.2"
} | {
"split": "train",
"index": 24748,
"solution": ""
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.