data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $(C)$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a > b > 0)$, the minor axis is $2 \\sqrt{3}$, and the eccentricity $e= \\frac{1}{2}$,\n(1) Find the standard equation of ellipse $(C)$;\n(2) If $F_{1}$ and $F_{2}$ are the left and right foci of ellipse $(C)... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 34449,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many positive integers $n$ is $\\frac{n}{30-n}$ also a positive integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 34450,
"solution": "\n#### Solution 1: Direct Substitution and Checking\n1. **Condition for Positive Fraction**: We need $\\frac{n}{30-n}$ to be a positive integer. For this to happen, both $n$ and $30-n$ must be positive. This implies $0 < n < 30$.\n2. **Condition for Integer Result*... |
deepscaler | [
{
"role": "user",
"content": "Adam and Simon start on bicycle trips from the same point at the same time. Adam travels north at 10 mph and Simon travels west at 12 mph. How many hours will it take for them to be 130 miles apart? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{65}{\\sqrt{61}}"
} | {
"split": "train",
"index": 34451,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers $b$ have the property that $\\log_{b} 729$ is a positive integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 34452,
"solution": "1. **Express the equation in exponential form**: Given $\\log_b 729 = n$, we can rewrite this as $b^n = 729$.\n\n2. **Factorize 729**: We know that $729 = 3^6$. Therefore, we can express $b^n = 3^6$.\n\n3. **Determine possible values of $b$**: Since $b^n = 3^6$, $b... |
deepscaler | [
{
"role": "user",
"content": "The shape of a bridge arch is a parabola. It is known that the width of the parabolic arch is 8 meters, and the area of the parabolic arch is 160 square meters. Then, the height of the parabolic arch is Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 34453,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( m = n^{4} + x \\), where \\( n \\) is a natural number and \\( x \\) is a two-digit positive integer, what value of \\( x \\) will make \\( m \\) a composite number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 34454,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each vertex of convex pentagon $ABCDE$ is to be assigned a color. There are $6$ colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "3120"
} | {
"split": "train",
"index": 34455,
"solution": "1. **Label the vertices and define the problem**: Let's label the vertices of the pentagon as $A, B, C, D, E$. We need to color each vertex such that no two vertices connected by a diagonal have the same color. There are 6 colors available.\n\n2. **Set up the probl... |
deepscaler | [
{
"role": "user",
"content": "Given that $BDEF$ is a square and $AB = BC = 1$, find the number of square units in the area of the regular octagon.\n\n[asy]\nreal x = sqrt(2);\npair A,B,C,D,E,F,G,H;\nF=(0,0); E=(2,0); D=(2+x,x); C=(2+x,2+x);\nB=(2,2+2x); A=(0,2+2x); H=(-x,2+x); G=(-x,x);\ndraw(A--B--C--D--E-... | math | {
"style": "rule",
"ground_truth": "4+4\\sqrt{2}"
} | {
"split": "train",
"index": 34456,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)$ and $g(x)$ be two monic cubic polynomials, and let $r$ be a real number. Two of the roots of $f(x)$ are $r + 2$ and $r + 4$. Two of the roots of $g(x)$ are $r + 3$ and $r + 5$, and\n\\[ f(x) - g(x) = 2r + 1 \\]\nfor all real numbers $x$. Find $r$. Let's think step by... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 34457,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a rectangular configuration $ABCD$, there are three squares with non-overlapping interiors. One of them, which is shaded, has an area of 4 square inches. The side length of the larger square is twice the side length of the shaded square. What is the area of rectangle $ABCD$,... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 34458,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circles $\\omega_1$ and $\\omega_2$ with radii $961$ and $625$, respectively, intersect at distinct points $A$ and $B$. A third circle $\\omega$ is externally tangent to both $\\omega_1$ and $\\omega_2$. Suppose line $AB$ intersects $\\omega$ at two points $P$ and $Q$ such that... | math | {
"style": "rule",
"ground_truth": "672"
} | {
"split": "train",
"index": 34459,
"solution": "Suppose we label the points as shown here. By radical axis, the tangents to $\\omega$ at $D$ and $E$ intersect on $AB$. Thus $PDQE$ is harmonic, so the tangents to $\\omega$ at $P$ and $Q$ intersect at $X \\in DE$. Moreover, $OX \\parallel O_1O_2$ because both $OX$... |
deepscaler | [
{
"role": "user",
"content": "Inside a non-isosceles acute triangle \\(ABC\\) with \\(\\angle ABC = 60^\\circ\\), point \\(T\\) is marked such that \\(\\angle ATB = \\angle BTC = \\angle ATC = 120^\\circ\\). The medians of the triangle intersect at point \\(M\\). The line \\(TM\\) intersects the circumcircl... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 34460,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\cos(x+y) \\cdot \\sin x - \\sin(x+y) \\cdot \\cos x = \\frac{12}{13}$, and $y$ is an angle in the fourth quadrant, express $\\tan \\frac{y}{2}$ in terms of a rational number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{3}"
} | {
"split": "train",
"index": 34461,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xoy$, point $P(0, \\sqrt{3})$ is given. The parametric equation of curve $C$ is $\\begin{cases} x = \\sqrt{2} \\cos \\varphi \\\\ y = 2 \\sin \\varphi \\end{cases}$ (where $\\varphi$ is the parameter). A polar coordinate system is established... | math | {
"style": "rule",
"ground_truth": "\\sqrt{14}"
} | {
"split": "train",
"index": 34462,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six students with distinct heights take a group photo, the photographer arranges them into two rows with three people each. What is the probability that every student in the back row is taller than the students in the front row? Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{20}"
} | {
"split": "train",
"index": 34463,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many four-digit numbers starting with the digit $2$ and having exactly three identical digits are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 34464,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Positive integers are put into the following table.\n\n\\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}\n\\hline\n1 & 3 & 6 & 10 & 15 & 21 & 28 & 36 & & \\\\ \\hline\n2 & 5 & 9 & 14 & 20 & 27 & 35 & 44 & & \\\\ \\hline\n4 & 8 & 13 & 19 & 26 & 34 & 43 & 53 & & \\\\ \\hli... | math | {
"style": "rule",
"ground_truth": "(62, 2)"
} | {
"split": "train",
"index": 34465,
"solution": "\nWe will analyze the pattern in the table to determine where the number \\(2015\\) is located. The arrangement of numbers in the table has a specific formulation in terms of line and column indices. We will first observe the pattern, derive the general formula for... |
deepscaler | [
{
"role": "user",
"content": "In a shooting match, eight clay targets are arranged in two hanging columns of three targets each and one column of two targets. A marksman is to break all the targets according to the following rules:\n1) The marksman first chooses a column from which a target is to be broken.... | math | {
"style": "rule",
"ground_truth": "560"
} | {
"split": "train",
"index": 34466,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a three-digit number \\(\\overline{a b c}\\) that satisfies \\(\\overline{a b c} = 37(a + b + c)\\), how many such three-digit numbers are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 34467,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point D is from AC of triangle ABC so that 2AD=DC. Let DE be perpendicular to BC and AE\nintersects BD at F. It is known that triangle BEF is equilateral. Find <ADB? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 34468,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a diagram, $\\triangle ABC$ and $\\triangle BDC$ are right-angled, with $\\angle ABC = \\angle BDC = 45^\\circ$, and $AB = 16$. Determine the length of $BC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8\\sqrt{2}"
} | {
"split": "train",
"index": 34469,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You have four tiles marked X and three tiles marked O. The seven tiles are randomly arranged in a row. What is the probability that no two X tiles are adjacent to each other? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{35}"
} | {
"split": "train",
"index": 34470,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the reciprocals of all the (positive) divisors of 144. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "403/144"
} | {
"split": "train",
"index": 34471,
"solution": "As $d$ ranges over the divisors of 144 , so does $144 / d$, so the sum of $1 / d$ is $1 / 144$ times the sum of the divisors of 144 . Using the formula for the sum of the divisors of a number (or just counting them out by hand), we get that this sum is 403, so the ... |
deepscaler | [
{
"role": "user",
"content": "(1) Calculate: $$\\lg 4 + 2\\lg 5 + (0.25)^{-\\frac{1}{2}} - 8^{\\frac{2}{3}}$$; \n(2) Given that $f(x)$ is an odd function on $\\mathbb{R}$ and $f(x+2) = -f(x)$, when $x \\in (0, 2)$, $f(x) = 2x^2$, find $f(2015)$. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 34472,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The volume of a sphere is $72\\pi$ cubic inches. A cylinder has the same height as the diameter of the sphere. The radius of the cylinder is equal to the radius of the sphere. Calculate the total surface area of the sphere plus the total surface area (including the top and bott... | math | {
"style": "rule",
"ground_truth": "90\\pi \\sqrt[3]{4}"
} | {
"split": "train",
"index": 34473,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $z$ be a complex number such that $|z| = 13.$ Find $z \\times \\overline{z}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "169"
} | {
"split": "train",
"index": 34474,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular dodecagon $P_{1} P_{2} \\cdots P_{12}$ is inscribed in a unit circle with center $O$. Let $X$ be the intersection of $P_{1} P_{5}$ and $O P_{2}$, and let $Y$ be the intersection of $P_{1} P_{5}$ and $O P_{4}$. Let $A$ be the area of the region bounded by $X Y, X P_{2}... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 34475,
"solution": "The area of sector $O P_{2} P_{4}$ is one sixth the area of the circle because its angle is $60^{\\circ}$. The desired area is just that of the sector subtracted by the area of equilateral triangle $O X Y$. Note that the altitude of this triangle is the distance fr... |
deepscaler | [
{
"role": "user",
"content": "Calculate the sum of the series $1-2-3+4+5-6-7+8+9-10-11+\\cdots+1998+1999-2000-2001$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2001"
} | {
"split": "train",
"index": 34476,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Smith family has four girls aged 5, 5, 5, and 12, and two boys aged 13 and 16. What is the mean (average) of the ages of the children? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9.33"
} | {
"split": "train",
"index": 34477,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$, the sum of the first $n$ terms is denoted as $S_n$. It is known that $a_1=9$, $a_2$ is an integer, and $S_n \\leqslant S_5$. The sum of the first $9$ terms of the sequence $\\left\\{ \\frac{1}{a_na_{n+1}} \\right\\}$ is ______. Let's th... | math | {
"style": "rule",
"ground_truth": "- \\frac{1}{9}"
} | {
"split": "train",
"index": 34478,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron with each edge measuring 1 meter. A bug, starting at vertex $A$, follows a rule: at each vertex, it randomly chooses one of the three edges with equal probability and crawls to the opposite vertex. Define $q = ... | math | {
"style": "rule",
"ground_truth": "547"
} | {
"split": "train",
"index": 34479,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A frustum of a right circular cone is formed by cutting a small cone off of the top of a larger cone. If a particular frustum has a lower base radius of 6 inches, an upper base radius of 3 inches, and a height of 4 inches, what is its lateral surface area? (The lateral surface... | math | {
"style": "rule",
"ground_truth": "45\\pi"
} | {
"split": "train",
"index": 34480,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "22. In the polar coordinate system, the polar equation of circle $C$ is $\\rho =4\\cos \\theta$. Taking the pole as the origin and the direction of the polar axis as the positive direction of the $x$-axis, a Cartesian coordinate system is established using the same unit length.... | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{46}}{2}"
} | {
"split": "train",
"index": 34481,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point is randomly dropped on the interval $[5 ; 7]$ and let $k$ be the resulting value. Find the probability that the roots of the equation $\\left(k^{2}-3 k-4\\right) x^{2}+(3 k-7) x+2=0$ satisfy the condition $x_{1} \\leq 2 x_{2}$. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"split": "train",
"index": 34482,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a trapezoid such that $|AC|=8$ , $|BD|=6$ , and $AD \\parallel BC$ . Let $P$ and $S$ be the midpoints of $[AD]$ and $[BC]$ , respectively. If $|PS|=5$ , find the area of the trapezoid $ABCD$ . Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 34483,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You have $7$ red shirts, $8$ green shirts, $10$ pairs of pants, $10$ blue hats, $10$ red hats, and $5$ scarves (each distinct). How many outfits can you make consisting of one shirt, one pair of pants, one hat, and one scarf, without having the same color of shirts and hats? Le... | math | {
"style": "rule",
"ground_truth": "7500"
} | {
"split": "train",
"index": 34484,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the original spherical dome has a height of $55$ meters and can be represented as holding $250,000$ liters of air, and Emily's scale model can hold only $0.2$ liters of air, determine the height, in meters, of the spherical dome in Emily's model. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"split": "train",
"index": 34485,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the graph of the power function $y=f(x)$ passes through the point $\\left(\\frac{1}{3},\\frac{\\sqrt{3}}{3}\\right)$, then the value of $\\log_{2}f(2)$ is \\_\\_\\_\\_. 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": 34486,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least positive integer $m$ such that the following is true?\n\n*Given $\\it m$ integers between $\\it1$ and $\\it{2023},$ inclusive, there must exist two of them $\\it a, b$ such that $1 < \\frac ab \\le 2.$* \\[\\mathrm a. ~ 10\\qquad \\mathrm b.~11\\qqu... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 34487,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider an equilateral triangle with a side length of 10 cm, where one side of the triangle is also the diameter of a semicircle. Find the sum of the areas of two small shaded regions formed inside the semicircle but outside the triangle. Express your answer in the form \\(a\\... | math | {
"style": "rule",
"ground_truth": "78"
} | {
"split": "train",
"index": 34488,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sin (2x+ \\frac {\\pi}{6})+\\cos 2x$. \n(I) Find the interval of monotonic increase for the function $f(x)$; \n(II) In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. Given that $f(A)= \\frac { \\... | math | {
"style": "rule",
"ground_truth": "\\frac {3+ \\sqrt {3}}{2}"
} | {
"split": "train",
"index": 34489,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\cos \\theta = \\frac{2}{3},$ then find $\\cos 2 \\theta.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{9}"
} | {
"split": "train",
"index": 34490,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest prime whose digits sum to \\(28\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1999"
} | {
"split": "train",
"index": 34491,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parabola $y^2=2x$, find the equation of its directrix. 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": 34492,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Memories all must have at least one out of five different possible colors, two of which are red and green. Furthermore, they each can have at most two distinct colors. If all possible colorings are equally likely, what is the probability that a memory is at least partly green g... | math | {
"style": "rule",
"ground_truth": "2/5"
} | {
"split": "train",
"index": 34493,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two cars covered the same distance. The speed of the first car was constant and three times less than the initial speed of the second car. The second car traveled the first half of the journey without changing speed, then its speed was suddenly halved, then traveled with consta... | math | {
"style": "rule",
"ground_truth": "5/3"
} | {
"split": "train",
"index": 34494,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $y = \\displaystyle\\frac{1}{3x+1}$, what is the value of $x$ when $y = 1$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 34495,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given sets $A=\\{0, |x|\\}$ and $B=\\{1, 0, -1\\}$. If $A \\subseteq B$, then find $A \\cap B$, $A \\cup B$, and the complement of $A$ in $B$, denoted by $\\complement_B A$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\{-1\\}"
} | {
"split": "train",
"index": 34496,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least three-digit whole number, the product of whose digits is 6? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "116"
} | {
"split": "train",
"index": 34497,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graphs of five functions are labelled from **(1) through (5)**. Provided below are descriptions of three:\n 1. The domain of function (2) is from $$\\{-6,-5,-4,-3,-2,-1,0,1,2,3\\}.$$ It is graphed as a set of discrete points.\n 2. Function (4) is defined by the equation... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 34498,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the following scenario where three stores offer different discounts on a television priced at $149.99$:\n\\begin{tabular}{|l|l|}\n\\hline\n\\textbf{Store} & \\textbf{Sale Price for Television $Y$} \\\\\n\\hline\nValue Market & $\\$10$~off the list price~$\\$149.99$ \\\... | math | {
"style": "rule",
"ground_truth": "3500"
} | {
"split": "train",
"index": 34499,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $F(n+1)=\\frac{2F(n)+1}{2}$ for $n=1,2,\\cdots$ and $F(1)=2$, then $F(101)$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 34500,
"solution": "1. **Identify the recurrence relation and initial condition**: \n Given the recurrence relation:\n \\[\n F(n+1) = \\frac{2F(n) + 1}{2}\n \\]\n and the initial condition:\n \\[\n F(1) = 2.\n \\]\n\n2. **Simplify the recurrence relation**:\n We can ... |
deepscaler | [
{
"role": "user",
"content": "Find all solutions to $x^{4}+2 x^{3}+2 x^{2}+2 x+1=0$ (including non-real solutions). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1, i, -i"
} | {
"split": "train",
"index": 34501,
"solution": "We can factor the polynomial as $(x+1)^{2}(x^{2}+1)$. Therefore, the solutions are $-1, i, -i$."
} |
deepscaler | [
{
"role": "user",
"content": "Rhombus $ABCD$ has side length $3$ and $\\angle B = 110$°. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$?\n**A)** $0.81$\n**B)** $1.62$\n**C)** $2.43$\n**D)** $2.16$\n**E)** $3.24... | math | {
"style": "rule",
"ground_truth": "2.16"
} | {
"split": "train",
"index": 34502,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "50 students with blond hair, brown hair, and red hair are sitting around a round table. It is known that in any group of students sitting consecutively, there is at least one student with brown hair between any two students with blond hair, and at least one student with red hai... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 34503,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The expression $\\frac{2021}{2020} - \\frac{2020}{2021}$ is equal to the fraction $\\frac{p}{q}$ in which $p$ and $q$ are positive integers whose greatest common divisor is 1. What is $p?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2021"
} | {
"split": "train",
"index": 34504,
"solution": "1. Let $a = 2020$. We start by rewriting the given expression using $a$:\n \\[\n \\frac{2021}{2020} - \\frac{2020}{2021} = \\frac{a+1}{a} - \\frac{a}{a+1}\n \\]\n\n2. To combine these fractions, find a common denominator, which is $a(a+1)$:\n \\[\n \\frac... |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $C$: $\\dfrac{x^{2}}{a^{2}} + \\dfrac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with its right focus at $F(\\sqrt{3}, 0)$, and point $M(-\\sqrt{3}, \\dfrac{1}{2})$ on ellipse $C$.\n\n(Ⅰ) Find the standard equation of ellipse $C$;\n\n(Ⅱ) Line $l$ passes through point $F$ ... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 34505,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Lauren solved the equation $|x-5| = 2$. Meanwhile Jane solved an equation of the form $x^2+ bx + c = 0$ that had the same two solutions for $x$ as Lauren's equation. What is the ordered pair $(b, c)$? Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "(-10,21)"
} | {
"split": "train",
"index": 34506,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\n58 balls of two colors - red and blue - are arranged in a circle. It is known that the number of consecutive triplets of balls with a majority of red balls is equal to the number of triplets with a majority of blue balls. What is the minimum possible number of red balls? Let'... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 34507,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a\\_n\\}$, where $a\\_n \\in \\mathbb{N}^*$, and $S\\_n = \\frac{1}{8}(a\\_n + 2)^2$. If $b\\_n = \\frac{1}{2}a\\_n - 30$, find the minimum value of the sum of the first $\\_\\_\\_\\_\\_\\_$ terms of the sequence $\\{b\\_n\\}$. Let's think step ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 34508,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ satisfies $a_1=3$, and $a_1+a_3+a_5=21$. Calculate the value of $a_3+a_5+a_7$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 34509,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow {a} = (1, -3)$ and $\\overrightarrow {b} = (-2, m)$, and $\\overrightarrow {a}$ is perpendicular to ($\\overrightarrow {a} - \\overrightarrow {b}$). \n(1) Find the value of the real number $m$; \n(2) When $k\\overrightarrow {a} + \\overrightarr... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 34510,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function \\( y = \\sqrt{2x^2 + 2} \\) with the graph represented by the curve \\( G \\). The curve \\( G \\) has a focus at \\( F \\). A line \\( l_1 \\) passing through \\( F \\) intersects the curve \\( G \\) at points \\( A \\) and \\( C \\), and another line \\( l... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 34511,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $1+2x+3x^2 + \\dotsb=9$, find $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 34512,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When Xiaohong was 3 years old, her mother's age was the same as Xiaohong's age this year. When her mother is 78 years old, Xiaohong's age is the same as her mother’s age this year. How old is Xiaohong’s mother this year? Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "53"
} | {
"split": "train",
"index": 34513,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the relationship $P={P}_{0}{e}^{-kt}$ between the concentration of toxic and harmful substances $P$ (unit: $mg/L$) in the exhaust gas and time $t$ (unit: $h$) during the filtration process, determine the percentage of the original toxic and harmful substances that will re... | math | {
"style": "rule",
"ground_truth": "57\\%"
} | {
"split": "train",
"index": 34514,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I had $\\$30$ in allowance money and spent it as indicated in the pie graph shown. How many dollars did I spend on burgers?\n\n[asy]\nsize(150);\npair A, B, C, D, O, W, X, Y, Z;\nO=(0,0);\nA=(.707,.707);\nB=(-.966,.259);\nC=(-.707,-.707);\nD=(.342,-.940);\ndraw(Circle(O, 1));\n... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 34515,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\frac{4}{3} (r + s + t) = 12$, what is the average of $r$, $s$, and $t$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 34516,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ f(x) = x^3 + x + 1$. Suppose $ g$ is a cubic polynomial such that $ g(0) = - 1$, and the roots of $ g$ are the squares of the roots of $ f$. Find $ g(9)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "899"
} | {
"split": "train",
"index": 34517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the volume of the set of points $(x, y, z)$ satisfying $$\\begin{array}{r} x, y, z \\geq 0 \\\\ x+y \\leq 1 \\\\ y+z \\leq 1 \\\\ z+x \\leq 1 \\end{array}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 34518,
"solution": "Without loss of generality, assume that $x \\geq y$ - half the volume of the solid is on this side of the plane $x=y$. For each value of $c$ from 0 to $\\frac{1}{2}$, the region of the intersection of this half of the solid with the plane $y=c$ is a trapezoid. The ... |
deepscaler | [
{
"role": "user",
"content": "Billy's age is three times Brenda's age and twice Joe's age. The sum of their ages is 72. How old is Billy? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{432}{11}"
} | {
"split": "train",
"index": 34519,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\left\\{a_{n}\\right\\}$ satisfying: $a_{1}=1, a_{2}=2, a_{2k+1}=\\frac{a_{2k}^{2}}{a_{2k-1}}$, and $a_{2k+2}=2a_{2k+1}-a_{2k}$ for $k \\in \\mathbf{N}^{*}$, find the last two digits of $a_{2022}$. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 34520,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular floor that is $10$ feet wide and $17$ feet long is tiled with $170$ one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit? Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 34521,
"solution": "To solve this problem, we need to determine how many tiles the bug crosses as it walks in a straight line from one corner of the rectangle to the opposite corner. The rectangle is $10$ feet wide and $17$ feet long, and the tiles are $1$ foot square each.\n\n1. **Un... |
deepscaler | [
{
"role": "user",
"content": "In the diagram, each circle is divided into two equal areas and $O$ is the center of the larger circle. The area of the larger circle is $64\\pi.$ What is the total area of the shaded regions? [asy]\nsize(100);\nimport graph;\nfill(Arc((0,0),2,180,360)--cycle,mediumgray);fill(A... | math | {
"style": "rule",
"ground_truth": "40\\pi"
} | {
"split": "train",
"index": 34522,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a triangle with $AB=9$ , $BC=10$ , $CA=11$ , and orthocenter $H$ . Suppose point $D$ is placed on $\\overline{BC}$ such that $AH=HD$ . Compute $AD$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{102}"
} | {
"split": "train",
"index": 34523,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many more digits does the base-3 representation of $987_{10}$ have than the base-8 representation of $987_{10}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 34524,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the fourth-largest divisor of $1,234,560,000$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "154,320,000"
} | {
"split": "train",
"index": 34525,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The supermarket sold two types of goods, both for a total of 660 yuan. One item made a profit of 10%, while the other suffered a loss of 10%. Express the original total price of these two items using a formula. Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "1333\\frac{1}{3}"
} | {
"split": "train",
"index": 34526,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using a permutation of the numbers $10, 20, 30, 40$ for $A, B, C, D$, maximize the value of the expression $\\frac{1}{A-\\frac{1}{B+\\frac{1}{C-\\frac{1}{D}}}}$. Then, find the value of $A+2B+3C+4D$. Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "290"
} | {
"split": "train",
"index": 34527,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers between $100$ and $999$, inclusive, have the property that some permutation of its digits is a multiple of $11$ between $100$ and $999?$ For example, both $121$ and $211$ have this property.\n$\\mathrm{\\textbf{(A)} \\ }226\\qquad \\mathrm{\\textbf{(B)} \\ } 2... | math | {
"style": "rule",
"ground_truth": "226"
} | {
"split": "train",
"index": 34528,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a circuit with two independently controlled automatic switches connected in parallel, the probability that the circuit can operate normally is given that one of the switches can close, and the probabilities of the switches being able to close are $0.5$ and $0.7$ respectively... | math | {
"style": "rule",
"ground_truth": "0.85"
} | {
"split": "train",
"index": 34529,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a circle of radius $42$, two chords of length $78$ intersect at a point whose distance from the center is $18$. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segments of unequal lengths, and the area of either of the... | math | {
"style": "rule",
"ground_truth": "378"
} | {
"split": "train",
"index": 34530,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain stationery store stipulates that if one purchases 250 or more exercise books at a time (including 250 books), they can pay at the wholesale price; if one purchases fewer than 250 books, they must pay at the retail price. Li, a teacher, went to the store to buy exercis... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 34531,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine all functions $f: \\mathbb{R} \\to \\mathbb{R}$ such that \n$$ f(x^3) + f(y)^3 + f(z)^3 = 3xyz $$\nfor all real numbers $x$, $y$ and $z$ with $x+y+z=0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x"
} | {
"split": "train",
"index": 34532,
"solution": "\nWe are tasked with finding all functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) that satisfy the equation \n\n\\[\nf(x^3) + f(y)^3 + f(z)^3 = 3xyz\n\\]\n\nfor all real numbers \\( x \\), \\( y \\), and \\( z \\) such that \\( x + y + z = 0 \\).\n\nFirst, we cons... |
deepscaler | [
{
"role": "user",
"content": "Person A starts traveling from point A to point B. Persons B and C start traveling from point B to point A. After person A has traveled 50 kilometers, persons B and C start traveling from point B. Person A and person B meet at point C, and person A and person C meet at point D.... | math | {
"style": "rule",
"ground_truth": "130"
} | {
"split": "train",
"index": 34533,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum $m + n$ where $m$ and $n$ are integers, such that the positive difference between the two roots of the quadratic equation $2x^2 - 5x - 12 = 0$ can be expressed as $\\frac{\\sqrt{m}}{n}$, and $m$ is not divisible by the square of any prime number. Let's think step b... | math | {
"style": "rule",
"ground_truth": "123"
} | {
"split": "train",
"index": 34534,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a right triangle $\\triangle STU$, where $\\angle S = 90^\\circ$, suppose $\\sin T = \\frac{3}{5}$. If the length of $SU$ is 15, find the length of $ST$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 34535,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Equilateral triangle \\( \\triangle ABC \\) and square \\( ABDE \\) have a common side \\( AB \\). The cosine of the dihedral angle \\( C-ABD \\) is \\(\\frac{\\sqrt{3}}{3}\\). If \\( M \\) and \\( N \\) are the midpoints of \\( AC \\) and \\( BC \\) respectively, then the cosi... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 34536,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $77\\cdot 88\\cdot 99 \\equiv n\\pmod{20}$, where $0\\le n<20$.\n\nWhat is the value of $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 34537,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $0 < \\beta < \\frac{\\pi}{2} < \\alpha < \\pi$ and $\\cos \\left(\\alpha- \\frac{\\beta}{2}\\right)=- \\frac{1}{9}, \\sin \\left( \\frac{\\alpha}{2}-\\beta\\right)= \\frac{2}{3}$, calculate the value of $\\cos (\\alpha+\\beta)$. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "-\\frac{239}{729}"
} | {
"split": "train",
"index": 34538,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $3a + 5b = 47$ and $7a + 2b = 52$, what is the value of $a + b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{35}{3}"
} | {
"split": "train",
"index": 34539,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point $Q$ is chosen in the interior of $\\triangle DEF$ such that when lines are drawn through $Q$ parallel to the sides of $\\triangle DEF$, the resulting smaller triangles $t_1$, $t_2$, and $t_3$ have areas of $16$, $25$, and $36$, respectively. Find the area of $\\triangle... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 34540,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape. The parallel sides of the trapezoid have lengths $18$ and $30$ meters. What fraction of the yard is occupied by the flower beds?\n... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 34541,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given points $A$, $B$, and $C$ are on the curve $y=\\sqrt{x}$ $(x \\geqslant 0)$, with x-coordinates $1$, $m$, and $4$ $(1 < m < 4)$, when the area of $\\triangle ABC$ is maximized, $m$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{4}"
} | {
"split": "train",
"index": 34542,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the measure of an orthogonal trihedral angle? What is the sum of the measures of polyhedral angles that share a common vertex, have no common interior points, and together cover the entire space? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"split": "train",
"index": 34543,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $100^a = 7$ and $100^b = 11,$ then find $20^{(1 - a - b)/(2(1 - b))}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{100}{77}"
} | {
"split": "train",
"index": 34544,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $n$ such that $2^6 \\cdot 3^3 \\cdot n = 10!$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1050"
} | {
"split": "train",
"index": 34545,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $w_1$ and $w_2$ denote the circles $x^2+y^2+10x-24y-87=0$ and $x^2 +y^2-10x-24y+153=0,$ respectively. Let $m$ be the smallest positive value of $a$ for which the line $y=ax$ contains the center of a circle that is externally tangent to $w_2$ and internally tangent to $w_1.$... | math | {
"style": "rule",
"ground_truth": "169"
} | {
"split": "train",
"index": 34546,
"solution": "We use the same reflection as in Solution 2. As $OF_1'=OF_2=13$, we know that $\\triangle OF_1'F_2$ is isosceles. Hence $\\angle F_2F_1'O=\\angle F_1'F_2O$. But by symmetry, we also know that $\\angle OF_1T=\\angle F_2F_1'O$. Hence $\\angle OF_1T=\\angle F_1'F_2O$.... |
deepscaler | [
{
"role": "user",
"content": "Given a trapezoid \\( ABCD \\) with \\( AD \\parallel BC \\). It turns out that \\( \\angle ABD = \\angle BCD \\). Find the length of segment \\( BD \\) if \\( BC = 36 \\) and \\( AD = 64 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 34547,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $\\dfrac {x^{2}}{a^{2}}- \\dfrac {y^{2}}{b^{2}}=1(a > 0,b > 0)$ with eccentricity $e= \\dfrac {2 \\sqrt {3}}{3}$, calculate the angle between the two asymptotes. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 34548,
"solution": ""
} |
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