data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "A circle with a radius of 3 units has its center at $(0, 0)$. A circle with a radius of 8 units has its center at $(18, 0)$. A line tangent to both circles intersects the $x$-axis at point $(x, 0)$ to the right of the origin. Find the value of $x$. Express your answer as a comm... | math | {
"style": "rule",
"ground_truth": "\\frac{54}{11}"
} | {
"split": "train",
"index": 25949,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A paper equilateral triangle $ABC$ has side length $12$. The paper triangle is folded so that vertex $A$ touches a point on side $\\overline{BC}$ a distance $9$ from point $B$. The length of the line segment along which the triangle is folded can be written as $\\frac{m\\sqrt{p... | math | {
"style": "rule",
"ground_truth": "113"
} | {
"split": "train",
"index": 25950,
"solution": "Let $P$ and $Q$ be the points on $\\overline{AB}$ and $\\overline{AC}$, respectively, where the paper is folded.\n Let $D$ be the point on $\\overline{BC}$ where the folded $A$ touches it.\n Let $a$, $b$, and $x$ be the lengths $AP$, $AQ$, and $PQ$, respectively.\n... |
deepscaler | [
{
"role": "user",
"content": "How many pairs of two-digit positive integers have a difference of 50? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 25951,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(Ⅰ) Find the equation of the line that passes through the intersection point of the two lines $2x-3y-3=0$ and $x+y+2=0$, and is perpendicular to the line $3x+y-1=0$. \n(Ⅱ) Given the equation of line $l$ in terms of $x$ and $y$ as $mx+y-2(m+1)=0$, find the maximum distance from... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt {2}"
} | {
"split": "train",
"index": 25952,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=2\\sin(\\omega x+\\varphi)$, where $(\\omega > 0, |\\varphi| < \\frac{\\pi}{2})$, the graph passes through the point $B(0,-1)$, and is monotonically increasing on the interval $\\left(\\frac{\\pi}{18}, \\frac{\\pi}{3}\\right)$. Additionally, the graph o... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 25953,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a function $f(x)$ satisfies $f(x) + f(4-x) = 4$, $f(x+2) - f(-x) = 0$, and $f(1) = a$, calculate the value of $f(1) + f(2) + f(3) + \\cdots + f(51)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "102"
} | {
"split": "train",
"index": 25954,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The magnitude of the vector $\\overset{→}{a} +2 \\overset{→}{b}$, where $\\overset{→}{a} =(2,0)$, $\\overset{→}{b}$ is a unit vector with a magnitude of 1 and the angle between the two vectors is $60^{\\circ}$. Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 25955,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be a point not on line $XY$, and $Q$ a point on line $XY$ such that $PQ \\perp XY.$ Meanwhile, $R$ is a point on line $PY$ such that $XR \\perp PY.$ If $XR = 3$, $PQ = 6$, and $XY = 7$, then what is the length of $PY?$ Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 25956,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On every card of a deck of cards a regular 17-gon is displayed with all sides and diagonals, and the vertices are numbered from 1 through 17. On every card all edges (sides and diagonals) are colored with a color 1,2,...,105 such that the following property holds: for every 15 ... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 25957,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pirate is counting the loot he has stolen from S.S. AOPS, on which everything is counted in base 5. A treasure chest contains $3124_{5}$ dollars worth of jewelry, $3122_{5}$ dollars worth of gold coins, and $124_{5}$ dollars worth of rubbing alcohol. What is the total dolla... | math | {
"style": "rule",
"ground_truth": "865"
} | {
"split": "train",
"index": 25958,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A hotel packed breakfast for each of three guests. Each breakfast should have consisted of three types of rolls, one each of nut, cheese, and fruit rolls. The preparer wrapped each of the nine rolls and once wrapped, the rolls were indistinguishable from one another. She then r... | math | {
"style": "rule",
"ground_truth": "79"
} | {
"split": "train",
"index": 25959,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a circle, there are 2018 points. Each of these points is labeled with an integer. Each number is greater than the sum of the two numbers immediately preceding it in a clockwise direction.\n\nDetermine the maximum possible number of positive numbers among these 2018 numbers.\... | math | {
"style": "rule",
"ground_truth": "1009"
} | {
"split": "train",
"index": 25960,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the the roots of the equation $4x^3 + 5x^2 - 8x = 0$? Express your answer as a decimal to the nearest hundredth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1.25"
} | {
"split": "train",
"index": 25961,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bowl contains 10 jellybeans (four red, one blue and five white). If you pick three jellybeans from the bowl at random and without replacement, what is the probability that exactly two will be red? Express your answer as a common fraction. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"split": "train",
"index": 25962,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point $P$ is a moving point on the parabola $y^{2}=4x$, the minimum value of the sum of the distance from point $P$ to line $l$: $2x-y+3=0$ and the $y$-axis is ___. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}-1"
} | {
"split": "train",
"index": 25963,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A geometric sequence of positive integers starts with the first term as 5, and the fifth term of the sequence is 320. Determine the second term of this sequence. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 25964,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The regular hexagon \\(ABCDEF\\) has diagonals \\(AC\\) and \\(CE\\). The internal points \\(M\\) and \\(N\\) divide these diagonals such that \\(AM: AC = CN: CE = r\\). Determine \\(r\\) if it is known that points \\(B\\), \\(M\\), and \\(N\\) are collinear. Let's think step b... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt{3}}"
} | {
"split": "train",
"index": 25965,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of the expression $\\frac{3\\sqrt{12}}{\\sqrt{3} + \\sqrt{4} + \\sqrt{6}}$.\nA) $\\sqrt{3} + 2\\sqrt{2} - \\sqrt{6}$\nB) $4 - \\sqrt{3} - 2\\sqrt{2}$\nC) $\\sqrt{3} - 2\\sqrt{2} + \\sqrt{6}$\nD) $\\frac{1}{2}(\\sqrt{3} + \\sqrt{6} - 2\\sqrt{2})$\nE) $\\frac{1}{3}... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3} + 2\\sqrt{2} - \\sqrt{6}"
} | {
"split": "train",
"index": 25966,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A hexagonal prism has a regular hexagonal base, and its lateral edges are perpendicular to the base. It is known that all the vertices of the hexagonal prism are on the same spherical surface, and the volume of the hexagonal prism is $\\frac{9}{8}$, with a base perimeter of 3. ... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\pi}{3}"
} | {
"split": "train",
"index": 25967,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the first $n$ terms of the sequence $\\{a_n\\}$ is denoted as $S_n$. Given $a_1= \\frac{1}{2}$ and $2a_{n+1}+S_n=0$ for $n=1, 2, 3, \\ldots$. If $S_n \\leqslant k$ always holds, then the minimum value of $k$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 25968,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest integer $n$ for which $2^n$ divides \\[ \\binom 21 \\binom 42 \\binom 63 \\dots \\binom {128}{64}. \\]*Proposed by Evan Chen* Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "193"
} | {
"split": "train",
"index": 25969,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the radius \\( AO \\) of a circle centered at \\( O \\), a point \\( M \\) is chosen. On one side of \\( AO \\), points \\( B \\) and \\( C \\) are chosen on the circle such that \\( \\angle AMB = \\angle OMC = \\alpha \\). Find the length of \\( BC \\) if the radius of the ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 25970,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $({\\frac{3}{{a+1}}-a+1})÷\\frac{{{a^2}-4}}{{{a^2}+2a+1}}$, then choose a suitable number from $-1$, $2$, $3$ to substitute and evaluate. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 25971,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the parallelogram $ABCD$ , a line through $C$ intersects the diagonal $BD$ at $E$ and $AB$ at $F$ . If $F$ is the midpoint of $AB$ and the area of $\\vartriangle BEC$ is $100$ , find the area of the quadrilateral $AFED$ . Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "250"
} | {
"split": "train",
"index": 25972,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $BCDK$ be a convex quadrilateral such that $BC=BK$ and $DC=DK$ . $A$ and $E$ are points such that $ABCDE$ is a convex pentagon such that $AB=BC$ and $DE=DC$ and $K$ lies in the interior of the pentagon $ABCDE$ . If $\\angle ABC=120^{\\circ}$ and $\\angl... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 25973,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the function $y= \\sqrt {3}\\cos x+\\sin x, (x\\in \\mathbb{R})$ is translated to the right by $\\theta$ ($\\theta>0$) units. Determine the minimum value of $\\theta$ such that the resulting graph is symmetric about the y-axis. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{6}"
} | {
"split": "train",
"index": 25974,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $2x + 6 = 16$, what is the value of $x + 4$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 25975,
"solution": "Solution 1: Since $2x + 6 = 16$, then $\\frac{2x + 6}{2} = \\frac{16}{2}$ and so $x + 3 = 8$. Since $x + 3 = 8$, then $x + 4 = (x + 3) + 1 = 8 + 1 = 9$. Solution 2: Since $2x + 6 = 16$, then $2x = 16 - 6 = 10$. Since $2x = 10$, then $\\frac{2x}{2} = \\frac{10}{2}$ ... |
deepscaler | [
{
"role": "user",
"content": "Let $a \\clubsuit b = \\frac{2a}{b} \\cdot \\frac{b}{a}$. What is $(5 \\clubsuit (3 \\clubsuit 6)) \\clubsuit 1$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 25976,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fair $6$ sided die is rolled twice. What is the probability that the first number that comes up is greater than or equal to the second number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{12}"
} | {
"split": "train",
"index": 25977,
"solution": "1. **Define the Event**: Let's define the event $A$ as the event where the number on the first die roll is greater than or equal to the number on the second die roll.\n\n2. **Total Possible Outcomes**: Since each die has 6 faces, and the die is rolled twice, the to... |
deepscaler | [
{
"role": "user",
"content": "Factor $9y^2-30y+25$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(3y - 5)^2"
} | {
"split": "train",
"index": 25978,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ivan Tsarevich is fighting the Dragon Gorynych on the Kalinov Bridge. The Dragon has 198 heads. With one swing of his sword, Ivan Tsarevich can cut off five heads. However, new heads immediately grow back, the number of which is equal to the remainder when the number of heads l... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 25979,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the $8$ vertices of a cube, select any $4$ vertices. The probability that these $4$ points lie in the same plane is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6}{35}"
} | {
"split": "train",
"index": 25980,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the largest prime factor of $18^4 + 12^5 - 6^6$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 25981,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If both $a$ and $b$ in the fraction $\\frac{a+b}{a^2+b^2}$ are enlarged by a factor of $2$, then calculate the new value of the fraction. 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": 25982,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P(x) = x^2 - 3x - 9$. A real number $x$ is chosen at random from the interval $5 \\le x \\le 15$. The probability that $\\lfloor\\sqrt{P(x)}\\rfloor = \\sqrt{P(\\lfloor x \\rfloor)}$ is equal to $\\frac{\\sqrt{a} + \\sqrt{b} + \\sqrt{c} - d}{e}$ , where $a$, $b$, $c$, $d$,... | math | {
"style": "rule",
"ground_truth": "850"
} | {
"split": "train",
"index": 25983,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is given polygon with $2013$ sides $A_{1}A_{2}...A_{2013}$ . His vertices are marked with numbers such that sum of numbers marked by any $9$ consecutive vertices is constant and its value is $300$ . If we know that $A_{13}$ is marked with $13$ and $A_{20}$ is mar... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 25984,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the value of\n\\[\\frac{\\frac{2016}{1} + \\frac{2015}{2} + \\frac{2014}{3} + \\dots + \\frac{1}{2016}}{\\frac{1}{2} + \\frac{1}{3} + \\frac{1}{4} + \\dots + \\frac{1}{2017}}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2017"
} | {
"split": "train",
"index": 25985,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two numbers in the $4 \\times 4$ grid can be swapped to create a Magic Square (in which all rows, all columns and both main diagonals add to the same total).\nWhat is the sum of these two numbers?\nA 12\nB 15\nC 22\nD 26\nE 28\n\n\\begin{tabular}{|c|c|c|c|}\n\\hline \n9 & 6 & 3... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 25986,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x)$ is a monic quartic polynomial such that $f(-1)=-1$, $f(2)=-4$, $f(-3)=-9$, and $f(4)=-16$, find $f(1)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 25987,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the determinant of a $2 \\times 2$ matrix $A = \\begin{vmatrix} a & b \\\\ c & d \\end{vmatrix}$ is 3, find the values of $\\begin{vmatrix} 3a & 3b \\\\ 3c & 3d \\end{vmatrix}$ and $\\begin{vmatrix} 4a & 2b \\\\ 4c & 2d \\end{vmatrix}$. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 25988,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider this histogram of the scores for $81$ students taking a test:\n\nThe median is in the interval labeled Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 25989,
"solution": "1. **Identify the position of the median**: \n Since there are $81$ students, the median will be the score of the middle student when all scores are arranged in order. The middle position is calculated as $\\frac{81+1}{2} = 41$. Thus, the median is the score of t... |
deepscaler | [
{
"role": "user",
"content": "Given triangle $\\triangle ABC$, $A=120^{\\circ}$, $D$ is a point on side $BC$, $AD\\bot AC$, and $AD=2$. Calculate the possible area of $\\triangle ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 25990,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x, y$ be two positive integers, with $x> y$ , such that $2n = x + y$ , where n is a number two-digit integer. If $\\sqrt{xy}$ is an integer with the digits of $n$ but in reverse order, determine the value of $x - y$ Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "66"
} | {
"split": "train",
"index": 25991,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Of the students attending a school party, $60\\%$ of the students are girls, and $40\\%$ of the students like to dance. After these students are joined by $20$ more boy students, all of whom like to dance, the party is now $58\\%$ girls. How many students now at the party like ... | math | {
"style": "rule",
"ground_truth": "252"
} | {
"split": "train",
"index": 25992,
"solution": "Let $p$ denote the total number of people at the party. Then, because we know the proportions of boys to $p$ both before and after 20 boys arrived, we can create the following equation: \\[0.4p+20 = 0.42(p+20)\\] Solving for p gives us $p=580$, so the solution is $... |
deepscaler | [
{
"role": "user",
"content": "Find \\(n\\) such that \\(2^6 \\cdot 3^3 \\cdot n = 10!\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2100"
} | {
"split": "train",
"index": 25993,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the manufacturing of a steel cable, it was found that the cable has the same length as the curve defined by the system of equations:\n\n$$\n\\left\\{\\begin{array}{l}\nx+y+z=10 \\\\\nx y+y z+x z=18\n\\end{array}\\right.\n$$\n\nFind the length of the cable.\n Let's think step... | math | {
"style": "rule",
"ground_truth": "4 \\pi \\sqrt{\\frac{23}{3}}"
} | {
"split": "train",
"index": 25994,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A toy store sells a type of building block set: each starship is priced at 8 yuan, and each mech is priced at 26 yuan. A starship and a mech can be combined to form an ultimate mech, which sells for 33 yuan per set. If the store owner sold a total of 31 starships and mechs in o... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 25995,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x) = 2x + 3$ and $g(x) = 3x - 2$ find $\\frac{f(g(f(2)))}{g(f(g(2)))}$. Express your answer in the form $\\frac{a}{b}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{41}{31}"
} | {
"split": "train",
"index": 25996,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $C$: $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$), with the circle centered at the right focus $F$ of $C$ ($c$, $0$) and with radius $a$ intersects one of the asymptotes of $C$ at points $A$ and $B$. If $|AB| = \\frac{2}{3}c$, determine the e... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{5}}{5}"
} | {
"split": "train",
"index": 25997,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bug crawls along a number line, starting at $-2$. It crawls to $-6$, then turns around and crawls to $5$. How many units does the bug crawl altogether? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 25998,
"solution": "1. **Calculate the distance from $-2$ to $-6$:**\n The distance on a number line is the absolute difference between the two points. Thus, the distance from $-2$ to $-6$ is:\n \\[\n |-6 - (-2)| = |-6 + 2| = |-4| = 4 \\text{ units}\n \\]\n\n2. **Calculate the... |
deepscaler | [
{
"role": "user",
"content": "Evaluate the following product of sequences: $\\frac{1}{3} \\cdot \\frac{9}{1} \\cdot \\frac{1}{27} \\cdot \\frac{81}{1} \\dotsm \\frac{1}{2187} \\cdot \\frac{6561}{1}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 25999,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A factory produced an original calculator that performs two operations:\n\n(a) the usual addition, denoted by \\( + \\) \n\n(b) an operation denoted by \\( \\circledast \\).\n\nWe know that, for any natural number \\( a \\), the following hold:\n\n\\[ \n(i) \\quad a \\circledas... | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"split": "train",
"index": 26000,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a right-angled triangle, the lengths of the two legs are 12 and 5, respectively. Find the length of the hypotenuse and the height from the right angle to the hypotenuse. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{60}{13}"
} | {
"split": "train",
"index": 26001,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Petrov booked an apartment in a newly-built house which has five identical entrances. Initially, the entrances were numbered from left to right, and Petrov's apartment had the number 636. Then the builder changed the numbering to the opposite direction (from right to left). Aft... | math | {
"style": "rule",
"ground_truth": "985"
} | {
"split": "train",
"index": 26002,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ satisfying $a_{1}=10$, $a_{n+1}=a_{n}+18n+10$ for $n \\in \\mathbb{N}^*$, find $\\lim_{n \\to \\infty} (\\sqrt{a_{n}} - [\\sqrt{a_{n}}])$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 26003,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On an island, there live three tribes: knights, who always tell the truth; liars, who always lie; and tricksters, who sometimes tell the truth and sometimes lie. At a round table sit 100 representatives of these tribes.\n\nEach person at the table said two sentences: 1) \"To my... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 26004,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many four-digit positive integers are multiples of 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1286"
} | {
"split": "train",
"index": 26005,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We have two concentric circles $C_{1}$ and $C_{2}$ with radii 1 and 2, respectively. A random chord of $C_{2}$ is chosen. What is the probability that it intersects $C_{1}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "N/A"
} | {
"split": "train",
"index": 26006,
"solution": "The question given at the beginning of the problem statement is a famous problem in probability theory widely known as Bertrand's paradox. Depending on the interpretation of the phrase \"random chord,\" there are at least three different possible answers to this qu... |
deepscaler | [
{
"role": "user",
"content": "Given a fair die is thrown twice, and let the numbers obtained be denoted as a and b respectively, find the probability that the equation ax^2 + bx + 1 = 0 has real solutions. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{19}{36}"
} | {
"split": "train",
"index": 26007,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a cube \\( ABCD-A_1B_1C_1D_1 \\) with edge length 1, a point \\( M \\) is taken on the diagonal \\( A_1D \\) and a point \\( N \\) is taken on \\( CD_1 \\) such that the line segment \\( MN \\) is parallel to the diagonal plane \\( A_1ACC_1 \\), find the minimum value of ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 26008,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An isosceles right triangle with side lengths in the ratio 1:1:\\(\\sqrt{2}\\) is inscribed in a circle with a radius of \\(\\sqrt{2}\\). What is the area of the triangle and the circumference of the circle? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "2\\pi\\sqrt{2}"
} | {
"split": "train",
"index": 26009,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all the roots of the equation \\( 4x^{2} - 58x + 190 = (29 - 4x - \\log_{2} x) \\cdot \\log_{2} x \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 26010,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the product $abc$ for the polynomial $Q(x) = x^3 + ax^2 + bx + c$ if its roots are $\\sin \\frac{\\pi}{6}, \\sin \\frac{\\pi}{3},$ and $\\sin \\frac{5\\pi}{6}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 26011,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Dana goes first and Carl's coin lands heads with probability $\\frac{2}{7}$, and Dana's coin lands heads with probability $\\frac{3}{8}$. Find the probability that Carl wins the game. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10}{31}"
} | {
"split": "train",
"index": 26012,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\tan \\alpha = 8$ and $\\tan \\beta = 7,$ then find $\\tan (\\alpha - \\beta).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{57}"
} | {
"split": "train",
"index": 26013,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $b$ if $-x^2+bx-5<0$ only when $x\\in (-\\infty, 1)\\cup(5,\\infty)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 26014,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The volume of a rectangular solid each of whose side, front, and bottom faces are $12\\text{ in}^{2}$, $8\\text{ in}^{2}$, and $6\\text{ in}^{2}$ respectively is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24\\text{ in}^{3}"
} | {
"split": "train",
"index": 26015,
"solution": "1. **Identify the equations based on the given areas of the faces:**\n Let the side lengths of the rectangular solid be $a$, $b$, and $c$. The areas of the side, front, and bottom faces are given as $12\\text{ in}^2$, $8\\text{ in}^2$, and $6\\text{ in}^2$ respec... |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system, points whose x and y coordinates are both integers are called lattice points. How many lattice points \\((x, y)\\) satisfy the inequality \\((|x|-1)^{2}+(|y|-1)^{2}<2\\)? Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 26016,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest prime divisor of $36^2 + 49^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 26017,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The arithmetic mean of the nine numbers in the set $\\{9, 99, 999, 9999, \\ldots, 999999999\\}$ is a $9$-digit number $M$, all of whose digits are distinct. The number $M$ doesn't contain the digit Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 26018,
"solution": "1. **Identify the pattern in the set**: The set given is $\\{9, 99, 999, 9999, \\ldots, 999999999\\}$. Each number in the set is composed of digits that are all 9s, and the number of 9s increases by one in each subsequent number.\n\n2. **Express each number as a su... |
deepscaler | [
{
"role": "user",
"content": "Let $x$, $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\\log_x w = 24$, $\\log_y w = 40$ and $\\log_{xyz} w = 12$. Find $\\log_z w$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 26019,
"solution": "Converting all of the logarithms to exponentials gives $x^{24} = w, y^{40} =w,$ and $x^{12}y^{12}z^{12}=w.$ Thus, we have $y^{40} = x^{24} \\Rightarrow z^3=y^2.$ We are looking for $\\log_z w,$ which by substitution, is $\\log_{y^{\\frac{2}{3}}} y^{40} = 40 \\div \... |
deepscaler | [
{
"role": "user",
"content": "A magician and their assistant plan to perform a trick. The spectator writes a sequence of $N$ digits on a board. The magician's assistant then covers two adjacent digits with a black dot. Next, the magician enters and has to guess both covered digits (including the order in wh... | math | {
"style": "rule",
"ground_truth": "101"
} | {
"split": "train",
"index": 26020,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the leftmost position can be occupied by student A or B, and the rightmost position cannot be occupied by student A, find the number of different arrangements of the six high-performing students from Class 1, Grade 12. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "216"
} | {
"split": "train",
"index": 26021,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let\n\\[f(x) = \\frac{x^2 - 6x + 6}{2x - 4}\\]and\n\\[g(x) = \\frac{ax^2 + bx + c}{x - d}.\\]You are given the following properties:\n\n$\\bullet$ The graphs of $f(x)$ and $g(x)$ have the same vertical asymptote.\n\n$\\bullet$ The oblique asymptotes of $f(x)$ and $g(x)$ are per... | math | {
"style": "rule",
"ground_truth": "\\left( 4, -\\frac{1}{2} \\right)"
} | {
"split": "train",
"index": 26022,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the equation of the line $y = mx + 3$, find the maximum possible value of $a$ such that the line passes through no lattice point with $0 < x \\leq 150$ for all $m$ satisfying $\\frac{2}{3} < m < a$. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "\\frac{101}{151}"
} | {
"split": "train",
"index": 26023,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $n$ such that $2^6 \\cdot 3^3 \\cdot n = 10!$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2100"
} | {
"split": "train",
"index": 26024,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The side length of square $A$ is 36 cm. The side length of square $B$ is 42 cm. What is the ratio of the area of square $A$ to the area of square $B$? 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{36}{49}"
} | {
"split": "train",
"index": 26025,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A person rolled a fair six-sided die $100$ times and obtained a $6$ $19$ times. What is the approximate probability of rolling a $6$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.19"
} | {
"split": "train",
"index": 26026,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The $\\$4.55$ in Carol's piggy bank consists of quarters and nickels. There are seven more nickels than quarters. How many nickels does Carol have in her bank? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 26027,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that complex numbers $a,$ $b,$ and $c$ are zeros of a polynomial $P(z) = z^3 + qz + r,$ and $|a|^2 + |b|^2 + |c|^2 = 300$. The points corresponding to $a,$ $b,$ and $c$ on the complex plane are the vertices of a right triangle. Find the square of the length of the hypoten... | math | {
"style": "rule",
"ground_truth": "450"
} | {
"split": "train",
"index": 26028,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the side lengths of a convex quadrilateral are $a=4, b=5, c=6, d=7$, find the radius $R$ of the circumscribed circle around this quadrilateral. Provide the integer part of $R^{2}$ as the answer. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 26029,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\\left(a-c\\right)\\left(a+c\\right)\\sin C=c\\left(b-c\\right)\\sin B$. \n$(1)$ Find angle $A$; \n$(2)$ If the area of $\\triangle ABC$ is $\\sqrt{3}$, $\\sin B... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 26030,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that three balls are randomly and independently tossed into bins numbered with the positive integers such that for each ball, the probability that it is tossed into bin i is $3^{-i}$ for i = 1,2,3,..., find the probability that all balls end up in consecutive bins. Let's ... | math | {
"style": "rule",
"ground_truth": "1/702"
} | {
"split": "train",
"index": 26031,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the 39th number in the row of Pascal's triangle that has 41 numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "780"
} | {
"split": "train",
"index": 26032,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sum_{k=1}^{35}\\sin 5k=\\tan \\frac mn,$ where angles are measured in degrees, and $m$ and $n$ are relatively prime positive integers that satisfy $\\frac mn<90,$ find $m+n.$\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "177"
} | {
"split": "train",
"index": 26033,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $u$ and $v$ be real numbers satisfying the inequalities $2u + 3v \\le 10$ and $4u + v \\le 9.$ Find the largest possible value of $u + 2v$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6.1"
} | {
"split": "train",
"index": 26034,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is preparing to offer a discount on the selling price, but the profit margin must not be less than $10\\%$. Find the maximum discount rate that can be offered. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "8.8"
} | {
"split": "train",
"index": 26035,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define $n!!$ to be $n(n-2)(n-4)\\cdots 3\\cdot 1$ for $n$ odd and $n(n-2)(n-4)\\cdots 4\\cdot 2$ for $n$ even. When $\\sum_{i=1}^{2009} \\frac{(2i-1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $2^ab$ with $b$ odd. Find $\\dfrac{ab}{10}$. Let's th... | math | {
"style": "rule",
"ground_truth": "401"
} | {
"split": "train",
"index": 26036,
"solution": "Using the steps of the previous solution we get $c = \\sum_{i=1}^{2009} {2i\\choose i} \\cdot 2^{2\\cdot 2009 - 2i}$ and if you do the small cases(like $1, 2, 3, 4, 5, 6$) you realize that you can \"thin-slice\" the problem and simply look at the cases where $i=200... |
deepscaler | [
{
"role": "user",
"content": "Suppose\n\\[\\frac{1}{x^3 - 3x^2 - 13x + 15} = \\frac{A}{x+3} + \\frac{B}{x-1} + \\frac{C}{(x-1)^2}\\]\nwhere $A$, $B$, and $C$ are real constants. What is $A$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"split": "train",
"index": 26037,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the circle $C: x^2 + y^2 - (6 - 2m)x - 4my + 5m^2 - 6m = 0$, and a fixed line $l$ passing through the point $A(1, 0)$, for any real number $m$, the chord intercepted by circle $C$ on line $l$ always has a constant length $A$. Find the constant value of $A$. Let's think st... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{145}}{5}"
} | {
"split": "train",
"index": 26038,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two identical cars are traveling in the same direction. The speed of one is $36 \\kappa \\mu / h$, and the other is catching up with a speed of $54 \\mathrm{kм} / h$. It is known that the reaction time of the driver in the rear car to the stop signals of the preceding car is 2 ... | math | {
"style": "rule",
"ground_truth": "42.5"
} | {
"split": "train",
"index": 26039,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(n)$ denotes the number of divisors of $2024^{2024}$ that are either less than $n$ or share at least one prime factor with $n$ , find the remainder when $$ \\sum^{2024^{2024}}_{n=1} f(n) $$ is divided by $1000$ . Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "224"
} | {
"split": "train",
"index": 26040,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( ABC \\), \\( AC = 3 AB \\). Let \\( AD \\) bisect angle \\( A \\) with \\( D \\) lying on \\( BC \\), and let \\( E \\) be the foot of the perpendicular from \\( C \\) to \\( AD \\). Find \\( \\frac{[ABD]}{[CDE]} \\). (Here, \\([XYZ]\\) denotes the area of trian... | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"split": "train",
"index": 26041,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $ABCD$ is a rectangle with sides of length $12$ and $18$ . Let $S$ be the region of points contained in $ABCD$ which are closer to the center of the rectangle than to any of its vertices. Find the area of $S$ . Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 26042,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $AHI$ is equilateral. We know $\\overline{BC}$, $\\overline{DE}$ and $\\overline{FG}$ are all parallel to $\\overline{HI}$ and $AB = BD = DF = FH$. What is the ratio of the area of trapezoid $FGIH$ to the area of triangle $AHI$? Express your answer as a common fraction... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{16}"
} | {
"split": "train",
"index": 26043,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the common ratio of the infinite geometric series: $$\\frac{4}{7} + \\frac{16}{21} + \\frac{64}{63} + \\dots$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 26044,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $2c-2a\\cos B=b$. \n$(1)$ Find the size of angle $A$; \n$(2)$ If the area of $\\triangle ABC$ is $\\frac{\\sqrt{3}}{4}$, and $c^{2}+ab\\cos ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{7}}{2}"
} | {
"split": "train",
"index": 26045,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The perimeter of an equilateral triangle exceeds the perimeter of a square by $1989 \\text{ cm}$. The length of each side of the triangle exceeds the length of each side of the square by $d \\text{ cm}$. The square has perimeter greater than 0. How many positive integers are NO... | math | {
"style": "rule",
"ground_truth": "663"
} | {
"split": "train",
"index": 26046,
"solution": "1. **Set up the equations based on the problem statement:**\n Let $t$ be the length of each side of the equilateral triangle, and $s$ be the length of each side of the square. The perimeter of the triangle is $3t$ and the perimeter of the square is $4s$. Accordin... |
deepscaler | [
{
"role": "user",
"content": "In the expansion of $(\\frac{3}{{x}^{2}}+x+2)^{5}$, the coefficient of the linear term in $x$ is ____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 26047,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $f(x)$ and $g(x)$ are functions which satisfy the equations $f(g(x)) = 2x^2$ and $g(f(x)) = x^4$ for all $x \\ge 1$. If $g(4) = 16$, compute $[g(2)]^4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 26048,
"solution": ""
} |
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