data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Given that Jackie has $40$ thin rods, one of each integer length from $1 \\text{ cm}$ through $40 \\text{ cm}$, with rods of lengths $5 \\text{ cm}$, $12 \\text{ cm}$, and $20 \\text{ cm}$ already placed on a table, find the number of the remaining rods that she can choose as t... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 25649,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the maximum length of a closed self-avoiding polygon that can travel along the grid lines of an $8 \\times 8$ square grid? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 25650,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When a car's brakes are applied, it travels 7 feet less in each second than the previous second until it comes to a complete stop. A car goes 28 feet in the first second after the brakes are applied. How many feet does the car travel from the time the brakes are applied to ... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 25651,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow {a}$ and $\\overrightarrow {b}$ that satisfy $| \\overrightarrow {a}|=2$, $| \\overrightarrow {b}|=1$, and $\\overrightarrow {b} \\perp ( \\overrightarrow {a}+ \\overrightarrow {b})$, determine the projection of vector $\\overrightarrow {a}$ onto... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 25652,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a^{2}+b^{2}+4 \\sqrt {2}=c^{2}$ and $ab=4$, find the minimum value of $\\frac {\\sin C}{\\tan ^{2}A\\cdot \\sin 2B}$. Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "\\frac {3 \\sqrt {2}}{2}+2"
} | {
"split": "train",
"index": 25653,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a square in the coordinate plane with vertices at $(2, 1)$, $(5, 1)$, $(5, 4)$, and $(2, 4)$. A line joining $(2, 3)$ and $(5, 1)$ divides the square shown into two parts. Determine the fraction of the area of the square that is above this line. Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"split": "train",
"index": 25654,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If two stagecoaches travel daily from Bratislava to Brașov, and likewise, two stagecoaches travel daily from Brașov to Bratislava, and considering that the journey takes ten days, how many stagecoaches will you encounter on your way when traveling by stagecoach from Bratislava ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 25655,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the volume of the tetrahedron \\(ABCD\\) is 40, \\(AB = 7\\), \\(BC = 8\\), \\(AC = 9\\), and the orthogonal projection of vertex \\(D\\) onto the plane \\(ABC\\) is precisely the incenter \\(H\\) of triangle \\(ABC\\), what is the surface area of the tetrahedron? Le... | math | {
"style": "rule",
"ground_truth": "60 + 12 \\sqrt{5}"
} | {
"split": "train",
"index": 25656,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a bag, there are 7 blue chips, 5 yellow chips, and 4 red chips. One chip is drawn from the bag and then replaced. A second chip is then drawn. What is the probability that the two selected chips are of different colors? Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "\\frac{83}{128}"
} | {
"split": "train",
"index": 25657,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given points $A$, $B$, $C$ with coordinates $(4,0)$, $(0,4)$, $(3\\cos \\alpha,3\\sin \\alpha)$ respectively, and $\\alpha\\in\\left( \\frac {\\pi}{2}, \\frac {3\\pi}{4}\\right)$. If $\\overrightarrow{AC} \\perp \\overrightarrow{BC}$, find the value of $\\frac {2\\sin ^{2}\\alp... | math | {
"style": "rule",
"ground_truth": "- \\frac {7 \\sqrt {23}}{48}"
} | {
"split": "train",
"index": 25658,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the quadratic polynomial $p(x)$ such that $p(-7) = 0,$ $p(4) = 0,$ and $p(5) = -36.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3x^2 - 9x + 84"
} | {
"split": "train",
"index": 25659,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Carmen takes a long bike ride on a hilly highway. The graph indicates the miles traveled during the time of her ride. What is Carmen's average speed for her entire ride in miles per hour? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 25660,
"solution": "To find Carmen's average speed for her entire ride, we need to use the formula for average speed, which is given by:\n\\[\n\\text{Average Speed} = \\frac{\\text{Total Distance}}{\\text{Total Time}}\n\\]\n\nFrom the problem statement, we are given:\n- Total Distance... |
deepscaler | [
{
"role": "user",
"content": "A necklace consists of 80 beads of red, blue, and green colors. It is known that in any segment of the necklace between two blue beads, there is at least one red bead, and in any segment of the necklace between two red beads, there is at least one green bead. What is the minimu... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 25661,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ with $a_1=1$, $0<q<\\frac{1}{2}$, and for any positive integer $k$, $a_k - (a_{k+1}+a_{k+2})$ is still an element of the sequence, find the common ratio $q$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} - 1"
} | {
"split": "train",
"index": 25662,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\mathbf{v}$ and $\\mathbf{w}$ such that $\\|\\mathbf{v}\\| = 3,$ $\\|\\mathbf{w}\\| = 7,$ and $\\mathbf{v} \\cdot \\mathbf{w} = 10,$ then find $\\|\\operatorname{proj}_{\\mathbf{w}} \\mathbf{v}\\|.$ Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "\\frac{10}{7}"
} | {
"split": "train",
"index": 25663,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Any six points are taken inside or on a rectangle with dimensions $1 \\times 2$. Let $b$ be the smallest possible value such that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Determine the va... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 25664,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Starting with 1, alternately add 4 and 3 to obtain the sequence $1, 5, 8, 12, 15, 19, 22, \\ldots \\ldots$. In this sequence, the number closest to 2013 is $\\qquad$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2014"
} | {
"split": "train",
"index": 25665,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each positive integer $p$, let $b(p)$ denote the unique positive integer $k$ such that $|k-\\sqrt{p}| < \\frac{1}{2}$. For example, $b(6) = 2$ and $b(23) = 5$. If $S = \\sum_{p=1}^{2007} b(p),$ find the remainder when $S$ is divided by 1000. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "955"
} | {
"split": "train",
"index": 25666,
"solution": "$\\left(k- \\frac 12\\right)^2=k^2-k+\\frac 14$ and $\\left(k+ \\frac 12\\right)^2=k^2+k+ \\frac 14$. Therefore $b(p)=k$ if and only if $p$ is in this range, or $k^2-k<p\\leq k^2+k$. There are $2k$ numbers in this range, so the sum of $b(p)$ over this range is $(2k... |
deepscaler | [
{
"role": "user",
"content": "Select 5 volunteers from 8 candidates, including A and B, to participate in community service activities from Monday to Friday, with one person arranged for each day, and each person participating only once. If at least one of A and B must participate, and when both A and B par... | math | {
"style": "rule",
"ground_truth": "5040"
} | {
"split": "train",
"index": 25667,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of pairs $(m,n)$ of positive integers with $1\\le m<n\\le 30$ such that there exists a real number $x$ satisfying \\[\\sin(mx)+\\sin(nx)=2.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 25668,
"solution": "We know that the range of $sin$ is between $-1$ and $1$.\n Thus, the only way for the sum to be $2$ is for $sin$ of $mx$ and $nx$ to both be $1$.\n The $sin$ of $(90+360k)$ is equal to 1.\n Assuming $mx$ and $nm$ are both positive, m and n could be $1,5,9,13,17,21,... |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the roots of $x^2 - 4x + 3 = 0$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 25669,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six congruent copies of the parabola $y = x^2$ are arranged in the plane so that each vertex is tangent to a circle, and each parabola is tangent to its two neighbors. Assume that each parabola is tangent to a line that forms a $45^\\circ$ angle with the x-axis. Find the radius... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 25670,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be the second smallest positive integer that is divisible by every positive integer less than $7$. What is the sum of the digits of $N$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 25671,
"solution": "1. **Identify the requirement for $N$:** \n $N$ must be divisible by every positive integer less than $7$. This includes the integers $1, 2, 3, 4, 5,$ and $6$.\n\n2. **Determine the least common multiple (LCM):** \n To find $N$, we need the least common multi... |
deepscaler | [
{
"role": "user",
"content": "John lists the integers from 1 to 20 in increasing order. He then erases the first half of the integers in the list and rewrites them in order at the end of the second half of the list. Which integer in the new list has exactly 12 integers to its left? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 25672,
"solution": "John first writes the integers from 1 to 20 in increasing order. When he erases the first half of the numbers, he erases the numbers from 1 to 10 and rewrites these at the end of the original list. Therefore, the number 1 has 10 numbers to its left. (These numbers ... |
deepscaler | [
{
"role": "user",
"content": "Given that $a_1$, $a_2$, $a_3$, $a_4$, $a_5$ are five different integers satisfying the condition $a_1 + a_2 + a_3 + a_4 + a_5 = 9$, if $b$ is an integer root of the equation $(x - a_1)(x - a_2)(x - a_3)(x - a_4)(x - a_5) = 2009$, then the value of $b$ is. Let's think step by s... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 25673,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two students, A and B, each select two courses from four elective courses. The probability that they share exactly one course in common is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 25674,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{D}$ be a matrix representing a dilation with scale factor $k > 0$, let $\\mathbf{R}$ be a matrix representing a rotation about the origin by an angle of $\\phi$ counter-clockwise, and let $\\mathbf{T}$ be a translation matrix that we will ignore in the computation... | math | {
"style": "rule",
"ground_truth": "\\frac{12}{5}"
} | {
"split": "train",
"index": 25675,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using the digits $1$, $2$, $3$, $5$, and $6$ exactly once, the five-digit positive integers are formed and arranged in ascending order. What is the $60^{\\text{th}}$ integer in this list? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32651"
} | {
"split": "train",
"index": 25676,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I have 7 books, two of which are identical copies of a science book and another two identical copies of a math book, while the rest of the books are all different. In how many ways can I arrange them on a shelf, and additionally, how many of these arrangements can be made if I ... | math | {
"style": "rule",
"ground_truth": "26460"
} | {
"split": "train",
"index": 25677,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the set $S={1,2,3,...,100}$, three numbers are randomly selected and arranged in ascending order. Find the probability that $50$ is exactly in the middle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{66}"
} | {
"split": "train",
"index": 25678,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Multiply the first eight positive composite integers, add the first prime number to this product, then divide by the product of the next eight positive composite integers after adding the second prime number to it. Express your answer as a common fraction. Let's think step by s... | math | {
"style": "rule",
"ground_truth": "\\frac{4 \\cdot 6 \\cdot 8 \\cdot 9 \\cdot 10 \\cdot 12 \\cdot 14 \\cdot 15 + 2}{16 \\cdot 18 \\cdot 20 \\cdot 21 \\cdot 22 \\cdot 24 \\cdot 25 \\cdot 26 + 3}"
} | {
"split": "train",
"index": 25679,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose the product $\\dfrac{4}{3}\\cdot \\dfrac{5}{4}\\cdot \\dfrac{6}{5}\\cdot \\ldots\\cdot \\dfrac{c}{d} = 16$, find the sum of $c$ and $d$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "95"
} | {
"split": "train",
"index": 25680,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 100 points marked on a circle, painted either red or blue. Some points are connected by segments, with each segment having one blue end and one red end. It is known that no two red points are connected to the same number of segments. What is the maximum possible numbe... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 25681,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area of the smallest square that can contain a circle of radius 6? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 25682,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A lattice point in the $x,y$-plane is a point both of whose coordinates are integers (not necessarily positive). How many lattice points lie on the graph of the equation $x^2-y^2=47$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 25683,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different ways can the five vertices S, A, B, C, and D of a square pyramid S-ABCD be colored using four distinct colors so that each vertex is assigned one color and no two vertices sharing an edge have the same color? Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 25684,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Huahua is writing letters to Yuanyuan with a pen. When she finishes the 3rd pen refill, she is working on the 4th letter; when she finishes the 5th letter, the 4th pen refill is not yet used up. If Huahua uses the same amount of ink for each letter, how many pen refills does sh... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 25685,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Moe has a new, larger rectangular lawn measuring 120 feet by 180 feet. He uses a mower with a swath width of 30 inches. However, he overlaps each cut by 6 inches to ensure no grass is missed. Moe walks at a rate of 6000 feet per hour while pushing the mower. What is the closest... | math | {
"style": "rule",
"ground_truth": "1.8"
} | {
"split": "train",
"index": 25686,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $y=f(x)$ is a quadratic function, and $f(0)=-5$, $f(-1)=-4$, $f(2)=-5$,\n\n(1) Find the analytical expression of this quadratic function. \n(2) Find the maximum and minimum values of the function $f(x)$ when $x \\in [0,5]$. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "- \\frac {16}{3}"
} | {
"split": "train",
"index": 25687,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\\overline{AB}$, and let $E$ be the midpoint of $\\overline{AC}$. The angle bisector of $\\angle BAC$ intersects $\\overline{DE}$ and $\\overline{BC}$ at $F$ and $G$, respectively. What is the a... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 25688,
"solution": "1. **Identify the areas of sub-triangles**: \n - The area of $\\triangle ABC$ is given as $120$.\n - Since $D$ and $E$ are midpoints of $AB$ and $AC$ respectively, $AD = DB = 25$ and $AE = EC = 5$.\n - The area of $\\triangle ADE$ can be calculated using the ... |
deepscaler | [
{
"role": "user",
"content": "There is a heads up coin on every integer of the number line. Lucky is initially standing on the zero point of the number line facing in the positive direction. Lucky performs the following procedure: he looks at the coin (or lack thereof) underneath him, and then, - If the coi... | math | {
"style": "rule",
"ground_truth": "6098"
} | {
"split": "train",
"index": 25689,
"solution": "We keep track of the following quantities: Let $N$ be the sum of $2^{k}$, where $k$ ranges over all nonnegative integers such that position $-1-k$ on the number line contains a tails-up coin. Let $M$ be the sum of $2^{k}$, where $k$ ranges over all nonnegative inte... |
deepscaler | [
{
"role": "user",
"content": "Regular octagon CHILDREN has area 1. Determine the area of quadrilateral LINE. 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": 25690,
"solution": "Suppose that the side length $C H=\\sqrt{2} a$, then the area of the octagon is $((2+\\sqrt{2}) a)^{2}-4 \\cdot \\frac{1}{2} a^{2}=$ $(4+4 \\sqrt{2}) a^{2}$, and the area of $L I N E$ is $(\\sqrt{2} a)((2+\\sqrt{2}) a)=(2+2 \\sqrt{2}) a^{2}$, which is exactly one-h... |
deepscaler | [
{
"role": "user",
"content": "Let $(2x+1)^6 = a_0x^6 + a_1x^5 + a_2x^4 + a_3x^3 + a_4x^2 + a_5x + a_6$, which is an identity in $x$ (i.e., it holds for any $x$). Try to find the values of the following three expressions: \n(1) $a_0 + a_1 + a_2 + a_3 + a_4 + a_5 + a_6$; \n(2) $a_1 + a_3 + a_5$; \n(3) $a_2... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 25691,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the acute triangle \\( \\triangle ABC \\),\n\\[\n\\sin(A+B) = \\frac{3}{5}, \\quad \\sin(A-B) = \\frac{1}{5}, \\quad AB = 3.\n\\]\n\nFind the area of \\( \\triangle ABC \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3(\\sqrt{6} + 2)}{2}"
} | {
"split": "train",
"index": 25692,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For some positive integer $n$ , the sum of all odd positive integers between $n^2-n$ and $n^2+n$ is a number between $9000$ and $10000$ , inclusive. Compute $n$ .\n\n*2020 CCA Math Bonanza Lightning Round #3.1* Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 25693,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangular pyramid $S-ABC$ whose base is an equilateral triangle, the projection of point $A$ on the face $SBC$ is the orthocenter $H$ of triangle $\\triangle SBC$. The dihedral angle between the planes $H-AB-C$ is $30^{\\circ}$, and $SA = 2\\sqrt{3}$. Find the volume o... | math | {
"style": "rule",
"ground_truth": "\\frac{9\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 25694,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Which digit will appear in the 534th place after the decimal point in the decimal representation of $\\frac{5}{13}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 25695,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the number of ordered pairs of integers $(x,y)$ with $1\\le x<y\\le 150$ such that $i^x+i^y$ is a real number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3515"
} | {
"split": "train",
"index": 25696,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each vertex of a convex hexagon $ABCDEF$ is to be assigned a color. There are $7$ colors to choose from, and no two adjacent vertices can have the same color, nor can the vertices at the ends of each diagonal. Calculate the total number of different colorings possible. Let's th... | math | {
"style": "rule",
"ground_truth": "5040"
} | {
"split": "train",
"index": 25697,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many eight-digit numbers can be written using only the digits 1, 2, and 3 such that the difference between any two adjacent digits is 1? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 25698,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers, not exceeding 100, are multiples of 2 or 3 but not 4? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 25699,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(\\mathbf{v}\\) be a vector such that\n\\[\n\\left\\| \\mathbf{v} + \\begin{pmatrix} 4 \\\\ 2 \\end{pmatrix} \\right\\| = 10.\n\\]\nFind the smallest possible value of \\(\\|\\mathbf{v}\\|\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10 - 2\\sqrt{5}"
} | {
"split": "train",
"index": 25700,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The positive four-digit integers that use each of the four digits $1,$ $2,$ $3,$ and $4$ exactly once are ordered from least to greatest. What is the $15^{\\text{th}}$ integer in the list? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3214"
} | {
"split": "train",
"index": 25701,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \\frac{3}{7}$, and\n\\[a_n=\\frac{a_{n-2} \\cdot a_{n-1}}{2a_{n-2} - a_{n-1}}\\]for all $n \\geq 3$. Then $a_{2019}$ can be written as $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Wha... | math | {
"style": "rule",
"ground_truth": "8078"
} | {
"split": "train",
"index": 25702,
"solution": "1. **Base Case Verification**: \n - We are given $a_1 = 1$ and $a_2 = \\frac{3}{7}$. \n - We check if these values satisfy the proposed formula $a_n = \\frac{3}{4n-1}$:\n - For $n=1$, $\\frac{3}{4(1)-1} = \\frac{3}{3} = 1$, which matches $a_1$.\n - For ... |
deepscaler | [
{
"role": "user",
"content": "Let $x$ and $y$ be positive real numbers. Find the minimum value of\n\\[x^2 + y^2 + \\frac{4}{(x + y)^2}.\\] 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": 25703,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To be able to walk to the center $C$ of a circular fountain, a repair crew places a 16-foot plank from $A$ to $B$ and then a 10-foot plank from $D$ to $C$, where $D$ is the midpoint of $\\overline{AB}$ . What is the area of the circular base of the fountain? Express your answer... | math | {
"style": "rule",
"ground_truth": "164 \\pi \\mbox{ square feet}"
} | {
"split": "train",
"index": 25704,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $z$ is a complex number such that $z^2 = 24-32i$. Find $|z|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{10}"
} | {
"split": "train",
"index": 25705,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A tank is to be filled with water. When the tank is one-sixth full, 130 gallons of water are added, making the tank three-fifths full. How many gallons does the tank contain when it is completely full? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 25706,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an isosceles right triangle $ABC$ with right angle at $C$ and an area of $18$ square units, the rays trisecting $\\angle ACB$ intersect $AB$ at points $D$ and $E$. Calculate the area of triangle $CDE$.\nA) 4.0\nB) 4.5\nC) 5.0\nD) 5.5 Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"split": "train",
"index": 25707,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the formula $d=\\sqrt[3]{\\frac{16}{9}V}$, where $d$ is the diameter of a sphere and $V$ is its volume, find the volume of a sphere with a radius of $\\frac{1}{3}$. 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": 25708,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Natural numbers \\(a, b, c\\) are chosen such that \\(a < b < c\\). It is also known that the system of equations \\(2x + y = 2027\\) and \\(y = |x - a| + |x - b| + |x - c|\\) has exactly one solution. Find the minimum possible value of \\(c\\). Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "1014"
} | {
"split": "train",
"index": 25709,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a computer screen is the single character a. The computer has two keys: c (copy) and p (paste), which may be pressed in any sequence. Pressing p increases the number of a's on screen by the number that were there the last time c was pressed. c doesn't change the number of a'... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 25710,
"solution": "The first keystroke must be c and the last keystroke must be p. If there are $k$ c's pressed in total, let $n_{i}$ denote one more than the number of p's pressed immediately following the $i$ 'th c , for $1 \\leq i \\leq k$. Then, we have that the total number of k... |
deepscaler | [
{
"role": "user",
"content": "A 9x9 chessboard has its squares labeled such that the label of the square in the ith row and jth column is given by $\\frac{1}{2 \\times (i + j - 1)}$. We need to select one square from each row and each column. Find the minimum sum of the labels of the nine chosen squares. Le... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 25711,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the number of positive integers less than or equal to $10000$ which are relatively prime to $2014$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4648"
} | {
"split": "train",
"index": 25712,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $P$ and $Q$ are both in the line segment $AB$ and on the same side of its midpoint. $P$ divides $AB$ in the ratio $2:3$, \nand $Q$ divides $AB$ in the ratio $3:4$. If $PQ=2$, then the length of $AB$ is: Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 25713,
"solution": "1. **Assign Variables:**\n Let $AP = x$ and $PB = y$. Since $P$ divides $AB$ in the ratio $2:3$, we have:\n \\[\n \\frac{x}{y} = \\frac{2}{3}\n \\]\n This implies:\n \\[\n 3x = 2y \\quad \\text{(Equation 1)}\n \\]\n\n2. **Express $AB$ in terms of $x... |
deepscaler | [
{
"role": "user",
"content": "Determine the volume of the original cube given that one dimension is increased by $3$, another is decreased by $2$, and the third is left unchanged, and the volume of the resulting rectangular solid is $6$ more than that of the original cube. Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "(3 + \\sqrt{15})^3"
} | {
"split": "train",
"index": 25714,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of positive integers $n,$ $1 \\le n \\le 2000,$ for which the polynomial $x^2 + 2x - n$ can be factored as the product of two linear factors with integer coefficients. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 25715,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero of the polynomial $P(x)={x}^{3}-a{x}^{2}+bx-65$. For each possible combination of $a$ and $b$, let ${p}_{a,b}$ be the sum of the zeros of $P(x)$. Find the sum of the ${p}_{a,b}$'s fo... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 25716,
"solution": "Since $r+si$ is a root, by the Complex Conjugate Root Theorem, $r-si$ must be the other imaginary root. Using $q$ to represent the real root, we have\n$(x-q)(x-r-si)(x-r+si) = x^3 -ax^2 + bx -65$\n Applying difference of squares, and regrouping, we have\n$(x-q)(x^2... |
deepscaler | [
{
"role": "user",
"content": "Let $AC$ and $CE$ be two diagonals of a regular hexagon $ABCDEF$. Points $M$ and $N$ divide $AC$ and $CE$ internally such that $AM:AC = CN:CE = r$. If $B$, $M$, and $N$ are collinear, find $r$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt{3}}"
} | {
"split": "train",
"index": 25717,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a trapezoid $ABCD$ with $\\angle A = \\angle B = 90^{\\circ}$, $|AB| = 5 \\text{cm}$, $|BC| = 1 \\text{cm}$, and $|AD| = 4 \\text{cm}$, point $M$ is taken on side $AB$ such that $2 \\angle BMC = \\angle AMD$. Find the ratio $|AM| : |BM|$. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "3/2"
} | {
"split": "train",
"index": 25718,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given sets \\( A = \\{ x \\mid 5x - a \\leq 0 \\} \\) and \\( B = \\{ x \\mid 6x - b > 0 \\} \\), where \\( a, b \\in \\mathbf{N} \\), and \\( A \\cap B \\cap \\mathbf{N} = \\{ 2, 3, 4 \\} \\), determine the number of integer pairs \\((a, b)\\). Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 25719,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Quadrilateral $ABCD$ is a square. A circle with center $D$ has arc $AEC$. A circle with center $B$ has arc $AFC$. If $AB = 4$ cm, determine the total area in square centimeters of the football-shaped area of regions II and III combined. Express your answer as a decimal to the n... | math | {
"style": "rule",
"ground_truth": "9.1"
} | {
"split": "train",
"index": 25720,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\cos(\\frac{\\pi}{6} - \\alpha) = \\frac{3}{5}$, find the value of $\\cos(\\frac{5\\pi}{6} + \\alpha)$:\n\nA) $\\frac{3}{5}$\nB) $-\\frac{3}{5}$\nC) $\\frac{4}{5}$\nD) $-\\frac{4}{5}$ 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": 25721,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nCalculate the limit of the function:\n\n\\[ \\lim _{x \\rightarrow \\frac{1}{2}} \\frac{\\sqrt[3]{\\frac{x}{4}}-\\frac{1}{2}}{\\sqrt{\\frac{1}{2}+x}-\\sqrt{2x}} \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{3}"
} | {
"split": "train",
"index": 25722,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a positive integer $n,$ let\n\\[G_n = 1^2 + \\frac{1}{2^2} + \\frac{1}{3^2} + \\dots + \\frac{1}{n^2}.\\]Compute\n\\[\\sum_{n = 1}^\\infty \\frac{1}{(n + 1) G_n G_{n + 1}}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1 - \\frac{6}{\\pi^2}"
} | {
"split": "train",
"index": 25723,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many distinct ways can I arrange six keys on a keychain, if I want to keep my house key adjacent to my car key and my office key adjacent to my mailbox key? Two arrangements are not considered different if the keys can be rotated or reflected into the same order. Let's t... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 25724,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nCalculate the areas of the regions bounded by the curves given in polar coordinates.\n\n$$\nr=\\cos 2 \\phi\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"split": "train",
"index": 25725,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many solutions does the equation \\[ \\frac{(x-1)(x-2)(x-3) \\dotsm (x-150)}{(x-1^3)(x-2^3)(x-3^3) \\dotsm (x-150^3)} = 0 \\] have for \\(x\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "145"
} | {
"split": "train",
"index": 25726,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$ABCD$ is a regular tetrahedron. Let $N$ be the midpoint of $\\overline{AB}$, and let $G$ be the centroid of triangle $ACD$. What is $\\cos \\angle BNG$? 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": 25727,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the result of subtracting $7.305$ from $-3.219$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-10.524"
} | {
"split": "train",
"index": 25728,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many positive integers $x$ is $\\log_{10}(x-40) + \\log_{10}(60-x) < 2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 25729,
"solution": "1. **Identify the domain of the function**: \n The expression $\\log_{10}(x-40) + \\log_{10}(60-x)$ is defined only when both $x-40$ and $60-x$ are positive. This implies:\n \\[\n x-40 > 0 \\quad \\text{and} \\quad 60-x > 0\n \\]\n Simplifying these inequ... |
deepscaler | [
{
"role": "user",
"content": "How many numbers are in the list starting from $-48$, increasing by $7$ each time, up to and including $119$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 25730,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( X = \\{0, a, b, c\\} \\) and \\( M(X) = \\{ f \\mid f: X \\rightarrow X \\} \\) be the set of all functions from \\( X \\) to itself. Define the addition operation \\( \\oplus \\) on \\( X \\) as given in the following table:\n\\[\n\\begin{array}{|c|c|c|c|c|}\n\\hline\n... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 25731,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the radius of the smallest circle inside which every planar closed polygonal line with a perimeter of 1 can be enclosed? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 25732,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the maximum and minimum values of the exponential function $f(x) = a^x$ on the interval $[1, 2]$ differ by $\\frac{a}{2}$, then find the value of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 25733,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Máté is always in a hurry. He observed that it takes 1.5 minutes to get to the subway when he stands on the moving escalator, while it takes 1 minute to run down the stationary stairs. How long does it take Máté to get down if he can run down the moving escalator? Let's think s... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 25734,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On an $8 \\times 8$ grid, 64 points are marked at the center of each square. What is the minimum number of lines needed to separate all of these points from each other? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 25735,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a tournament, there are 16 chess players. Determine the number of different possible schedules for the first round (schedules are considered different if they differ by the participants of at least one match; the color of the pieces and the board number are not considered). ... | math | {
"style": "rule",
"ground_truth": "2027025"
} | {
"split": "train",
"index": 25736,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the coefficient of $x^2y^6$ in the expansion of $\\left(\\frac{3}{5}x-\\frac{y}{2}\\right)^8$? 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{63}{400}"
} | {
"split": "train",
"index": 25737,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Connecting the right-angled vertex of a right triangle and the two trisection points on the hypotenuse, the lengths of the two resulting line segments are $\\sin \\alpha$ and $\\cos \\alpha$ (where $0 < \\alpha < \\frac{\\pi}{2}$). What is the length of the hypotenuse? Let's th... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{\\sqrt{5}}"
} | {
"split": "train",
"index": 25738,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The foci of the ellipse $\\frac{x^2}{16} + \\frac{y^2}{b^2} = 1$ and the foci of the hyperbola\n\\[\\frac{x^2}{144} - \\frac{y^2}{81} = \\frac{1}{25}\\]coincide. Find $b^2.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 25739,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $a, b$ and $c$ are integers such that the greatest common divisor of $x^{2}+a x+b$ and $x^{2}+b x+c$ is $x+1$ (in the ring of polynomials in $x$ with integer coefficients), and the least common multiple of $x^{2}+a x+b$ and $x^{2}+b x+c$ is $x^{3}-4 x^{2}+x+6$. Find $a+... | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 25740,
"solution": "Since $x+1$ divides $x^{2}+a x+b$ and the constant term is $b$, we have $x^{2}+a x+b=(x+1)(x+b)$, and similarly $x^{2}+b x+c=(x+1)(x+c)$. Therefore, $a=b+1=c+2$. Furthermore, the least common multiple of the two polynomials is $(x+1)(x+b)(x+b-1)=x^{3}-4 x^{2}+x+6$,... |
deepscaler | [
{
"role": "user",
"content": "A rectangular piece of paper measures 4 units by 5 units. Several lines are drawn parallel to the edges of the paper. A rectangle determined by the intersections of some of these lines is called basic if\n (i) all four sides of the rectangle are segments of drawn line segments,... | math | {
"style": "rule",
"ground_truth": "896"
} | {
"split": "train",
"index": 25741,
"solution": "Denote the number of horizontal lines drawn as $x$, and the number of vertical lines drawn as $y$. The number of basic rectangles is $(x - 1)(y - 1)$. $5x + 4y = 2007 \\Longrightarrow y = \\frac{2007 - 5x}{4}$. Substituting, we find that $(x - 1)\\left(-\\frac 54x ... |
deepscaler | [
{
"role": "user",
"content": "Two congruent cones with radius 12 cm and height 12 cm are enclosed within a cylinder. The base of each cone is a base of the cylinder, and the height of the cylinder is 24 cm. What is the number of cubic centimeters in the volume of the cylinder not occupied by the cones? Expr... | math | {
"style": "rule",
"ground_truth": "2304\\pi"
} | {
"split": "train",
"index": 25742,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$, $Q$, and $R$ be points on a circle of radius $24$. If $\\angle PRQ = 40^\\circ$, what is the circumference of the minor arc $PQ$? Express your answer in terms of $\\pi$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{32\\pi}{3}"
} | {
"split": "train",
"index": 25743,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many multiples of 5 are between 80 and 375? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 25744,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC^{}_{}$, $A'$, $B'$, and $C'$ are on the sides $BC$, $AC^{}_{}$, and $AB^{}_{}$, respectively. Given that $AA'$, $BB'$, and $CC'$ are concurrent at the point $O^{}_{}$, and that $\\frac{AO^{}_{}}{OA'}+\\frac{BO}{OB'}+\\frac{CO}{OC'}=92$, find $\\frac{AO}{OA'}\\c... | math | {
"style": "rule",
"ground_truth": "94"
} | {
"split": "train",
"index": 25745,
"solution": "Using mass points, let the weights of $A$, $B$, and $C$ be $a$, $b$, and $c$ respectively.\n Then, the weights of $A'$, $B'$, and $C'$ are $b+c$, $c+a$, and $a+b$ respectively.\n Thus, $\\frac{AO^{}_{}}{OA'} = \\frac{b+c}{a}$, $\\frac{BO^{}_{}}{OB'} = \\frac{c+a}{b... |
deepscaler | [
{
"role": "user",
"content": "Evaluate the infinite geometric series:\n$$\\frac{5}{3} - \\frac{5}{4} + \\frac{25}{48} - \\frac{125}{384} + \\dots$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{20}{21}"
} | {
"split": "train",
"index": 25746,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the non-decreasing sequence of positive integers\n\\[1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,\\cdots\\]\nin which the $n^{th}$ positive integer appears $n$ times. The remainder when the $1993^{rd}$ term is divided by $5$ is Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 25747,
"solution": "1. **Identify the sequence pattern**: The given sequence is such that each integer $n$ appears $n$ times. This means the sequence looks like:\n \\[1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \\ldots\\]\n\n2. **Determine the position of each integer**: The position where each i... |
deepscaler | [
{
"role": "user",
"content": "In rectangle \\( ABCD \\), a circle \\(\\omega\\) is constructed using side \\( AB \\) as its diameter. Let \\( P \\) be the second intersection point of segment \\( AC \\) with circle \\(\\omega\\). The tangent to \\(\\omega\\) at point \\( P \\) intersects segment \\( BC \\) ... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 25748,
"solution": ""
} |
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