data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length 2 so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region ---- inside the hexagon but outside all of the semici... | math | {
"style": "rule",
"ground_truth": "6\\sqrt{3} - 3\\pi"
} | {
"split": "train",
"index": 8200,
"solution": "1. **Identify the Geometry of the Problem:**\n - The problem involves a regular hexagon with side length 2.\n - Six semicircles are inscribed such that their diameters coincide with the sides of the hexagon.\n\n2. **Calculate the Area of the Regular Hexagon:**\n... |
deepscaler | [
{
"role": "user",
"content": "\nBrothers Lyosha and Sasha decided to get from home to the skate park. They left at the same time, but Lyosha walked with the skateboard in his hands, while Sasha rode the skateboard. It is known that Sasha rides the skateboard 3 times faster than Lyosha walks with the skatebo... | math | {
"style": "rule",
"ground_truth": "1100"
} | {
"split": "train",
"index": 8201,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain middle school, 500 eighth-grade students took the biology and geography exam. There were a total of 180 students who scored between 80 and 100 points. What is the frequency of this score range? Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "0.36"
} | {
"split": "train",
"index": 8202,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to put 5 balls in 3 boxes if the balls are not distinguishable but the boxes are? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 8203,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the value of\n\\[\n3003 + \\frac{1}{3} \\left( 3002 + \\frac{1}{3} \\left( 3001 + \\dots + \\frac{1}{3} \\left( 4 + \\frac{1}{3} \\cdot 3 \\right) \\right) \\dotsb \\right).\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4503.75"
} | {
"split": "train",
"index": 8204,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least positive integer with exactly $12$ positive factors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 8205,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The first three numbers of a sequence are \\(1, 7, 8\\). Every subsequent number is the remainder obtained when the sum of the previous three numbers is divided by 4. Find the sum of the first 2011 numbers in this sequence. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "3028"
} | {
"split": "train",
"index": 8206,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Right triangle $ABC$ (hypotenuse $\\overline{AB}$) is inscribed in equilateral triangle $PQR,$ as shown. If $PC = 3$ and $BP = CQ = 2,$ compute $AQ.$\n\n[asy]\nunitsize(0.8 cm);\n\npair A, B, C, P, Q, R;\n\nP = (0,0);\nQ = (5,0);\nR = 5*dir(60);\nA = Q + 8/5*dir(120);\nB = 2*... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{5}"
} | {
"split": "train",
"index": 8207,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An amusement park has a series of miniature buildings and landscape models from various places in the United States, scaled at $1: 20$. If the height of the United States Capitol Building is 289 feet, what is the nearest integer height of its model in feet? Let's think step by ... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 8208,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Zhang Bing was born in 1953. In a certain year before this year, his age was a multiple of 9 and equal to the sum of the digits of that year. How old was he that year? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 8209,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is\n$$\n\\begin{cases}\nx=2+t\\cos \\alpha \\\\\ny=1+t\\sin \\alpha\n\\end{cases}\n(t \\text{ is the parameter}),\n$$\nIn the polar coordinate system (which uses the same unit length as the Cartesian ... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{7}"
} | {
"split": "train",
"index": 8210,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Count the number of permutations $a_{1} a_{2} \\ldots a_{7}$ of 1234567 with longest decreasing subsequence of length at most two (i.e. there does not exist $i<j<k$ such that $a_{i}>a_{j}>a_{k}$ ). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "429"
} | {
"split": "train",
"index": 8211,
"solution": "$C(7)=429$."
} |
deepscaler | [
{
"role": "user",
"content": "From a batch of parts, 50 are drawn, and then 40 out of these 50 are inspected. It is found that there are 38 qualified products. Calculate the pass rate of this batch of products as a percentage. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "95\\%"
} | {
"split": "train",
"index": 8212,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the first three digits to the right of the decimal point in the decimal representation of $\\left(10^{987} + 1\\right)^{8/3}$ using the Binomial Expansion. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "666"
} | {
"split": "train",
"index": 8213,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a regular tetrahedron O-ABC with each edge length equal to 1, if point P satisfies $$\\overrightarrow {OP}=x \\overrightarrow {OA}+y \\overrightarrow {OB}+z \\overrightarrow {OC}(x+y+z=1)$$, find the minimum value of $$| \\overrightarrow {OP}|$$. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{3}"
} | {
"split": "train",
"index": 8214,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $x$ in the diagram?\n\n[asy]\n\nimport olympiad;\n\ndraw((0,0)--(sqrt(3),0)--(0,sqrt(3))--cycle);\n\ndraw((0,0)--(-1,0)--(0,sqrt(3))--cycle);\n\nlabel(\"8\",(-1/2,sqrt(3)/2),NW);\n\nlabel(\"$x$\",(sqrt(3)/2,sqrt(3)/2),NE);\n\ndraw(\"$45^{\\circ}$\",(1.5,0),... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{6}"
} | {
"split": "train",
"index": 8215,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The polynomial $2x^3 + bx + 7$ has a factor of the form $x^2 + px + 1.$ Find $b.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{45}{2}"
} | {
"split": "train",
"index": 8216,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, each of the three identical circles touch the other two. The circumference of each circle is 36. What is the perimeter of the shaded region? [asy]\n\ndefaultpen(1);\n\npath p = (1, 0){down}..{-dir(30)}dir(-60){dir(30)}..{dir(-30)}((2, 0) + dir(-120)){-dir(-30)... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 8217,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the following diagram, \\(\\angle ACB = 90^\\circ\\), \\(DE \\perp BC\\), \\(BE = AC\\), \\(BD = \\frac{1}{2} \\mathrm{~cm}\\), and \\(DE + BC = 1 \\mathrm{~cm}\\). Suppose \\(\\angle ABC = x^\\circ\\). Find the value of \\(x\\). Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 8218,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mark's cousin has $10$ identical stickers and $5$ identical sheets of paper. How many ways are there for him to distribute all of the stickers on the sheets of paper, given that each sheet must have at least one sticker, and only the number of stickers on each sheet matters? Le... | math | {
"style": "rule",
"ground_truth": "126"
} | {
"split": "train",
"index": 8219,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $|\\overrightarrow{a}|=1$, $(\\overrightarrow{a}+\\overrightarrow{b}) \\perp \\overrightarrow{a}$, and $(2\\overrightarrow{a}+\\overrightarrow{b}) \\perp \\overrightarrow{b}$, calculate the angle between vect... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\pi}{4}"
} | {
"split": "train",
"index": 8220,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Twelve million added to twelve thousand equals what number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12012000"
} | {
"split": "train",
"index": 8221,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the 3rd grade, the boys wear blue swim caps, and the girls wear red swim caps. The male sports commissioner says, \"I see 1 more blue swim cap than 4 times the number of red swim caps.\" The female sports commissioner says, \"I see 24 more blue swim caps than red swim caps.\... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 8222,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S_n$ and $T_n$ respectively be the sum of the first $n$ terms of the arithmetic sequences $\\{a_n\\}$ and $\\{b_n\\}$. Given that $\\frac{S_n}{T_n} = \\frac{n}{2n+1}$ for $n \\in \\mathbb{N}^*$, find the value of $\\frac{a_5}{b_5}$. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{19}"
} | {
"split": "train",
"index": 8223,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the expression $(-\\frac{1}{343})^{-2/3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 8224,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $3+a=4-b$ and $4+b=7+a$, what is $3-a$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 8225,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Divide the product of the first five positive composite integers by the product of the next five composite integers. 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{1}{42}"
} | {
"split": "train",
"index": 8226,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ y_0$ be chosen randomly from $ \\{0, 50\\}$ , let $ y_1$ be chosen randomly from $ \\{40, 60, 80\\}$ , let $ y_2$ be chosen randomly from $ \\{10, 40, 70, 80\\}$ , and let $ y_3$ be chosen randomly from $ \\{10, 30, 40, 70, 90\\}$ . (In each choice, the possib... | math | {
"style": "rule",
"ground_truth": "107"
} | {
"split": "train",
"index": 8227,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\overrightarrow{a}=(\\sin x, \\frac{3}{4})$, $\\overrightarrow{b}=( \\frac{1}{3}, \\frac{1}{2}\\cos x )$, and $\\overrightarrow{a} \\parallel \\overrightarrow{b}$. Find the acute angle $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 8228,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\sin\\left( \\frac{\\pi}{3} + a \\right) = \\frac{5}{13}$, and $a \\in \\left( \\frac{\\pi}{6}, \\frac{2\\pi}{3} \\right)$, find the value of $\\sin\\left( \\frac{\\pi}{12} + a \\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17\\sqrt{2}}{26}"
} | {
"split": "train",
"index": 8229,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a circle with a diameter of $1$ unit, if a point $P$ on the circle starts from point $A$ representing $3$ on the number line and rolls one round to the left along the number line, find the real number represented by the point $B$ where $P$ arrives on the number line. Let'... | math | {
"style": "rule",
"ground_truth": "3 - \\pi"
} | {
"split": "train",
"index": 8230,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a triangle, the larger angle at the base is $45^{\\circ}$, and the altitude divides the base into segments of 20 and 21. Find the length of the larger lateral side. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 8231,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The slope of a line is $-2$ and its $x$-intercept is $(5,0).$ What is the $y$-intercept point of the line? Express your answer as an ordered pair. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(0,10)"
} | {
"split": "train",
"index": 8232,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A function $f$ is defined by $f(z) = (4 + i) z^2 + \\alpha z + \\gamma$ for all complex numbers $z$, where $\\alpha$ and $\\gamma$ are complex numbers and $i^2 = - 1$. Suppose that $f(1)$ and $f(i)$ are both real. What is the smallest possible value of $| \\alpha | + |\\gamma |... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 8233,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Person A, Person B, and Person C start from point $A$ to point $B$. Person A starts at 8:00, Person B starts at 8:20, and Person C starts at 8:30. They all travel at the same speed. After Person C has been traveling for 10 minutes, Person A is exactly halfway between Person B a... | math | {
"style": "rule",
"ground_truth": "2418"
} | {
"split": "train",
"index": 8234,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bus arrives randomly between 3:30 pm and 4:30 pm, waits for 40 minutes, and then departs. If Sara also arrives randomly between 3:30 pm and 4:30 pm, what is the probability that the bus will still be there when she arrives? Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 8235,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of: \\(\\frac{\\left(\\sqrt{3} \\cdot \\tan 12^{\\circ} - 3\\right) \\cdot \\csc 12^{\\circ}}{4 \\cos ^{2} 12^{\\circ} - 2}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4 \\sqrt{3}"
} | {
"split": "train",
"index": 8236,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is given that \\( a = 103 \\times 97 \\times 10009 \\). Find \\( a \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "99999919"
} | {
"split": "train",
"index": 8237,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two 8-sided dice are tossed (each die has faces numbered 1 to 8). What is the probability that the sum of the numbers shown on the dice is either a prime or a multiple of 4? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{39}{64}"
} | {
"split": "train",
"index": 8238,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Brand X soda advertises, ``We will give you 20$\\%$ more soda than Brand Y for a total price that is 10$\\%$ less than Brand Y's price!'' What is the ratio of the unit price of Brand X soda to the unit price of Brand Y soda? Express your answer as a common fraction. Let's think... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 8239,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of the floor in a square room is 225 square feet. The homeowners plan to cover the floor with rows of 6-inch by 6-inch tiles. How many tiles will be in each row? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 8240,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/19 of the original integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "950"
} | {
"split": "train",
"index": 8241,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Masha talked a lot on the phone with her friends, and the charged battery discharged exactly after a day. It is known that the charge lasts for 5 hours of talk time or 150 hours of standby time. How long did Masha talk with her friends? Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "126/29"
} | {
"split": "train",
"index": 8242,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line passing through the point P(3/2, 1/2) intersects the ellipse x^2/6 + y^2/2 = 1 at points A and B, satisfying PA + PB = 0. If M is any point on the line AB and O is the origin, find the minimum value of |OM|. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 8243,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area, in square units, of a triangle that has sides of $4,3$ and $3$ units? Express your answer in simplest radical form. 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": 8244,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\sin^2(wx) - \\sin^2(wx - \\frac{\\pi}{6})$ ($x \\in \\mathbb{R}$, $w$ is a constant and $\\frac{1}{2} < w < 1$), the graph of function $f(x)$ is symmetric about the line $x = \\pi$.\n\n(I) Find the smallest positive period of the function $f(x)$;\n\... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 8245,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive multiples of 6 that are less than 150 have a units digit of 6? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 8246,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to line up $19$ girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "262144"
} | {
"split": "train",
"index": 8247,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $A_1,A_2,\\cdots ,A_n \\subseteq \\left \\{ 1,2,\\cdots ,2018 \\right \\}$ and $\\left | A_i \\right |=2, i=1,2,\\cdots ,n$, satisfying that $$A_i + A_j, \\; 1 \\le i \\le j \\le n ,$$ are distinct from each other. $A + B = \\left \\{ a+b|a\\in A,\\,b\\in B \\right \\}$... | math | {
"style": "rule",
"ground_truth": "4033"
} | {
"split": "train",
"index": 8248,
"solution": "\nSuppose \\( A_1, A_2, \\ldots, A_n \\subseteq \\{1, 2, \\ldots, 2018\\} \\) and \\( |A_i| = 2 \\) for \\( i = 1, 2, \\ldots, n \\), satisfying that \\( A_i + A_j \\), \\( 1 \\leq i \\leq j \\leq n \\), are distinct from each other. Here, \\( A + B = \\{a + b \\mid... |
deepscaler | [
{
"role": "user",
"content": "What is the sum and the product of the values of $x$ that satisfy the equation $x^2 - 7x + 12 = 0$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 8249,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the sum of the number $10 - \\sqrt{2018}$ and its radical conjugate. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 8250,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Karen has seven envelopes and seven letters of congratulations to various HMMT coaches. If she places the letters in the envelopes at random with each possible configuration having an equal probability, what is the probability that exactly six of the letters are in the correct ... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 8251,
"solution": "0, since if six letters are in their correct envelopes the seventh is as well."
} |
deepscaler | [
{
"role": "user",
"content": "Given that the coordinate of one focus of the ellipse $3x^{2} + ky^{2} = 1$ is $(0, 1)$, determine its eccentricity. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 8252,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nAn investor has an open brokerage account with an investment company. In 2021, the investor received the following income from securities:\n\n- Dividends from shares of the company PAO “Winning” amounted to 50,000 rubles.\n- Coupon income from government bonds OFZ amounted to... | math | {
"style": "rule",
"ground_truth": "11050"
} | {
"split": "train",
"index": 8253,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fair six-sided die is rolled twice. Let $a$ and $b$ be the numbers obtained from the first and second roll respectively. Determine the probability that three line segments of lengths $a$, $b$, and $5$ can form an isosceles triangle. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{18}"
} | {
"split": "train",
"index": 8254,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has an area 1. Points $E,F,G$ lie, respectively, on sides $BC$, $CA$, $AB$ such that $AE$ bisects $BF$ at point $R$, $BF$ bisects $CG$ at point $S$, and $CG$ bisects $AE$ at point $T$. Find the area of the triangle $RST$. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "\\frac{7 - 3 \\sqrt{5}}{4}"
} | {
"split": "train",
"index": 8255,
"solution": "Choose $r,s,t$ so that $EC = rBC, FA = sCA, GB = tCB$, and let $[XYZ]$ denote the area of triangle $XYZ$. Then $[ABE] = [AFE]$ since the triangles have the same altitude and base. Also $[ABE] = (BE/BC) [ABC] = 1-r$, and $[ECF] = (EC/BC)(CF/CA)[ABC] = r(1-s)$ (e.g., ... |
deepscaler | [
{
"role": "user",
"content": "The increasing sequence of positive integers $b_1, b_2, b_3, \\ldots$ follows the rule:\n\\[ b_{n + 2} = b_{n + 1} + b_n \\]\nfor all $n \\geq 1$. If $b_5 = 55$, find $b_6$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 8256,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiao Ming must stand in the very center, and Xiao Li and Xiao Zhang must stand together in a graduation photo with seven students. Find the number of different arrangements. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 8257,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=-x^{3}+ax^{2}+bx$ in the interval $(-2,1)$. The function reaches its minimum value when $x=-1$ and its maximum value when $x=\\frac{2}{3}$.\n(1) Find the equation of the tangent line to the function $y=f(x)$ at $x=-2$.\n(2) Find the maximum and minimum ... | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"split": "train",
"index": 8258,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an isosceles trapezoid with bases \\(a = 21\\), \\(b = 9\\) and height \\(h = 8\\), find the radius of the circumscribed circle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{85}{8}"
} | {
"split": "train",
"index": 8259,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line is parameterized by\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ -2 \\end{pmatrix} + t \\begin{pmatrix} 3 \\\\ 4 \\end{pmatrix}.\\]A second line is parameterized by\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} -8 \\\\ 12 \\end... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{\\sqrt{10}}"
} | {
"split": "train",
"index": 8260,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Values for $A, B, C,$ and $D$ are to be selected from $\\{1, 2, 3, 4, 5, 6\\}$ without replacement (i.e. no two letters have the same value). How many ways are there to make such choices so that the two curves $y=Ax^2+B$ and $y=Cx^2+D$ intersect? (The order in which the curves ... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 8261,
"solution": "1. **Understanding the Problem**: We need to find the number of ways to choose values for $A, B, C, D$ from the set $\\{1, 2, 3, 4, 5, 6\\}$ without replacement such that the curves $y = Ax^2 + B$ and $y = Cx^2 + D$ intersect.\n\n2. **Condition for Intersection**: T... |
deepscaler | [
{
"role": "user",
"content": "In a regular hexagon \\(ABCDEF\\), points \\(M\\) and \\(K\\) are taken on the diagonals \\(AC\\) and \\(CE\\) respectively, such that \\(AM : AC = CK : CE = n\\). Points \\(B, M,\\) and \\(K\\) are collinear. Find \\(n\\). Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 8262,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation \\(x^{2}+5x+1=0\\) has roots \\(x_{1}\\) and \\(x_{2}\\). Find the value of the expression\n\n\\[ \n\\left(\\frac{x_{1} \\sqrt{6}}{1+x_{2}}\\right)^{2}+\\left(\\frac{x_{2} \\sqrt{6}}{1+x_{1}}\\right)^{2} \n\\] Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "220"
} | {
"split": "train",
"index": 8263,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the radius \\( AO \\) of a circle with center \\( O \\), point \\( M \\) is chosen. On one side of \\( AO \\) on the circle, points \\( B \\) and \\( C \\) are chosen such that \\(\\angle AMB = \\angle OMC = \\alpha\\). Find the length of \\( BC \\), given that the radius of... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 8264,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve in positive real numbers: $n+ \\lfloor \\sqrt{n} \\rfloor+\\lfloor \\sqrt[3]{n} \\rfloor=2014$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1958"
} | {
"split": "train",
"index": 8265,
"solution": "\nWe are asked to solve the equation \\( n+ \\lfloor \\sqrt{n} \\rfloor + \\lfloor \\sqrt[3]{n} \\rfloor = 2014 \\) for positive real numbers \\( n \\).\n\nTo begin, we denote:\n- \\( x = \\lfloor \\sqrt{n} \\rfloor \\),\n- \\( y = \\lfloor \\sqrt[3]{n} \\rfloor \\)... |
deepscaler | [
{
"role": "user",
"content": "Real numbers \\(a, b, c\\) and a positive number \\(\\lambda\\) satisfy \\(f(x) = x^3 + a x^2 + b x + c\\), which has 3 real roots \\(x_1, x_2, x_3\\), such that:\n(1) \\(x_2 - x_1 = \\lambda\\);\n(2) \\(x_3 > \\frac{1}{2}(x_1 + x_2)\\).\nFind the maximum value of \\(\\frac{2a^... | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{3}}{2}"
} | {
"split": "train",
"index": 8266,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cyclic quadrilateral $A B C D$ has side lengths $A B=1, B C=2, C D=3$ and $D A=4$. Points $P$ and $Q$ are the midpoints of $\\overline{B C}$ and $\\overline{D A}$. Compute $P Q^{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{116}{35}"
} | {
"split": "train",
"index": 8267,
"solution": "Construct $\\overline{A C}, \\overline{A Q}, \\overline{B Q}, \\overline{B D}$, and let $R$ denote the intersection of $\\overline{A C}$ and $\\overline{B D}$. Because $A B C D$ is cyclic, we have that $\\triangle A B R \\sim \\triangle D C R$ and $\\triangle A D R ... |
deepscaler | [
{
"role": "user",
"content": "Two workers started paving paths in a park simultaneously from point $A$. The first worker paves the section $A-B-C$ and the second worker paves the section $A-D-E-F-C$. They worked at constant speeds and finished at the same time, having spent 9 hours on the job. It is known t... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 8268,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $α∈(0, \\dfrac{π}{2})$, $\\cos ( \\dfrac{π}{4}-α)=2 \\sqrt{2}\\cos 2α$, then $\\sin 2α=$____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{15}{16}"
} | {
"split": "train",
"index": 8269,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\tan \\theta =2$, find $\\cos 2\\theta =$____ and $\\tan (\\theta -\\frac{π}{4})=$____. 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": 8270,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the general term of the sequence $\\{a_n\\}$ is $a_n=n^2\\left(\\cos^2 \\frac{n\\pi}{3}-\\sin^2 \\frac{n\\pi}{3}\\right)$, calculate the value of $S_{30}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "470"
} | {
"split": "train",
"index": 8271,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the equation $x^2 + y^2 = |x| + 2|y|$, calculate the area enclosed by the graph of this equation. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{4}"
} | {
"split": "train",
"index": 8272,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The integer $m$ is the largest positive multiple of $18$ such that every digit of $m$ is either $9$ or $0$. Compute $\\frac{m}{18}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "555"
} | {
"split": "train",
"index": 8273,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $(21 \\div (6 + 1 - 4)) \\cdot 5.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 8274,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A student types the following pattern on a computer (where '〇' represents an empty circle and '●' represents a solid circle): 〇●〇〇●〇〇〇●〇〇〇〇●... If this pattern of circles continues, what is the number of solid circles among the first 2019 circles? Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "62"
} | {
"split": "train",
"index": 8275,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the internal angles $A$, $B$, and $C$ satisfy the equation $$2(\\tan B + \\tan C) = \\frac{\\tan B}{\\cos C} + \\frac{\\tan C}{\\cos B}$$. Find the minimum value of $\\cos A$. 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": 8276,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area enclosed by the region defined by the equation $x^2+y^2+6x+8y=0$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25\\pi"
} | {
"split": "train",
"index": 8277,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a seminar recording of 495 minutes that needs to be divided into multiple USB sticks, each capable of holding up to 65 minutes of audio, and the minimum number of USB sticks is used, calculate the length of audio that each USB stick will contain. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "61.875"
} | {
"split": "train",
"index": 8278,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Fill the digits 1 to 9 into the boxes in the following equation such that each digit is used exactly once and the equality holds true. It is known that the two-digit number $\\overrightarrow{\\mathrm{DE}}$ is not a multiple of 3. Determine the five-digit number $\\overrightarro... | math | {
"style": "rule",
"ground_truth": "85132"
} | {
"split": "train",
"index": 8279,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{m}=(\\sin x,-1)$ and $\\overrightarrow{n}=(\\sqrt{3}\\cos x,-\\frac{1}{2})$, and the function $f(x)=(\\overrightarrow{m}+\\overrightarrow{n})\\cdot\\overrightarrow{m}$.\n- (I) Find the interval where $f(x)$ is monotonically decreasing;\n- (II) Gi... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 8280,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given points O(0,0) and A(1,1), the slope angle of line OA is $\\boxed{\\text{answer}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 8281,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nFind all roots of the equation \\((x-a)(x-b)=(x-c)(x-d)\\), given that \\(a+d=b+c=2016\\) and \\(a \\neq c\\) (numerical values are not given). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1008"
} | {
"split": "train",
"index": 8282,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $AC=5 \\sqrt {2}$, $\\cos C= \\frac {3}{5}$, $B= \\frac {\\pi}{4}$.\n(1) Find the length of $AB$;\n(2) Find the area of $\\triangle ABC$, denoted as $S_{\\triangle ABC}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 8283,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $F$ be the focus of the parabola $C: y^2=4x$, point $A$ lies on $C$, and point $B(3,0)$. If $|AF|=|BF|$, then calculate the distance of point $A$ from point $B$. 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": 8284,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the power function $f(x) = (m^2 - m - 1)x^{m^2 + m - 3}$ on the interval $(0, +\\infty)$, determine the value of $m$ that makes it a decreasing function. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 8285,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The café has enough chairs to seat $310_5$ people. If $3$ people are supposed to sit at one table, how many tables does the café have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 8286,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nOn the lateral edges \\(AA_1\\), \\(BB_1\\), and \\(CC_1\\) of a triangular prism \\(ABC A_1 B_1 C_1\\), points \\(M\\), \\(N\\), and \\(P\\) are located respectively such that \\(AM: AA_1 = B_1N: BB_1 = C_1P: CC_1 = 3:4\\). On the segments \\(CM\\) and \\(A_1N\\), points \\(... | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"split": "train",
"index": 8287,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 54 chips in a box. Each chip is either small or large. If the number of small chips is greater than the number of large chips by a prime number of chips, what is the greatest possible number of large chips? Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 8288,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all positive integers $n$ such that the inequality $$\\left( \\sum\\limits_{i=1}^n a_i^2\\right) \\left(\\sum\\limits_{i=1}^n a_i \\right) -\\sum\\limits_{i=1}^n a_i^3 \\geq 6 \\prod\\limits_{i=1}^n a_i$$ holds for any $n$ positive numbers $a_1, \\dots, a_n$. Let's think s... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 8289,
"solution": "\nTo find all positive integers \\( n \\) such that the given inequality:\n\n\\[\n\\left( \\sum_{i=1}^n a_i^2\\right) \\left(\\sum_{i=1}^n a_i \\right) -\\sum_{i=1}^n a_i^3 \\geq 6 \\prod_{i=1}^n a_i\n\\]\n\nholds for any \\( n \\) positive numbers \\( a_1, \\dots, ... |
deepscaler | [
{
"role": "user",
"content": "There are 3 rods with several golden disks of different sizes placed on them. Initially, 5 disks are arranged on the leftmost rod (A) in descending order of size. According to the rule that only one disk can be moved at a time and a larger disk can never be placed on top of a s... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 8290,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle $ABCD$, $AB=8$ and $BC=6$. Points $F$ and $G$ are on $\\overline{CD}$ such that $DF=3$ and $GC=1$. Lines $AF$ and $BG$ intersect at $E$. Find the area of $\\triangle AEB$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 8291,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\frac{4}{3x^{-3}} \\cdot \\frac{3x^{2}}{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2x^5"
} | {
"split": "train",
"index": 8292,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Player A and player B are preparing for a badminton match. The rules state that the winner of a round will serve in the next round. If player A serves, the probability of player A winning the round is $\\frac{3}{4}$; if player B serves, the probability of player A winning the r... | math | {
"style": "rule",
"ground_truth": "\\frac{21}{32}"
} | {
"split": "train",
"index": 8293,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\\(PQRS\\) is a square. The points \\(T\\) and \\(U\\) are the midpoints of \\(QR\\) and \\(RS\\) respectively. The line \\(QS\\) cuts \\(PT\\) and \\(PU\\) at \\(W\\) and \\(V\\) respectively. What fraction of the total area of the square \\(PQRS\\) is the area of the pentagon... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 8294,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $$ 1$ to $$ 9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three pric... | math | {
"style": "rule",
"ground_truth": "420"
} | {
"split": "train",
"index": 8295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $$\\frac{1}{C_5^m} - \\frac{1}{C_6^m} = \\frac{7}{10C_7^m}$$, find $C_{21}^m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "210"
} | {
"split": "train",
"index": 8296,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an office at various times during the day, the boss gives the secretary a letter to type, each time putting the letter on top of the pile in the secretary's in-box. When there is time, the secretary takes the top letter off the pile and types it. There are nine letters to be... | math | {
"style": "rule",
"ground_truth": "704"
} | {
"split": "train",
"index": 8297,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain store sells a type of handbag, and it is known that the cost price of this handbag is $50$ yuan per unit. Market research shows that the daily sales quantity $y$ (unit: units) of this handbag is related to the selling price $x$ (unit: yuan) as follows: $y=-x+80$ $(50 ... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 8298,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $M = 1764 \\div 4$, $N = M \\div 4$, and $X = M - N$, what is the value of $X$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "330.75"
} | {
"split": "train",
"index": 8299,
"solution": ""
} |
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