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 the domain of $f(x)$ is $\\mathbb{R}$, $f(1)=\\frac{1}{4}$, and it satisfies the equation $4f(x)f(y) = f(x+y) + f(x-y)$, find the value of $f(2016)$. 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": 27049,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two spherical balls of different sizes placed in two corners of a rectangular room, where each ball touches two walls and the floor, and there is a point on each ball such that the distance from the two walls it touches to that point is 5 inches and the distance from the ... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 27050,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six regular hexagons surround a regular hexagon of side length $1$ as shown. What is the area of $\\triangle{ABC}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3}"
} | {
"split": "train",
"index": 27051,
"solution": "1. **Understanding the Configuration**: We are given a central regular hexagon surrounded by six regular hexagons, all of the same side length $1$. We need to find the area of $\\triangle{ABC}$, which is formed by connecting the centers of three adjacent outer hexa... |
deepscaler | [
{
"role": "user",
"content": "At what point does the line containing the points $(1, 7)$ and $(3, 11)$ intersect the $y$-axis? Express your answer as an ordered pair. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(0,5)"
} | {
"split": "train",
"index": 27052,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the remainder when $123456789012$ is divided by $240$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "132"
} | {
"split": "train",
"index": 27053,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice and Bob each arrive at a meeting at a random time between 8:00 and 9:00 AM. If Alice arrives after Bob, what is the probability that Bob arrived before 8:45 AM? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{16}"
} | {
"split": "train",
"index": 27054,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\\{a_n\\}$. If $S_7=3(a_1+a_9)$, then the value of $\\frac{a_5}{a_4}$ is \\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{6}"
} | {
"split": "train",
"index": 27055,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the general term of the sequence \\\\(\\{a_n\\}\\) is \\\\(a_n = (-1)^{n+1} \\cdot (3n-2)\\\\), then \\\\(a_1 + a_2 + \\cdots + a_{20} = \\)____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-30"
} | {
"split": "train",
"index": 27056,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation of curve $C$ is $\\frac{x^2}{m^2} + \\frac{y^2}{n^2} = 1$, where $m$ and $n$ are the numbers obtained by rolling a die twice in succession. Let event $A$ be \"The equation $\\frac{x^2}{m^2} + \\frac{y^2}{n^2} = 1$ represents an ellipse with foci on the $x$-axis\". ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{12}"
} | {
"split": "train",
"index": 27057,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $15$ people, including Petruk, Gareng, and Bagong, which will be partitioned into $6$ groups, randomly, that consists of $3, 3, 3, 2, 2$ , and $2$ people (orders are ignored). Determine the probability that Petruk, Gareng, and Bagong are in a group. Let's think... | math | {
"style": "rule",
"ground_truth": "3/455"
} | {
"split": "train",
"index": 27058,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the 6 finalists, 1 first prize, 2 second prizes, and 3 third prizes are to be awarded. Calculate the total number of possible outcomes. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 27059,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $MPQ$, a line parallel to side $MQ$ intersects side $MP$, the median $MM_1$, and side $PQ$ at points $D$, $E$, and $F$ respectively. It is known that $DE = 5$ and $EF = 7$. What is the length of $MQ$? Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 27060,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The lines $-2x + y = k$ and $0.5x + y = 14$ intersect when $x = -8.4$. What is the value of $k$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 27061,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point $P$ moves on the ellipse $\\frac{x^{2}}{4}+y^{2}=1$, find the minimum distance from point $P$ to line $l$: $x+y-2\\sqrt{5}=0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{2}"
} | {
"split": "train",
"index": 27062,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a set of data $x_1, x_2, x_3, x_4, x_5$ with a mean of 8 and variance of 2, find the mean and variance of a new set of data: $4x_1+1, 4x_2+1, 4x_3+1, 4x_4+1, 4x_5+1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 27063,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, with $a=6$ and $b\\sin\\frac{B+C}{2}=a\\sin B$. Find:\n1. The measure of angle $A$.\n2. Let $M$ be a point inside triangle $\\triangle ABC$. Extend $AM... | math | {
"style": "rule",
"ground_truth": "9\\sqrt{3}"
} | {
"split": "train",
"index": 27064,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest constant $N$, such that for any triangle with sides $a, b,$ and $c$, and perimeter $p = a + b + c$, the inequality holds:\n\\[\n\\frac{a^2 + b^2 + k}{c^2} > N\n\\]\nwhere $k$ is a constant. 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": 27065,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The real numbers \\(a, b, c\\) satisfy the following system of equations:\n\n$$\n\\left\\{\\begin{array}{l}\n\\frac{a b}{a+b}=4 \\\\\n\\frac{b c}{b+c}=5 \\\\\n\\frac{c a}{c+a}=7\n\\end{array}\\right.\n$$\n\nFind the value of the expression \\(\\frac{a b c}{a b + b c + c a}\\). ... | math | {
"style": "rule",
"ground_truth": "280/83"
} | {
"split": "train",
"index": 27066,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mary and Pat play the following number game. Mary picks an initial integer greater than $2017$ . She then multiplies this number by $2017$ and adds $2$ to the result. Pat will add $2019$ to this new number and it will again be Mary’s turn. Both players will continue to t... | math | {
"style": "rule",
"ground_truth": "2022"
} | {
"split": "train",
"index": 27067,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a set of 150 cards numbered from 1 to 150. Each card is randomly placed in a box. A card is selected at random from this box. What is the probability that the number on the card is a multiple of 4, 5, or 6? Express your answer as a common fraction. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{15}"
} | {
"split": "train",
"index": 27068,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose an integer $x$ , a natural number $n$ and a prime number $p$ satisfy the equation $7x^2-44x+12=p^n$ . Find the largest value of $p$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 27069,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be a triangle with $A B=13, B C=14$, and $C A=15$. Let $D$ be the foot of the altitude from $A$ to $B C$. The inscribed circles of triangles $A B D$ and $A C D$ are tangent to $A D$ at $P$ and $Q$, respectively, and are tangent to $B C$ at $X$ and $Y$, respectively.... | math | {
"style": "rule",
"ground_truth": "\\frac{25}{4}"
} | {
"split": "train",
"index": 27070,
"solution": "First, note that $A D=12, B D=5, C D=9$. By equal tangents, we get that $P D=D X$, so $P D X$ is isosceles. Because $D$ is a right angle, we get that $\\angle P X D=45^{\\circ}$. Similarly, $\\angle X Y Z=45^{\\circ}$, so $X Y Z$ is an isosceles right triangle with... |
deepscaler | [
{
"role": "user",
"content": "Positive integers $a$ and $b$ are such that the graphs of $y=ax+5$ and $y=3x+b$ intersect the $x$-axis at the same point. What is the sum of all possible $x$-coordinates of these points of intersection? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"split": "train",
"index": 27071,
"solution": "1. **Identify the points of intersection with the x-axis**: \n The equations of the lines are given as $y = ax + 5$ and $y = 3x + b$. The points where these lines intersect the x-axis occur when $y = 0$.\n\n2. **Set up equations for x-intercepts**:\n For the li... |
deepscaler | [
{
"role": "user",
"content": "Given that the side lengths of triangle \\( \\triangle ABC \\) are 6, \\( x \\), and \\( 2x \\), find the maximum value of its area \\( S \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 27072,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence consists of 2010 terms. Each term after the first is 1 larger than the previous term. The sum of the 2010 terms is 5307. What is the sum when every second term is added up, starting with the first term and ending with the second last term? Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "2151"
} | {
"split": "train",
"index": 27073,
"solution": "We label the terms $x_{1}, x_{2}, x_{3}, \\ldots, x_{2009}, x_{2010}$. Suppose that $S$ is the sum of the odd-numbered terms in the sequence; that is, $S=x_{1}+x_{3}+x_{5}+\\cdots+x_{2007}+x_{2009}$. We know that the sum of all of the terms is 5307; that is, $x_{1}... |
deepscaler | [
{
"role": "user",
"content": "Let $\\overline{MN}$ be a diameter of a circle with diameter 1. Let $A$ and $B$ be points on one of the semicircular arcs determined by $\\overline{MN}$ such that $A$ is the midpoint of the semicircle and $MB=\\frac{3}5$. Point $C$ lies on the other semicircular arc. Let $d$ be... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 27074,
"solution": "By Pythagoras in $\\triangle BMN,$ we get $BN=\\dfrac{4}{5}.$\n Since cross ratios are preserved upon projecting, note that $(M,Y;X,N)\\stackrel{C}{=}(M,B;A,N).$ By definition of a cross ratio, this becomes \\[\\dfrac{XM}{NY}:\\dfrac{NM}{NY}=\\dfrac{AM}{AB}:\\dfrac... |
deepscaler | [
{
"role": "user",
"content": "Given a point M$(x_0, y_0)$ moves on the circle $x^2+y^2=4$, and N$(4, 0)$, the point P$(x, y)$ is the midpoint of the line segment MN. \n(1) Find the trajectory equation of point P$(x, y)$. \n(2) Find the maximum and minimum distances from point P$(x, y)$ to the line $3x+4y-... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 27075,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jia and Yi are dividing 999 playing cards numbered 001, 002, 003, ..., 998, 999. All the cards whose numbers have all three digits not greater than 5 belong to Jia; cards whose numbers have one or more digits greater than 5 belong to Yi. \n(1) How many cards does Jia get? \n(... | math | {
"style": "rule",
"ground_truth": "59940"
} | {
"split": "train",
"index": 27076,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\left\\lfloor -\\frac{5}{3}\\right\\rfloor.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 27077,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of the two circles? Express your answer in fully expanded form in terms of $\\pi$. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "\\frac{9\\pi - 18}{2}"
} | {
"split": "train",
"index": 27078,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When n takes all values from 1, 2, 3, ..., to 2015, how many of the numbers in the form of \\(3^{n} + n^{3}\\) are divisible by 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "335"
} | {
"split": "train",
"index": 27079,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An isosceles trapezoid \\(ABCD\\) is circumscribed around a circle. The lateral sides \\(AB\\) and \\(CD\\) are tangent to the circle at points \\(M\\) and \\(N\\), respectively, and \\(K\\) is the midpoint of \\(AD\\). In what ratio does the line \\(BK\\) divide the segment \\... | math | {
"style": "rule",
"ground_truth": "1:3"
} | {
"split": "train",
"index": 27080,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cylinder has a radius of 5 cm and a height of 12 cm. What is the longest segment, in centimeters, that would fit inside the cylinder? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{61}"
} | {
"split": "train",
"index": 27081,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\cos (\\pi+\\alpha)=- \\frac { \\sqrt {10}}{5}$, and $\\alpha\\in(- \\frac {\\pi}{2},0)$, determine the value of $\\tan \\alpha$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{\\sqrt{6}}{2}"
} | {
"split": "train",
"index": 27082,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If six people decide to come to a basketball game, but three of them are only 2/5 sure that they will stay for the entire time (the other three are sure they'll stay the whole time), what is the probability that at the end, at least 5 people stayed the entire time? Let's think ... | math | {
"style": "rule",
"ground_truth": "\\frac{44}{125}"
} | {
"split": "train",
"index": 27083,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using only the paths and the directions shown, how many different routes are there from $\\text{M}$ to $\\text{N}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 27084,
"solution": "To solve this problem, we will use a systematic approach by counting the number of ways to reach $\\text{N}$ from each point, starting from the points closest to $\\text{N}$ and working backwards.\n\n1. **From $\\text{C}$ to $\\text{N}$**: There is only one direct ... |
deepscaler | [
{
"role": "user",
"content": "Determine the largest positive integer $n$ such that there exist positive integers $x, y, z$ so that \\[\nn^2 = x^2+y^2+z^2+2xy+2yz+2zx+3x+3y+3z-6\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 27085,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Riley has 64 cubes with dimensions \\(1 \\times 1 \\times 1\\). Each cube has its six faces labeled with a 2 on two opposite faces and a 1 on each of its other four faces. The 64 cubes are arranged to build a \\(4 \\times 4 \\times 4\\) cube. Riley determines the total of the n... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 27086,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest positive integer \\(n\\) for which there exist \\(n\\) finite sets \\(X_{1}, X_{2}, \\ldots, X_{n}\\) with the property that for every \\(1 \\leq a<b<c \\leq n\\), the equation \\(\\left|X_{a} \\cup X_{b} \\cup X_{c}\\right|=\\lceil\\sqrt{a b c}\\rceil\\) holds... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 27087,
"solution": "First, we construct an example for \\(N=4\\). Let \\(X_{1}, X_{2}, X_{3}, X_{4}\\) be pairwise disjoint sets such that \\(X_{1}=\\varnothing,\\left|X_{2}\\right|=1,\\left|X_{3}\\right|=2\\), and \\(\\left|X_{4}\\right|=2\\). It is straightforward to verify the cond... |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be an isosceles trapezoid with $\\overline{AD}||\\overline{BC}$ whose angle at the longer base $\\overline{AD}$ is $\\dfrac{\\pi}{3}$. The diagonals have length $10\\sqrt {21}$, and point $E$ is at distances $10\\sqrt {7}$ and $30\\sqrt {7}$ from vertices $A$ and $D$... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 27088,
"solution": "Extend $\\overline {AB}$ through $B$, to meet $\\overline {DC}$ (extended through $C$) at $G$. $ADG$ is an equilateral triangle because of the angle conditions on the base.\n If $\\overline {GC} = x$ then $\\overline {CD} = 40\\sqrt{7}-x$, because $\\overline{AD}$ ... |
deepscaler | [
{
"role": "user",
"content": "Let $N = 34 \\cdot 34 \\cdot 63 \\cdot 270$. What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1 : 14"
} | {
"split": "train",
"index": 27089,
"solution": "1. **Prime Factorization of \\(N\\):**\n Given \\(N = 34 \\cdot 34 \\cdot 63 \\cdot 270\\), we start by factorizing each number:\n - \\(34 = 2 \\cdot 17\\)\n - \\(63 = 3^2 \\cdot 7\\)\n - \\(270 = 2 \\cdot 3^3 \\cdot 5\\)\n\n Therefore, \\(N = (2 \\cdot 1... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $BC= \\sqrt {5}$, $AC=3$, $\\sin C=2\\sin A$. Find:\n1. The value of $AB$.\n2. The value of $\\sin(A- \\frac {\\pi}{4})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac { \\sqrt {10}}{10}"
} | {
"split": "train",
"index": 27090,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}=(1,3)$ and $\\overrightarrow{b}=(-2,4)$, calculate the projection of $\\overrightarrow{a}$ in the direction of $\\overrightarrow{b}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 27091,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You are given a set of cards labeled from 1 to 100. You wish to make piles of three cards such that in any pile, the number on one of the cards is the product of the numbers on the other two cards. However, no card can be in more than one pile. What is the maximum number of pil... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 27092,
"solution": "Certainly, the two factors in any pile cannot both be at least 10, since then the product would be at least $10 \\times 11>100$. Also, the number 1 can not appear in any pile, since then the other two cards in the pile would have to be the same. So each pile must u... |
deepscaler | [
{
"role": "user",
"content": "Given that \\(a - b = 2 + \\sqrt{3}\\) and \\(b - c = 2 - \\sqrt{3}\\), find the value of \\(a^2 + b^2 + c^2 - ab - bc - ca\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 27093,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "All the complex roots of $(z + 2)^6 = 64z^6$, when plotted in the complex plane, lie on a circle. Find the radius of this circle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{\\sqrt{3}}"
} | {
"split": "train",
"index": 27094,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34$. What is the distance between two adjacent parallel lines? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 27095,
"solution": "1. **Identify the setup and apply Stewart's Theorem**: We are given three equally spaced parallel lines intersecting a circle, creating three chords of lengths 38, 38, and 34. Let's denote the center of the circle as $O$, and the points where the chords intersect t... |
deepscaler | [
{
"role": "user",
"content": "Find the number of integers $n$ with $1\\le n\\le 100$ for which $n-\\phi(n)$ is prime. Here $\\phi(n)$ denotes the number of positive integers less than $n$ which are relatively prime to $n$ .\n\n*Proposed by Mehtaab Sawhney* Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 27096,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point O is the circumcenter of acute triangle ABC, with AB=6, AC=2. Determine the value of $\\overrightarrow {AO}\\cdot ( \\overrightarrow {AB}+ \\overrightarrow {AC})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 27097,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If there exists a permutation $a_{1}, a_{2}, \\cdots, a_{n}$ of $1, 2, \\cdots, n$ such that $k + a_{k}$ $(k = 1, 2, \\cdots, n)$ are all perfect squares, then $n$ is called a \"good number\". Among the set $\\{11, 13, 15, 17, 19\\}$, which numbers are \"good numbers\" and whic... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 27098,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the coordinate plane, the points \\(A(0, 2)\\), \\(B(1, 7)\\), \\(C(10, 7)\\), and \\(D(7, 1)\\) are given. Find the area of the pentagon \\(A B C D E\\), where \\(E\\) is the intersection point of the lines \\(A C\\) and \\(B D\\). Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 27099,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider that for integers from 1 to 1500, $x_1+2=x_2+4=x_3+6=\\cdots=x_{1500}+3000=\\sum_{n=1}^{1500}x_n + 3001$. Find the value of $\\left\\lfloor|S|\\right\\rfloor$, where $S=\\sum_{n=1}^{1500}x_n$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "1500"
} | {
"split": "train",
"index": 27100,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express $0.3\\overline{206}$ as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5057}{9990}"
} | {
"split": "train",
"index": 27101,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ms.Osborne asks each student in her class to draw a rectangle with integer side lengths and a perimeter of $50$ units. All of her students calculate the area of the rectangle they draw. What is the difference between the largest and smallest possible areas of the rectangles? Le... | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 27102,
"solution": "1. **Understanding the Problem:**\n - We are given a rectangle with a perimeter of 50 units.\n - The side lengths of the rectangle are integers.\n - We need to find the difference between the largest and smallest possible areas of such rectangles.\n\n2. **Exp... |
deepscaler | [
{
"role": "user",
"content": "Given that $a\\in (\\frac{\\pi }{2},\\pi )$ and $\\sin \\alpha =\\frac{1}{3}$,\n\n(1) Find the value of $\\sin 2\\alpha$;\n\n(2) If $\\sin (\\alpha +\\beta )=-\\frac{3}{5}$, $\\beta \\in (0,\\frac{\\pi }{2})$, find the value of $\\sin \\beta$. Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "\\frac{6\\sqrt{2}+4}{15}"
} | {
"split": "train",
"index": 27103,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The arithmetic mean of these six expressions is 30. What is the value of $y$? $$y + 10 \\hspace{.5cm} 20 \\hspace{.5cm} 3y \\hspace{.5cm} 18 \\hspace{.5cm} 3y + 6 \\hspace{.5cm} 12$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{114}{7}"
} | {
"split": "train",
"index": 27104,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c,$ $d$ be real numbers, none of which are equal to $-1,$ and let $\\omega$ be a complex number such that $\\omega^3 = 1$ and $\\omega \\neq 1.$ If\n\\[\\frac{1}{a + \\omega} + \\frac{1}{b + \\omega} + \\frac{1}{c + \\omega} + \\frac{1}{d + \\omega} = \\frac{2}{... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 27105,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the graph of the function y = a^(x-2) (where a > 0 and a ≠ 1) always passes through a fixed point A, which lies on the graph of the linear function y = mx + 4n (where m, n > 0), find the minimum value of $\\frac{1}{m} + \\frac{2}{n}$. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 27106,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Equilateral triangle $ABC$ has circumcircle $\\Omega$. Points $D$ and $E$ are chosen on minor arcs $AB$ and $AC$ of $\\Omega$ respectively such that $BC=DE$. Given that triangle $ABE$ has area 3 and triangle $ACD$ has area 4, find the area of triangle $ABC$. Let's think step by... | math | {
"style": "rule",
"ground_truth": "\\frac{37}{7}"
} | {
"split": "train",
"index": 27107,
"solution": "A rotation by $120^{\\circ}$ about the center of the circle will take $ABE$ to $BCD$, so $BCD$ has area 3. Let $AD=x, BD=y$, and observe that $\\angle ADC=\\angle CDB=60^{\\circ}$. By Ptolemy's Theorem, $CD=x+y$. We have $$4=[ACD]=\\frac{1}{2} AD \\cdot CD \\cdot \... |
deepscaler | [
{
"role": "user",
"content": "A natural number $n$ is called a \"good number\" if the column addition of $n$, $n+1$, and $n+2$ does not produce any carry-over. For example, 32 is a \"good number\" because $32+33+34$ does not result in a carry-over; however, 23 is not a \"good number\" because $23+24+25$ doe... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 27108,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square has sides of length 10, and a circle centered at one of its vertices has radius 10. What is the area of the union of the regions enclosed by the square and the circle? Express your answer in terms of $\\pi$. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "100+75\\pi"
} | {
"split": "train",
"index": 27109,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest possible value of $x$ such that $2x^2+24x-60=x(x+13)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-15"
} | {
"split": "train",
"index": 27110,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. It is known that $c^{2}=a^{2}+b^{2}-4bc\\cos C$, and $A-C= \\frac {\\pi}{2}$. \n(Ⅰ) Find the value of $\\cos C$; \n(Ⅱ) Find the value of $\\cos \\left(B+ \\frac {\\pi}{3}\... | math | {
"style": "rule",
"ground_truth": "\\frac {4-3 \\sqrt {3}}{10}"
} | {
"split": "train",
"index": 27111,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_{n}\\}$ with the sum of its first $n$ terms denoted as $A_{n}$, and $a_{1}+a_{2}=3$, $A_{5}=15$, a sequence $\\{b_{n}\\}$ satisfies ${b}_{n}={a}_{n}•[1+{(-1)}^{n}n]$ for $n\\in{N}^{*}$.<br/>$(1)$ Find the general formula for the sequence $\\{a... | math | {
"style": "rule",
"ground_truth": "2520"
} | {
"split": "train",
"index": 27112,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a parallelepiped \\(A B C D A_{1} B_{1} C_{1} D_{1}\\). Point \\(X\\) is chosen on edge \\(A_{1} D_{1}\\) and point \\(Y\\) is chosen on edge \\(B C\\). It is known that \\(A_{1} X = 5\\), \\(B Y = 3\\), and \\(B_{1} C_{1} = 14\\). The plane \\(C_{1} X Y\\) intersects the... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 27113,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\tan \\alpha = 2$, find the value of $\\frac{2 \\sin \\alpha - \\cos \\alpha}{\\sin \\alpha + 2 \\cos \\alpha}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 27114,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x<y$ be positive real numbers such that $\\sqrt{x}+\\sqrt{y}=4$ and $\\sqrt{x+2}+\\sqrt{y+2}=5$. Compute $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{49}{36}"
} | {
"split": "train",
"index": 27115,
"solution": "Adding and subtracting both equations gives $$\\begin{aligned} & \\sqrt{x+2}+\\sqrt{x}+\\sqrt{y+2}+\\sqrt{y}=9 \\\\ & \\sqrt{x+2}-\\sqrt{x}+\\sqrt{y+2}-\\sqrt{y}=1 \\end{aligned}$$ Substitute $a=\\sqrt{x}+\\sqrt{x+2}$ and $b=\\sqrt{y}+\\sqrt{y+2}$. Then since $(\\s... |
deepscaler | [
{
"role": "user",
"content": "Each vertex of this parallelogram has integer coordinates. The perimeter of the parallelogram is \\( p \\) units, and the area is \\( a \\) square units. If the parallelogram is defined by vertices \\((2, 3)\\), \\((7, 3)\\), \\((x, y)\\), and \\((x-5, y)\\), where \\(x\\) and ... | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 27116,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the function $f(x) = 2\\cos^2x + 2\\sqrt{3}\\sin x\\cos x + m$.\n\n(1) Find the smallest positive period of the function $f(x)$ and its intervals of monotonic decrease; \n(2) If $x \\in \\left[0, \\frac{\\pi}{2}\\right]$, does there exist a real number $m$ such that the ra... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 27117,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integer divisors of $201^9$ are perfect squares or perfect cubes (or both)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 27118,
"solution": "To find the number of positive integer divisors of $201^9$ that are perfect squares or perfect cubes (or both), we start by factoring $201$:\n\\[ 201 = 3 \\times 67. \\]\nThus, \n\\[ 201^9 = (3 \\times 67)^9 = 3^9 \\times 67^9. \\]\n\nA divisor of $201^9$ can be wr... |
deepscaler | [
{
"role": "user",
"content": "A line $l$ with an inclination angle of $45^\\circ$ passes through the focus $F$ of the parabola $y^2=8x$ and intersects the parabola at points $A$ and $B$. Find the value of $|\\overrightarrow{FA}| \\cdot |\\overrightarrow{FB}|$. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 27119,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=A\\sin^2(\\omega x+\\frac{\\pi}{8})$ ($A>0, \\omega>0$), the graph of which is symmetric with respect to the point $({\\frac{\\pi}{2},1})$, and its minimum positive period is $T$, where $\\frac{\\pi}{2}<T<\\frac{3\\pi}{2}$. Find the value of $\\omega$. ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"split": "train",
"index": 27120,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid $KLMN$ with bases $KN$ and $LN$, the angle $\\angle LMN$ is $60^{\\circ}$. A circle is circumscribed around triangle $KLN$ and is tangent to the lines $LM$ and $MN$. Find the radius of the circle, given that the perimeter of triangle $KLN$ is 12. Let's think step b... | math | {
"style": "rule",
"ground_truth": "\\frac{4 \\sqrt{3}}{3}"
} | {
"split": "train",
"index": 27121,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCDEF$ be a regular hexagon. Let $G$, $H$, $I$, $J$, $K$, and $L$ be the midpoints of sides $AB$, $BC$, $CD$, $DE$, $EF$, and $AF$, respectively. The segments $\\overline{AH}$, $\\overline{BI}$, $\\overline{CJ}$, $\\overline{DK}$, $\\overline{EL}$, and $\\overline{FG}$ bo... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 27122,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle XYZ$, we have $\\angle X = 90^\\circ$ and $\\tan Z = 3$. What is $\\cos Z$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{10}"
} | {
"split": "train",
"index": 27123,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right triangle has a side length of 21 inches and a hypotenuse of 29 inches. A second triangle is similar to the first and has a hypotenuse of 87 inches. What is the length of the shortest side of the second triangle? Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "60\\text{ inches}"
} | {
"split": "train",
"index": 27124,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system $(xOy)$, the curve $C\\_1$: $ \\begin{cases} x=a\\cos φ \\ y=b\\sin φ\\end{cases}(φ)$ is a parameter, where $(a > b > 0)$, and in the polar coordinate system with $O$ as the pole and the positive semi-axis of $x$ as the polar axis, the curve... | math | {
"style": "rule",
"ground_truth": "\\dfrac {3 \\sqrt {30}}{20}"
} | {
"split": "train",
"index": 27125,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the 343 smallest positive integers written in base 7 use 4 or 5 (or both) as a digit? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "218"
} | {
"split": "train",
"index": 27126,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \\cdots$ defined by \\[ a_0 = M + \\frac{1}{2} \\qquad \\textrm{and} \\qquad a_{k+1} = a_k\\lfloor a_k \\rfloor \\quad \\textrm{for} \\, k = 0, 1, 2, \\cdots \\] contains at least one integer ter... | math | {
"style": "rule",
"ground_truth": "M > 1"
} | {
"split": "train",
"index": 27127,
"solution": "\nConsider the sequence \\( a_0, a_1, a_2, \\ldots \\) defined by:\n\n\\[\na_0 = M + \\frac{1}{2}\n\\]\n\nand\n\n\\[\na_{k+1} = a_k \\lfloor a_k \\rfloor \\quad \\text{for} \\quad k = 0, 1, 2, \\ldots\n\\]\n\nWe are tasked with finding all positive integers \\( M \... |
deepscaler | [
{
"role": "user",
"content": "Let $A B C D$ be a rectangle with $A B=6$ and $B C=4$. Let $E$ be the point on $B C$ with $B E=3$, and let $F$ be the point on segment $A E$ such that $F$ lies halfway between the segments $A B$ and $C D$. If $G$ is the point of intersection of $D F$ and $B C$, find $B G$. Let'... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 27128,
"solution": "Note that since $F$ is a point halfway between $A B$ and $A C$, the diagram must be symmetric about the line through $F$ parallel to $A B$. Hence, G must be the reflection of $E$ across the midpoint of $B C$. Therefore, $B G=E C=1$."
} |
deepscaler | [
{
"role": "user",
"content": "How many six-digit multiples of 27 have only 3, 6, or 9 as their digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 27129,
"solution": "Divide by 3. We now want to count the number of six-digit multiples of 9 that only have 1, 2, or 3 as their digits. Due to the divisibility rule for 9, we only need to consider when the digit sum is a multiple of 9. Note that $3 \\cdot 6=18$ is the maximum digit su... |
deepscaler | [
{
"role": "user",
"content": "Find $k$ where $2^k$ is the largest power of $2$ that divides the product \\[2008\\cdot 2009\\cdot 2010\\cdots 4014.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2007"
} | {
"split": "train",
"index": 27130,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the first and third of three consecutive integers is 118. What is the value of the second integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 27131,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The three sides of a triangle are 25, 39, and 40. Find the diameter of its circumscribed circle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{125}{3}"
} | {
"split": "train",
"index": 27132,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the possible value of $x + y$ given that $x^3 + 6x^2 + 16x = -15$ and $y^3 + 6y^2 + 16y = -17$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 27133,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC, D$ and $E$ are the midpoints of $BC$ and $CA$, respectively. $AD$ and $BE$ intersect at $G$. Given that $GEC$D is cyclic, $AB=41$, and $AC=31$, compute $BC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 27134,
"solution": "By Power of a Point, $$\\frac{2}{3}AD^{2}=AD \\cdot AG=AE \\cdot AC=\\frac{1}{2} \\cdot 31^{2}$$ so $AD^{2}=\\frac{3}{4} \\cdot 31^{2}$. The median length formula yields $$AD^{2}=\\frac{1}{4}\\left(2AB^{2}+2AC^{2}-BC^{2}\\right)$$ whence $$BC=\\sqrt{2AB^{2}+2AC^{2}... |
deepscaler | [
{
"role": "user",
"content": "Given that \\(\\theta\\) is an angle in the third quadrant, and \\(\\sin^{4}\\theta + \\cos^{4}\\theta = \\frac{5}{9}\\), determine the value of \\(\\sin 2\\theta\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2\\sqrt{2}}{3}"
} | {
"split": "train",
"index": 27135,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Uncle Wang has some chickens, rabbits, and geese. Geese stand on two legs during the day and on one leg at night; chickens tuck their heads under their wings when sleeping. Careful Yue Yue discovered that the difference between the number of legs and the number of heads is alwa... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 27136,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There exist vectors $\\mathbf{a}$ and $\\mathbf{b}$ such that\n\\[\\mathbf{a} + \\mathbf{b} = \\begin{pmatrix} 6 \\\\ -3 \\\\ -6 \\end{pmatrix},\\]where $\\mathbf{a}$ is parallel to $\\begin{pmatrix} 1 \\\\ 1 \\\\ 1 \\end{pmatrix},$ and $\\mathbf{b}$ is orthogonal to $\\begin{p... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 7 \\\\ -2 \\\\ -5 \\end{pmatrix}"
} | {
"split": "train",
"index": 27137,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $A B C$ obeys $A B=2 A C$ and $\\angle B A C=120^{\\circ}$. Points $P$ and $Q$ lie on segment $B C$ such that $$\\begin{aligned} A B^{2}+B C \\cdot C P & =B C^{2} \\\\ 3 A C^{2}+2 B C \\cdot C Q & =B C^{2} \\end{aligned}$$ Find $\\angle P A Q$ in degrees. Let's think s... | math | {
"style": "rule",
"ground_truth": "40^{\\circ}"
} | {
"split": "train",
"index": 27138,
"solution": "We have $A B^{2}=B C(B C-C P)=B C \\cdot B P$, so triangle $A B C$ is similar to triangle $P B A$. Also, $A B^{2}=B C(B C-2 C Q)+A C^{2}=(B C-C Q)^{2}-C Q^{2}+A C^{2}$, which rewrites as $A B^{2}+C Q^{2}=$ $B Q^{2}+A C^{2}$. We deduce that $Q$ is the foot of the al... |
deepscaler | [
{
"role": "user",
"content": "A certain high school is planning to hold a coming-of-age ceremony for senior students on the \"May Fourth\" Youth Day to motivate the seniors who are preparing for the college entrance examination. The Student Affairs Office has prepared five inspirational songs, a video speec... | math | {
"style": "rule",
"ground_truth": "20160"
} | {
"split": "train",
"index": 27139,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let's define a calendar week as even or odd according to whether the sum of the day numbers within the month in that week is even or odd. Out of the 52 consecutive weeks starting from the first Monday of January, how many can be even? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 27140,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The opposite number of $2- \\sqrt{3}$ is ______, and its absolute value is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2- \\sqrt{3}"
} | {
"split": "train",
"index": 27141,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area enclosed by the curves $y=e^{x}$, $y=e^{-x}$, and the line $x=1$ is $e^{1}-e^{-1}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "e+e^{-1}-2"
} | {
"split": "train",
"index": 27142,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider all 1000-element subsets of the set $\\{1, 2, 3, ... , 2015\\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.\nHint\n Use th... | math | {
"style": "rule",
"ground_truth": "431"
} | {
"split": "train",
"index": 27143,
"solution": "Let $p$ be the size of the large set and $q$ be the size of the subset (i.e. in this problem, $p = 2015$ and $q = 1000$). We can easily find the answers for smaller values of $p$ and $q$:\n For $p = 2$ and $q = 2$, the answer is $1$.\n For $p = 3$ and $q = 2$, the ... |
deepscaler | [
{
"role": "user",
"content": "If $y<0$, find the range of all possible values of $y$ such that $\\lceil{y}\\rceil\\cdot\\lfloor{y}\\rfloor=110$. Express your answer using interval notation. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-11, -10)"
} | {
"split": "train",
"index": 27144,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On Marika's $8^{\\text{th}}$ birthday, in 2004, her father said, ''My age is now four times your age.'' In what year will Marika's father be able to say, ''My age is now three times your age,'' on Marika's birthday? Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "2008"
} | {
"split": "train",
"index": 27145,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At the beginning of the school year, Lisa's goal was to earn an $A$ on at least $80\\%$ of her $50$ quizzes for the year. She earned an $A$ on $22$ of the first $30$ quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lowe... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 27146,
"solution": "1. **Determine the total number of quizzes Lisa needs to score an A on to meet her goal:** \n Lisa's goal is to earn an A on at least 80% of her 50 quizzes. Therefore, the number of quizzes she needs to score an A on is:\n \\[\n 0.8 \\times 50 = 40\n \\]\n... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $AB = 86$, and $AC=97$. A circle with center $A$ and radius $AB$ intersects $\\overline{BC}$ at points $B$ and $X$. Moreover $\\overline{BX}$ and $\\overline{CX}$ have integer lengths. What is $BC$? Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"split": "train",
"index": 27147,
"solution": "1. **Assign Variables:**\n Let $x$ represent $CX$, and let $y$ represent $BX$. Since the circle with center $A$ and radius $AB$ intersects $\\overline{BC}$ at points $B$ and $X$, we have $AB = AX = 86$.\n\n2. **Apply Stewart's Theorem:**\n Stewart's Theorem sta... |
deepscaler | [
{
"role": "user",
"content": "Let $\\{x\\}$ denote the smallest integer not less than the real number \\(x\\). Find the value of the expression $\\left\\{\\log _{2} 1\\right\\}+\\left\\{\\log _{2} 2\\right\\}+\\left\\{\\log _{2} 3\\right\\}+\\cdots+\\left\\{\\log _{2} 1991\\right\\}$. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "19854"
} | {
"split": "train",
"index": 27148,
"solution": ""
} |
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