data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "In $\\vartriangle ABC, AB=AC=14 \\sqrt2 , D$ is the midpoint of $CA$ and $E$ is the midpoint of $BD$ . Suppose $\\vartriangle CDE$ is similar to $\\vartriangle ABC$ . Find the length of $BD$ . Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 8500,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( ABC \\) be a triangle in which \\( \\angle ABC = 60^\\circ \\). Let \\( I \\) and \\( O \\) be the incentre and circumcentre of \\( ABC \\), respectively. Let \\( M \\) be the midpoint of the arc \\( BC \\) of the circumcircle of \\( ABC \\), which does not contain the ... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 8501,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given numbers \\( x_{1}, \\ldots, x_{n} \\in (0,1) \\), find the maximum value of the expression\n\n$$\nA = \\frac{\\sqrt[4]{1-x_{1}} + \\ldots + \\sqrt[4]{1-x_{n}}}{\\frac{1}{\\sqrt[4]{x_{1}}} + \\ldots + \\frac{1}{\\sqrt[4]{x_{n}}}}\n$$ Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 8502,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find and describe the pattern by which the sequence of numbers is formed. Determine the next number in this sequence.\n\n$$\n112, 224, 448, 8816, 6612\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "224"
} | {
"split": "train",
"index": 8503,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a $12$-hour clock, an elapsed time of four hours looks the same as an elapsed time of $16$ hours. Because of this, we can say that four hours is \"clock equivalent'' to its square number of hours. What is the least whole number of hours that is greater than $4$ hours and is ... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 8504,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, given $A(1,4)$, $B(4,1)$, $C(0,-4)$, find the minimum value of $\\overrightarrow{PA} \\cdot \\overrightarrow{PB} + \\overrightarrow{PB} \\cdot \\overrightarrow{PC} + \\overrightarrow{PC} \\cdot \\overrightarrow{PA}$. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "- \\dfrac {62}{3}"
} | {
"split": "train",
"index": 8505,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Pedro must choose two irreducible fractions, each with a positive numerator and denominator such that:\n\n\n- The sum of the fractions is equal to $2$ .\n- The sum of the numerators of the fractions is equal to $1000$ .\n\nIn how many ways can Pedro do this?\n Let's think ste... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 8506,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An equilateral triangle is inscribed in the ellipse whose equation is $x^2+4y^2=4$. One vertex of the triangle is $(0,1)$, one altitude is contained in the y-axis, and the square of the length of each side is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integ... | math | {
"style": "rule",
"ground_truth": "937"
} | {
"split": "train",
"index": 8507,
"solution": "Solving for $y$ in terms of $x$ gives $y=\\sqrt{4-x^2}/2$, so the two other points of the triangle are $(x,\\sqrt{4-x^2}/2)$ and $(-x,\\sqrt{4-x^2}/2)$, which are a distance of $2x$ apart. Thus $2x$ equals the distance between $(x,\\sqrt{4-x^2}/2)$ and $(0,1)$, so b... |
deepscaler | [
{
"role": "user",
"content": "It is possible to arrange eight of the nine numbers $2, 3, 4, 7, 10, 11, 12, 13, 15$ in the vacant squares of the $3$ by $4$ array shown on the right so that the arithmetic average of the numbers in each row and in each column is the same integer. Exhibit such an arrange... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 8508,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of the following 15 cards has a letter on one side and a positive integer on the other side. What is the minimum number of cards that need to be turned over to check if the following statement is true? 'If a card has a lower case letter on one side, then it has an odd inte... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 8509,
"solution": "Each card fits into exactly one of the following categories: (A) lower case letter on one side, even integer on the other side (B) lower case letter on one side, odd integer on the other side (C) upper case letter on one side, even integer on the other side (D) uppe... |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be real numbers such that $a^5b^8=12$ and $a^8b^{13}=18$ . Find $ab$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{128}{3}"
} | {
"split": "train",
"index": 8510,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points \\( M, N, \\) and \\( K \\) are located on the lateral edges \\( A A_{1}, B B_{1}, \\) and \\( C C_{1} \\) of the triangular prism \\( A B C A_{1} B_{1} C_{1} \\) such that \\( \\frac{A M}{A A_{1}} = \\frac{5}{6}, \\frac{B N}{B B_{1}} = \\frac{6}{7}, \\) and \\( \\frac{C... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 8511,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the country of Francisca, there are 2010 cities, some of which are connected by roads. Between any two cities, there is a unique path which runs along the roads and which does not pass through any city twice. What is the maximum possible number of cities in Francisca which h... | math | {
"style": "rule",
"ground_truth": "1004"
} | {
"split": "train",
"index": 8512,
"solution": "The restrictions on how roads connect cities directly imply that the graph of the cities of Francisca with the roads as edges is a tree. Therefore the sum of the degrees of all the vertices is $2009 \\cdot 2=4018$. Suppose that $b$ vertices have degree \\geq 3. The ... |
deepscaler | [
{
"role": "user",
"content": "The roots of the equation $x^2 + kx + 8 = 0$ differ by $\\sqrt{72}$. Find the greatest possible value of $k$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{26}"
} | {
"split": "train",
"index": 8513,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line is expressed in the form\n\\[\\begin{pmatrix} 1 \\\\ 3 \\end{pmatrix} \\cdot \\left( \\begin{pmatrix} x \\\\ y \\end{pmatrix} - \\begin{pmatrix} -2 \\\\ 8 \\end{pmatrix} \\right) = 0.\\]The equation of the line can be expressed in the form $y = mx + b.$ Enter the ordere... | math | {
"style": "rule",
"ground_truth": "\\left( -\\frac{1}{3}, \\frac{22}{3} \\right)"
} | {
"split": "train",
"index": 8514,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer that satisfies the congruence $4x \\equiv 13 \\pmod{27}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 8515,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the function $f(x)$ is an odd function defined on $\\mathbb{R}$, and when $x \\in [0,1]$, $f(x)=x^{3}$. Also, for all $x \\in \\mathbb{R}$, $f(x)=f(2-x)$. Determine the value of $f(2017.5)$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{8}"
} | {
"split": "train",
"index": 8516,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(D\\) be a point on side \\(BC\\) of \\(\\triangle ABC\\). Points \\(E\\) and \\(F\\) are the centroids of \\(\\triangle ABD\\) and \\(\\triangle ACD\\), respectively. The line segment \\(EF\\) intersects \\(AD\\) at point \\(G\\). Find the value of \\(\\frac{DG}{GA}\\). ... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 8517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of two-digit numbers that can be formed using the digits 0, 1, 2, 3, 4 without repeating any digit must be calculated. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 8518,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point \\( P \\) lies on the hyperbola \\(\\frac{x^{2}}{16} - \\frac{y^{2}}{9} = 1\\), and the distance from \\( P \\) to the right directrix of the hyperbola is the arithmetic mean of the distances from \\( P \\) to the two foci of the hyperbola, find the x-coordinat... | math | {
"style": "rule",
"ground_truth": "-\\frac{64}{5}"
} | {
"split": "train",
"index": 8519,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pedestrian and a cyclist set off from point $A$ to point $B$ simultaneously. At point $B$, the cyclist turns back and meets the pedestrian 20 minutes after starting. Without stopping, the cyclist continues to point $A$, turns around, and catches up with the pedestrian 10 minu... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 8520,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Diagonals $A C$ and $C E$ of a regular hexagon $A B C D E F$ are divided by points $M$ and $N$ such that $A M : A C = C N : C E = \\lambda$. Find $\\lambda$ if it is known that points $B, M$, and $N$ are collinear. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt{3}}"
} | {
"split": "train",
"index": 8521,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If we exchange a 10-dollar bill into dimes and quarters, what is the total number \\( n \\) of different ways to have two types of coins? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 8522,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $f(x) = 6x - 9$ and $g(x) = \\frac{x}{3} + 2$. Find $f(g(x)) - g(f(x))$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 8523,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) Find the domain of the function $y= \\sqrt {\\sin x}+ \\sqrt { \\frac{1}{2}-\\cos x}$.\n(2) Find the range of the function $y=\\cos ^{2}x-\\sin x$, where $x\\in\\left[-\\frac {\\pi}{4}, \\frac {\\pi}{4}\\right]$. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "\\frac{1-\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 8524,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When you simplify $\\left[ \\sqrt [3]{\\sqrt [6]{a^9}} \\right]^4\\left[ \\sqrt [6]{\\sqrt [3]{a^9}} \\right]^4$, the result is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "a^4"
} | {
"split": "train",
"index": 8525,
"solution": "1. **Simplify each term inside the brackets:**\n - For the first term $\\sqrt[3]{\\sqrt[6]{a^9}}$, we simplify the exponents:\n \\[\n \\sqrt[3]{\\sqrt[6]{a^9}} = a^{\\frac{9}{6} \\cdot \\frac{1}{3}} = a^{\\frac{9}{18}} = a^{\\frac{1}{2}}\n \\]\n - Fo... |
deepscaler | [
{
"role": "user",
"content": "The side lengths of a triangle are distinct positive integers. One of the side lengths is a multiple of 42, and another is a multiple of 72. What is the minimum possible length of the third side? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 8526,
"solution": "Suppose that two of the side lengths are $42 a$ and $72 b$, for some positive integers $a$ and $b$. Let $c$ be the third side length. We know that $42 a$ is not equal to $72 b$, since the side lengths are distinct. Also, $6 \\mid 42 a-72 b$. Therefore, by the triang... |
deepscaler | [
{
"role": "user",
"content": "Given the convex pentagon $ABCDE$, where each pair of neighboring vertices must have different colors and vertices at the ends of each diagonal must not share the same color, determine the number of possible colorings using 5 available colors. Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 8527,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a given list of three numbers, the operation \"changesum\" replaces each number in the list with the sum of the other two. For example, applying \"changesum\" to \\(3,11,7\\) gives \\(18,10,14\\). Arav starts with the list \\(20,2,3\\) and applies the operation \"changesum\... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 8528,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A survey of $150$ teachers determined the following:\n- $90$ had high blood pressure\n- $60$ had heart trouble\n- $50$ had diabetes\n- $30$ had both high blood pressure and heart trouble\n- $20$ had both high blood pressure and diabetes\n- $10$ had both heart trouble and diabet... | math | {
"style": "rule",
"ground_truth": "3.33\\%"
} | {
"split": "train",
"index": 8529,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of $y = f(x)$ is shown below.\n\n[asy]\nunitsize(0.5 cm);\n\nreal func(real x) {\n real y;\n if (x >= -3 && x <= 0) {y = -2 - x;}\n if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}\n if (x >= 2 && x <= 3) {y = 2*(x - 2);}\n return(y);\n}\n\nint i, n;\n\nfor (... | math | {
"style": "rule",
"ground_truth": "\\text{C}"
} | {
"split": "train",
"index": 8530,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) Solve the inequality $$\\frac {2x+1}{3-x}≥1$$ \n(2) Given $x>0$, $y>0$, and $x+y=1$, find the minimum value of $$\\frac {4}{x} + \\frac {9}{y}$$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 8531,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the trapezoid \\(ABCD \\) with \\( AD \\parallel BC \\), the angle \\( \\angle ADB \\) is twice the angle \\( \\angle ACB \\). It is known that \\( BC = AC = 5 \\) and \\( AD = 6 \\). Find the area of the trapezoid. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 8532,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$, $b$, $c$ be the three sides of a triangle, and let $\\alpha$, $\\beta$, $\\gamma$ be the angles opposite them. If $a^2+b^2=1989c^2$, find the value of\n\\[\\frac{\\cot \\gamma}{\\cot \\alpha+\\cot \\beta}.\\] Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "994"
} | {
"split": "train",
"index": 8533,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate: \\( 4\\left(\\sin ^{3} \\frac{49 \\pi}{48} \\cos \\frac{49 \\pi}{16} + \\cos ^{3} \\frac{49 \\pi}{48} \\sin \\frac{49 \\pi}{16}\\right) \\cos \\frac{49 \\pi}{12} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.75"
} | {
"split": "train",
"index": 8534,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From a six-digit phone number, how many seven-digit numbers can be obtained by removing one digit? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 8535,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A unit square $A B C D$ and a circle $\\Gamma$ have the following property: if $P$ is a point in the plane not contained in the interior of $\\Gamma$, then $\\min (\\angle A P B, \\angle B P C, \\angle C P D, \\angle D P A) \\leq 60^{\\circ}$. The minimum possible area of $\\Ga... | math | {
"style": "rule",
"ground_truth": "106"
} | {
"split": "train",
"index": 8536,
"solution": "Note that the condition for $\\Gamma$ in the problem is equivalent to the following condition: if $\\min (\\angle A P B, \\angle B P C, \\angle C P D, \\angle D P A)>60^{\\circ}$, then $P$ is contained in the interior of $\\Gamma$. Let $X_{1}, X_{2}, X_{3}$, and $X_... |
deepscaler | [
{
"role": "user",
"content": "Consider $\\triangle \\natural\\flat\\sharp$ . Let $\\flat\\sharp$ , $\\sharp\\natural$ and $\\natural\\flat$ be the answers to problems $4$ , $5$ , and $6$ , respectively. If the incircle of $\\triangle \\natural\\flat\\sharp$ touches $\\natural\\flat$ at $\\od... | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"split": "train",
"index": 8537,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\(\\alpha\\), \\(\\beta\\), and \\(\\gamma\\) are acute angles such that \\(\\cos^2 \\alpha + \\cos^2 \\beta + \\cos^2 \\gamma = 1\\), find the minimum value of \\(\\tan \\alpha \\cdot \\tan \\beta \\cdot \\tan \\gamma\\). Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 8538,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The tetrahedron $ S.ABC$ has the faces $ SBC$ and $ ABC$ perpendicular. The three angles at $ S$ are all $ 60^{\\circ}$ and $ SB \\equal{} SC \\equal{} 1$ . Find the volume of the tetrahedron. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "1/8"
} | {
"split": "train",
"index": 8539,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The acute angles of a right triangle are $a^{\\circ}$ and $b^{\\circ}$, where $a>b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 8540,
"solution": "1. **Identify the relationship between the angles**: In a right triangle, the sum of the angles is $180^\\circ$. Given one angle is $90^\\circ$, the sum of the other two angles, $a^\\circ$ and $b^\\circ$, must be $90^\\circ$. Thus, we have:\n \\[\n a + b = 90^\\... |
deepscaler | [
{
"role": "user",
"content": "Four congruent rectangles and a square are assembled without overlapping to form a large square. Each of the rectangles has a perimeter of 40 cm. What is the total area of the large square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "400 \\mathrm{~cm}^{2}"
} | {
"split": "train",
"index": 8541,
"solution": "Suppose that each of the smaller rectangles has a longer side of length $x \\mathrm{~cm}$ and a shorter side of length $y \\mathrm{~cm}$. Since the perimeter of each of the rectangles is 40 cm, then $2x+2y=40$ or $x+y=20$. But the side length of the large square is ... |
deepscaler | [
{
"role": "user",
"content": "Given that $x = \\frac{3}{4}$ is a solution to the equation $108x^2 - 35x - 77 = 0$, what is the other value of $x$ that will solve the equation? 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{23}{54}"
} | {
"split": "train",
"index": 8542,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given rectangle $ABCD$ with $E$ the midpoint of diagonal $BD$. Point $E$ is connected to a point $F$ on $DA$ so that $DF=\\frac{1}{4}DA$. Find the ratio of the area of $\\triangle DFE$ to the area of quadrilateral $ABEF$. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{7}"
} | {
"split": "train",
"index": 8543,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any permutation $p$ of set $\\{1, 2, \\ldots, n\\}$, define $d(p) = |p(1) - 1| + |p(2) - 2| + \\ldots + |p(n) - n|$. Denoted by $i(p)$ the number of integer pairs $(i, j)$ in permutation $p$ such that $1 \\leqq < j \\leq n$ and $p(i) > p(j)$. Find all the real numbers $c$, ... | math | {
"style": "rule",
"ground_truth": "p=(1 \\; n)"
} | {
"split": "train",
"index": 8544,
"solution": "\nTo solve this problem, we need to understand the relationship between \\(d(p)\\) and \\(i(p)\\) for any permutation \\(p\\) of the set \\(\\{1, 2, \\ldots, n\\}\\).\n\n### Definitions:\n- A permutation \\(p\\) of a set \\(\\{1, 2, \\ldots, n\\}\\) is a bijection f... |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, it is known that $\\overrightarrow{CD}=2\\overrightarrow{DB}$, $P$ is a point on segment $AD$, and satisfies $\\overrightarrow{CP}=\\frac{1}{2}\\overrightarrow{CA}+m\\overrightarrow{CB}$. If the area of $\\triangle ABC$ is $\\sqrt{3}$ and $∠ACB=\\f... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 8545,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a particular street in Waterloo, there are exactly 14 houses, each numbered with an integer between 500 and 599, inclusive. The 14 house numbers form an arithmetic sequence in which 7 terms are even and 7 terms are odd. One of the houses is numbered 555 and none of the remai... | math | {
"style": "rule",
"ground_truth": "506"
} | {
"split": "train",
"index": 8546,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The year 2002 is a palindrome (a number that reads the same from left to right as it does from right to left). What is the product of the digits of the next year after 2002 that is a palindrome? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 8547,
"solution": "To find the next palindromic year after 2002, we need to identify a year that reads the same forwards and backwards. We start by checking the years sequentially from 2003 onwards.\n\n1. **Check year 2003**: This is not a palindrome.\n2. **Check year 2004**: This is ... |
deepscaler | [
{
"role": "user",
"content": "Compute $\\sqrt{(41)(40)(39)(38) + 1}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1559"
} | {
"split": "train",
"index": 8548,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $10\\cot(\\cot^{-1}3+\\cot^{-1}7+\\cot^{-1}13+\\cot^{-1}21).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 8549,
"solution": "On the coordinate plane, let $O=(0,0)$, $A_1=(3,0)$, $A_2=(3,1)$, $B_1=(21,7)$, $B_2=(20,10)$, $C_1=(260,130)$, $C_2=(250,150)$, $D_1=(5250,3150)$, $D_2=(5100,3400)$, and $H=(5100,0)$. We see that $\\cot^{-1}(\\angle A_2OA_1)=3$, $\\cot^{-1}(\\angle B_2OB_1)=7$, $\\... |
deepscaler | [
{
"role": "user",
"content": "Assume $x$, $y$, $z$, and $w$ are positive integers such that $x^3 = y^2$, $z^5 = w^4$, and $z - x = 31$. Determine $w - y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2351"
} | {
"split": "train",
"index": 8550,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a 12-hour digital clock with a glitch where every '2' is displayed as a '7', determine the fraction of the day that the clock shows the correct time. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{55}{72}"
} | {
"split": "train",
"index": 8551,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the sequence $\\{a_n\\}$ satisfy $a_1=2$, $a_2=6$, and $a_{n+2}-2a_{n+1}+a_n=2$. If $\\lfloor x \\rfloor$ represents the greatest integer not exceeding $x$, calculate $$\\left\\lfloor \\frac{2017}{a_1} + \\frac{2017}{a_2} + \\ldots + \\frac{2017}{a_{2017}} \\right\\rfloor.$... | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"split": "train",
"index": 8552,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\cos x\\cos \\left( x+\\dfrac{\\pi}{3} \\right)$.\n\n(1) Find the smallest positive period of $f(x)$;\n\n(2) In $\\triangle ABC$, angles $A$, $B$, $C$ correspond to sides $a$, $b$, $c$, respectively. If $f(C)=-\\dfrac{1}{4}$, $a=2$, and the area of $\\... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt {3}"
} | {
"split": "train",
"index": 8553,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Factor completely: $x^6 - 3x^4 + 3x^2 - 1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(x-1)^3(x+1)^3"
} | {
"split": "train",
"index": 8554,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $\\sin N = \\frac{2}{3}$ in the diagram below. What is $LN$?\n\n\n\n[asy]\npair L,M,N;\nM = (0,0);\nN = (17.89,0);\nL = (0,16);\ndraw(L--M--N--L);\ndraw(rightanglemark(L,M,N,18));\nlabel(\"$M$\",M,SW);\nlabel(\"$N$\",N,SE);\nlabel(\"$L$\",L,NE);\nlabel(\"$16$\",L/2,W);... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 8555,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": " $ABCD$ forms a rhombus. $E$ is the intersection of $AC$ and $BD$ . $F$ lie on $AD$ such that $EF$ is perpendicular to $FD$ . Given $EF=2$ and $FD=1$ . Find the area of the rhombus $ABCD$ Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 8556,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When Ma Xiaohu was doing a subtraction problem, he mistakenly wrote the units digit of the minuend as 5 instead of 3, and the tens digit as 0 instead of 6. Additionally, he wrote the hundreds digit of the subtrahend as 2 instead of 7. The resulting difference was 1994. What sho... | math | {
"style": "rule",
"ground_truth": "1552"
} | {
"split": "train",
"index": 8557,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\cos \\alpha + \\sin \\alpha = \\frac{2}{3}$, find the value of $\\frac{\\sqrt{2}\\sin(2\\alpha - \\frac{\\pi}{4}) + 1}{1 + \\tan \\alpha}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{5}{9}"
} | {
"split": "train",
"index": 8558,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Which number from the set $\\{1,2,3,4,5,6,7,8,9,10,11\\}$ must be removed so that the mean (average) of the numbers remaining in the set is 6.1? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 8559,
"solution": "The original set contains 11 elements whose sum is 66. When one number is removed, there will be 10 elements in the set. For the average of these elements to be 6.1, their sum must be $10 \\times 6.1=61$. Since the sum of the original 11 elements is 66 and the sum o... |
deepscaler | [
{
"role": "user",
"content": "A youth radio station, to attract listeners' attention, gives away gifts and grand prizes among them. Gifts are given away hourly over sixteen hours (one gift per hour), and grand prizes are given away during four evening hours (one grand prize per hour). The probability that l... | math | {
"style": "rule",
"ground_truth": "0.862"
} | {
"split": "train",
"index": 8560,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\( N \\) is the smallest positive integer whose digits have a product of 2700, what is the sum of the digits of \\( N \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 8561,
"solution": "In order to find \\( N \\), which is the smallest possible integer whose digits have a fixed product, we must first find the minimum possible number of digits with this product. Once we have determined the digits that form \\( N \\), then the integer \\( N \\) itsel... |
deepscaler | [
{
"role": "user",
"content": "How many four-digit numbers are composed of four distinct digits such that one digit is the average of any two other digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "216"
} | {
"split": "train",
"index": 8562,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two students, A and B, each choose 2 out of 6 extracurricular reading materials. Calculate the number of ways in which the two students choose extracurricular reading materials such that they have exactly 1 material in common. Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 8563,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be a polynomial such that $P(x)=P(0)+P(1) x+P(2) x^{2}$ and $P(-1)=1$. Compute $P(3)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 8564,
"solution": "Plugging in $x=-1,1,2$ results in the trio of equations $1=P(-1)=P(0)-P(1)+P(2)$, $P(1)=P(0)+P(1)+P(2) \\Rightarrow P(1)+P(2)=0$, and $P(2)=P(0)+2 P(1)+4 P(2)$. Solving these as a system of equations in $P(0), P(1), P(2)$ gives $P(0)=-1, P(1)=-1, P(2)=1$. Consequent... |
deepscaler | [
{
"role": "user",
"content": "An angle $x$ is chosen at random from the interval $0^\\circ < x < 90^\\circ.$ Let $p$ be the probability that the numbers $\\sin^2 x, \\cos^2 x,$ and $\\sin x \\cos x$ are not the lengths of the sides of a triangle. Given that $p = d/n,$ where $d$ is the number of degrees in $... | math | {
"style": "rule",
"ground_truth": "92"
} | {
"split": "train",
"index": 8565,
"solution": "Note that the three expressions are symmetric with respect to interchanging $\\sin$ and $\\cos$, and so the probability is symmetric around $45^\\circ$. Thus, take $0 < x < 45$ so that $\\sin x < \\cos x$. Then $\\cos^2 x$ is the largest of the three given expressio... |
deepscaler | [
{
"role": "user",
"content": "Completely factor the following expression: \\[(9x^5+25x^3-4)-(x^5-3x^3-4).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4x^3(2x^2+7)"
} | {
"split": "train",
"index": 8566,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Denote by $\\mathbb{Z}^2$ the set of all points $(x,y)$ in the plane with integer coordinates. For each integer $n \\geq 0$, let $P_n$ be the subset of $\\mathbb{Z}^2$ consisting of the point $(0,0)$ together with all points $(x,y)$ such that $x^2 + y^2 = 2^k$ for some integer ... | math | {
"style": "rule",
"ground_truth": "5n+1"
} | {
"split": "train",
"index": 8567,
"solution": "The answer is $5n+1$.\n\nWe first determine the set $P_n$. Let $Q_n$ be the set of points in $\\mathbb{Z}^2$ of the form $(0, \\pm 2^k)$ or $(\\pm 2^k, 0)$ for some $k \\leq n$. Let $R_n$ be the set of points in $\\mathbb{Z}^2$ of the form $(\\pm 2^k, \\pm 2^k)$ for... |
deepscaler | [
{
"role": "user",
"content": "Let $A = \\{1, 2, 3, 4, 5, 6, 7\\}$, and let $N$ be the number of functions $f$ from set $A$ to set $A$ such that $f(f(x))$ is a constant function. 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": "399"
} | {
"split": "train",
"index": 8568,
"solution": "Any such function can be constructed by distributing the elements of $A$ on three tiers.\n The bottom tier contains the constant value, $c=f(f(x))$ for any $x$. (Obviously $f(c)=c$.)\n The middle tier contains $k$ elements $x\\ne c$ such that $f(x)=c$, where $1\\le ... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\cos(75^\\circ + \\alpha) = \\frac{1}{3}$, where $\\alpha$ is an angle in the third quadrant, find the value of $\\cos(105^\\circ - \\alpha) + \\sin(\\alpha - 105^\\circ)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{2} - 1}{3}"
} | {
"split": "train",
"index": 8569,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fair 10-sided die is rolled repeatedly until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number?\n- **A** $\\frac{1}{300}$\n- **B** $\\frac{1}{252}$\n- **C** $\\frac{1}{500}$\n- **D** $\\fra... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{252}"
} | {
"split": "train",
"index": 8570,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A severe earthquake in Madrid caused a total of €50 million in damages. At the time of the earthquake, the exchange rate was such that 2 Euros were worth 3 American dollars. How much damage did the earthquake cause in American dollars? Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "75,000,000"
} | {
"split": "train",
"index": 8571,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the expression \\( \\left(1-\\frac{1}{2^{2}}\\right)\\left(1-\\frac{1}{3^{2}}\\right)\\ldots\\left(1-\\frac{1}{12^{2}}\\right) \\), compute its value. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{24}"
} | {
"split": "train",
"index": 8572,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There exist positive integers $a,$ $b,$ and $c$ such that\n\\[3 \\sqrt{\\sqrt[3]{5} - \\sqrt[3]{4}} = \\sqrt[3]{a} + \\sqrt[3]{b} - \\sqrt[3]{c}.\\]Find $a + b + c.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 8573,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function f(x) = a^x (a > 0, a ≠ 1).\n(I) If $f(1) + f(-1) = \\frac{5}{2}$, find the value of f(2) + f(-2).\n(II) If the difference between the maximum and minimum values of the function f(x) on [-1, 1] is $\\frac{8}{3}$, find the value of the real number a. Let's thin... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 8574,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all of the two-digit primes that are greater than 12 but less than 99 and are still prime when their two digits are interchanged? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "418"
} | {
"split": "train",
"index": 8575,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a$, $b$, $c$, $d$, $e$, and $f$ are all positive numbers, and $\\frac{bcdef}{a}=\\frac{1}{2}$, $\\frac{acdef}{b}=\\frac{1}{4}$, $\\frac{abdef}{c}=\\frac{1}{8}$, $\\frac{abcef}{d}=2$, $\\frac{abcdf}{e}=4$, $\\frac{abcde}{f}=8$, find $a^{2}+b^{2}+c^{2}+d^{2}+e^{2}+f^{... | math | {
"style": "rule",
"ground_truth": "\\frac{119}{8}"
} | {
"split": "train",
"index": 8576,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the equation $y = \\frac{x}{x^3 + Ax^2 + Bx + C}$, where $A,B,C$ are integers, is shown below. Find $A + B + C$. [asy]\nimport graph; size(8.14cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.52,xmax=4.62,ymin=-3.6... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 8577,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\(x\\) satisfies \\(\\log _{5x} (2x) = \\log _{625x} (8x)\\), find the value of \\(\\log _{2} x\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\ln 5}{2 \\ln 2 - 3 \\ln 5}"
} | {
"split": "train",
"index": 8578,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On graph paper, a stepwise right triangle was drawn with legs equal to 6 cells each. Then, all grid lines inside the triangle were outlined. What is the maximum number of rectangles that can be found in this drawing? Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "126"
} | {
"split": "train",
"index": 8579,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Record the outcome of hitting or missing for 6 consecutive shots in order. \n① How many possible outcomes are there? \n② How many outcomes are there where exactly 3 shots hit the target? \n③ How many outcomes are there where 3 shots hit the target, and exactly two of those h... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 8580,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of the expression \\(\\sum_{i=0}^{1009}(2 k+1)-\\sum_{i=1}^{1009} 2 k\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"split": "train",
"index": 8581,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In acute $\\triangle A B C$ with centroid $G, A B=22$ and $A C=19$. Let $E$ and $F$ be the feet of the altitudes from $B$ and $C$ to $A C$ and $A B$ respectively. Let $G^{\\prime}$ be the reflection of $G$ over $B C$. If $E, F, G$, and $G^{\\prime}$ lie on a circle, compute $B ... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 8582,
"solution": "Note that $B, C, E, F$ lie on a circle. Moreover, since $B C$ bisects $G G^{\\prime}$, the center of the circle that goes through $E, F, G, G^{\\prime}$ must lie on $B C$. Therefore, $B, C, E, F, G, G^{\\prime}$ lie on a circle. Specifically, the center of this circ... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle PQR$, where $PQ=7$, $PR=9$, $QR=12$, and $S$ is the midpoint of $\\overline{QR}$. What is the sum of the radii of the circles inscribed in $\\triangle PQS$ and $\\triangle PRS$?\nA) $\\frac{14\\sqrt{5}}{13}$\nB) $\\frac{14\\sqrt{5}}{6.5 + \\sqrt{29}}$\nC) $\\frac... | math | {
"style": "rule",
"ground_truth": "\\frac{14\\sqrt{5}}{6.5 + \\sqrt{29}}"
} | {
"split": "train",
"index": 8583,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Nine copies of a certain pamphlet cost less than $10.00 while ten copies of the same pamphlet (at the same price) cost more than $11.00. How much does one copy of this pamphlet cost? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$1.11"
} | {
"split": "train",
"index": 8584,
"solution": "1. **Define the variable**: Let $p$ be the cost of one pamphlet in dollars.\n\n2. **Translate the conditions into inequalities**:\n - From the first condition, nine copies of the pamphlet cost less than $10.00. This can be expressed as:\n \\[\n 9p < 10\n ... |
deepscaler | [
{
"role": "user",
"content": "Three people, A, B, and C, are playing a game: each person chooses a real number from the interval $[0,1]$. The person whose chosen number is between the numbers chosen by the other two wins. A randomly chooses a number from the interval $[0,1]$, B randomly chooses a number fro... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{24}"
} | {
"split": "train",
"index": 8585,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The \"One Helmet, One Belt\" safety protection campaign is a safety protection campaign launched by the Ministry of Public Security nationwide. It is also an important standard for creating a civilized city and being a civilized citizen. \"One helmet\" refers to a safety helmet... | math | {
"style": "rule",
"ground_truth": "6200"
} | {
"split": "train",
"index": 8586,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the minimum value of prime $p> 3$ for which there is no natural number $n> 0$ such that $2^n+3^n\\equiv 0\\pmod{p} $ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 8587,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An infinite arithmetic progression of positive integers contains the terms 7, 11, 15, 71, 75, and 79. The first term in the progression is 7. Kim writes down all the possible values of the one-hundredth term in the progression. What is the sum of the numbers Kim writes down? Le... | math | {
"style": "rule",
"ground_truth": "714"
} | {
"split": "train",
"index": 8588,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Use the method of random simulation to estimate the probability that it will rain on exactly two of the three days. Using a calculator, generate random integer values between 0 and 9, where 1, 2, 3, and 4 represent raining days and 5, 6, 7, 8, 9, and 0 signify non-raining days.... | math | {
"style": "rule",
"ground_truth": "0.25"
} | {
"split": "train",
"index": 8589,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The conference has 12 teams. Each team plays every other team twice and an additional 6 games against non-conference opponents. Calculate the total number of games in a season involving the conference teams. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "204"
} | {
"split": "train",
"index": 8590,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of real solutions of the equation\n\\[\\frac{4x}{x^2 + x + 3} + \\frac{5x}{x^2 - 5x + 3} = -\\frac{3}{2}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 8591,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S=\\{(x,y) : x\\in \\{0,1,2,3,4\\}, y\\in \\{0,1,2,3,4,5\\},\\text{ and } (x,y)\\ne (0,0)\\}$. \nLet $T$ be the set of all right triangles whose vertices are in $S$. For every right triangle $t=\\triangle{ABC}$ with vertices $A$, $B$, and $C$ in counter-clockwise order and... | math | {
"style": "rule",
"ground_truth": "\\frac{625}{144}"
} | {
"split": "train",
"index": 8592,
"solution": "1. **Define the Set $S$ and $T$:**\n - $S$ is defined as the set of all points $(x,y)$ where $x \\in \\{0,1,2,3,4\\}$ and $y \\in \\{0,1,2,3,4,5\\}$, excluding the point $(0,0)$.\n - $T$ is the set of all right triangles whose vertices are points in $S$.\n\n2. *... |
deepscaler | [
{
"role": "user",
"content": "Two cyclists, $k$ miles apart, and starting at the same time, would be together in $r$ hours if they traveled in the same direction, but would pass each other in $t$ hours if they traveled in opposite directions. The ratio of the speed of the faster cyclist to that of the slowe... | math | {
"style": "rule",
"ground_truth": "\\frac {r + t}{r - t}"
} | {
"split": "train",
"index": 8593,
"solution": "Let's denote the speeds of the faster and slower cyclists as $v_1$ and $v_2$ respectively, where $v_1 > v_2$. The distance between the cyclists is $k$ miles.\n\n1. **Traveling in the same direction**: When the cyclists travel in the same direction, the relative spee... |
deepscaler | [
{
"role": "user",
"content": "Find the area of triangle $ABC$ given that $AB=8$, $AC=3$, and $\\angle BAC=60^{\\circ}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6 \\sqrt{3}"
} | {
"split": "train",
"index": 8594,
"solution": "Using the law of cosines gives: $$\\begin{aligned} x^{2}+(11-x)^{2}-2x(11-x) \\cos 60^{\\circ} & =7^{2} \\\\ 3x^{2}-33x+72 & =0 \\\\ x & =3 \\text{ or } 8. \\end{aligned}$$ Therefore, $AB=8$ and $AC=3$ or $AB=3$ and $AC=8$. In both cases, the area of the triangle is... |
deepscaler | [
{
"role": "user",
"content": "A positive integer is called primer if it has a prime number of distinct prime factors. A positive integer is called primest if it has a primer number of distinct primer factors. Find the smallest primest number. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 8595,
"solution": "We claim the answer is 72 , as it has 6 primer factors: $6,12,24,18,36,72$, and 6 is a primer. We now prove that there is no smaller primer. Suppose there were a smaller primer $r<72$. We do casework on the number of distinct prime factors of $r$. - $r$ has \\geq 4$... |
deepscaler | [
{
"role": "user",
"content": "Let $T$ be the set of all ordered triples of integers $(b_1, b_2, b_3)$ with $1 \\leq b_1, b_2, b_3 \\leq 20$. Each ordered triple in $T$ generates a sequence according to the rule $b_n = b_{n-1} \\cdot |b_{n-2} - b_{n-3}|$ for all $n \\geq 4$. Find the number of such sequences... | math | {
"style": "rule",
"ground_truth": "780"
} | {
"split": "train",
"index": 8596,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P(n)$ and $S(n)$ denote the product and the sum, respectively, of the digits of the integer $n$. For example, $P(23) = 6$ and $S(23) = 5$. Suppose $N$ is a two-digit number such that $N = P(N) +\nS(N)$. What is the units digit of $N$? Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 8597,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all positive values of $c$ so that the inequality $x^2-6x+c<0$ has real solutions for $x$. 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": "(0,9)"
} | {
"split": "train",
"index": 8598,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)= \\begin{cases} |\\ln x|, & (0 < x\\leqslant e^{3}) \\\\ e^{3}+3-x, & (x > e^{3})\\end{cases}$, there exist $x\\_1 < x\\_2 < x\\_3$ such that $f(x\\_1)=f(x\\_2)=f(x\\_3)$. Find the maximum value of $\\frac{f(x\\_3)}{x\\_2}$. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{e}"
} | {
"split": "train",
"index": 8599,
"solution": ""
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.