data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Arrange the positive odd numbers as shown in the pattern below. What is the 5th number from the left in the 21st row?\n\n$$\n1 \\\\\n3 \\quad 5 \\quad 7 \\\\\n9 \\quad 11 \\quad 13 \\quad 15 \\quad 17 \\\\\n19 \\quad 21 \\quad 23 \\quad 25 \\quad 27 \\quad 29 \\quad 31 \\\\\n\\... | math | {
"style": "rule",
"ground_truth": "809"
} | {
"split": "train",
"index": 0,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the expression $x \\cdot y^z - w$, the values of $x$, $y$, $z$, and $w$ are 1, 2, 3, and 4, although not necessarily in that order. What is the maximum possible value of the expression? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "161"
} | {
"split": "train",
"index": 1,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sets:\n$$\nU = \\{1, 2, 3, 4, 5\\}, \\quad I = \\{X \\mid X \\subseteq U\\}\n$$\n\nIf two different elements \\( A \\) and \\( B \\) are randomly selected from the set \\( I \\), what is the probability that \\( A \\cap B \\) contains exactly three elements? Let's thi... | math | {
"style": "rule",
"ground_truth": "5/62"
} | {
"split": "train",
"index": 2,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two congruent cylinders each have radius 8 inches and height 3 inches. The radius of one cylinder and the height of the other are both increased by the same nonzero number of inches. The resulting volumes are equal. How many inches is the increase? Express your answer as a comm... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{3}"
} | {
"split": "train",
"index": 3,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a sequence $\\left\\{a_{n}\\right\\}$ by $a_{1}=1$ and $a_{n}=\\left(a_{n-1}\\right)!+1$ for every $n>1$. Find the least $n$ for which $a_{n}>10^{10}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 4,
"solution": "We have $a_{2}=2, a_{3}=3, a_{4}=7, a_{5}=7!+1=5041$, and $a_{6}=5041!+1$. But $$5041!+1 \\gg 5041 \\cdot 5040 \\cdot 5039>10^{10}$$ Hence, the answer is 6."
} |
deepscaler | [
{
"role": "user",
"content": "At time $t=0,$ a ball is thrown downward at 24 feet per second from a height of 160 feet above the ground. The equation $h = -16t^2 - 24t +160$ describes the height (in feet) of the ball. In how many seconds will the ball hit the ground? Express your answer as a decimal. Let's... | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"split": "train",
"index": 5,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $z$ be a complex number with $|z|=2014$. Let $P$ be the polygon in the complex plane whose vertices are $z$ and every $w$ such that $\\frac{1}{z+w}=\\frac{1}{z}+\\frac{1}{w}$. Then the area enclosed by $P$ can be written in the form $n\\sqrt{3}$, where $n$ is an integer. Fi... | math | {
"style": "rule",
"ground_truth": "147"
} | {
"split": "train",
"index": 6,
"solution": "Notice that \\[\\frac1{w+z} = \\frac{w+z}{wz} \\implies 0 = w^2 + wz + z^2 = \\frac{w^3-z^3}{w-z}.\\] Hence, $w=ze^{2\\pi i/3},ze^{4\\pi i/3}$, and $P$ is an equilateral triangle with circumradius $2014$. Then, \\[[P]=\\frac{3}{2}\\cdot 2014^2\\cdot\\sin\\frac{\\pi}3=3... |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ with vertices of $A(6,2)$, $B(2,5)$, and $C(2,2)$ is reflected over the x-axis to triangle $A'B'C'$. This triangle is reflected over the y-axis to triangle $A''B''C''$. What are the coordinates of point $C''$? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "(-2, -2)"
} | {
"split": "train",
"index": 7,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. It is known that $\\overrightarrow{a}=(\\cos A,\\cos B)$, $\\overrightarrow{b}=(a,2c-b)$, and $\\overrightarrow{a} \\parallel \\overrightarrow{b}$.\n(Ⅰ) Find the magnitude of... | math | {
"style": "rule",
"ground_truth": "\\sqrt {13}"
} | {
"split": "train",
"index": 8,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.... | math | {
"style": "rule",
"ground_truth": "1680"
} | {
"split": "train",
"index": 9,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has $\\angle BAC=90^\\circ$ . A semicircle with diameter $XY$ is inscribed inside $\\triangle ABC$ such that it is tangent to a point $D$ on side $BC$ , with $X$ on $AB$ and $Y$ on $AC$ . Let $O$ be the midpoint of $XY$ . Given that $AB=3$ , ... | math | {
"style": "rule",
"ground_truth": "39/32"
} | {
"split": "train",
"index": 10,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let points $A = (0,0)$, $B = (2,3)$, $C = (5,4)$, and $D = (6,0)$. Quadrilateral $ABCD$ is divided into two equal area pieces by a line passing through $A$. This line intersects $\\overline{CD}$ at point $\\left (\\frac{p}{q}, \\frac{r}{s} \\right )$, where these fractions are ... | math | {
"style": "rule",
"ground_truth": "58"
} | {
"split": "train",
"index": 11,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a solid iron cone with a base radius of $3cm$ and a slant height of $5cm$. After melting it at high temperature and casting it into a solid iron sphere (without considering any loss), the radius of this iron sphere is _______ $cm$. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "\\sqrt[3]{9}"
} | {
"split": "train",
"index": 12,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The dimensions of a part on a drawing are $7{}_{-0.02}^{+0.05}$ (unit: $mm$), indicating that the maximum requirement for processing this part should not exceed ______, and the minimum should not be less than ______. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "6.98"
} | {
"split": "train",
"index": 13,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Pyramid $OABCD$ has square base $ABCD,$ congruent edges $\\overline{OA}, \\overline{OB}, \\overline{OC},$ and $\\overline{OD},$ and $\\angle AOB=45^\\circ.$ Let $\\theta$ be the measure of the dihedral angle formed by faces $OAB$ and $OBC.$ Given that $\\cos \\theta=m+\\sqrt{n}... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 14,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When 1524 shi of rice is mixed with an unknown amount of wheat, and in a sample of 254 grains, 28 are wheat grains, calculate the estimated amount of wheat mixed with this batch of rice. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "168"
} | {
"split": "train",
"index": 15,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the expression: $$\\dfrac{\\sqrt{450}}{\\sqrt{288}} + \\dfrac{\\sqrt{245}}{\\sqrt{96}}.$$ Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{30 + 7\\sqrt{30}}{24}"
} | {
"split": "train",
"index": 16,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the square of 1017 without a calculator. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1034289"
} | {
"split": "train",
"index": 17,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The hypotenuse of a right triangle, where the legs are consecutive whole numbers, is 53 units long. What is the sum of the lengths of the two legs? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 18,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An equilateral triangle $PQR$ is inscribed in the ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1,$ so that $Q$ is at $(0,b),$ and $\\overline{PR}$ is parallel to the $x$-axis, as shown below. Also, foci $F_1$ and $F_2$ lie on sides $\\overline{QR}$ and $\\overline{PQ},$ resp... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{5}"
} | {
"split": "train",
"index": 19,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The cheetah takes strides of 2 meters each and the fox takes strides of 1 meter each. The time it takes for the cheetah to run 2 strides is the same as the time it takes for the fox to run 3 strides. Given that the distance between the cheetah and the fox is 30 meters, find the... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 20,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. One day the boat captain notes that on the 10 AM trip there were 100 tourists on the ferry boat, and that on each successive trip, the number of tourists was 1 fe... | math | {
"style": "rule",
"ground_truth": "585"
} | {
"split": "train",
"index": 21,
"solution": "1. **Identify the number of trips**: The ferry starts at 10 AM and makes its last trip at 3 PM. Since the trips are hourly, we can calculate the number of trips as follows:\n - 10 AM, 11 AM, 12 PM, 1 PM, 2 PM, 3 PM\n - This gives us a total of 6 trips.\n\n2. **Cal... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)={(3\\ln x-x^{2}-a-2)}^{2}+{(x-a)}^{2}$ $(a\\in \\mathbb{R})$, determine the value of the real number $a$ such that the inequality $f(x)\\leqslant 8$ has solutions for $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 22,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a graph with 30 vertices. If any of 26 of its vertices with their outgoiing edges are deleted, then the remained graph is a connected graph with 4 vertices.\nWhat is the smallest number of the edges in the initial graph with 30 vertices? Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "405"
} | {
"split": "train",
"index": 23,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $\\left(\\frac{2}{3}\\right)\\left(\\frac{3}{4}\\right)\\left(\\frac{4}{5}\\right)\\left(\\frac{5}{6}\\right)$? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 24,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six straight lines are drawn in a plane with no two parallel and no three concurrent. The number of regions into which they divide the plane is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 25,
"solution": "\n#### Detailed Analysis:\n\n1. **Understanding the Problem:**\n We are given six straight lines in a plane, with the condition that no two lines are parallel and no three lines are concurrent. We need to determine how many regions these lines divide the plane into.... |
deepscaler | [
{
"role": "user",
"content": "You are the general of an army. You and the opposing general both have an equal number of troops to distribute among three battlefields. Whoever has more troops on a battlefield always wins (you win ties). An order is an ordered triple of non-negative real numbers $(x, y, z)$ s... | math | {
"style": "rule",
"ground_truth": "\\sqrt[5]{8}"
} | {
"split": "train",
"index": 26,
"solution": "Let $x$ be the portion of soldiers the opposing general sends to the first battlefield, and $y$ the portion he sends to the second. Then $1-x-y$ is the portion he sends to the third. Then $x \\geq 0$, $y \\geq 0$, and $x+y \\leq 1$. Furthermore, you win if one of the ... |
deepscaler | [
{
"role": "user",
"content": "Calculate the sum of the geometric series $1+\\left(\\frac{1}{5}\\right)+\\left(\\frac{1}{5}\\right)^2 + \\left(\\frac{1}{5}\\right)^3 + \\dots$. Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"split": "train",
"index": 27,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cone is perfectly fitted inside a cube such that the cone's base is one face of the cube and its vertex touches the opposite face. A sphere is inscribed in the same cube. Given that one edge of the cube is 8 inches, calculate:\n1. The volume of the inscribed sphere.\n2. The v... | math | {
"style": "rule",
"ground_truth": "\\frac{128}{3}\\pi"
} | {
"split": "train",
"index": 28,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose \\(2+\\frac{1}{1+\\frac{1}{2+\\frac{2}{3+x}}}=\\frac{144}{53}.\\)What is the value of \\(x?\\) 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": 29,
"solution": "1. **Start with the given equation:**\n \\[\n 2 + \\frac{1}{1 + \\frac{1}{2 + \\frac{2}{3+x}}} = \\frac{144}{53}\n \\]\n\n2. **Isolate the fraction by subtracting 2 from both sides:**\n \\[\n \\frac{1}{1 + \\frac{1}{2 + \\frac{2}{3+x}}} = \\frac{144}{53} - 2... |
deepscaler | [
{
"role": "user",
"content": "Given a 2x3 rectangle with six unit squares, the lower left corner at the origin, find the value of $c$ such that a slanted line extending from $(c,0)$ to $(4,4)$ divides the entire region into two regions of equal area. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"split": "train",
"index": 30,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $θ$ is an angle in the second quadrant and $\\tan(\\begin{matrix}θ+ \\frac{π}{4}\\end{matrix}) = \\frac{1}{2}$, find the value of $\\sin(θ) + \\cos(θ)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{10}}{5}"
} | {
"split": "train",
"index": 31,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Medians are drawn from point $A$ and point $B$ in this right triangle to divide segments $\\overline{BC}$ and $\\overline{AC}$ in half, respectively. The lengths of the medians are 6 and $2\\sqrt{11}$ units, respectively. How many units are in the length of segment $\\overline{... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 32,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A TV station is broadcasting 5 different advertisements, among which there are 3 different commercial advertisements and 2 different Olympic promotional advertisements. It is required that the last advertisement must be an Olympic promotional advertisement, and the two Olympic ... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 33,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many digits are there in the base-7 representation of $956$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 34,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\tan\\left(\\frac{\\pi}{9}\\right)\\tan\\left(\\frac{2\\pi}{9}\\right)\\tan\\left(\\frac{4\\pi}{9}\\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 35,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be a triangle with $A B=A C=5$ and $B C=6$. Denote by $\\omega$ the circumcircle of $A B C$. We draw a circle $\\Omega$ which is externally tangent to $\\omega$ as well as to the lines $A B$ and $A C$ (such a circle is called an $A$-mixtilinear excircle). Find the r... | math | {
"style": "rule",
"ground_truth": "\\frac{75}{8}"
} | {
"split": "train",
"index": 36,
"solution": "Let $M$ be the midpoint of $B C$. Let $D$ be the point diametrically opposite $A$ on the circumcircle, and let the $A$-mixtilinear excircle be tangent to lines $A B$ and $A C$ at $X$ and $Y$. Let $O$ be the center of the $A$-mixtilinear excircle. Notice that $\\triang... |
deepscaler | [
{
"role": "user",
"content": "During the fight against the epidemic, a certain store purchased a type of disinfectant product at a cost of $8$ yuan per item. It was found during the sales process that there is a linear relationship between the daily sales quantity $y$ (items) and the selling price per item ... | math | {
"style": "rule",
"ground_truth": "525"
} | {
"split": "train",
"index": 37,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all values of $x$ with $0 \\le x < \\pi$ that satisfy $\\sin x - \\cos x = 1$. Enter all the solutions, separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"split": "train",
"index": 38,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of ordered pairs $(a,b)$ of complex numbers such that\n\\[a^3 b^5 = a^7 b^2 = 1.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 39,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Twelve tiles numbered $1$ through $12$ are turned up at random, and an eight-sided die is rolled. Calculate the probability that the product of the numbers on the tile and the die will be a perfect square. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{96}"
} | {
"split": "train",
"index": 40,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A copper cube with an edge length of $l = 5 \\text{ cm}$ is heated to a temperature of $t_{1} = 100^{\\circ} \\text{C}$. Then, it is placed on ice, which has a temperature of $t_{2} = 0^{\\circ} \\text{C}$. Determine the maximum depth the cube can sink into the ice. The specifi... | math | {
"style": "rule",
"ground_truth": "0.06"
} | {
"split": "train",
"index": 41,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), \\(\\angle A = 45^\\circ\\) and \\(M\\) is the midpoint of \\(\\overline{BC}\\). \\(\\overline{AM}\\) intersects the circumcircle of \\(ABC\\) for the second time at \\(D\\), and \\(AM = 2MD\\). Find \\(\\cos \\angle AOD\\), where \\(O\\) is the circumcen... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{8}"
} | {
"split": "train",
"index": 42,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A train arrives randomly some time between 2:00 and 4:00 PM, waits for 30 minutes, and then leaves. If Maria also arrives randomly between 2:00 and 4:00 PM, what is the probability that the train will be there when Maria arrives? Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{32}"
} | {
"split": "train",
"index": 43,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If any two adjacent digits of a three-digit number have a difference of at most 1, it is called a \"steady number\". How many steady numbers are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 44,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A massive vertical plate is fixed to a car moving at a speed of $5 \\, \\text{m/s}$. A ball is flying towards it at a speed of $6 \\, \\text{m/s}$ with respect to the ground. Determine the speed of the ball with respect to the ground after a perfectly elastic normal collision. ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 45,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given angles $α$ and $β$ satisfy $\\frac{\\tan α}{\\tan β} = \\frac{7}{13}$, and $\\sin(α+β) = \\frac{2}{3}$, find the value of $\\sin(α-β)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{5}"
} | {
"split": "train",
"index": 46,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sides of rectangle $ABCD$ have lengths $10$ and $11$. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$. The maximum possible area of such a triangle can be written in the form $p\\sqrt{q}-r$, where $p$, $q$, and $r$ are positive integers... | math | {
"style": "rule",
"ground_truth": "554"
} | {
"split": "train",
"index": 47,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The complex numbers \\( \\alpha_{1}, \\alpha_{2}, \\alpha_{3}, \\) and \\( \\alpha_{4} \\) are the four distinct roots of the equation \\( x^{4}+2 x^{3}+2=0 \\). Determine the unordered set \\( \\left\\{\\alpha_{1} \\alpha_{2}+\\alpha_{3} \\alpha_{4}, \\alpha_{1} \\alpha_{3}+\\... | math | {
"style": "rule",
"ground_truth": "\\{1 \\pm \\sqrt{5},-2\\}"
} | {
"split": "train",
"index": 48,
"solution": "Employing the elementary symmetric polynomials \\( \\left(s_{1}=\\alpha_{1}+\\alpha_{2}+\\alpha_{3}+\\alpha_{4}=\\right. -2, s_{2}=\\alpha_{1} \\alpha_{2}+\\alpha_{1} \\alpha_{3}+\\alpha_{1} \\alpha_{4}+\\alpha_{2} \\alpha_{3}+\\alpha_{2} \\alpha_{4}+\\alpha_{3} \\alp... |
deepscaler | [
{
"role": "user",
"content": "Find the number of positive integers \\( x \\), where \\( x \\neq 9 \\), such that\n\\[\n\\log _{\\frac{x}{9}}\\left(\\frac{x^{2}}{3}\\right)<6+\\log _{3}\\left(\\frac{9}{x}\\right) .\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "223"
} | {
"split": "train",
"index": 49,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p,$ $q,$ $r,$ $s$ be real numbers such that\n\\[\\frac{(p - q)(r - s)}{(q - r)(s - p)} = \\frac{3}{7}.\\]Find the sum of all possible values of\n\\[\\frac{(p - r)(q - s)}{(p - q)(r - s)}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{3}"
} | {
"split": "train",
"index": 50,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the matrix that corresponds to projecting onto the vector $\\begin{pmatrix} 2 \\\\ -3 \\end{pmatrix}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 4/13 & -6/13 \\\\ -6/13 & 9/13 \\end{pmatrix}"
} | {
"split": "train",
"index": 51,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square $PQRS$ has sides of length 1. Points $T$ and $U$ are on $\\overline{QR}$ and $\\overline{RS}$, respectively, so that $\\triangle PTU$ is equilateral. A square with vertex $Q$ has sides that are parallel to those of $PQRS$ and a vertex on $\\overline{PT}.$ The length of a... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 52,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $α$ and $β ∈ ( \\frac{π}{2},π)$, and $sinα + cosα = a$, $cos(β - α) = \\frac{3}{5}$.\n(1) If $a = \\frac{1}{3}$, find the value of $sinαcosα + tanα - \\frac{1}{3cosα}$;\n(2) If $a = \\frac{7}{13}$, find the value of $sinβ$. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{65}"
} | {
"split": "train",
"index": 53,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A farmer contracted several acres of fruit trees. This year, he invested 13,800 yuan, and the total fruit yield was 18,000 kilograms. The fruit sells for a yuan per kilogram in the market and b yuan per kilogram when sold directly from the orchard (b < a). The farmer transports... | math | {
"style": "rule",
"ground_truth": "20\\%"
} | {
"split": "train",
"index": 54,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose six points are taken inside or on a rectangle with dimensions $1 \\times 2$. Let $b$ be the smallest possible number with the property that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 55,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the product of $218_9 \\cdot 5_9$. Express your answer in base 9. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1204_9"
} | {
"split": "train",
"index": 56,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the first $n$ terms of the sequence $\\{a_n\\}$ is $S_n=n^2+n+1$, and $b_n=(-1)^n(a_n-2)$ $(n\\in\\mathbb{N}^*)$, then the sum of the first $50$ terms of the sequence $\\{b_n\\}$ is $\\_\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 57,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( x \\) be a real number with the property that \\( x+\\frac{1}{x} = 4 \\). Define \\( S_m = x^m + \\frac{1}{x^m} \\). Determine the value of \\( S_6 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2702"
} | {
"split": "train",
"index": 58,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a finite increasing sequence \\(a_{1}, a_{2}, \\ldots, a_{n}\\) of natural numbers (with \\(n \\geq 3\\)), and the recurrence relation \\(a_{k+2} = 3a_{k+1} - 2a_{k} - 2\\) holds for all \\(\\kappa \\leq n-2\\). The sequence must contain \\(a_{k} = 2022\\). Determine the ... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 59,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A school selects 4 teachers from 8 to teach in 4 remote areas at the same time (one person per area), where A and B cannot go together, and A and C can only go together or not go at all. The total number of different dispatch plans is \\_\\_\\_\\_\\_\\_ (answer in numbers). Let... | math | {
"style": "rule",
"ground_truth": "600"
} | {
"split": "train",
"index": 60,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $(2x)_((-1)^{5}=a_0+a_1x+a_2x^2+...+a_5x^5$, find:\n(1) $a_0+a_1+...+a_5$;\n(2) $|a_0|+|a_1|+...+|a_5|$;\n(3) $a_1+a_3+a_5$;\n(4) $(a_0+a_2+a_4)^2-(a_1+a_3+a_5)^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-243"
} | {
"split": "train",
"index": 61,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 2019 numbers written on the board. One of them occurs more frequently than the others - 10 times. What is the minimum number of different numbers that could be written on the board? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 62,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Place several small circles with a radius of 1 inside a large circle with a radius of 11, so that each small circle is tangentially inscribed in the large circle and these small circles do not overlap. What is the maximum number of small circles that can be placed? Let's think ... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 63,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $PQRS$ is a trapezoid with an area of $12.$ $RS$ is twice the length of $PQ.$ What is the area of $\\triangle PQS?$\n\n[asy]\ndraw((0,0)--(1,4)--(7,4)--(12,0)--cycle);\ndraw((7,4)--(0,0));\nlabel(\"$S$\",(0,0),W);\nlabel(\"$P$\",(1,4),NW);\nlabel(\"$Q$\",(7,4),N... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 64,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S(n)$ equal the sum of the digits of positive integer $n$. For example, $S(1507) = 13$. For a particular positive integer $n$, $S(n) = 1274$. Which of the following could be the value of $S(n+1)$?\n$\\textbf{(A)}\\ 1 \\qquad\\textbf{(B)}\\ 3\\qquad\\textbf{(C)}\\ 12\\qquad... | math | {
"style": "rule",
"ground_truth": "1239"
} | {
"split": "train",
"index": 65,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tom's age is $T$ years, which is also the sum of the ages of his three children. His age $N$ years ago was twice the sum of their ages then. What is $T/N$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 66,
"solution": "1. **Define the variables and set up the equation:**\n - Let $T$ be Tom's current age.\n - Let $N$ be the number of years ago when Tom's age was twice the sum of his children's ages.\n - The sum of the ages of Tom's three children is also $T$.\n\n2. **Calculate ... |
deepscaler | [
{
"role": "user",
"content": "Positive integers $a$ and $b$ satisfy the condition \\[\\log_2(\\log_{2^a}(\\log_{2^b}(2^{1000}))) = 0.\\] Find the sum of all possible values of $a+b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "881"
} | {
"split": "train",
"index": 67,
"solution": "To simplify, we write this logarithmic expression as an exponential one. Just looking at the first log, it has a base of 2 and an argument of the expression in parenthesis. Therefore, we can make 2 the base, 0 the exponent, and the argument the result. That means $\\l... |
deepscaler | [
{
"role": "user",
"content": "Let $M$ denote the number of positive integers which divide 2014!, and let $N$ be the integer closest to $\\ln (M)$. Estimate the value of $N$. If your answer is a positive integer $A$, your score on this problem will be the larger of 0 and $\\left\\lfloor 20-\\frac{1}{8}|A-N|\... | math | {
"style": "rule",
"ground_truth": "439"
} | {
"split": "train",
"index": 68,
"solution": "Combining Legendre's Formula and the standard prime approximations, the answer is $$\\prod_{p}\\left(1+\\frac{2014-s_{p}(2014)}{p-1}\\right)$$ where $s_{p}(n)$ denotes the sum of the base $p$-digits of $n$. Estimate $\\ln 1000 \\approx 8$, and $\\ln 2014 \\approx 9$. ... |
deepscaler | [
{
"role": "user",
"content": "Among the 2019 natural numbers from 1 to 2019, how many of them, when added to the four-digit number 8866, result in at least one carry? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1956"
} | {
"split": "train",
"index": 69,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the cross-section of a cylinder is a square and the height of the cylinder is equal to the diameter of a sphere, calculate the ratio of the total surface area of the cylinder to the surface area of the sphere. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 70,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is known that the ellipse $C_1$ and the parabola $C_2$ have a common focus $F(1,0)$. The center of $C_1$ and the vertex of $C_2$ are both at the origin. A line $l$ passes through point $M(4,0)$ and intersects the parabola $C_2$ at points $A$ and $B$ (with point $A$ in the fo... | math | {
"style": "rule",
"ground_truth": "\\sqrt{34}"
} | {
"split": "train",
"index": 71,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In quadrilateral $ABCD$, sides $\\overline{AB}$ and $\\overline{BC}$ both have length 10, sides $\\overline{CD}$ and $\\overline{DA}$ both have length 17, and the measure of angle $ADC$ is $60^\\circ$. What is the length of diagonal $\\overline{AC}$? [asy]\ndraw((0,0)--(17,0));... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 72,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x^2+bx+9$ has two non-real roots, find all real possible values of $b$. Express your answer in interval notation. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-6, 6)"
} | {
"split": "train",
"index": 73,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A and B are running on a circular track at their respective constant speeds. If both start running from point A in opposite directions, and after their first meeting, B takes another 8 minutes to return to the starting point. Given that A takes 6 minutes to complete a lap, how ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 74,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a local government meeting, leaders from five different companies are present. It is known that two representatives are from Company A, and each of the remaining four companies has one representative attending. If three individuals give a speech at the meeting, how many poss... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 75,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six rectangles each with a common base width of $2$ have lengths of $1, 4, 9, 16, 25$, and $36$. What is the sum of the areas of the six rectangles? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "182"
} | {
"split": "train",
"index": 76,
"solution": "1. **Identify the dimensions of each rectangle**: Each rectangle has a common base width of $2$. The lengths of the rectangles are given as $1, 4, 9, 16, 25$, and $36$.\n\n2. **Calculate the area of each rectangle**: The area of a rectangle is calculated by multiplyin... |
deepscaler | [
{
"role": "user",
"content": "Given the set of 10 integers {1, 2, 3, ..., 9, 10}, choose any 3 distinct numbers to be the coefficients of the quadratic function f(x) = ax^2 + bx + c. Determine the number of ways to choose the coefficients such that f(1)/3 is an integer. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "252"
} | {
"split": "train",
"index": 77,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of $3^{12} \\cdot 3^3$ and express it as some integer raised to the third power. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "243"
} | {
"split": "train",
"index": 78,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer $N$ with three digits in its base ten representation is chosen at random, with each three digit number having an equal chance of being chosen. The probability that $\\log_2 N$ is an integer is Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{300}"
} | {
"split": "train",
"index": 79,
"solution": "1. **Identify the Condition for $\\log_2 N$ to be an Integer:**\n To have $\\log_2 N$ as an integer, $N$ must be a power of $2$, i.e., $N = 2^k$ for some integer $k$.\n\n2. **Determine the Range of $k$:**\n Since $N$ is a three-digit number, $100 \\leq N \\leq 999... |
deepscaler | [
{
"role": "user",
"content": "The first 14 terms of the sequence $\\left\\{a_{n}\\right\\}$ are $4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, \\ldots$. Following this pattern, what is $a_{18}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 80,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Patrick and Anderson are having a snowball fight. Patrick throws a snowball at Anderson which is shaped like a sphere with a radius of 10 centimeters. Anderson catches the snowball and uses the snow from the snowball to construct snowballs with radii of 4 centimeters. Given tha... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 81,
"solution": "$$\\left\\lfloor\\left(\\frac{10}{4}\\right)^{3}\\right\\rfloor=\\left\\lfloor\\frac{125}{8}\\right\\rfloor=15$$"
} |
deepscaler | [
{
"role": "user",
"content": "Kevin colors a ninja star on a piece of graph paper where each small square has area $1$ square inch. Find the area of the region colored, in square inches.\n Let's think s... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 82,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A loonie is a $\\$ 1$ coin and a dime is a $\\$ 0.10$ coin. One loonie has the same mass as 4 dimes. A bag of dimes has the same mass as a bag of loonies. The coins in the bag of loonies are worth $\\$ 400$ in total. How much are the coins in the bag of dimes worth? Let's think... | math | {
"style": "rule",
"ground_truth": "\\$ 160"
} | {
"split": "train",
"index": 83,
"solution": "Since the coins in the bag of loonies are worth $\\$ 400$, then there are 400 coins in the bag. Since 1 loonie has the same mass as 4 dimes, then 400 loonies have the same mass as $4(400)$ or 1600 dimes. Therefore, the bag of dimes contains 1600 dimes, and so the coin... |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\{a_n\\}$ satisfies $a_1=1$, $a_2=2$, $a_{n+2}=(1+\\cos ^{2} \\frac {nπ}{2})a_{n}+\\sin ^{2} \\frac {nπ}{2}$, find the sum of the first 12 terms of the sequence. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "147"
} | {
"split": "train",
"index": 84,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find \\( g(2021) \\) if for any real numbers \\( x \\) and \\( y \\) the following equality holds:\n\\[ g(x-y) = 2021(g(x) + g(y)) - 2022xy \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2043231"
} | {
"split": "train",
"index": 85,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $17^9 \\div 17^7$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "289"
} | {
"split": "train",
"index": 86,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid \\(ABCD\\), \\(BC\\) is parallel to \\(AD\\), \\(AB = AD\\), \\(\\angle ABC = \\frac{2\\pi}{3}\\), and \\(\\angle BCD = \\frac{\\pi}{2}\\). \\(\\triangle ABD\\) is folded along \\(BD\\) such that point \\(A\\)'s projection onto plane \\(BCD\\) is point \\(P\\). Giv... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 87,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Reading material: For non-zero real numbers $a$ and $b$, if the value of the fraction $\\frac{(x-a)(x-b)}{x}$ with respect to $x$ is zero, then the solutions are $x_{1}=a$ and $x_{2}=b$. Also, because $\\frac{(x-a)(x-b)}{x}=\\frac{{x}^{2}-(a+b)x+ab}{x}=x+\\frac{ab}{x}-\\left(a+... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 88,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two friends agree to meet at a specific place between 12:00 PM and 12:30 PM. The first one to arrive waits for the other for 20 minutes before leaving. Find the probability that the friends will meet, assuming each chooses their arrival time randomly (between 12:00 PM and 12:30... | math | {
"style": "rule",
"ground_truth": "8/9"
} | {
"split": "train",
"index": 89,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the product of the least common multiple and the greatest common factor of $20$ and $90$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1800"
} | {
"split": "train",
"index": 90,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The polar equation of circle C is $\\rho = 2\\cos(θ + \\frac{π}{4})$. The parametric equation of line l is \n$$\n\\begin{cases}\n x= \\sqrt{2}t \\\\\n y= \\sqrt{2}t+4\\sqrt{2}\n\\end{cases}\n$$\n(where t is the parameter). A tangent is drawn from a point P on line l to circle... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{6}"
} | {
"split": "train",
"index": 91,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The exterior angles of a triangle are proportional to the numbers 5: 7: 8. Find the angle between the altitudes of this triangle drawn from the vertices of its smaller angles. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 92,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three spheres with radii $11,$ $13,$ and $19$ are mutually externally tangent. A plane intersects the spheres in three congruent circles centered at $A,$ $B,$ and $C,$ respectively, and the centers of the spheres all lie on the same side of this plane. Suppose that $AB^2 = 560.... | math | {
"style": "rule",
"ground_truth": "756"
} | {
"split": "train",
"index": 93,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An equiangular hexagon has side lengths $1,1, a, 1,1, a$ in that order. Given that there exists a circle that intersects the hexagon at 12 distinct points, we have $M<a<N$ for some real numbers $M$ and $N$. Determine the minimum possible value of the ratio $\\frac{N}{M}$. Let's... | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{3}+3}{2}"
} | {
"split": "train",
"index": 94,
"solution": "We claim that the greatest possible value of $M$ is $\\sqrt{3}-1$, whereas the least possible value of $N$ is 3 . To begin, note that the condition requires the circle to intersect each side of the hexagon at two points on its interior. This implies that the center mu... |
deepscaler | [
{
"role": "user",
"content": "We roll a fair die consecutively until the sum of the numbers obtained, \\( S \\), exceeds 100. What is the most probable value of \\( S \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "101"
} | {
"split": "train",
"index": 95,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Bricklayer Brenda takes $9$ hours to build a chimney alone, and bricklayer Brandon takes $10$ hours to build it alone. When they work together, they talk a lot, and their combined output decreases by $10$ bricks per hour. Working together, they build the chimney in $5$ hours. H... | math | {
"style": "rule",
"ground_truth": "900"
} | {
"split": "train",
"index": 96,
"solution": "1. **Define the Variables:**\n Let $x$ be the total number of bricks in the chimney.\n\n2. **Set Up Individual Rates:**\n - Brenda's rate of building is $\\frac{x}{9}$ bricks per hour (since she can build the chimney alone in 9 hours).\n - Brandon's rate of buil... |
deepscaler | [
{
"role": "user",
"content": "Simplify the expression and then evaluate: $(a-2b)(a^2+2ab+4b^2)-a(a-5b)(a+3b)$, where $a=-1$ and $b=1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-21"
} | {
"split": "train",
"index": 97,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle XOY$ be a right-angled triangle with $m\\angle XOY = 90^{\\circ}$. Let $M$ and $N$ be the midpoints of legs $OX$ and $OY$, respectively. Given that $XN=19$ and $YM=22$, find $XY$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 98,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The three roots of the cubic $ 30 x^3 - 50x^2 + 22x - 1$ are distinct real numbers strictly between $ 0$ and $ 1$. If the roots are $p$, $q$, and $r$, what is the sum\n\\[ \\frac{1}{1-p} + \\frac{1}{1-q} +\\frac{1}{1-r} ?\\] Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 99,
"solution": ""
} |
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