data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Eight distinct integers are picked at random from $\\{1,2,3,\\ldots,15\\}$. What is the probability that, among those selected, the third smallest is $5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{17}"
} | {
"split": "train",
"index": 7200,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A belt is installed on two pulleys with radii of 14 inches and 4 inches respectively. The belt is taut and does not intersect itself. If the distance between the points where the belt touches the two pulleys is 24 inches, what is the distance (in inches) between the centers of ... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 7201,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), given that\n$$\n\\angle A = 30^{\\circ}, \\quad 2 \\overrightarrow{AB} \\cdot \\overrightarrow{AC} = 3 \\overrightarrow{BC}^2,\n$$\nfind the cosine of the largest angle of \\( \\triangle ABC \\). Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 7202,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nGiven \\( x_1, x_2, \\cdots, x_{1993} \\) that satisfy\n\\[\n\\begin{array}{l}\n\\left|x_1 - x_2\\right| + \\left|x_2 - x_3\\right| + \\cdots + \\left|x_{1992} - x_{1993}\\right| = 1993, \\\\\ny_k = \\frac{x_1 + x_2 + \\cdots + x_k}{k}, \\quad (k=1, 2, \\cdots, 1993)\n\\end{a... | math | {
"style": "rule",
"ground_truth": "1992"
} | {
"split": "train",
"index": 7203,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least positive integer greater than 1 that leaves a remainder of 2 when divided by each of 3, 4, 5, 6, 7, 8, 9, and 11? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27722"
} | {
"split": "train",
"index": 7204,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are three boxes \\(A\\), \\(B\\), and \\(C\\) containing 100, 50, and 80 balls respectively. Each box has a certain number of black balls. It is known that box \\(A\\) contains 15 black balls. If a box is chosen randomly and then a ball is randomly drawn from that box, th... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 7205,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct arrangements of the letters in the word \"balloon\" are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1260"
} | {
"split": "train",
"index": 7206,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Carl decided to fence his rectangular flowerbed using 24 fence posts, including one on each corner. He placed the remaining posts spaced exactly 3 yards apart along the perimeter of the bed. The bed’s longer side has three times as many posts compared to the shorter side, inclu... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 7207,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For every $m \\geq 2$, let $Q(m)$ be the least positive integer with the following property: For every $n \\geq Q(m)$, there is always a perfect cube $k^3$ in the range $n < k^3 \\leq m \\cdot n$. Find the remainder when \\[\\sum_{m = 2}^{2017} Q(m)\\]is divided by 1000. Let's ... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 7208,
"solution": "We claim that $Q(m) = 1$ when $m \\ge 8$.\n When $m \\ge 8$, for every $n \\ge Q(m) = 1$, we need to prove there exists an integer $k$, such that $n < k^3 \\le m*n$.\n That because $\\sqrt[3]{m*n} - \\sqrt[3]{n} \\ge 2\\sqrt[3]{n} - \\sqrt[3]{n} = \\sqrt[3]{n} \\ge ... |
deepscaler | [
{
"role": "user",
"content": "A four-digit number \\((xyzt)_B\\) is called a stable number in base \\(B\\) if \\((xyzt)_B = (dcba)_B - (abcd)_B\\), where \\(a \\leq b \\leq c \\leq d\\) are the digits \\(x, y, z, t\\) arranged in ascending order. Determine all the stable numbers in base \\(B\\).\n\n(Problem... | math | {
"style": "rule",
"ground_truth": "(1001)_2, (3021)_4, (3032)_5, (3B/5, B/5-1, 4B/5-1, 2B/5)_B, 5 | B"
} | {
"split": "train",
"index": 7209,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kelly is attempting to unlock her electronic device with a four-digit password. She remembers that she only used digits from 1 to 6, each digit possibly being repeated, and that each odd digit must be followed by an even digit, with no specific rule for the sequences following ... | math | {
"style": "rule",
"ground_truth": "648"
} | {
"split": "train",
"index": 7210,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the remainder when $2^{19}$ is divided by $7$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 7211,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\log (xy^3)= 1$ and $\\log (x^2y) = 1$, what is $\\log (xy)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 7212,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two circles intersecting at points A(1, 3) and B(m, -1), where the centers of both circles lie on the line $x - y + c = 0$, find the value of $m + c$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 7213,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An 8 by 8 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 5 black squares, can be drawn on the checkerboard?\n\n[asy]\ndraw((0,0)--(8,0)--(8,8)--(... | math | {
"style": "rule",
"ground_truth": "73"
} | {
"split": "train",
"index": 7214,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The average age of 8 people in a room is 35 years. A 22-year-old person leaves the room. Calculate the average age of the seven remaining people. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{258}{7}"
} | {
"split": "train",
"index": 7215,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nThe number \\( 20! = 1 \\cdot 2 \\cdot \\ldots \\cdot 20 = 2432902008176640000 \\) has 41,040 natural divisors. How many of them are odd? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2160"
} | {
"split": "train",
"index": 7216,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sixteen 6-inch wide square posts are evenly spaced with 4 feet between them to enclose a square field. What is the outer perimeter, in feet, of the fence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 7217,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Twelve delegates, four each from three different countries, randomly select chairs at a round table that seats twelve people. Calculate the probability that each delegate sits next to at least one delegate from another country, and express this probability as a fraction $\\frac... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 7218,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area of the region defined by the equation $x^2+y^2 - 7 = 4y-14x+3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "63\\pi"
} | {
"split": "train",
"index": 7219,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many four-digit positive integers have at least one digit that is a 2 or a 3? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5416"
} | {
"split": "train",
"index": 7220,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x) = 2\\cos^{2}x + \\sqrt{3}\\sin2x + a$, where $a$ is a real constant, find the value of $a$, given that the function has a minimum value of $-4$ on the interval $\\left[0, \\frac{\\pi}{2}\\right]$. Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 7221,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The perimeter of a triangle is 30, and all sides are different integers. There are a total of triangles. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 7222,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle has a radius of 3 units. There are many line segments of length 4 units that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments.\nA) $3\\pi$\nB) $5\\pi$\nC) $4\\pi$\nD) $7\\pi$\nE) $6\\pi$ Let's think step by ... | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"split": "train",
"index": 7223,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Charles has two six-sided die. One of the die is fair, and the other die is biased so that it comes up six with probability $\\frac{2}{3}$ and each of the other five sides has probability $\\frac{1}{15}$. Charles chooses one of the two dice at random and rolls it three times. G... | math | {
"style": "rule",
"ground_truth": "167"
} | {
"split": "train",
"index": 7224,
"solution": "The probability that he rolls a six twice when using the fair die is $\\frac{1}{6}\\times \\frac{1}{6}=\\frac{1}{36}$. The probability that he rolls a six twice using the biased die is $\\frac{2}{3}\\times \\frac{2}{3}=\\frac{4}{9}=\\frac{16}{36}$. Given that Charle... |
deepscaler | [
{
"role": "user",
"content": "In how many ways can 10 people be seated in a row of chairs if four of the people, Alice, Bob, Cindy, and Dave, refuse to sit in four consecutive seats? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3507840"
} | {
"split": "train",
"index": 7225,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $8$ balls of the same size, including $4$ different black balls, $2$ different red balls, and $2$ different yellow balls.<br/>$(1)$ Arrange these $8$ balls in a line, with the black balls together, the 2 red balls adjacent, and the 2 yellow balls not adjacent. Find th... | math | {
"style": "rule",
"ground_truth": "490"
} | {
"split": "train",
"index": 7226,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads, and 1 if the second flip is tails. On the other hand, if the first coin flip ... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{16}"
} | {
"split": "train",
"index": 7227,
"solution": "To solve the problem, we need to consider the probability that the absolute difference between two independently chosen numbers $x$ and $y$ is greater than $\\frac{1}{2}$. We can express this as $P(|x-y| > \\frac{1}{2}) = 2P(x-y > \\frac{1}{2})$ due to symmetry.\n\n... |
deepscaler | [
{
"role": "user",
"content": "A belt is placed without slack around two non-crossing circular pulleys which have radii of $15$ inches and $5$ inches respectively. The distance between the points where the belt contacts the pulleys is $30$ inches. Determine the distance between the centers of the two pulleys... | math | {
"style": "rule",
"ground_truth": "10\\sqrt{10}"
} | {
"split": "train",
"index": 7228,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number \\( N = 3^{16} - 1 \\) has a divisor of 193. It also has some divisors between 75 and 85 inclusive. What is the sum of these divisors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "247"
} | {
"split": "train",
"index": 7229,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a circle with center $O$, $AD$ is a diameter, $ABC$ is a chord, $BO = 6$, and $\\angle ABO = \\text{arc } CD = 45^\\circ$. Find the length of $BC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4.6"
} | {
"split": "train",
"index": 7230,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the 20 numbers 11, 12, 13, 14, ... 30, how many numbers must be chosen to ensure that there are at least two numbers whose sum is a multiple of 10? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 7231,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many three-digit positive integers $x$ satisfy $3874x+481\\equiv 1205 \\pmod{23}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 7232,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Call a positive integer $N \\geq 2$ \"special\" if for every $k$ such that $2 \\leq k \\leq N, N$ can be expressed as a sum of $k$ positive integers that are relatively prime to $N$ (although not necessarily relatively prime to each other). How many special integers are there l... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 7233,
"solution": "We claim that all odd numbers are special, and the only special even number is 2. For any even $N>2$, the numbers relatively prime to $N$ must be odd. When we consider $k=3$, we see that $N$ can't be expressed as a sum of 3 odd numbers. Now suppose that $N$ is odd, ... |
deepscaler | [
{
"role": "user",
"content": "Find the maximum of\n\\[\n\\sqrt{x + 31} + \\sqrt{17 - x} + \\sqrt{x}\n\\]\nfor $0 \\le x \\le 17$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 7234,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( a, b, c, d \\) are prime numbers (they can be the same), and \\( abcd \\) is the sum of 35 consecutive positive integers, find the minimum value of \\( a + b + c + d \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 7235,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the maximum number of L-shaped 3-cell pieces that can be cut out from a rectangular grid of size: \na) $5 \\times 10$ cells; \nb) $5 \\times 9$ cells? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 7236,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$C$ is a point on the extension of diameter $A B$, $C D$ is a tangent, and the angle $A D C$ is $110^{\\circ}$. Find the angular measure of arc $B D$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 7237,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The positive integers \\(a\\), \\(b\\) are such that \\(15a + 16b\\) and \\(16a - 15b\\) are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "481"
} | {
"split": "train",
"index": 7238,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A wooden cube with edges of length $3$ meters has square holes, of side one meter, centered in each face, cut through to the opposite face. Find the entire surface area, including the inside, of this cube in square meters. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 7239,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Robinson had 200,000 strands of hair when he ended up on the deserted island. At that time, his hair strands were $5 \\mathrm{~cm}$ long. The hair strands grew $0.5 \\mathrm{~mm}$ per day, but Robinson did not cut his hair because he did not have the proper tools, and he lost 5... | math | {
"style": "rule",
"ground_truth": "1950"
} | {
"split": "train",
"index": 7240,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A$ be a point on the parabola $y = x^2 - 4x + 4,$ and let $B$ be a point on the line $y = 2x - 3.$ Find shortest possible distance $AB.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{5}}{5}"
} | {
"split": "train",
"index": 7241,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$, $b$, $c$, $d$, and $e$ be positive integers with $a+b+c+d+e=2020$ and let $M$ be the largest of the sum $a+b$, $b+c$, $c+d$ and $d+e$. What is the smallest possible value of $M$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"split": "train",
"index": 7242,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the radius of a hemisphere is 2, calculate the maximum lateral area of the inscribed cylinder. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"split": "train",
"index": 7243,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $689\\Box\\Box\\Box20312 \\approx 69$ billion (rounded), find the number of ways to fill in the three-digit number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "500"
} | {
"split": "train",
"index": 7244,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As $x$ ranges over all real numbers, find the range of\n\\[f(x) = \\sin^4 x + \\cos ^2 x.\\]Enter your answer using interval notation. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left[ \\frac{3}{4}, 1 \\right]"
} | {
"split": "train",
"index": 7245,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a circle of radius 3, find the area of the region consisting of all line segments of length 6 that are tangent to the circle at their midpoints.\nA) $3\\pi$ \nB) $6\\pi$ \nC) $9\\pi$ \nD) $12\\pi$ \nE) $15\\pi$ Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "9\\pi"
} | {
"split": "train",
"index": 7246,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the triangular pyramid $SABC$, the height $SO$ passes through point $O$ - the center of the circle inscribed in the base $ABC$ of the pyramid. It is known that $\\angle SAC = 60^\\circ$, $\\angle SCA = 45^\\circ$, and the ratio of the area of triangle $AOB$ to the area of tr... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 7247,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that bricklayer Alice can build a wall alone in 8 hours, and bricklayer Bob can build it alone in 12 hours, and they complete the wall in 6 hours when working together with a 15-brick-per-hour decrease in productivity, determine the number of bricks in the wall. Let's thi... | math | {
"style": "rule",
"ground_truth": "360"
} | {
"split": "train",
"index": 7248,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For what base is the representation of $285_{10}$ a four digit number whose final digit is odd? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 7249,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of integers $x$ such that the following three conditions all hold: - $x$ is a multiple of 5 - $121<x<1331$ - When $x$ is written as an integer in base 11 with no leading 0 s (i.e. no 0 s at the very left), its rightmost digit is strictly greater than its leftmos... | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 7250,
"solution": "We will work in base 11, so let $x=\\overline{\\operatorname{def}}_{11}$ such that $d>0$. Then, based on the first two conditions, we aim to find multiples of 5 between $100_{11}$ and $1000_{11}$. We note that $$\\overline{d e f}_{11} \\equiv 11^{2} \\cdot d+11 \\cd... |
deepscaler | [
{
"role": "user",
"content": "We flip a fair coin 10 times. What is the probability that we get heads in at least 8 of the 10 flips? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{7}{128}"
} | {
"split": "train",
"index": 7251,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer \\( m \\) such that the equation regarding \\( x, y, \\) and \\( z \\):\n\\[ 2^x + 3^y - 5^z = 2m \\]\nhas no positive integer solutions. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 7252,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Which of the following integers cannot be written as a product of two integers, each greater than 1: 6, 27, 53, 39, 77? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "53"
} | {
"split": "train",
"index": 7253,
"solution": "We note that $6=2 \times 3$ and $27=3 \times 9$ and $39=3 \times 13$ and $77=7 \times 11$, which means that each of $6,27,39$, and 77 can be written as the product of two integers, each greater than 1. Thus, 53 must be the integer that cannot be written in this way.... |
deepscaler | [
{
"role": "user",
"content": "Another trapezoid \\(ABCD\\) has \\(AD\\) parallel to \\(BC\\). \\(AC\\) and \\(BD\\) intersect at \\(P\\). If \\(\\frac{[ADP]}{[BCP]} = \\frac{1}{2}\\), find \\(\\frac{[ADP]}{[ABCD]}\\). (Here, the notation \\(\\left[P_1 \\cdots P_n\\right]\\) denotes the area of the polygon \... | math | {
"style": "rule",
"ground_truth": "3 - 2\\sqrt{2}"
} | {
"split": "train",
"index": 7254,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $F\\_1$ and $F\\_2$ are the left and right foci of the hyperbola $C: \\frac{x^{2}}{a^{2}}- \\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$, if a point $P$ on the right branch of the hyperbola $C$ satisfies $|PF\\_1|=3|PF\\_2|$ and $\\overrightarrow{PF\\_1} \\cdot \\overrighta... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 7255,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\\(1.25 \\times 67.875 + 125 \\times 6.7875 + 1250 \\times 0.053375\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"split": "train",
"index": 7256,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( z \\) be a complex number with a modulus of 1. Then the maximum value of \\(\\left|\\frac{z+\\mathrm{i}}{z+2}\\right|\\) is \\(\\ \\ \\ \\ \\ \\ \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{5}}{3}"
} | {
"split": "train",
"index": 7257,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the two lines $ax+2y+6=0$ and $x+(a-1)y+(a^{2}-1)=0$ are parallel, determine the set of possible values for $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\{-1\\}"
} | {
"split": "train",
"index": 7258,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer sequence has its first term as 8 and its second term as 1. From the third term onwards, each term is the sum of the two preceding terms. What is the remainder when the 2013th term in this sequence is divided by 105? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 7259,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Lesha's summer cottage has the shape of a nonagon with three pairs of equal and parallel sides. Lesha knows that the area of the triangle with vertices at the midpoints of the remaining sides of the nonagon is 12 sotkas. Help him find the area of the entire summer cottage. Let'... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 7260,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "This month, I spent 26 days exercising for 20 minutes or more, 24 days exercising 40 minutes or more, and 4 days of exercising 2 hours exactly. I never exercise for less than 20 minutes or for more than 2 hours. What is the minimum number of hours I could have exercised this mo... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 7261,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five people of different heights are standing in line from shortest to tallest. As it happens, the tops of their heads are all collinear; also, for any two successive people, the horizontal distance between them equals the height of the shorter person. If the shortest person is... | math | {
"style": "rule",
"ground_truth": "\\sqrt{21}"
} | {
"split": "train",
"index": 7262,
"solution": "If $A, B$, and $C$ are the tops of the heads of three successive people and $D, E$, and $F$ are their respective feet, let $P$ be the foot of the perpendicular from $A$ to $B E$ and let $Q$ be the foot of the perpendicular from $B$ to $C F$. Then, by equal angles, $... |
deepscaler | [
{
"role": "user",
"content": "A student reads a book, reading 35 pages on the first day and then 5 more pages each subsequent day, until only 35 pages are left on the last day. The second time he reads it, he reads 45 pages on the first day and then 5 more pages each subsequent day, until only 40 pages are ... | math | {
"style": "rule",
"ground_truth": "385"
} | {
"split": "train",
"index": 7263,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\cos 150^\\circ$. 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": 7264,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the remainder when 1,493,824 is divided by 4? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 7265,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a wooden block shaped like a cube, all the vertices and edge midpoints are marked. The cube is cut along all possible planes that pass through at least four marked points. Let \\(N\\) be the number of pieces the cube is cut into. Estimate \\(N\\). An estimate of \\(E>0\\) ea... | math | {
"style": "rule",
"ground_truth": "15600"
} | {
"split": "train",
"index": 7266,
"solution": "Answer: 15600"
} |
deepscaler | [
{
"role": "user",
"content": "The hyperbola $C: x^{2}-y^{2}=2$ has its right focus at $F$. Let $P$ be any point on the left branch of the hyperbola, and point $A$ has coordinates $(-1,1)$. Find the minimum perimeter of $\\triangle A P F$. Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2} + \\sqrt{10}"
} | {
"split": "train",
"index": 7267,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A national team needs to select 4 out of 6 sprinters to participate in the 4×100m relay at the Asian Games. If one of them, A, cannot run the first leg, and another, B, cannot run the fourth leg, how many different methods are there to select the team? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "252"
} | {
"split": "train",
"index": 7268,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose one of the eight lettered identical squares is included with the four squares in the T-shaped figure outlined. How many of the resulting figures can be folded into a topless cubical box? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 7269,
"solution": "To solve this problem, we need to determine how many of the eight lettered squares can be added to the T-shaped figure to create a figure that can be folded into a topless cubical box. The T-shaped figure already consists of four squares that can be thought of as fo... |
deepscaler | [
{
"role": "user",
"content": "If \\( x \\) and \\( y \\) are real numbers such that \\( x + y = 4 \\) and \\( xy = -2 \\), then the value of \\( x + \\frac{x^3}{y^2} + \\frac{y^3}{x^2} + y \\) is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "440"
} | {
"split": "train",
"index": 7270,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find in explicit form all ordered pairs of positive integers $(m, n)$ such that $mn-1$ divides $m^2 + n^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(2, 1), (3, 1), (1, 2), (1, 3)"
} | {
"split": "train",
"index": 7271,
"solution": "\nTo find all ordered pairs of positive integers \\((m, n)\\) such that \\(mn-1\\) divides \\(m^2 + n^2\\), we start by considering the condition:\n\\[\n\\frac{m^2 + n^2}{mn - 1} = c \\quad \\text{where} \\quad c \\in \\mathbb{Z}.\n\\]\nThis implies:\n\\[\nm^2 + n^2... |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the fractions of the form $\\frac{2}{n(n+2)}$, where $n$ takes on odd positive integers from 1 to 2011? Express your answer as a decimal to the nearest thousandth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.999"
} | {
"split": "train",
"index": 7272,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that 600 athletes are numbered from 001 to 600 and divided into three color groups (red: 001 to 311, white: 312 to 496, and yellow: 497 to 600), calculate the probability of randomly drawing an athlete wearing white clothing. Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{25}"
} | {
"split": "train",
"index": 7273,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of the $5$ sides and the $5$ diagonals of a regular pentagon are randomly and independently colored red or blue with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the pentagon such that all of its sides ha... | math | {
"style": "rule",
"ground_truth": "\\frac{253}{256}"
} | {
"split": "train",
"index": 7274,
"solution": "To solve this problem, we need to calculate the probability that there exists a triangle (a $K_3$ subgraph) in a complete graph $K_5$ (a pentagon with all diagonals drawn) such that all its edges are colored the same color (either all red or all blue).\n\n#### Step ... |
deepscaler | [
{
"role": "user",
"content": "What is the positive difference of the solutions of $\\dfrac{r^2-3r-17}{r+4}=2r+7$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 7275,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bicycle trip is 30 km long. Ari rides at an average speed of 20 km/h. Bri rides at an average speed of 15 km/h. If Ari and Bri begin at the same time, how many minutes after Ari finishes the trip will Bri finish? Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 7276,
"solution": "Riding at 15 km/h, Bri finishes the 30 km in $\\frac{30 \\text{ km}}{15 \\text{ km/h}} = 2 \\text{ h}$. Riding at 20 km/h, Ari finishes the 30 km in $\\frac{30 \\text{ km}}{20 \\text{ km/h}} = 1.5 \\text{ h}$. Therefore, Bri finishes 0.5 h after Ari, which is 30 min... |
deepscaler | [
{
"role": "user",
"content": "Let $x=1$ be an extremum point of the function $f(x)=a_{n+1}x^{3}-a_{n}x^{2}-a_{n+2}x+1(n\\in N_{+})$, where the sequence $\\{a_{n}\\}$ is given by $a_{1}=1$, $a_{2}=2$, and $b_{n}=\\log _{2}a_{2n}$. Determine the value of $\\left\\lfloor \\dfrac {2018}{b_{1}b_{2}}+ \\dfrac {20... | math | {
"style": "rule",
"ground_truth": "1008"
} | {
"split": "train",
"index": 7277,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two cones share a common base and the vertices of both cones and the circumference of the base are all on the same sphere. If the area of the base of the cone is $\\frac{3}{16}$ of the area of the sphere, calculate the ratio of the volumes of the two cones. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "1:3"
} | {
"split": "train",
"index": 7278,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A school plans to purchase two brands of soccer balls, brand A and brand B. It is known that the unit price of brand A soccer balls is $30 less than the unit price of brand B soccer balls. The quantity of brand A soccer balls that can be purchased with $1000 is the same as the ... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 7279,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A lattice point in the plane is a point of the form $(n, m)$, where $n$ and $m$ are integers. Consider a set $S$ of lattice points. We construct the transform of $S$, denoted by $S^{\\prime}$, by the following rule: the pair $(n, m)$ is in $S^{\\prime}$ if and only if any of $(... | math | {
"style": "rule",
"ground_truth": "421"
} | {
"split": "train",
"index": 7280,
"solution": "Transforming it $k \\geq 1$ times yields the 'diamond' of points $(n, m)$ such that $|n|+|m| \\leq k$. The diamond contains $(k+1)^{2}+k^{2}$ lattice points (this can be seen by rotating the plane 45 degrees and noticing the lattice points in the transforms form two... |
deepscaler | [
{
"role": "user",
"content": "Given two vectors in the plane, $\\mathbf{a} = (2m+1, 3)$ and $\\mathbf{b} = (2, m)$, and $\\mathbf{a}$ is in the opposite direction to $\\mathbf{b}$, calculate the magnitude of $\\mathbf{a} + \\mathbf{b}$. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 7281,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the equations:<br/>$(1)2\\left(x-1\\right)^{2}=1-x$;<br/>$(2)4{x}^{2}-2\\sqrt{3}x-1=0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3} - \\sqrt{7}}{4}"
} | {
"split": "train",
"index": 7282,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be the number of ordered pairs of nonempty sets $\\mathcal{A}$ and $\\mathcal{B}$ that have the following properties:\n$\\mathcal{A} \\cup \\mathcal{B} = \\{1,2,3,4,5,6,7,8,9,10,11,12\\}$,\n$\\mathcal{A} \\cap \\mathcal{B} = \\emptyset$,\nThe number of elements of $\\ma... | math | {
"style": "rule",
"ground_truth": "772"
} | {
"split": "train",
"index": 7283,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A barn with a roof is rectangular in shape, $10$ yd. wide, $13$ yd. long and $5$ yd. high. It is to be painted inside and outside, and on the ceiling, but not on the roof or floor. The total number of sq. yd. to be painted is: Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "490"
} | {
"split": "train",
"index": 7284,
"solution": "To find the total area to be painted, we need to calculate the areas of the walls and the ceiling that will be painted. The barn is a rectangular prism with dimensions:\n- Width = $10$ yd\n- Length = $13$ yd\n- Height = $5$ yd\n\n#### Step 1: Calculate the area of e... |
deepscaler | [
{
"role": "user",
"content": "Expand and find the sum of the coefficients of the expression $-(2x - 5)(4x + 3(2x - 5))$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-15"
} | {
"split": "train",
"index": 7285,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Esquecinaldo has a poor memory for storing numbers but an excellent memory for remembering sequences of operations. To remember his five-digit bank code, he can recall that:\n\n1. The code has no repeated digits.\n2. None of the digits are zero.\n3. The first two digits form a ... | math | {
"style": "rule",
"ground_truth": "25916"
} | {
"split": "train",
"index": 7286,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence of length 15 composed of zeros and ones, find the number of sequences where all zeros are consecutive, all ones are consecutive, or both. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "270"
} | {
"split": "train",
"index": 7287,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of the expression $\frac{3}{10}+\frac{3}{100}+\frac{3}{1000}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.333"
} | {
"split": "train",
"index": 7288,
"solution": "Evaluating, $\frac{3}{10}+\frac{3}{100}+\frac{3}{1000}=0.3+0.03+0.003=0.333$."
} |
deepscaler | [
{
"role": "user",
"content": "The reciprocal of $\\frac{2}{3}$ is ______, the opposite of $-2.5$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"split": "train",
"index": 7289,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a strip of size \\(1 \\times N\\), 25 checkers are placed on the first 25 squares on the left. A checker can move to the adjacent right empty square or jump over the adjacent right checker to the next square (if that square is empty). Movement to the left is not allowed. Wha... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 7290,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = x^2 - 3x - 1$, find the derivative of $f(2)$ and $f'(1)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 7291,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A dice is thrown twice. Let $a$ be the number of dots that appear on the first throw and $b$ be the number of dots that appear on the second throw. Given the system of equations $\\begin{cases} ax+by=2 \\\\\\\\ 2x+y=3\\\\end{cases}$.\n\n(I) Find the probability that the system ... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{12}"
} | {
"split": "train",
"index": 7292,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be the number of consecutive $0$'s at the right end of the decimal representation of the product $1!2!3!4!\\cdots99!100!.$ Find the remainder when $N$ is divided by $1000$.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "124"
} | {
"split": "train",
"index": 7293,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A play has two different male roles, two different female roles and two different roles that can be either gender. Only a man can be assigned to a male role, and only a woman can be assigned to a female role. If five men and six women audition, in how many ways can the six role... | math | {
"style": "rule",
"ground_truth": "25200"
} | {
"split": "train",
"index": 7294,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that points $E$ and $F$ are on the same side of diameter $\\overline{GH}$ in circle $P$, $\\angle GPE = 60^\\circ$, and $\\angle FPH = 90^\\circ$, find the ratio of the area of the smaller sector $PEF$ to the area of the circle. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"split": "train",
"index": 7295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system xOy, the parametric equation of curve C1 is given by $$\\begin{cases} x=5cos\\alpha \\\\ y=5+5sin\\alpha \\end{cases}$$ (where α is the parameter). Point M is a moving point on curve C1. When the line segment OM is rotated counterclockwise b... | math | {
"style": "rule",
"ground_truth": "15-5 \\sqrt {3}"
} | {
"split": "train",
"index": 7296,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The remainder when 111 is divided by 10 is 1. The remainder when 111 is divided by the positive integer $n$ is 6. How many possible values of $n$ are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 7297,
"solution": "Since the remainder when 111 is divided by $n$ is 6, then $111-6=105$ is a multiple of $n$ and $n>6$ (since, by definition, the remainder must be less than the divisor). Since $105=3 \\cdot 5 \\cdot 7$, the positive divisors of 105 are $1,3,5,7,15,21,35,105$. Theref... |
deepscaler | [
{
"role": "user",
"content": "Circles $A, B,$ and $C$ each have radius 1. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\\overline{AB}.$ What is the area inside circle $C$ but outside circle $A$ and circle $B?$ Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 7298,
"solution": "1. **Identify the Configuration**: Circles $A$, $B$, and $C$ each have a radius of 1. Circles $A$ and $B$ are tangent to each other, and circle $C$ is tangent to the midpoint $M$ of line segment $\\overline{AB}$. \n\n2. **Positioning of Circle $C$**: Since circle $C... |
deepscaler | [
{
"role": "user",
"content": "Circles $C_1$ and $C_2$ are externally tangent, and they are both internally tangent to circle $C_3.$ The radii of $C_1$ and $C_2$ are 4 and 10, respectively, and the centers of the three circles are all collinear. A chord of $C_3$ is also a common external tangent of $C_1$ and... | math | {
"style": "rule",
"ground_truth": "405"
} | {
"split": "train",
"index": 7299,
"solution": ""
} |
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