data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "64 people are in a single elimination rock-paper-scissors tournament, which consists of a 6-round knockout bracket. Each person has a different rock-paper-scissors skill level, and in any game, the person with the higher skill level will always win. For how many players $P$ is ... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 19149,
"solution": "Note that a sub-bracket, that is, a subset of games of the tournament that themselves constitute a bracket, is always won by the person with the highest skill level. Therefore, a person wins her first four rounds if and only if she has the highest skill level among... |
deepscaler | [
{
"role": "user",
"content": "Let the function $f(x) = (\\sin x + \\cos x)^2 - \\sqrt{3}\\cos 2x$.\n\n(Ⅰ) Find the smallest positive period of $f(x)$; \n(Ⅱ) Find the maximum value of $f(x)$ on the interval $\\left[0, \\frac{\\pi}{2}\\right]$ and the corresponding value of $x$ when the maximum value is atta... | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{12}"
} | {
"split": "train",
"index": 19150,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For the sequence of numbers $n_{1}, n_{2}, n_{3}, \\ldots$, the relation $n_{i}=2 n_{i-1}+a$ holds for all $i>1$. If $n_{2}=5$ and $n_{8}=257$, what is $n_{5}$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 19151,
"solution": "33."
} |
deepscaler | [
{
"role": "user",
"content": "One of Euler's conjectures was disproved in the 1960s by three American mathematicians when they showed there was a positive integer such that $133^5+110^5+84^5+27^5=n^{5}$. Find the value of $n$.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 19152,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $1.4 \\mathrm{~m}$ long rod has $3 \\mathrm{~kg}$ masses at both ends. Where should the rod be pivoted so that, when released from a horizontal position, the mass on the left side passes under the pivot with a speed of $1.6 \\mathrm{~m} /\\mathrm{s}$? Let's think step by step... | math | {
"style": "rule",
"ground_truth": "0.8"
} | {
"split": "train",
"index": 19153,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given real numbers $b_0, b_1, \\dots, b_{2019}$ with $b_{2019} \\neq 0$, let $z_1,z_2,\\dots,z_{2019}$ be the roots in the complex plane of the polynomial \\[ P(z) = \\sum_{k=0}^{2019} b_k z^k. \\] Let $\\mu = (|z_1| + \\cdots + |z_{2019}|)/2019$ be the average of the distances... | math | {
"style": "rule",
"ground_truth": "2019^{-1/2019}"
} | {
"split": "train",
"index": 19154,
"solution": "The answer is $M = 2019^{-1/2019}$. For any choices of $b_0,\\ldots,b_{2019}$ as specified, AM-GM gives \\[ \\mu \\geq |z_1\\cdots z_{2019}|^{1/2019} = |b_0/b_{2019}|^{1/2019} \\geq 2019^{-1/2019}. \\] To see that this is best possible, consider $b_0,\\ldots,b_{201... |
deepscaler | [
{
"role": "user",
"content": "Given a regular triangular pyramid $P-ABC$ (the base triangle is an equilateral triangle, and the vertex $P$ is the center of the base) with all vertices lying on the same sphere, and $PA$, $PB$, $PC$ are mutually perpendicular, and the side length of the base equilateral trian... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}\\pi}{2}"
} | {
"split": "train",
"index": 19155,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The house number. A person mentioned that his friend's house is located on a long street (where the houses on the side of the street with his friend's house are numbered consecutively: $1, 2, 3, \\ldots$), and that the sum of the house numbers from the beginning of the street t... | math | {
"style": "rule",
"ground_truth": "204"
} | {
"split": "train",
"index": 19156,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers $1,2,\\ldots,64$ are written in the squares of an $8\\times 8$ chessboard, one number to each square. Then $2\\times 2$ tiles are placed on the chessboard (without overlapping) so that each tile covers exactly four squares whose numbers sum to less than $100$. Find,... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 19157,
"solution": "\nTo solve this problem, we need to maximize the number of \\(2 \\times 2\\) tiles that can be placed on a \\(8 \\times 8\\) chessboard, such that the sum of the numbers in each tile is less than 100. The numbers \\(1, 2, \\ldots, 64\\) are written on the chessboar... |
deepscaler | [
{
"role": "user",
"content": "Given two fixed points $A(-1,0)$ and $B(1,0)$, and a moving point $P(x,y)$ on the line $l$: $y=x+3$, an ellipse $C$ has foci at $A$ and $B$ and passes through point $P$. Find the maximum value of the eccentricity of ellipse $C$. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{5}}{5}"
} | {
"split": "train",
"index": 19158,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A majority of the $30$ students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than $1$. The cost of a pencil in cents was greater than the number of pencils each student boug... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 19159,
"solution": "1. **Define Variables:**\n Let $s$ be the number of students that bought pencils, $c$ be the cost of each pencil in cents, and $n$ be the number of pencils each student bought. We need to find $c$.\n\n2. **Analyze Given Information:**\n - Since a majority of th... |
deepscaler | [
{
"role": "user",
"content": "There are 37 people lined up in a row, and they are counting off one by one. The first person says 1, and each subsequent person says the number obtained by adding 3 to the previous person’s number. At one point, someone makes a mistake and subtracts 3 from the previous person'... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 19160,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest six-digit number in which all digits are distinct, and each digit, except for the extreme ones, is equal either to the sum or the difference of its neighboring digits. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "972538"
} | {
"split": "train",
"index": 19161,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $\\sqrt{10p} \\cdot \\sqrt{5p^2} \\cdot \\sqrt{6p^4}$ . Express your answer in simplest radical form in terms of $p$.\n\nNote: When entering a square root with more than one character, you must use parentheses or brackets. For example, you should enter $\\sqrt{14}$ a... | math | {
"style": "rule",
"ground_truth": "10p^3 \\sqrt{3p}"
} | {
"split": "train",
"index": 19162,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$BL$ is the angle bisector of triangle $ABC$. Find its area, given that $|AL| = 2$, $|BL| = 3\\sqrt{10}$, and $|CL| = 3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15 \\sqrt{15}}{4}"
} | {
"split": "train",
"index": 19163,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles are externally tangent. Lines $\\overline{PAB}$ and $\\overline{PA'B'}$ are common tangents with points $A$, $A'$ on the smaller circle and $B$, $B'$ on the larger circle. If $PA=AB=5$ and the radius of the larger circle is 3 times the radius of the smaller circle, ... | math | {
"style": "rule",
"ground_truth": "5\\pi"
} | {
"split": "train",
"index": 19164,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the result of the expression \\( 2015 \\frac{1999}{2015} \\times \\frac{1}{4} - \\frac{2011}{2015} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "503"
} | {
"split": "train",
"index": 19165,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(n)$ be the base-10 logarithm of the sum of the elements of the $n$th row in Pascal's triangle. Express $\\frac{f(n)}{\\log_{10} 2}$ in terms of $n$. Recall that Pascal's triangle begins\n\n\\begin{tabular}{rccccccccc}\n$n=0$:& & & & & 1\\\\\\noalign{\\smallskip\\smallski... | math | {
"style": "rule",
"ground_truth": "n"
} | {
"split": "train",
"index": 19166,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence is defined by $A_{0}=0, A_{1}=1, A_{2}=2$, and, for integers $n \\geq 3$, $$A_{n}=\\frac{A_{n-1}+A_{n-2}+A_{n-3}}{3}+\\frac{1}{n^{4}-n^{2}}$$ Compute $\\lim _{N \\rightarrow \\infty} A_{N}$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{6}-\\frac{\\pi^{2}}{12}"
} | {
"split": "train",
"index": 19167,
"solution": "If we sum the given equation for $n=3,4,5, \\ldots, N$, we obtain $$\\sum_{n=3}^{N} A_{n}=\\sum_{n=3}^{N} \\frac{A_{n-1}+A_{n-2}+A_{n-3}}{3}+\\frac{1}{n^{4}-n^{2}}$$ This reduces dramatically to $$\\begin{equation*} A_{N}+\\frac{2 A_{N-1}}{3}+\\frac{A_{N-2}}{3}=A_{... |
deepscaler | [
{
"role": "user",
"content": "Neo has an infinite supply of red pills and blue pills. When he takes a red pill, his weight will double, and when he takes a blue pill, he will lose one pound. If Neo originally weighs one pound, what is the minimum number of pills he must take to make his weight 2015 pounds? ... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 19168,
"solution": "Suppose instead Neo started at a weight of 2015 pounds, instead had green pills, which halve his weight, and purple pills, which increase his weight by a pound, and he wished to reduce his weight to one pound. It is clear that, if Neo were able to find such a seque... |
deepscaler | [
{
"role": "user",
"content": "Alex and Katy play a game on an \\(8 \\times 8\\) grid made of 64 unit cells. They take turns to play, with Alex going first. On Alex's turn, he writes 'A' in an empty cell. On Katy's turn, she writes 'K' in two empty cells that share an edge. The game ends when one player cann... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 19169,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $(a^b)^a - (b^a)^b$ for $a = 2$ and $b = 3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-665"
} | {
"split": "train",
"index": 19170,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the hyperbola $C:\\frac{x^2}{a^2}-y^2=1\\;(a>0)$ intersect the line $l:x+y=1$ at two distinct points $A$ and $B$. \n$(1)$ Find the range of real numbers for $a$. \n$(2)$ If the intersection point of the line $l$ and the $y$-axis is $P$, and $\\overrightarrow{PA}=\\frac{5}... | math | {
"style": "rule",
"ground_truth": "a = \\frac{17}{13}"
} | {
"split": "train",
"index": 19171,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A machine can operate at different speeds, and some of the items it produces will have defects. The number of defective items produced per hour varies with the machine's operating speed. Let $x$ represent the speed (in revolutions per second), and $y$ represent the number of de... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 19172,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle{PQR}$ be a right triangle with $PQ = 90$, $PR = 120$, and $QR = 150$. Let $C_{1}$ be the inscribed circle. Construct $\\overline{ST}$ with $S$ on $\\overline{PR}$ and $T$ on $\\overline{QR}$, such that $\\overline{ST}$ is perpendicular to $\\overline{PR}$ and ta... | math | {
"style": "rule",
"ground_truth": "725"
} | {
"split": "train",
"index": 19173,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A student, Liam, wants to earn a total of 30 homework points. For earning the first four homework points, he has to do one homework assignment each; for the next four points, he has to do two homework assignments each; and so on, such that for every subsequent set of four point... | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 19174,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a configuration of four unit squares arranged in a 2x2 grid, find the area of triangle $\\triangle ABC$, where $A$ is the midpoint of the top side of the top-left square, $B$ is the bottom-right corner of the bottom-right square, and $C$ is the midpoint of the right side ... | math | {
"style": "rule",
"ground_truth": "0.375"
} | {
"split": "train",
"index": 19175,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$A$, $B$, $C$, and $D$ are points on a circle, and segments $\\overline{AC}$ and $\\overline{BD}$ intersect at $P$, such that $AP=8$, $PC=1$, and $BD=6$. Find $BP$, given that $BP < DP.$\n\n[asy]\n\nunitsize(0.6 inch);\n\ndraw(circle((0,0),1));\ndraw((-0.3,0.94)--(0.3,-0.94));\... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 19176,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that real numbers x and y satisfy x + y = 5 and x * y = -3, find the value of x + x^4 / y^3 + y^4 / x^3 + y. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5 + \\frac{2829}{27}"
} | {
"split": "train",
"index": 19177,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circles of radii $5, 5, 8,$ and $\\frac mn$ are mutually externally tangent, where $m$ and $n$ are relatively prime positive integers. Find $m + n.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 19178,
"solution": "We may also use Descartes' theorem, $k_4=k_1+k_2+k_3\\pm 2\\sqrt{k_1k_2+k_2k_3+k_3k_1}$ where each of $k_i$ is the curvature of a circle with radius $r_i$, and the curvature is defined as $k_i=\\frac{1}{r_i}$. The larger solution for $k_4$ will give the curvature o... |
deepscaler | [
{
"role": "user",
"content": "The given arithmetic sequences $\\{a_{n}\\}$ and $\\{b_{n}\\}$ have respective sums of the first $n$ terms, denoted by $S_{n}$ and $T_{n}$. The ratio $\\frac{S_{n}}{T_{n}} = \\frac{3n + 1}{n + 3}$. Determine the value of $\\frac{a_{2} + a_{20}}{b_{7} + b_{15}}$. Let's think ste... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}"
} | {
"split": "train",
"index": 19179,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the distinct prime factors of $7^7 - 7^4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 19180,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px + 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$ Find the value of $p + q + r + s.$ Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "2028"
} | {
"split": "train",
"index": 19181,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\\(\\triangle ABC\\) is isosceles with base \\(AC\\). Points \\(P\\) and \\(Q\\) are respectively in \\(CB\\) and \\(AB\\) and such that \\(AC=AP=PQ=QB\\). \nThe number of degrees in \\(\\angle B\\) is: Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "25\\frac{5}{7}"
} | {
"split": "train",
"index": 19182,
"solution": "1. **Identify the properties of the triangles**: Given that $\\triangle ABC$ is isosceles with base $AC$, and $AC = AP = PQ = QB$, we can infer that $\\triangle APQ$ and $\\triangle BQP$ are also isosceles. Additionally, $\\triangle APC$ and $\\triangle BQC$ are is... |
deepscaler | [
{
"role": "user",
"content": "What expression is never a prime number when $p$ is a prime number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "p^2+26"
} | {
"split": "train",
"index": 19183,
"solution": "We need to determine which of the given expressions is never a prime number when $p$ is a prime number. To do this, we will analyze each expression modulo a small integer to see if it can be factored or if it always results in a composite number.\n\n1. **Expression... |
deepscaler | [
{
"role": "user",
"content": "If $a$ and $b$ are randomly selected real numbers between 0 and 1, find the probability that the nearest integer to $\\frac{a-b}{a+b}$ is odd. 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": 19184,
"solution": "The only reasonable way I know of to do this problem is geometrically (yes, you can use integrals to find the areas of the triangles involved, but I don't consider that reasonable). First let us find the points $(a, b)$ in the plane for which the nearest integer to... |
deepscaler | [
{
"role": "user",
"content": "How many unordered pairs of prime numbers have a sum of 40? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 19185,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all pairs of integers $a$ and $b$ for which\n\\[7a+14b=5a^2+5ab+5b^2\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(0,0),(1,2),(-1,3)"
} | {
"split": "train",
"index": 19186,
"solution": "\nWe are given the equation:\n\n\\[\n7a + 14b = 5a^2 + 5ab + 5b^2\n\\]\n\nand tasked with finding all integer pairs \\((a, b)\\) satisfying it. Let's begin by simplifying and solving the equation.\n\n### Step 1: Simplify the Equation\n\nFirst, divide the entire equ... |
deepscaler | [
{
"role": "user",
"content": "Let \\( \\triangle DEF \\) be a triangle and \\( H \\) the foot of the altitude from \\( D \\) to \\( EF \\). If \\( DE = 60 \\), \\( DF = 35 \\), and \\( DH = 21 \\), what is the difference between the minimum and the maximum possible values for the area of \\( \\triangle DEF ... | math | {
"style": "rule",
"ground_truth": "588"
} | {
"split": "train",
"index": 19187,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "During a secret meeting, 20 trainees elect their favorite supervisor. Each trainee votes for two supervisors. It is known that for any two trainees, there is always at least one supervisor for whom both have voted. What is the minimum number of votes received by the supervisor ... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 19188,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the perimeter of a square is 28, what is the side length of the square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 19189,
"solution": "Since a square has four equal sides, the side length of a square equals one-quarter of the perimeter of the square. Thus, the side length of a square with perimeter 28 is $28 \\div 4 = 7$."
} |
deepscaler | [
{
"role": "user",
"content": "Sally the snail sits on the $3 \\times 24$ lattice of points $(i, j)$ for all $1 \\leq i \\leq 3$ and $1 \\leq j \\leq 24$. She wants to visit every point in the lattice exactly once. In a move, Sally can move to a point in the lattice exactly one unit away. Given that Sally st... | math | {
"style": "rule",
"ground_truth": "4096"
} | {
"split": "train",
"index": 19190,
"solution": "On her first turn, Sally cannot continue moving down the middle row. She must turn either to the bottom row or the top row. WLOG, she turns to the top row, and enters the cell $(3,1)$ and we will multiply by 2 later. Then, we can see that the path must finish in $(... |
deepscaler | [
{
"role": "user",
"content": "Find the product of the roots of the equation \\[(3x^4 + 2x^3 - 9x + 30)(4x^3 - 20x^2 + 24x - 36) = 0.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-90"
} | {
"split": "train",
"index": 19191,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\\\(f(x)=ax^{2}-b\\sin x\\\\) and \\\\(f′(0)=1\\\\), \\\\(f′\\left( \\dfrac {π}{3}\\right)= \\dfrac {1}{2}\\\\). Find the values of \\\\(a\\\\) and \\\\(b\\\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 19192,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a circle $C: (x-3)^{2}+y^{2}=25$ and a line $l: (m+1)x+(m-1)y-2=0$ (where $m$ is a parameter), the minimum length of the chord intercepted by the circle $C$ and the line $l$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{5}"
} | {
"split": "train",
"index": 19193,
"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": "150"
} | {
"split": "train",
"index": 19194,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)= \\dfrac {2-\\cos \\left( \\dfrac {\\pi}{4}(1-x)\\right)+\\sin \\left( \\dfrac {\\pi}{4}(1-x)\\right)}{x^{2}+4x+5}(-4\\leqslant x\\leqslant 0)$, find the maximum value of $f(x)$. 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": 19195,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sequentially throw a die and record the numbers obtained as $a$ and $b$. Perform the following operations: \n```\nINPUT \"a, b=\"; a, b \nIF a >= b THEN \ny = a - b \nELSE \ny = b - a \nEND IF \nPRINT y \nEND \n```\n(1) If $a=3$, $b=6$, find the value of $y$ output by ... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"split": "train",
"index": 19196,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given 2009 points, where no three points are collinear, along with three vertices, making a total of 2012 points, and connecting these 2012 points to form non-overlapping small triangles, calculate the total number of small triangles that can be formed. Let's think step by step... | math | {
"style": "rule",
"ground_truth": "4019"
} | {
"split": "train",
"index": 19197,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If an arc of $45^{\\circ}$ on circle $A$ has the same length as an arc of $30^{\\circ}$ on circle $B$, then what is the ratio of the area of circle $A$ to the area of circle $B$? Express your answer as a common fraction. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"split": "train",
"index": 19198,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a regional science fair, 25 participants each have their own room in the same hotel, with room numbers from 1 to 25. All participants have arrived except those assigned to rooms 14 and 20. What is the median room number of the other 23 participants? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 19199,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form\n\\[a + b \\sqrt{2} + c \\sqrt{3} + d \\sqrt{6},\\]where $a$, $b$, $c$, and $d$ are positive integers. Find $a+b+c+d$. Let's think ... | math | {
"style": "rule",
"ground_truth": "720"
} | {
"split": "train",
"index": 19200,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T$ be the sum of all the real coefficients of the expansion of $(1 + ix)^{2018}$. What is $\\log_2(T)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1009"
} | {
"split": "train",
"index": 19201,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bowling ball must have a diameter of 9 inches. Calculate both the surface area and the volume of the bowling ball before the finger holes are drilled. Express your answers as common fractions in terms of \\(\\pi\\). Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "\\frac{729\\pi}{6}"
} | {
"split": "train",
"index": 19202,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $\\sin C+\\cos C=1-\\sin \\frac{C}{2}$.\n\n$(1)$ Find the value of $\\sin C$.\n\n$(2)$ If $a^{2}+b^{2}=4(a+b)-8$, find the value of side $c$. Let'... | math | {
"style": "rule",
"ground_truth": "1+ \\sqrt{7}"
} | {
"split": "train",
"index": 19203,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To qualify for the competition, wrestler Vladimir must have three matches and win at least two of them consecutively. His opponents are Andrei (A) and Boris (B). Vladimir can choose to schedule the matches in the order ABA or BAB. The probability of Vladimir losing one match to... | math | {
"style": "rule",
"ground_truth": "0.742"
} | {
"split": "train",
"index": 19204,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Distinct planes $p_1, p_2, \\dots, p_k$ intersect the interior of a cube $Q$. Let $S$ be the union of the faces of $Q$ and let $P = \\bigcup_{j=1}^{k} p_j$. The intersection of $P$ and $S$ consists of the union of all segments joining the midpoints of every pair of edges belong... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 19205,
"solution": "To solve this problem, we need to analyze the different types of planes that can intersect the cube $Q$ such that their intersection with the surface $S$ of the cube consists of segments joining the midpoints of edges on the same face of $Q$. We categorize these pl... |
deepscaler | [
{
"role": "user",
"content": "For any positive real number $x, \\lfloor x \\rfloor$ denotes the largest integer less than or equal to $x$. If $\\lfloor x \\rfloor \\cdot x = 36$ and $\\lfloor y \\rfloor \\cdot y = 71$ where $x, y > 0$, what is $x + y$ equal to? Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "\\frac{119}{8}"
} | {
"split": "train",
"index": 19206,
"solution": "For any positive real number $x, \\lfloor x \\rfloor$ equals the largest integer less than or equal to $x$ and so $\\lfloor x \\rfloor \\leq x$. In particular, $\\lfloor x \\rfloor \\cdot x \\leq x \\cdot x = x^{2}$. Thus, if $\\lfloor x \\rfloor \\cdot x = 36$, th... |
deepscaler | [
{
"role": "user",
"content": "In the 2013 Zhejiang College Entrance Examination, arrange the six letters A, B, C, D, E, F in a row, with both A and B on the same side of C. How many different arrangements are there? (Answer with a number.) Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "480"
} | {
"split": "train",
"index": 19207,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer that is both an integer power of 13 and is not a palindrome. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "169"
} | {
"split": "train",
"index": 19208,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the sum of distances from point $P$ to the two points $\\left(0, -\\sqrt{3}\\right)$ and $\\left(0, \\sqrt{3}\\right)$ is $4$. Let the trajectory of point $P$ be $C$.\n\n(Ⅰ) Find the equation of curve $C$;\n\n(Ⅱ) Find the coordinates of... | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 19209,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle is divided into 1000 smaller triangles. What is the minimum number of distinct points that can be the vertices of these triangles? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "503"
} | {
"split": "train",
"index": 19210,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An omino is a 1-by-1 square or a 1-by-2 horizontal rectangle. An omino tiling of a region of the plane is a way of covering it (and only it) by ominoes. How many omino tilings are there of a 2-by-10 horizontal rectangle? Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "7921"
} | {
"split": "train",
"index": 19211,
"solution": "There are exactly as many omino tilings of a 1-by-$n$ rectangle as there are domino tilings of a 2-by-$n$ rectangle. Since the rows don't interact at all, the number of omino tilings of an $m$-by-$n$ rectangle is the number of omino tilings of a 1-by-$n$ rectangle ... |
deepscaler | [
{
"role": "user",
"content": "Luke wants to fence a rectangular piece of land with an area of at least 450 square feet. The length of the land is 1.5 times the width. What should the width of the rectangle be if he wants to use the least amount of fencing? Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "10\\sqrt{3}"
} | {
"split": "train",
"index": 19212,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $a$ and $b$ are nonzero integers such that two of the roots of\n\\[x^3 + ax^2 + bx + 9a\\]coincide, and all three roots are integers. Find $|ab|.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1344"
} | {
"split": "train",
"index": 19213,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\Diamond4_7=\\Diamond1_{8}$ and $\\Diamond$ represents a digit, solve for $\\Diamond$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 19214,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has $\\angle{A}=90^{\\circ}$ , $AB=2$ , and $AC=4$ . Circle $\\omega_1$ has center $C$ and radius $CA$ , while circle $\\omega_2$ has center $B$ and radius $BA$ . The two circles intersect at $E$ , different from point $A$ . Point $M$ is on $\... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 19215,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $b$ for which $x^2 + bx + 1760$ factors into a product of two polynomials, each having integer coefficients. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 19216,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An isosceles triangle, a square, and a regular pentagon each have a perimeter of 20 inches. What is the ratio of the side length of the triangle (assuming both equal sides for simplicity) to the side length of the square? Express your answer as a common fraction. Let's think st... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 19217,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many terms of the arithmetic sequence 88, 85, 82, $\\dots$ appear before the number $-17$ appears? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 19218,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( f(x) = a^x - 1 \\). Find the largest value of \\( a > 1 \\) such that if \\( 0 \\leq x \\leq 3 \\), then \\( 0 \\leq f(x) \\leq 3 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[3]{4}"
} | {
"split": "train",
"index": 19219,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle A B C$ be an acute triangle, with $M$ being the midpoint of $\\overline{B C}$, such that $A M=B C$. Let $D$ and $E$ be the intersection of the internal angle bisectors of $\\angle A M B$ and $\\angle A M C$ with $A B$ and $A C$, respectively. Find the ratio of t... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"split": "train",
"index": 19220,
"solution": "Let $[X Y Z]$ denote the area of $\\triangle X Y Z$. Solution 1: Let $A M=\\ell$, let $D E=d$, and let the midpoint of $\\overline{D E}$ be $F$. Since $\\frac{A D}{A B}=\\frac{A E}{A C}=\\frac{2}{3}$ by the angle bisector theorem, $F$ lies on $\\overline{A M}$ and ... |
deepscaler | [
{
"role": "user",
"content": "The graph of the polynomial\n\n$P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e$\n\nhas five distinct $x$-intercepts, one of which is at $(0,0)$. Which of the following coefficients cannot be zero?\n\n$\\textbf{(A)}\\ a \\qquad \\textbf{(B)}\\ b \\qquad \\textbf{(C)}\\ c \\qquad \\tex... | math | {
"style": "rule",
"ground_truth": "\\text{(D)}"
} | {
"split": "train",
"index": 19221,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that we are given 40 points equally spaced around the perimeter of a square, so that four of them are located at the vertices and the remaining points divide each side into ten congruent segments. If $P$, $Q$, and $R$ are chosen to be any three of these points which ar... | math | {
"style": "rule",
"ground_truth": "841"
} | {
"split": "train",
"index": 19222,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four different balls are to be placed into five boxes numbered 1, 2, 3, 4, and 5. (Write the result as a number) \n(1) How many ways can there be with at least one ball in box number 1? \n(2) How many ways are there to have exactly two empty boxes? \n(3) How many ways are th... | math | {
"style": "rule",
"ground_truth": "375"
} | {
"split": "train",
"index": 19223,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the system of equations\n\n\\begin{align*}\n8x - 6y &= c, \\\\\n10y - 15x &= d.\n\\end{align*}\n\nhas a solution $(x,y)$ where $x$ and $y$ are both nonzero, find $\\frac{c}{d},$ assuming $d$ is nonzero. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{5}"
} | {
"split": "train",
"index": 19224,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of positive integers $n$ that satisfy\n\\[(n - 1)(n - 5)(n - 9) \\dotsm (n - 101) < 0.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 19225,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cagney can frost a cupcake every 25 seconds and Lacey can frost a cupcake every 35 seconds. They work together for 7 minutes, but there is a 1-minute period where only Cagney is frosting because Lacey takes a break. What is the number of cupcakes they can frost in these 7 minut... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 19226,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The shelf life $y$ (in hours) of a certain food product and its storage temperature $x$ (in °C) satisfy the function relationship $y=e^{kx+b}$ (where $e=2.718\\ldots$ is the base of the natural logarithm, and $k$, $b$ are constants). It is known that the shelf life of this food... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 19227,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What common fraction (that is, a fraction reduced to its lowest terms) is equivalent to $0.4\\overline{13}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{409}{990}"
} | {
"split": "train",
"index": 19228,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Fibonacci sequence is the sequence 1, 1, 2, 3, 5, $\\ldots$ where the first and second terms are 1 and each term after that is the sum of the previous two terms. What is the remainder when the $100^{\\mathrm{th}}$ term of the sequence is divided by 8? Let's think step by st... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 19229,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest real number $x$ such that\n\\[\\frac{\\lfloor x \\rfloor}{x} = \\frac{9}{10}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{80}{9}"
} | {
"split": "train",
"index": 19230,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a row with 120 seats, some of the seats are already occupied. If a new person arrives and must sit next to someone regardless of their choice of seat, what is the minimum number of people who were already seated? Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 19231,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The line $ax+(a+1)y=a+2$ passes through the point $(4,-8)$. Find $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 19232,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\n\\log_2 \\left( \\prod_{a=1}^{2015} \\prod_{b=1}^{2015} (1+e^{2\\pi i a b/2015}) \\right)\n\\]\nHere $i$ is the imaginary unit (that is, $i^2=-1$). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13725"
} | {
"split": "train",
"index": 19233,
"solution": "The answer is $13725$.\nWe first claim that if $n$ is odd, then $\\prod_{b=1}^{n} (1+e^{2\\pi i ab/n}) = 2^{\\gcd(a,n)}$. To see this, write $d = \\gcd(a,n)$ and $a = da_1$, $n=dn_1$ with $\\gcd(a_1,n_1) = 1$. Then\n$a_1, 2a_1,\\dots,n_1 a_1$ modulo $n_1$ is a perm... |
deepscaler | [
{
"role": "user",
"content": "Let $A B C D$ be a rectangle with $A B=3$ and $B C=7$. Let $W$ be a point on segment $A B$ such that $A W=1$. Let $X, Y, Z$ be points on segments $B C, C D, D A$, respectively, so that quadrilateral $W X Y Z$ is a rectangle, and $B X<X C$. Determine the length of segment $B X$.... | math | {
"style": "rule",
"ground_truth": "\\frac{7-\\sqrt{41}}{2}"
} | {
"split": "train",
"index": 19234,
"solution": "We note that $$\\angle Y X C=90-\\angle W X B=\\angle X W B=90-\\angle A W Z=\\angle A Z W$$ gives us that $X Y C \\cong Z W A$ and $X Y Z \\sim W X B$. Consequently, we get that $Y C=A W=1$. From $X Y Z \\sim W X B$, we get that $$\\frac{B X}{B W}=\\frac{C Y}{C X}... |
deepscaler | [
{
"role": "user",
"content": "Compute the positive real number $x$ satisfying $x^{\\left(2 x^{6}\\right)}=3$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[6]{3}"
} | {
"split": "train",
"index": 19235,
"solution": "Let $t=x^{6}$, so $x^{2 t}=3$. Taking this to the third power gives $x^{6 t}=27$, or equivalently $t^{t}=3^{3}$. We can see by inspection that $t=3$, and this is the only solution as for $t>1$, the function $t^{t}$ is monotonically increasing, and if $0<t<1, t^{t}<... |
deepscaler | [
{
"role": "user",
"content": "You have 128 teams in a single elimination tournament. The Engineers and the Crimson are two of these teams. Each of the 128 teams in the tournament is equally strong, so during each match, each team has an equal probability of winning. Now, the 128 teams are randomly put into ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{64}"
} | {
"split": "train",
"index": 19236,
"solution": "There are $\\binom{128}{2}=127 \\cdot 64$ pairs of teams. In each tournament, 127 of these pairs play. By symmetry, the answer is $\\frac{127}{127 \\cdot 64}=\\frac{1}{64}$."
} |
deepscaler | [
{
"role": "user",
"content": "When the product of $\\underbrace{\\frac{2}{3} \\times \\frac{2}{3} \\times \\cdots \\times \\frac{2}{3}}_{10}$ is written as a decimal, what are the first two decimal places? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.01"
} | {
"split": "train",
"index": 19237,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nIn two days, 50 financiers raised funds to combat a new virus. Each of them contributed a one-time whole number amount in thousands of rubles, not exceeding 100. Each contribution on the first day did not exceed 50 thousand rubles, while each on the second day was greater tha... | math | {
"style": "rule",
"ground_truth": "2525"
} | {
"split": "train",
"index": 19238,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square $EFGH$ has one vertex on each side of square $ABCD$. Point $E$ is on $AB$ with $AE=7\\cdot EB$. What is the ratio of the area of $EFGH$ to the area of $ABCD$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{25}{32}"
} | {
"split": "train",
"index": 19239,
"solution": "1. **Assigning Side Lengths:**\n Let the side length of square $ABCD$ be $8s$. This choice is made to simplify calculations, as $AE = 7 \\cdot EB$ implies a division of side $AB$ into 8 equal parts.\n\n2. **Locating Point E:**\n Since $AE = 7 \\cdot EB$, and if... |
deepscaler | [
{
"role": "user",
"content": "If $g(x) = 3x + 7$ and $f(x) = 5x - 9$, what is the value of $f(g(8))$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "146"
} | {
"split": "train",
"index": 19240,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can I choose a 4-person committee from a club of 9 people? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "126"
} | {
"split": "train",
"index": 19241,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In \\( \\triangle ABC \\), \\( AB = 4 \\), \\( BC = 7 \\), \\( CA = 5 \\). Let \\(\\angle BAC = \\alpha\\). Find the value of \\( \\sin^6 \\frac{\\alpha}{2} + \\cos^6 \\frac{\\alpha}{2} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7/25"
} | {
"split": "train",
"index": 19242,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three coplanar circles intersect as shown. What is the maximum number of points on the circles that a line passing through all three circles can touch?\n\n[asy]import graph;\ndraw(Circle((-9,9),15));\ndraw(Circle((0,-9),15));\ndraw(Circle((9,9),15));\n[/asy] Let's think step by... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 19243,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the unit circle shown in the figure, chords $PQ$ and $MN$ are parallel to the unit radius $OR$ of the circle with center at $O$. Chords $MP$, $PQ$, and $NR$ are each $s$ units long and chord $MN$ is $d$ units long.\n\nOf the three equations\n\\begin{equation*} \\label{eq:1} ... | math | {
"style": "rule",
"ground_truth": "I, II and III"
} | {
"split": "train",
"index": 19244,
"solution": "1. **Lemma 1: $SM = QN = s$**\n - Since $OM = ON$, by the Base Angle Theorem, $\\angle OMN = \\angle ONM$.\n - By the Alternate Interior Angles Theorem, $\\angle MOS = \\angle NMO = \\angle MNO = \\angle NOR$, making $\\triangle MOS \\cong \\triangle NOR$ by SA... |
deepscaler | [
{
"role": "user",
"content": "The graph of the function $f(x)=\\sin({ωx-\\frac{π}{6}})$, where $0<ω<6$, is shifted to the right by $\\frac{π}{6}$ units to obtain the graph of the function $g(x)$. If $\\left(0,\\frac{π}{ω}\\right)$ is a monotone interval of $g(x)$, and $F(x)=f(x)+g(x)$, determine the maximum... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 19245,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $|\\overrightarrow{a}|=1$, $|\\overrightarrow{b}|=2$, and $\\overrightarrow{a}⊥(\\overrightarrow{a}+\\overrightarrow{b})$, determine the projection of $\\overrightarrow{a}$ onto $\\overrightarrow{b}$. 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": 19246,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the first 9 positive multiples of 5? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 19247,
"solution": "Since $1+2+3+4+5+6+7+8+9=45$ then $5+10+15+\\cdots+40+45=5(1+2+3+\\cdots+8+9)=5(45)=225$"
} |
deepscaler | [
{
"role": "user",
"content": "Compute $10^{-1}\\pmod{1001}$. Express your answer as a residue from $0$ to $1000$, inclusive. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "901"
} | {
"split": "train",
"index": 19248,
"solution": ""
} |
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