data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "If \\( p \\) and \\( q \\) are positive integers and \\(\\frac{2008}{2009} < \\frac{p}{q} < \\frac{2009}{2010} \\), what is the minimum value of \\( p \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4017"
} | {
"split": "train",
"index": 1400,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The quadrilateral $ABCD$ has the following equality $\\angle ABC=\\angle BCD=150^{\\circ}$. Moreover, $AB=18$ and $BC=24$, the equilateral triangles $\\triangle APB,\\triangle BQC,\\triangle CRD$ are drawn outside the quadrilateral. If $P(X)$ is the perimeter of the polygon $X$... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 1401,
"solution": "\nGiven that the quadrilateral \\(ABCD\\) satisfies \\(\\angle ABC = \\angle BCD = 150^\\circ\\), and that equilateral triangles \\(\\triangle APB\\), \\(\\triangle BQC\\), and \\(\\triangle CRD\\) are drawn outside the quadrilateral. We are provided with the length... |
deepscaler | [
{
"role": "user",
"content": "Petya and Vasya are playing the following game. Petya thinks of a natural number \\( x \\) with a digit sum of 2012. On each turn, Vasya chooses any natural number \\( a \\) and finds out the digit sum of the number \\( |x-a| \\) from Petya. What is the minimum number of turns ... | math | {
"style": "rule",
"ground_truth": "2012"
} | {
"split": "train",
"index": 1402,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that D is a point on the hypotenuse BC of right triangle ABC, and $AC= \\sqrt {3}DC$, $BD=2DC$. If $AD=2 \\sqrt {3}$, then $DC=\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt {6}"
} | {
"split": "train",
"index": 1403,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A shooter fires 5 shots in succession, hitting the target with scores of: $9.7$, $9.9$, $10.1$, $10.2$, $10.1$. The variance of this set of data is __________. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.032"
} | {
"split": "train",
"index": 1404,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a trapezoid \\(ABCD\\). A point \\(M\\) is chosen on its lateral side \\(CD\\) such that \\( \\frac{CM}{MD} = \\frac{4}{3} \\). It turns out that segment \\( BM \\) divides the diagonal \\( AC \\) into two segments, the ratio of the lengths of which is also \\( \\frac{4}{... | math | {
"style": "rule",
"ground_truth": "7/12"
} | {
"split": "train",
"index": 1405,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be a regular $2006$-gon. A diagonal is called [i]good[/i] if its endpoints divide the boundary of $P$ into two parts, each composed of an odd number of sides of $P$. The sides of $P$ are also called [i]good[/i].\nSuppose $P$ has been dissected into triangles by $2003$ d... | math | {
"style": "rule",
"ground_truth": "1003"
} | {
"split": "train",
"index": 1406,
"solution": "\nLet \\( P \\) be a regular 2006-gon. We are tasked with finding the maximum number of isosceles triangles that can be formed by dissecting \\( P \\) using 2003 diagonals such that each triangle has two good sides, where a side is called good if it divides the boun... |
deepscaler | [
{
"role": "user",
"content": "$8[x]$ represents the greatest integer not exceeding the real number $x$. Then, $\\left[\\log _{2} 1\\right]+\\left[\\log _{2} 2\\right]+\\left[\\log _{2} 3\\right]+\\cdots+\\left[\\log _{2} 2012\\right]=$ $\\qquad$ . Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "18084"
} | {
"split": "train",
"index": 1407,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangle whose three sides are all positive integers, with only one side length equal to 5 and not the shortest side, find the number of such triangles. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 1408,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given three points \\(A, B, C\\) forming a triangle with angles \\(30^{\\circ}\\), \\(45^{\\circ}\\), and \\(105^{\\circ}\\). Two of these points are chosen, and the perpendicular bisector of the segment connecting them is drawn. The third point is then reflected across this pe... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 1409,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Equilateral triangle $ABC$ and square $BCDE$ are coplanar, as shown. What is the number of degrees in the measure of angle $CAD$?\n\n[asy]\nsize(70);\ndraw((0,0)--(20,0)--(20,20)--(0,20)--cycle);\ndraw((0,20)--(10,37.3)--(20,20));\ndraw((10,37.3)--(20,0));\nlabel(\"$A$\",(10,37... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 1410,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a quadratic equation \\( x^{2} + bx + c = 0 \\) with roots 98 and 99, within the quadratic function \\( y = x^{2} + bx + c \\), if \\( x \\) takes on values 0, 1, 2, 3, ..., 100, how many of the values of \\( y \\) are divisible by 6? Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 1411,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid \\( KLMN \\), the lengths of the bases are \\( KN = 25 \\), \\( LM = 15 \\), and the lengths of the legs are \\( KL = 6 \\), \\( MN = 8 \\). Find the length of the segment connecting the midpoints of the bases. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 1412,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A convex polyhedron \\( P \\) has 2021 edges. By cutting off a pyramid at each vertex, which uses one edge of \\( P \\) as a base edge, a new convex polyhedron \\( Q \\) is obtained. The planes of the bases of the pyramids do not intersect each other on or inside \\( P \\). How... | math | {
"style": "rule",
"ground_truth": "6063"
} | {
"split": "train",
"index": 1413,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers less than or equal to 5689 contain either the digit '6' or the digit '0'? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2545"
} | {
"split": "train",
"index": 1414,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, $sinB=\\sqrt{2}sinA$, $∠C=105°$, and $c=\\sqrt{3}+1$. Calculate the area of the triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3} + 1}{2}"
} | {
"split": "train",
"index": 1415,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $g(x)$ be a function piecewise defined as \\[g(x) = \\left\\{\n\\begin{array}{cl}\n-x & x\\le 0, \\\\\n2x-41 & x>0.\n\\end{array}\n\\right.\\] If $a$ is negative, find $a$ so that $g(g(g(10.5)))=g(g(g(a)))$. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "a=-30.5"
} | {
"split": "train",
"index": 1416,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alicia has $n$ candies, where $n$ is a positive integer with three digits. If she buys $5$ more, she will have a multiple of $8$. If she loses $8$, she will have a multiple of $5$. What is the smallest possible value of $n$? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "123"
} | {
"split": "train",
"index": 1417,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $2\\frac{1}{4}$ divided by $\\frac{3}{5}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 \\frac{3}{4}"
} | {
"split": "train",
"index": 1418,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the equation $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1$ $(a > b > 0)$, where $M$ and $N$ are the left and right vertices of the ellipse, and $P$ is any point on the ellipse. The slopes of the lines $PM$ and $PN$ are ${k_{1}}$ and ${k_{2}}$ respectively, and ${k_{1}}{k... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 1419,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $f(x)=\\frac{1}{2x+b}$. For what value of $b$ does $f^{-1}(x)=\\frac{1-2x}{2x}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 1420,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the equation of the line parallel to $4x+2y=8$ and passing through the point $(0,1)$? Write the equation in slope-intercept form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "y=-2x+1"
} | {
"split": "train",
"index": 1421,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Along a straight alley, there are 400 streetlights placed at equal intervals, numbered consecutively from 1 to 400. Alla and Boris start walking towards each other from opposite ends of the alley at the same time but with different constant speeds (Alla from the first streetlig... | math | {
"style": "rule",
"ground_truth": "163"
} | {
"split": "train",
"index": 1422,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $x$: $100^3 = 10^x$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 1423,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer divisor of $10!$ is chosen at random. Calculate the probability that the divisor chosen is a perfect square, expressed as a simplified fraction $\\frac{m}{n}$, and find the sum of the numerator and denominator. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 1424,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Any seven points are taken inside or on a square with side length $2$. Determine $b$, the smallest possible number with the property that it is always possible to select one pair of points from these seven such that the distance between them is equal to or less than $b$. Let's ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 1425,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the infinite sum $\\sum_{n=0}^{\\infty}\\binom{2 n}{n} \\frac{1}{5^{n}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 1426,
"solution": "First Solution: Note that $\\binom{2 n}{n} =\\frac{(2 n)!}{n!\\cdot n!}=\\frac{(2 n)(2 n-2)(2 n-4) \\cdots(2)}{n!} \\cdot \\frac{(2 n-1)(2 n-3)(2 n-5) \\cdots(1)}{n!} =2^{n} \\cdot \\frac{(-2)^{n}}{n!}(-\\frac{1}{2})(-\\frac{1}{2}-1)(-\\frac{1}{2}-2) \\cdots(-\\frac... |
deepscaler | [
{
"role": "user",
"content": "Find $a$ if the remainder is constant when $10x^3-7x^2+ax+6$ is divided by $2x^2-3x+1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"split": "train",
"index": 1427,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sphere is inscribed in a cube, and the cube has a surface area of 54 square meters. A second cube is then inscribed within the sphere. A third, smaller sphere is then inscribed within this second cube. What is the surface area of the second cube and the volume of the third sp... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}\\pi}{2}"
} | {
"split": "train",
"index": 1428,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Add 53.463 to 12.9873 and round to the nearest thousandth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "66.450"
} | {
"split": "train",
"index": 1429,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many possible distinct arrangements are there of the letters in the word DOG? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 1430,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of solutions in natural numbers for the equation \\(\\left\\lfloor \\frac{x}{10} \\right\\rfloor = \\left\\lfloor \\frac{x}{11} \\right\\rfloor + 1\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "110"
} | {
"split": "train",
"index": 1431,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A mixture of $30$ liters of paint is $25\\%$ red tint, $30\\%$ yellow tint and $45\\%$ water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint in the new mixture? Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 1432,
"solution": "1. **Calculate the amount of yellow tint in the original mixture:**\n The original mixture contains $30\\%$ yellow tint in $30$ liters. Therefore, the amount of yellow tint is:\n \\[\n 0.30 \\times 30 = 9 \\text{ liters}\n \\]\n\n2. **Add the additional yell... |
deepscaler | [
{
"role": "user",
"content": "Let $T_1$ and $T_2$ be the points of tangency of the excircles of a triangle $ABC$ with its sides $BC$ and $AC$ respectively. It is known that the reflection of the incenter of $ABC$ across the midpoint of $AB$ lies on the circumcircle of triangle $CT_1T_2$ . Fin... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 1433,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex $W$?\n\n[... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"split": "train",
"index": 1434,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of the expression $[ a-(b-c) ] - [(a-b) - c ]$ when $a = 17$, $b=21$ and $c=5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 1435,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Trapezoid $EFGH$ has base $EF = 15$ units and base $GH = 25$ units. Diagonals $EG$ and $FH$ intersect at $Y$. If the area of trapezoid $EFGH$ is $200$ square units, what is the area of triangle $FYG$? Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "46.875"
} | {
"split": "train",
"index": 1436,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cryptographer devises the following method for encoding positive integers. First, the integer is expressed in base $5$. \nSecond, a 1-to-1 correspondence is established between the digits that appear in the expressions in base $5$ and the elements of the set \n$\\{V, W, X, Y,... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 1437,
"solution": "1. **Identify the pattern and the base-5 system**: The problem states that three consecutive integers are coded as $VYZ, VYX, VVW$. We know that in base-5, each digit represents powers of 5, starting from the rightmost digit (units place) to the leftmost digit.\n\n2... |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow {a}$ and $\\overrightarrow {b}$ that satisfy $| \\overrightarrow {a} |=5$, $| \\overrightarrow {b} |=3$, and $( \\overrightarrow {a} - \\overrightarrow {b} )(2 \\overrightarrow {a} + 3 \\overrightarrow {b} )=13$.\n\n1. Find the cosine value of th... | math | {
"style": "rule",
"ground_truth": "\\sqrt{21}"
} | {
"split": "train",
"index": 1438,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a sample of size \\( n = 41 \\), a biased estimate \\( D_{\\text{в}} = 3 \\) of the population variance is found. Find the unbiased estimate of the population variance. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3.075"
} | {
"split": "train",
"index": 1439,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P_1P_2\\ldots P_{24}$ be a regular $24$-sided polygon inscribed in a circle $\\omega$ with circumference $24$. Determine the number of ways to choose sets of eight distinct vertices from these $24$ such that none of the arcs has length $3$ or $8$. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "258"
} | {
"split": "train",
"index": 1440,
"solution": "\nLet \\( P_1P_2\\ldots P_{24} \\) be a regular 24-sided polygon inscribed in a circle \\(\\omega\\) with circumference 24. We aim to determine the number of ways to choose sets of eight distinct vertices from these 24 such that none of the arcs has length 3 or 8.\n... |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle XYZ$ be a right triangle with $\\angle Y$ as the right angle. A circle with a diameter of $YZ$ intersects side $XZ$ at point $W$. Given that $XW = 2$ and $YW = 3$, find the length of $ZW$. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"split": "train",
"index": 1441,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A chord AB of length 6 passes through the left focus F<sub>1</sub> of the hyperbola $$\\frac {x^{2}}{16}- \\frac {y^{2}}{9}=1$$. Find the perimeter of the triangle ABF<sub>2</sub> (where F<sub>2</sub> is the right focus). Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 1442,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse with its foci on the x-axis and its lower vertex at D(0, -1), the eccentricity of the ellipse is $e = \\frac{\\sqrt{6}}{3}$. A line L passes through the point P(0, 2).\n(Ⅰ) Find the standard equation of the ellipse.\n(Ⅱ) If line L is tangent to the ellipse, fin... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 1443,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The edge of a regular tetrahedron is equal to \\(\\sqrt{2}\\). Find the radius of the sphere whose surface touches all the edges of the tetrahedron. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 1444,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $P$ is outside circle $C$ on the plane. At most how many points on $C$ are $3$ cm from $P$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 1445,
"solution": "1. **Identify the Geometric Configuration**: We are given a circle $C$ and a point $P$ outside this circle. We need to find the maximum number of points on circle $C$ that are exactly $3$ cm away from point $P$.\n\n2. **Construct a Circle Around $P$**: Consider a ci... |
deepscaler | [
{
"role": "user",
"content": "A 50-card deck consists of 4 cards labeled \" $i$ \" for $i=1,2, \\ldots, 12$ and 2 cards labeled \" 13 \". If Bob randomly chooses 2 cards from the deck without replacement, what is the probability that his 2 cards have the same label? Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "\\frac{73}{1225}"
} | {
"split": "train",
"index": 1446,
"solution": "All pairs of distinct cards (where we distinguish cards even with the same label) are equally likely. There are $\\binom{2}{2}+12\\binom{4}{2}=73$ pairs of cards with the same label and $\\binom{50}{2}=100 \\cdot \\frac{49}{4}=1225$ pairs of cards overall, so the de... |
deepscaler | [
{
"role": "user",
"content": "Compute the product of the roots of the equation \\[x^3 - 12x^2 + 48x + 28 = 0.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-28"
} | {
"split": "train",
"index": 1447,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? $$\n\\text A. \\ \\ 21 \\qquad \\text B. \\ \\ 60 \\qquad \\text C. \\ \\ 119 \\qquad \\text D. \\ \\ 180 \\qquad \\te... | math | {
"style": "rule",
"ground_truth": "119"
} | {
"split": "train",
"index": 1448,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find a sequence of maximal length consisting of non-zero integers in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-7,-7,18,-7,-7,-7,18,-7,-7,18,-7,-7,-7,18,-7,-7)"
} | {
"split": "train",
"index": 1449,
"solution": "Suppose it is possible to have more than 16 terms in the sequence. Let $a_{1}, a_{2}, \\ldots, a_{17}$ be the first 17 terms of the sequence. Consider the following array of terms in the sequence: \\begin{tabular}{lllllllllll} $a_{1}$ & $a_{2}$ & $a_{3}$ & $a_{4}$ &... |
deepscaler | [
{
"role": "user",
"content": "In isosceles $\\triangle ABC$ where $AB = AC = 2$ and $BC = 1$, equilateral triangles $ABD$, $BCE$, and $CAF$ are constructed outside the triangle. Calculate the area of polygon $DEF$.\nA) $3\\sqrt{3} - \\sqrt{3.75}$\nB) $3\\sqrt{3} + \\sqrt{3.75}$\nC) $2\\sqrt{3} - \\sqrt{3.75... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3} - \\sqrt{3.75}"
} | {
"split": "train",
"index": 1450,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\{a_n\\}$ satisfies $a_1=\\frac{1}{2}$, $a_{n+1}=1-\\frac{1}{a_n} (n\\in N^*)$, find the maximum positive integer $k$ such that $a_1+a_2+\\cdots +a_k < 100$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "199"
} | {
"split": "train",
"index": 1451,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We write one of the numbers $0$ and $1$ into each unit square of a chessboard with $40$ rows and $7$ columns. If any two rows have different sequences, at most how many $1$ s can be written into the unit squares? Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "198"
} | {
"split": "train",
"index": 1452,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers $2^{1989}$ and $5^{1989}$ are written out one after the other (in decimal notation). How many digits are written altogether?\n\n(G. Galperin) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1990"
} | {
"split": "train",
"index": 1453,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A function $f(x, y)$ is linear in $x$ and in $y . f(x, y)=\\frac{1}{x y}$ for $x, y \\in\\{3,4\\}$. What is $f(5,5)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{36}"
} | {
"split": "train",
"index": 1454,
"solution": "The main fact that we will use in solving this problem is that $f(x+2, y)-f(x+1, y)=f(x+1, y)-f(x, y)$ whenever $f$ is linear in $x$ and $y$. Suppose that $f(x, y)=a x y+b y+c x+d=x(a y+c)+(b y+d)$ for some constants $a, b, c$, and $d$. Then it is easy to see that $... |
deepscaler | [
{
"role": "user",
"content": "How many natural numbers greater than 10 but less than 100 are relatively prime to 21? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 1455,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{P}_1$ and $\\mathcal{P}_2$ be two parabolas with distinct directrices $\\ell_1$ and $\\ell_2$ and distinct foci $F_1$ and $F_2$ respectively. It is known that $F_1F_2||\\ell_1||\\ell_2$ , $F_1$ lies on $\\mathcal{P}_2$ , and $F_2$ lies on $\\ma... | math | {
"style": "rule",
"ground_truth": "1504"
} | {
"split": "train",
"index": 1456,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $x$: $2(3^x) = 162$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 1457,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many ordered pairs $(x,y)$ of integers is it true that $0 < x < y < 10^6$ and that the arithmetic mean of $x$ and $y$ is exactly $2$ more than the geometric mean of $x$ and $y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "997"
} | {
"split": "train",
"index": 1458,
"solution": "\\begin{eqnarray*} \\frac{x+y}{2} &=& \\sqrt{xy} + 2\\\\ x+y-4 &=& 2\\sqrt{xy}\\\\ y - 2\\sqrt{xy} + x &=& 4\\\\ \\sqrt{y} - \\sqrt{x} &=& \\pm 2\\end{eqnarray*}\n Because $y > x$, we only consider $+2$.\n For simplicity, we can count how many valid pairs of $(\\sqr... |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $\\dfrac {x^{2}}{a^{2}}- \\dfrac {y^{2}}{b^{2}}=1$ ($a > 0$, $b > 0$), its left and right vertices are $A$ and $B$, respectively. The right focus is $F$, and the line $l$ passing through point $F$ and perpendicular to the $x$-axis intersects the hyperbola at... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 1459,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the function $f(x)= \\begin{cases} \\log_{2}(1-x) & (x < 0) \\\\ g(x)+1 & (x > 0) \\end{cases}$. If $f(x)$ is an odd function, determine the value of $g(3)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 1460,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x)=\\log \\left(\\frac{1+x}{1-x}\\right)$ for $-1<x<1$, then $f\\left(\\frac{3x+x^3}{1+3x^2}\\right)$ in terms of $f(x)$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3f(x)"
} | {
"split": "train",
"index": 1461,
"solution": "Given the function \\( f(x) = \\log \\left(\\frac{1+x}{1-x}\\right) \\) for \\( -1 < x < 1 \\), we need to find \\( f\\left(\\frac{3x+x^3}{1+3x^2}\\right) \\) in terms of \\( f(x) \\).\n\n1. **Substitute \\( y = \\frac{3x+x^3}{1+3x^2} \\) into \\( f(y) \\):**\n \\... |
deepscaler | [
{
"role": "user",
"content": "The polar coordinate equation of curve C is given by C: ρ² = $\\frac{12}{5 - \\cos(2\\theta)}$, and the parametric equations of line l are given by $\\begin{cases} x = 1 + \\frac{\\sqrt{2}}{2}t \\\\ y = \\frac{\\sqrt{2}}{2}t \\end{cases}$ (where t is the parameter).\n\n1. Write... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 1462,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Altitudes $\\overline{AX}$ and $\\overline{BY}$ of acute triangle $ABC$ intersect at $H$. If $\\angle BAC = 61^\\circ$ and $\\angle ABC = 73^\\circ$, then what is $\\angle CHX$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "73^\\circ"
} | {
"split": "train",
"index": 1463,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two players in turn play a game. First Player has cards with numbers $2, 4, \\ldots, 2000$ while Second Player has cards with numbers $1, 3, \\ldots, 2001$ . In each his turn, a player chooses one of his cards and puts it on a table; the opponent sees it and puts his card ne... | math | {
"style": "rule",
"ground_truth": "999"
} | {
"split": "train",
"index": 1464,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the set of integers $\\{1,2,3,\\dots,3009\\}$, choose $k$ pairs $\\{a_i,b_i\\}$ such that $a_i < b_i$ and no two pairs have a common element. Assume all the sums $a_i+b_i$ are distinct and less than or equal to 3009. Determine the maximum possible value of $k$. Let's think... | math | {
"style": "rule",
"ground_truth": "1203"
} | {
"split": "train",
"index": 1465,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the set S={1, 2, 3, ..., 40}, and a subset A⊆S containing three elements, find the number of such sets A that can form an arithmetic progression. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "380"
} | {
"split": "train",
"index": 1466,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine into how many parts a regular tetrahedron is divided by six planes, each passing through one edge and the midpoint of the opposite edge of the tetrahedron. Find the volume of each part if the volume of the tetrahedron is 1. Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "1/24"
} | {
"split": "train",
"index": 1467,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC,$ it is given that angles $B$ and $C$ are congruent. Points $P$ and $Q$ lie on $\\overline{AC}$ and $\\overline{AB},$ respectively, so that $AP = PQ = QB = BC.$ Angle $ACB$ is $r$ times as large as angle $APQ,$ where $r$ is a positive real number. Find $\\lfloo... | math | {
"style": "rule",
"ground_truth": "571"
} | {
"split": "train",
"index": 1468,
"solution": "Let $\\angle BAC= 2\\theta$ and $AP=PQ=QB=BC=x$. $\\triangle APQ$ is isosceles, so $AQ=2x\\cos 2\\theta =2x(1-2\\sin^2\\theta)$ and $AB= AQ+x=x\\left(3-4\\sin^2\\theta\\right)$. $\\triangle{ABC}$ is isosceles too, so $x=BC=2AB\\sin\\theta$. Using the expression for ... |
deepscaler | [
{
"role": "user",
"content": "Sides $\\overline{AH}$ and $\\overline{CD}$ of regular octagon $ABCDEFGH$ are extended to meet at point $P$. What is the degree measure of angle $P$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45^\\circ"
} | {
"split": "train",
"index": 1469,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a cyclic quadrilateral with $AB=4,BC=5,CD=6,$ and $DA=7.$ Let $A_1$ and $C_1$ be the feet of the perpendiculars from $A$ and $C,$ respectively, to line $BD,$ and let $B_1$ and $D_1$ be the feet of the perpendiculars from $B$ and $D,$ respectively, to line $AC.$ Th... | math | {
"style": "rule",
"ground_truth": "301"
} | {
"split": "train",
"index": 1470,
"solution": "The angle $\\theta$ between diagonals satisfies \\[\\tan{\\frac{\\theta}{2}}=\\sqrt{\\frac{(s-b)(s-d)}{(s-a)(s-c)}}\\] (see https://en.wikipedia.org/wiki/Cyclic_quadrilateral#Angle_formulas). Thus, \\[\\tan{\\frac{\\theta}{2}}=\\sqrt{\\frac{(11-4)(11-6)}{(11-5)(11-7... |
deepscaler | [
{
"role": "user",
"content": "Given sets $A=\\{2,3,4\\}$ and $B=\\{a+2,a\\}$, if $A \\cap B = B$, find $A^cB$ ___. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\{3\\}"
} | {
"split": "train",
"index": 1471,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A hexagonal lattice now consists of 13 points where each point is one unit distance from its nearest neighbors. Points are arranged with one center point, six points forming a hexagon around the center, and an additional outer layer of six points forming a larger hexagon. How m... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 1472,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the expression first, then evaluate it: \\\\(5(3a^{2}b-ab^{2})-(ab^{2}+3a^{2}b)\\\\), where \\\\(a= \\frac {1}{2}\\\\), \\\\(b= \\frac {1}{3}\\\\). 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": 1473,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cylinder with radius 15 and height 16 is inscribed in a sphere. Three congruent smaller spheres of radius $x$ are externally tangent to the base of the cylinder, externally tangent to each other, and internally tangent to the large sphere. What is the value of $x$? Let's thin... | math | {
"style": "rule",
"ground_truth": "\\frac{15 \\sqrt{37}-75}{4}"
} | {
"split": "train",
"index": 1474,
"solution": "Let $O$ be the center of the large sphere, and let $O_{1}, O_{2}, O_{3}$ be the centers of the small spheres. Consider $G$, the center of equilateral $\\triangle O_{1} O_{2} O_{3}$. Then if the radii of the small spheres are $r$, we have that $O G=8+r$ and $O_{1} O_... |
deepscaler | [
{
"role": "user",
"content": "The value of $\\log_{10}{17}$ is between the consecutive integers $a$ and $b$. Find $a+b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 1475,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $AB=2$, $AC=5$, and $\\cos A= \\frac {4}{5}$. A point $P$ is chosen randomly inside triangle $ABC$. The probability that the area of triangle $PAB$ is greater than $1$ and less than or equal to $2$ is __________. Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "\\frac {1}{3}"
} | {
"split": "train",
"index": 1476,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, with $C= \\dfrac {\\pi}{3}$, $b=8$. The area of $\\triangle ABC$ is $10 \\sqrt {3}$.\n(I) Find the value of $c$;\n(II) Find the value of $\\cos (B-C)$. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "\\dfrac {13}{14}"
} | {
"split": "train",
"index": 1477,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle \\( ABC \\) is given. It is known that \\( AB=4 \\), \\( AC=2 \\), and \\( BC=3 \\). The angle bisector of \\( \\angle BAC \\) intersects side \\( BC \\) at point \\( K \\). A line passing through point \\( B \\) parallel to \\( AC \\) intersects the extension of the... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{6}"
} | {
"split": "train",
"index": 1478,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that our number system has a base of eight, determine the fifteenth number in the sequence. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 1479,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all real solutions to $x^4+(2-x)^4=34$. 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": "1 + \\sqrt{2}, 1 - \\sqrt{2}"
} | {
"split": "train",
"index": 1480,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the complex numbers \\( z, -z, z^2 - z + 1, z^2 + z + 1 \\) correspond to the points \\( A, B, C, D \\) respectively in the complex plane. Given that \\( |z| = 2 \\) and the quadrilateral \\( ABCD \\) is a rhombus, and let such \\( z = a + bi \\) ( \\( \\mathrm{i} \\) is th... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{7}+3}{2}"
} | {
"split": "train",
"index": 1481,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to put 7 balls in 4 boxes if the balls are not distinguishable and neither are the boxes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 1482,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$\\frac{2}{1-\\frac{2}{3}}=$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 1483,
"solution": "1. **Simplify the denominator**: Start by simplifying the expression inside the denominator:\n \\[\n 1 - \\frac{2}{3} = \\frac{3}{3} - \\frac{2}{3} = \\frac{1}{3}\n \\]\n\n2. **Substitute back into the fraction**: Replace the simplified denominator back into t... |
deepscaler | [
{
"role": "user",
"content": "On a party with 99 guests, hosts Ann and Bob play a game (the hosts are not regarded as guests). There are 99 chairs arranged in a circle; initially, all guests hang around those chairs. The hosts take turns alternately. By a turn, a host orders any standing guest to sit on an ... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 1484,
"solution": "Answer. $k=34$. Solution. Preliminary notes. Let $F$ denote the number of occupied chairs at the current position in the game. Notice that, on any turn, $F$ does not decrease. Thus, we need to determine the maximal value of $F$ Ann can guarantee after an arbitrary m... |
deepscaler | [
{
"role": "user",
"content": "Given the line $y = \\frac{3}{4}x + 6$ and a line $L$ parallel to the given line and $4$ units from it. A possible equation for $L$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "y =\\frac{3}{4}x+1"
} | {
"split": "train",
"index": 1485,
"solution": "1. **Identify the slope of the given line**: The line given is $y = \\frac{3}{4}x + 6$. The slope of this line, $m$, is $\\frac{3}{4}$.\n\n2. **Understand the relationship between parallel lines**: Lines that are parallel have the same slope. Therefore, the slope of... |
deepscaler | [
{
"role": "user",
"content": "Calculate the lengths of arcs of curves given by equations in polar coordinates.\n\n$$\n\\rho = 3(1 + \\sin \\varphi), -\\frac{\\pi}{6} \\leq \\varphi \\leq 0\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6(\\sqrt{3} - \\sqrt{2})"
} | {
"split": "train",
"index": 1486,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that four people A, B, C, D are randomly selected for a volunteer activity, find the probability that A is selected and B is not. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 1487,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x+y=10$ and $xy=12$, what is the value of $x^3-y^3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "176\\sqrt{13}"
} | {
"split": "train",
"index": 1488,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $(3,17)$ and $(9,-4)$ are the coordinates of two opposite vertices of a rectangle, what is the sum of the $y$-coordinates of the other two vertices? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 1489,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the coefficient of $x^2$ when $-5x^3 - 5x^2 - 7x + 1$ is multiplied by $-x^2 - 6x + 1$ and the like terms are combined? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 1490,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose the function $f$ has all real numbers in its domain and range and is invertible. Some values of $f$ are given by the following table: $$\\begin{array}{c || c | c | c | c | c}\nx & 1 & 2 & 3 & 4 & 5 \\\\\n\\hline\nf(x) & 2 & 3 & 5 & 7 & 8\n\\end{array}$$What is the value... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 1491,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the numbers 1, 2, 3, 4, 5, two numbers are randomly selected to be the base and the true number (antilogarithm) of a logarithm, respectively. The total number of different logarithmic values that can be obtained is ___. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 1492,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the multiplicative inverse of $101$ modulo $401$. Express your answer as an integer from $0$ to $400$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "135"
} | {
"split": "train",
"index": 1493,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiao Ming, Xiao Hong, and Xiao Gang are three people whose ages are three consecutive even numbers. Their total age is 48 years old. What is the youngest age? What is the oldest age? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 1494,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A club consists of three board members and a certain number of regular members. Every year, the board members retire and are not replaced. Each regular member recruits one new person to join as a regular member. Initially, there are nine people in the club total. How many peopl... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 1495,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Monica is trying to solve the following equation by completing the square: $$x^2-8x+8 = 0.$$She successfully rewrites the above equation in the following form: $$(x + b)^2 = c,$$where $b$ and $c$ are integers. What is the value of $b + c$? Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 1496,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC,$ $M$ is the midpoint of $\\overline{BC},$ $AB = 12,$ and $AC = 16.$ Let $E$ be on $\\overline{AC},$ and $F$ be on $\\overline{AB},$ and let $G$ be the intersection of $\\overline{EF}$ and $\\overline{AM}.$ If $AE = 2AF,$ then find $\\frac{EG}{GF}.$\n\n[asy]\... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 1497,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An integer is called snakelike if its decimal representation $a_1a_2a_3\\cdots a_k$ satisfies $a_i<a_{i+1}$ if $i$ is odd and $a_i>a_{i+1}$ if $i$ is even. How many snakelike integers between 1000 and 9999 have four distinct digits? Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "882"
} | {
"split": "train",
"index": 1498,
"solution": "Let's create the snakelike number from digits $a < b < c < d$, and, if we already picked the digits there are 5 ways to do so, as said in the first solution. And, let's just pick the digits from 0-9. This get's a total count of $5\\cdot{10 \\choose 4}$ But, this ove... |
deepscaler | [
{
"role": "user",
"content": "Suppose $b$ is an integer such that $1 \\le b \\le 30$, and $524123_{81}-b$ is a multiple of $17$. What is $b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 1499,
"solution": ""
} |
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