data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "A biologist sequentially placed 150 beetles into ten jars. In each subsequent jar, he placed more beetles than in the previous one. The number of beetles in the first jar is at least half of the number of beetles in the tenth jar. How many beetles are in the sixth jar? Let's th... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 1200,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many squares of side at least $8$ have their four vertices in the set $H$, where $H$ is defined by the points $(x,y)$ with integer coordinates, $-8 \\le x \\le 8$ and $-8 \\le y \\le 8$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "285"
} | {
"split": "train",
"index": 1201,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The symbol \\( R_{k} \\) represents an integer whose decimal representation consists of \\( k \\) consecutive 1s. For example, \\( R_{3} = 111 \\), \\( R_{5} = 11111 \\), and so on. If \\( R_{4} \\) divides \\( R_{24} \\), the quotient \\( Q = \\frac{R_{24}}{R_{4}} \\) is an in... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 1202,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A set S contains triangles whose sides have integer lengths less than 7, and no two elements of S are congruent or similar. Calculate the largest number of elements that S can have. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 1203,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system, a polar coordinate system is established with the origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. Given the curve $C:ρ\\sin^2θ=2a\\cos θ (a > 0)$, the line $l:\\begin{cases}x=-2+\\frac{\\sqrt{2}}{2}t\\\\y=-4... | math | {
"style": "rule",
"ground_truth": "a=1"
} | {
"split": "train",
"index": 1204,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $F=\\log\\dfrac{1+x}{1-x}$. Find a new function $G$ by replacing each $x$ in $F$ by $\\dfrac{3x+x^3}{1+3x^2}$, and simplify.\nThe simplified expression $G$ is equal to: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3F"
} | {
"split": "train",
"index": 1205,
"solution": "1. **Substitute $x$ with $\\frac{3x+x^3}{1+3x^2}$ in $F$:**\n \\[\n F = \\log \\frac{1+x}{1-x} \\quad \\text{becomes} \\quad G = \\log \\frac{1 + \\frac{3x+x^3}{1+3x^2}}{1 - \\frac{3x+x^3}{1+3x^2}}\n \\]\n\n2. **Simplify the expression inside the logarithm:**\... |
deepscaler | [
{
"role": "user",
"content": "The radius of the Earth measures approximately $6378 \\mathrm{~km}$ at the Equator. Suppose that a wire is adjusted exactly over the Equator.\n\nNext, suppose that the length of the wire is increased by $1 \\mathrm{~m}$, so that the wire and the Equator are concentric circles a... | math | {
"style": "rule",
"ground_truth": "0.159"
} | {
"split": "train",
"index": 1206,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle \\( \\triangle ABC \\) where \\(\\angle ACB = 90^\\circ\\), \\(CA = 3\\), and \\(CB = 4\\), there is a point \\(P\\) inside \\(\\triangle ABC\\) such that the sum of the distances from \\(P\\) to the three sides is \\(\\frac{13}{5}\\). Find the length of the l... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 1207,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The integer 2019 can be formed by placing two consecutive two-digit positive integers, 19 and 20, in decreasing order. What is the sum of all four-digit positive integers greater than 2019 that can be formed in this way? Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "478661"
} | {
"split": "train",
"index": 1208,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$ the medians $\\overline{AD}$ and $\\overline{CE}$ have lengths $18$ and $27$, respectively, and $AB=24$. Extend $\\overline{CE}$ to intersect the circumcircle of $ABC$ at $F$. The area of triangle $AFB$ is $m\\sqrt{n}$, where $m$ and $n$ are positive integers ... | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 1209,
"solution": "Use the same diagram as in Solution 1. Call the centroid $P$. It should be clear that $PE=9$, and likewise $AP=12$, $AE=12$. Then, $\\sin \\angle AEP = \\frac{\\sqrt{55}}{8}$. Power of a Point on $E$ gives $FE=\\frac{16}{3}$, and the area of $AFB$ is $AE * EF* \\sin... |
deepscaler | [
{
"role": "user",
"content": "Find the $3 \\times 3$ matrix $\\mathbf{M}$ such that for a $3 \\times 3$ matrix $\\mathbf{N},$ $\\mathbf{M} \\mathbf{N}$ is the result of swapping the first row and second row of $\\mathbf{N},$ and doubling the third row of $\\mathbf{N}.$ In other words,\n\\[\\mathbf{M} \\beg... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 0 & 1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 2 \\end{pmatrix}"
} | {
"split": "train",
"index": 1210,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A quadrilateral is inscribed in a circle with a radius of 13. The diagonals of the quadrilateral are perpendicular to each other. One of the diagonals is 18, and the distance from the center of the circle to the point where the diagonals intersect is \\( 4 \\sqrt{6} \\).\n\nFin... | math | {
"style": "rule",
"ground_truth": "18 \\sqrt{161}"
} | {
"split": "train",
"index": 1211,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An enterprise has four employees participating in a vocational skills assessment. Each employee can draw any one of the four available assessment projects to participate in. Calculate the probability that exactly one project is not selected. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{16}"
} | {
"split": "train",
"index": 1212,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The endpoints of a line segment \\\\(AB\\\\) with a fixed length of \\\\(3\\\\) move on the parabola \\\\(y^{2}=x\\\\). Let \\\\(M\\\\) be the midpoint of the line segment \\\\(AB\\\\). The minimum distance from \\\\(M\\\\) to the \\\\(y\\\\)-axis is \\_\\_\\_\\_\\_\\_. Let's t... | math | {
"style": "rule",
"ground_truth": "\\dfrac{5}{4}"
} | {
"split": "train",
"index": 1213,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circle $A$ has its center at $A(4, 4)$ and has a radius of 4 units. Circle $B$ has its center at $B(12, 4)$ and has a radius of 4 units. What is the area of the gray region bound by the circles and the $x$-axis? Express your answer in terms of $\\pi$. [asy]\nimport olympiad; si... | math | {
"style": "rule",
"ground_truth": "32 - 8\\pi"
} | {
"split": "train",
"index": 1214,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a triangle in which $AB=AC$ . Suppose the orthocentre of the triangle lies on the incircle. Find the ratio $\\frac{AB}{BC}$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3/4"
} | {
"split": "train",
"index": 1215,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The English college entrance examination consists of two parts: listening and speaking test with a full score of $50$ points, and the English written test with a full score of $100$ points. The English listening and speaking test is conducted twice. If a student takes both test... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{10}"
} | {
"split": "train",
"index": 1216,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The polynomial $f(x)=x^{3}-3 x^{2}-4 x+4$ has three real roots $r_{1}, r_{2}$, and $r_{3}$. Let $g(x)=x^{3}+a x^{2}+b x+c$ be the polynomial which has roots $s_{1}, s_{2}$, and $s_{3}$, where $s_{1}=r_{1}+r_{2} z+r_{3} z^{2}$, $s_{2}=r_{1} z+r_{2} z^{2}+r_{3}, s_{3}=r_{1} z^{2}... | math | {
"style": "rule",
"ground_truth": "-26"
} | {
"split": "train",
"index": 1217,
"solution": "Note that $z=e^{\\frac{2 \\pi}{3} i}=\\cos \\frac{2 \\pi}{3}+i \\sin \\frac{2 \\pi}{3}$, so that $z^{3}=1$ and $z^{2}+z+1=0$. Also, $s_{2}=s_{1} z$ and $s_{3}=s_{1} z^{2}$. Then, the sum of the coefficients of $g(x)$ is $g(1)=\\left(1-s_{1}\\right)\\left(1-s_{2}\\ri... |
deepscaler | [
{
"role": "user",
"content": "Find the number of terms $n \\leq 2012$ such that $a_{n}=\\frac{3^{n+1}-1}{2}$ is 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": 1218,
"solution": "To demonstrate that $7 \\mid a_{n}=\\frac{3^{n+1}-1}{2}$ is equivalent to showing that $3^{n+1}-1 \\equiv 0 \\bmod 7$. However $3^{n+1}-1=0 \\bmod 7 \\Longrightarrow n+1 \\equiv 0 \\bmod 6$, hence the values of $n \\leq 2012$ are $5,11,17, \\ldots, 2009$, which is i... |
deepscaler | [
{
"role": "user",
"content": "The square root of $x$ is greater than 2 and less than 4. How many integer values of $x$ satisfy this condition? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 1219,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the expansion of $(a + b)^n$ there are $n + 1$ dissimilar terms. The number of dissimilar terms in the expansion of $(a + b + c)^{10}$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "66"
} | {
"split": "train",
"index": 1220,
"solution": "To find the number of dissimilar terms in the expansion of $(a + b + c)^{10}$, we can use the concept of combinations to determine the number of distinct terms generated by the expansion.\n\n1. **Understanding the Expansion**:\n The expression $(a + b + c)^{10}$ c... |
deepscaler | [
{
"role": "user",
"content": "Circles with centers $P, Q$ and $R$, having radii $1, 2$ and $3$, respectively, lie on the same side of line $l$ and are tangent to $l$ at $P', Q'$ and $R'$, respectively, with $Q'$ between $P'$ and $R'$. The circle with center $Q$ is externally tangent to each of the other two... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 1221,
"solution": "To solve this problem, we need to determine the positions of the centers $P$, $Q$, and $R$ of the circles relative to each other and then calculate the area of triangle $PQR$.\n\n1. **Positioning the Circles:**\n - Since the circles are tangent to line $l$ at poin... |
deepscaler | [
{
"role": "user",
"content": "Find the remainder when $$88134 + 88135 + 88136 + 88137 + 88138 + 88139$$is divided by 9. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 1222,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A granite pedestal. When constructing a square foundation and a cubic pedestal for a monument, granite cubic blocks of size \\(1 \\times 1\\) meter were used. The pedestal used exactly as many blocks as the square foundation upon which it stood. All the blocks were used whole a... | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 1223,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $x,$ $y,$ and $z$ are three positive numbers that satisfy the equations $xyz = 1,$ $x + \\frac {1}{z} = 5,$ and $y + \\frac {1}{x} = 29.$ Find $z + \\frac {1}{y}.$ 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": 1224,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An ellipse has foci at $(9, 20)$ and $(49, 55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its major axis? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 1225,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is preparing to offer a discount on the selling price, but the profit margin must not be less than $10\\%$. Find the maximum discount rate that can be offered. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "8.8\\%"
} | {
"split": "train",
"index": 1226,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $AB$ be a diameter of a circle and let $C$ be a point on the segement $AB$ such that $AC : CB = 6 : 7$ . Let $D$ be a point on the circle such that $DC$ is perpendicular to $AB$ . Let $DE$ be the diameter through $D$ . If $[XYZ]$ denotes the area of the tr... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 1227,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $S_{n}$ is the sum of the first $n$ terms of the sequence ${a_{n}}$, and $S_{n}=n^{2}-4n+4$.\n(1) Find the general term formula of the sequence ${a_{n}}$;\n(2) Let ${c_{n}}$ be a sequence where all $c_{n}$ are non-zero, and the number of positive integers $k$ satisfy... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 1228,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Little Twelve Basketball Conference has two divisions, with six teams in each division. Each team plays each of the other teams in its own division twice and every team in the other division once. How many conference games are scheduled? Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 1229,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of natural numbers that cannot be written as the sum of three squares. Legendre's three-square theorem states that $S$ consists of precisely the integers of the form $4^a(8b+7)$ where $a$ and $b$ are nonnegative integers. Find the smallest $n\\in\\... | math | {
"style": "rule",
"ground_truth": "111"
} | {
"split": "train",
"index": 1230,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ with the first term being a positive number, and the sum of the first $n$ terms is $S_n$. If $a_{1006}$ and $a_{1007}$ are the two roots of the equation $x^2 - 2012x - 2011 = 0$, then the maximum value of the positive integer $n$ for whi... | math | {
"style": "rule",
"ground_truth": "2011"
} | {
"split": "train",
"index": 1231,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( b \\) is an even number between 1 and 11 (inclusive), and \\( c \\) is any natural number, determine the number of quadratic equations \\( x^{2} + b x + c = 0 \\) that have two distinct real roots. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 1232,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A flock of geese is flying, and a lone goose flies towards them and says, \"Hello, a hundred geese!\" The leader of the flock responds, \"No, we are not a hundred geese! If there were as many of us as there are now, plus the same amount, plus half of that amount, plus a quarter... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 1233,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A malfunctioning digital clock shows the time $9: 57 \\mathrm{AM}$; however, the correct time is $10: 10 \\mathrm{AM}$. There are two buttons on the clock, one of which increases the time displayed by 9 minutes, and another which decreases the time by 20 minutes. What is the mi... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 1234,
"solution": "We need to increase the time by 13 minutes. If we click the 9 minute button $a$ times and the 20 minute button $b$ times, then we must have $9 a-20 b=13$. Note that if this equation is satisfied, then $b$ increases as $a$ increases, so it suffices to minimize $a$. T... |
deepscaler | [
{
"role": "user",
"content": "Let $x,$ $y,$ $z$ be positive real numbers such that $x + y + z = 1.$ Find the minimum value of\n\\[\\frac{1}{x + y} + \\frac{1}{x + z} + \\frac{1}{y + z}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"split": "train",
"index": 1235,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In writing the integers from 10 through 99 inclusive, how many times is the digit 6 written? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 1236,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let tetrahedron $ABCD$ have $AD=BC=30$, $AC=BD=40$, and $AB=CD=50$. For any point $X$ in space, suppose $g(X)=AX+BX+CX+DX$. Determine the least possible value of $g(X)$, expressed as $p\\sqrt{q}$ where $p$ and $q$ are positive integers with $q$ not divisible by the square of an... | math | {
"style": "rule",
"ground_truth": "101"
} | {
"split": "train",
"index": 1237,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all positive real numbers \\(c\\) such that the graph of \\(f: \\mathbb{R} \\rightarrow \\mathbb{R}\\) given by \\(f(x) = x^3 - cx\\) has the property that the circle of curvature at any local extremum is centered at a point on the \\(x\\)-axis. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 1238,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The line \\(y = -\\frac{2}{3}x + 10\\) crosses the \\(x\\)-axis at point \\(P\\) and the \\(y\\)-axis at point \\(Q\\). Point \\(T(r, s)\\) is on the segment \\(PQ\\). If the area of \\(\\triangle POQ\\) is four times the area of \\(\\triangle TOP\\), find the value of \\(r + s... | math | {
"style": "rule",
"ground_truth": "13.75"
} | {
"split": "train",
"index": 1239,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\frac{1}{4}$ of all ninth graders are paired with $\\frac{1}{3}$ of all sixth graders, what fraction of the total number of sixth and ninth graders are paired? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{7}"
} | {
"split": "train",
"index": 1240,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In Pascal's triangle, compute the seventh element in Row 20. Afterward, determine how many times greater this element is compared to the third element in the same row. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "204"
} | {
"split": "train",
"index": 1241,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an acute-angled triangle $ABC$ , the point $O$ is the center of the circumcircle, and the point $H$ is the orthocenter. It is known that the lines $OH$ and $BC$ are parallel, and $BC = 4OH $ . Find the value of the smallest angle of triangle $ ABC $ .\n\n(Black Ma... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 1242,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{m}=(1,3\\cos \\alpha)$ and $\\overrightarrow{n}=(1,4\\tan \\alpha)$, where $\\alpha \\in (-\\frac{\\pi}{2}, \\;\\;\\frac{\\pi}{2})$, and their dot product $\\overrightarrow{m} \\cdot \\overrightarrow{n} = 5$.\n(I) Find the magnitude of $|\\overri... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 1243,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When the height of a cylinder is doubled and its radius is increased by $200\\%$, the cylinder's volume is multiplied by a factor of $X$. 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": "18"
} | {
"split": "train",
"index": 1244,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the sequence\n\n$$\na_{n}=\\cos (\\underbrace{100 \\ldots 0^{\\circ}}_{n-1})\n$$\n\nFor example, $a_{1}=\\cos 1^{\\circ}, a_{6}=\\cos 100000^{\\circ}$.\n\nHow many of the numbers $a_{1}, a_{2}, \\ldots, a_{100}$ are positive? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 1245,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many prime positive integers are divisors of 555? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 1246,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, a line passing through the midpoint $E$ of the median $AD$ intersects sides $AB$ and $AC$ at points $M$ and $N$ respectively. Let $\\overrightarrow{AM} = x\\overrightarrow{AB}$ and $\\overrightarrow{AN} = y\\overrightarrow{AC}$ ($x, y \\neq 0$), th... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{4}"
} | {
"split": "train",
"index": 1247,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x + 2y= 4$ and $xy = -8$, what is the value of $x^2 + 4y^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 1248,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the focus of the parabola $C: y^{2}=4x$ is $F$, two lines $l_{1}$ and $l_{2}$ are drawn passing through point $F$. Line $l_{1}$ intersects the parabola $C$ at points $A$ and $B$, while line $l_{2}$ intersects the parabola $C$ at points $M$ and $N$. If the product of ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 1249,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive $x$-axis as the polar axis, the parametric equation of curve $C_1$ is \n$$\n\\begin{cases} \nx=2+ \\sqrt {3}\\cos \\theta \\\\ \ny= \\sqrt {3}\\sin \\theta\n\\end{cases}\n(\\theta \\text{ is ... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt {2}"
} | {
"split": "train",
"index": 1250,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sally's salary in 2006 was $\\$ 37,500 $. For 2007 she got a salary increase of $ x $ percent. For 2008 she got another salary increase of $ x $ percent. For 2009 she got a salary decrease of $ 2x $ percent. Her 2009 salary is $ \\ $34,825$ . Suppose instead, Sally had gotten ... | math | {
"style": "rule",
"ground_truth": "34825"
} | {
"split": "train",
"index": 1251,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A random permutation $a=\\left(a_{1}, a_{2}, \\ldots, a_{40}\\right)$ of $(1,2, \\ldots, 40)$ is chosen, with all permutations being equally likely. William writes down a $20 \\times 20$ grid of numbers $b_{i j}$ such that $b_{i j}=\\max \\left(a_{i}, a_{j+20}\\right)$ for all ... | math | {
"style": "rule",
"ground_truth": "\\frac{10}{13}"
} | {
"split": "train",
"index": 1252,
"solution": "We can deduce information about $a$ from the grid $b$ by looking at the largest element of it, say $m$. If $m$ fills an entire row, then the value of $a$ corresponding to this row must be equal to $m$. Otherwise, $m$ must fill an entire column, and the value of $a$ ... |
deepscaler | [
{
"role": "user",
"content": "If $\\sqrt[3]{3}$ is approximated as $1.442$, calculate the value of $\\sqrt[3]{3}-3\\sqrt[3]{3}-98\\sqrt[3]{3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-144.2"
} | {
"split": "train",
"index": 1253,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a circle with center \\(O\\) and radius \\(OD\\) perpendicular to chord \\(AB\\), intersecting \\(AB\\) at point \\(C\\). Line segment \\(AO\\) is extended to intersect the circle at point \\(E\\). If \\(AB = 8\\) and \\(CD = 2\\), what is the area of \\(\\triangle BCE\\)... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 1254,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a nonnegative integer \\(n\\), let \\(r_{11}(7n)\\) stand for the remainder left when \\(n\\) is divided by \\(11.\\) For example, \\(r_{11}(7 \\cdot 3) = 10.\\)\n\nWhat is the \\(15^{\\text{th}}\\) entry in an ordered list of all nonnegative integers \\(n\\) that satisfy $... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 1255,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rationalize the denominator of $\\displaystyle \\frac{1}{\\sqrt{2} + \\sqrt{3} + \\sqrt{7}}$, and write your answer in the form \\[\n\\frac{A\\sqrt{2} + B\\sqrt{3} + C\\sqrt{7} + D\\sqrt{E}}{F},\n\\]where everything is in simplest radical form and the fraction is in lowest term... | math | {
"style": "rule",
"ground_truth": "57"
} | {
"split": "train",
"index": 1256,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point $(x, y)$ is randomly selected such that $0 \\leq x \\leq 3$ and $0 \\leq y \\leq 3$. What is the probability that $x + 2y \\leq 6$? 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}{4}"
} | {
"split": "train",
"index": 1257,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An equilateral triangle has an area of $64\\sqrt{3}$ $\\text{cm}^2$. If each side of the triangle is decreased by 4 cm, by how many square centimeters is the area decreased? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28\\sqrt{3}"
} | {
"split": "train",
"index": 1258,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $5^{35}-6^{21}$ is evaluated, what is the units (ones) digit? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 1259,
"solution": "First, we note that $5^{35}-6^{21}$ is a positive integer. Second, we note that any positive integer power of 5 has a units digit of 5. Similarly, each power of 6 has a units digit of 6. Therefore, $5^{35}$ has a units digit of 5 and $6^{21}$ has a units digit of 6.... |
deepscaler | [
{
"role": "user",
"content": "Using the six digits 0, 1, 2, 3, 4, 5 to form a six-digit number without repeating any digit, how many such numbers are there where the unit digit is less than the ten's digit? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 1260,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In \\(\\triangle ABC\\), \\(BC = a\\), \\(CA = b\\), \\(AB = c\\). If \\(2a^{2} + b^{2} + c^{2} = 4\\), then the maximum area of \\(\\triangle ABC\\) is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{5}"
} | {
"split": "train",
"index": 1261,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find \\[\\left|\\left(\\frac 35+\\frac 45i\\right)^6\\right|\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 1262,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of all positive integers from 1 through 1000 that are not perfect squares. What is the length of the longest, non-constant, arithmetic sequence that consists of elements of $S$ ? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "333"
} | {
"split": "train",
"index": 1263,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( p, q, r, s, t, u, v, \\) and \\( w \\) be real numbers such that \\( pqrs = 16 \\) and \\( tuvw = 25 \\). Find the minimum value of\n\\[\n(pt)^2 + (qu)^2 + (rv)^2 + (sw)^2.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 1264,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $\\frac{a^{8}-6561}{81 a^{4}} \\cdot \\frac{3 a}{a^{2}+9}$, given that $\\frac{a}{3}-\\frac{3}{a}=4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 1265,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fair standard six-sided dice is tossed three times. Given that the sum of the first two tosses equal the third, what is the probability that at least one \"2\" is tossed? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{15}"
} | {
"split": "train",
"index": 1266,
"solution": "1. **Identify possible outcomes for the third toss**: The third toss, denoted as $X_3$, must be at least 2 because the minimum sum of two dice ($X_1$ and $X_2$) is 2. Thus, $X_3$ can be 2, 3, 4, 5, or 6.\n\n2. **List possible combinations for each value of $X_3$**:\... |
deepscaler | [
{
"role": "user",
"content": "A $\\textit{composite number}$ is a number that has two or more prime factors. The number 87 can be expressed as the sum of two composite numbers in many ways. What is the minimum positive difference between two such numbers? Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 1267,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the maximal size of a set of positive integers with the following properties:\n\n1. The integers consist of digits from the set {1,2,3,4,5,6}.\n2. No digit occurs more than once in the same integer.\n3. The digits in each integer are in increasing order.\n4. Any two i... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 1268,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pentagon is formed by placing an equilateral triangle on top of a square. Calculate the percentage of the pentagon's total area that is made up by the equilateral triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25.4551\\%"
} | {
"split": "train",
"index": 1269,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the positive divisors of 3960 are multiples of 5? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 1270,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider an isosceles right triangle with leg lengths of 1 each. Inscribed in this triangle is a square in such a way that one vertex of the square coincides with the right-angle vertex of the triangle. Another square with side length $y$ is inscribed in an identical isosceles ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 1271,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$A B C D$ is a cyclic quadrilateral in which $A B=3, B C=5, C D=6$, and $A D=10 . M, I$, and $T$ are the feet of the perpendiculars from $D$ to lines $A B, A C$, and $B C$ respectively. Determine the value of $M I / I T$. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "\\frac{25}{9}"
} | {
"split": "train",
"index": 1272,
"solution": "Quadrilaterals $A M I D$ and $D I C T$ are cyclic, having right angles $\\angle A M D, \\angle A I D$, and $\\angle C I D, \\angle C T D$ respectively. We see that $M, I$, and $T$ are collinear. For, $m \\angle M I D=\\pi-m \\angle D A M=$ $\\pi-m \\angle D A B=m \\... |
deepscaler | [
{
"role": "user",
"content": "Let S<sub>n</sub> represent the sum of the first n terms of the arithmetic sequence {a<sub>n</sub>}. If S<sub>5</sub> = 2S<sub>4</sub> and a<sub>2</sub> + a<sub>4</sub> = 8, find the value of a<sub>5</sub>. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 1273,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cube, all of whose surfaces are painted, is cut into $1000$ smaller cubes of the same size. Find the expected value $E(X)$, where $X$ denotes the number of painted faces of a small cube randomly selected. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 1274,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many distinct ways can I arrange my six keys on a keychain, if I want to put my house key next to my car key and additionally ensure my bike key is next to my mailbox key? Two arrangements are not considered different if the keys are in the same order (or can be made to ... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 1275,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Fully simplify the following expression: $[(2+3+4+5)\\div2] + [(2\\cdot5+8)\\div3]$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 1276,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that in triangle ABC, the lengths of the sides opposite to angles A, B, and C are a, b, and c respectively, and b = 3, c = 1, and A = 2B.\n(1) Find the value of a;\n(2) Find the value of sin(A + $\\frac{\\pi}{4}$). Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "\\frac{4 - \\sqrt{2}}{6}"
} | {
"split": "train",
"index": 1277,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse E: $$\\frac {x^{2}}{a^{2}}+ \\frac {y^{2}}{b^{2}} = 1$$ ($a > b > 0$) with a focal length of $2\\sqrt{3}$, and the ellipse passes through the point $(\\sqrt{3}, \\frac{1}{2})$.\n(Ⅰ) Find the equation of ellipse E;\n(Ⅱ) Through point P$(-2, 0)$, draw two lines w... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 1278,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\{a_{n}\\}_{n \\geq 1}$ be an arithmetic sequence and \\{g_{n}\\}_{n \\geq 1}$ be a geometric sequence such that the first four terms of \\{a_{n}+g_{n}\\}$ are $0,0,1$, and 0 , in that order. What is the 10th term of \\{a_{n}+g_{n}\\}$ ? Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "-54"
} | {
"split": "train",
"index": 1279,
"solution": "Let the terms of the geometric sequence be $a, r a, r^{2} a, r^{3} a$. Then, the terms of the arithmetic sequence are $-a,-r a,-r^{2} a+1,-r^{3} a$. However, if the first two terms of this sequence are $-a,-r a$, the next two terms must also be $(-2 r+1) a,(-3 r+2) ... |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{u},$ $\\mathbf{v},$ and $\\mathbf{w}$ be nonzero vectors, no two of which are parallel, such that\n\\[(\\mathbf{u} \\times \\mathbf{v}) \\times \\mathbf{w} = \\frac{1}{4} \\|\\mathbf{v}\\| \\|\\mathbf{w}\\| \\mathbf{u}.\\]\nLet $\\phi$ be the angle between $\\math... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{15}}{4}"
} | {
"split": "train",
"index": 1280,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $x, y \\in (-1, 1)$, find the minimum value of the expression $$\\sqrt {(x+1)^{2}+(y-1)^{2}}+\\sqrt {(x+1)^{2}+(y+1)^{2}}+\\sqrt {(x-1)^{2}+(y+1)^{2}}+\\sqrt {(x-1)^{2}+(y-1)^{2}}.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{2}"
} | {
"split": "train",
"index": 1281,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_0$, $a_1$, $a_2$, $\\dots$ be an infinite sequence of real numbers such that $a_0 = \\frac{5}{13}$ and\n\\[\n a_{n} = 2 a_{n-1}^2 - 1\n\\]for every positive integer $n$. Let $c$ be the smallest number such that for every positive integer $n$, the product of the first $... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 1282,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A teacher offers candy to her class of 50 students, with the mean number of pieces taken by each student being 7. If every student takes at least one candy but no more than 20 candies, what is the greatest number of pieces one student could have taken? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 1283,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\lfloor x\\rfloor$ denote the largest integer less than or equal to $x$, and let $\\{x\\}$ denote the fractional part of $x$. For example, $\\lfloor\\pi\\rfloor=3$, and $\\{\\pi\\}=0.14159 \\ldots$, while $\\lfloor 100\\rfloor=100$ and $\\{100\\}=0$. If $n$ is the largest... | math | {
"style": "rule",
"ground_truth": "\\frac{2014}{2015}"
} | {
"split": "train",
"index": 1284,
"solution": "Note that $n=\\lfloor n\\rfloor+\\{n\\}$, so $$\\frac{\\lfloor n\\rfloor}{n} =\\frac{\\lfloor n\\rfloor}{\\lfloor n\\rfloor+\\{n\\}} =\\frac{2015}{2016} \\Longrightarrow 2016\\lfloor n\\rfloor =2015\\lfloor n\\rfloor+2015\\{n\\} \\Longrightarrow\\lfloor n\\rfloor =2... |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $(xOy)$, let $l$ be a line with an angle of inclination $\\alpha$ and parametric equations $\\begin{cases} x = 3 + t\\cos\\alpha \\\\ y = t\\sin\\alpha \\end{cases}$ (where $t$ is a parameter). The line $l$ intersects the curve $C$: $\\begin{c... | math | {
"style": "rule",
"ground_truth": "\\frac{40}{3}"
} | {
"split": "train",
"index": 1285,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let **v** be a vector such that\n\\[\n\\left\\| \\mathbf{v} + \\begin{pmatrix} 4 \\\\ 2 \\end{pmatrix} \\right\\| = 10.\n\\]\nFind the smallest possible value of $\\|\\mathbf{v}\\|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10 - 2\\sqrt{5}"
} | {
"split": "train",
"index": 1286,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( AE \\) and \\( BD \\) are straight lines that intersect at \\( C \\), \\( BD = 16 \\), \\( AB = 9 \\), \\( CE = 5 \\), and \\( DE = 3 \\), calculate the length of \\( AC \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 1287,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A football association stipulates that in the league, a team earns $a$ points for a win, $b$ points for a draw, and 0 points for a loss, where $a$ and $b$ are real numbers such that $a > b > 0$. If a team has 2015 possible total scores after $n$ games, find the minimum value of... | math | {
"style": "rule",
"ground_truth": "62"
} | {
"split": "train",
"index": 1288,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $x, y$, and $z$ are non-negative real numbers such that $x+y+z=1$. What is the maximum possible value of $x+y^{2}+z^{3}$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 1289,
"solution": "Since $0 \\leq y, z \\leq 1$, we have $y^{2} \\leq y$ and $z^{3} \\leq z$. Therefore $x+y^{2}+z^{3} \\leq x+y+z=1$. We can get $x+y^{2}+z^{3}=1$ by setting $(x, y, z)=(1,0,0)$."
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $\\alpha,\\beta,\\gamma\\in\\{-2,3\\}$ are chosen such that\n\\[M=\\max_{x\\in\\mathbb{R}}\\min_{y\\in\\mathbb{R}_{\\ge0}}\\alpha x+\\beta y+\\gamma xy\\]\nis finite and positive (note: $\\mathbb{R}_{\\ge0}$ is the set of nonnegative real numbers). What is the sum o... | math | {
"style": "rule",
"ground_truth": "13/2"
} | {
"split": "train",
"index": 1290,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Elena earns $\\$ 13.25$ per hour working at a store. How much does Elena earn in 4 hours? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\$53.00"
} | {
"split": "train",
"index": 1291,
"solution": "Elena works for 4 hours and earns $\\$ 13.25$ per hour. This means that she earns a total of $4 \\times \\$ 13.25=\\$ 53.00$."
} |
deepscaler | [
{
"role": "user",
"content": "On the sides \\( BC \\) and \\( AC \\) of triangle \\( ABC \\), points \\( M \\) and \\( N \\) are taken respectively such that \\( CM:MB = 1:3 \\) and \\( AN:NC = 3:2 \\). Segments \\( AM \\) and \\( BN \\) intersect at point \\( K \\). Find the area of quadrilateral \\( CMKN ... | math | {
"style": "rule",
"ground_truth": "3/20"
} | {
"split": "train",
"index": 1292,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( A \\), \\( B \\), and \\( C \\) are any three non-collinear points on a plane, and point \\( O \\) is inside \\( \\triangle ABC \\) such that:\n\\[\n\\angle AOB = \\angle BOC = \\angle COA = 120^\\circ.\n\\]\nFind the maximum value of \\( \\frac{OA + OB + OC}{AB ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 1293,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four balls numbered $1, 2, 3$, and $4$ are placed in an urn. One ball is drawn, its number noted, and then returned to the urn. This process is repeated three times. If the sum of the numbers noted is $9$, determine the probability that the ball numbered $3$ was drawn all three... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{10}"
} | {
"split": "train",
"index": 1294,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ben rolls four fair 10-sided dice, each numbered with numbers from 1 to 10. What is the probability that exactly two of the dice show a prime number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{216}{625}"
} | {
"split": "train",
"index": 1295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be the number of complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. Find the remainder when $N$ is divided by $1000$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "440"
} | {
"split": "train",
"index": 1296,
"solution": "As mentioned in solution one, for the difference of two complex numbers to be real, their imaginary parts must be equal. We use exponential form of complex numbers. Let $z = e^{i \\theta}$. We have two cases to consider. Either $z^{6!} = z^{5!}$, or $z^{6!}$ and $z^... |
deepscaler | [
{
"role": "user",
"content": "Find the number of square units in the area of the shaded region. [asy]\nsize(200);\ndraw((0,0)--(30,0)--(30,30)--(0, 30)--cycle);\ndot((30,0));dot((0,0));dot((10,0));dot((30,20));dot((30,30)); dot((20,30));dot((0,30));dot((0,10));\nfilldraw((0,0)--(10,0)--(30,20)--(30,30)--(20... | math | {
"style": "rule",
"ground_truth": "500"
} | {
"split": "train",
"index": 1297,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In quadrilateral \\(ABCD\\), \\(AB = BC = 9\\) cm, \\(AD = DC = 8\\) cm, \\(AB\\) is perpendicular to \\(BC\\), and \\(AD\\) is perpendicular to \\(DC\\). What is the area of quadrilateral \\(ABCD\\) in square centimeters? Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "82.5"
} | {
"split": "train",
"index": 1298,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a point P on the parabola $y^2=4x$, let the distance from point P to the directrix of this parabola be $d_1$, and the distance from point P to the line $x+2y-12=0$ be $d_2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11 \\sqrt{5}}{5}"
} | {
"split": "train",
"index": 1299,
"solution": ""
} |
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