data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Susie was given $\\$1,500$ for her birthday. She decides to invest the money in a bank account that earns $12\\%$ interest, compounded quarterly. How much total interest will Susie have earned 4 years later? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "901.55"
} | {
"split": "train",
"index": 2500,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the equations $z^2 = 1 + 3\\sqrt{10}i$ and $z^2 = 2 - 2\\sqrt{2}i$, where $i = \\sqrt{-1}$, find the vertices formed by the solutions of these equations on the complex plane and compute the area of the quadrilateral they form.\nA) $17\\sqrt{6} - 2\\sqrt{2}$\nB) $18\\sqrt{... | math | {
"style": "rule",
"ground_truth": "19\\sqrt{6} - 2\\sqrt{2}"
} | {
"split": "train",
"index": 2501,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$.\n[asy] import three; im... | math | {
"style": "rule",
"ground_truth": "384"
} | {
"split": "train",
"index": 2502,
"solution": "We can use the same method as in Solution 2 to find the side length of the equilateral triangle, which is $4\\sqrt3$. From here, its area is \\[\\dfrac{\\bigl(4\\sqrt3\\bigr)^2\\sqrt3}4=12\\sqrt3.\\] The leg of the isosceles right triangle is $\\dfrac{4\\sqrt3}{\\sq... |
deepscaler | [
{
"role": "user",
"content": "Let \\\\(x,y\\\\) satisfy the constraint conditions \\\\(\\begin{cases} & x+y-2\\geqslant 0 \\\\\\\\ & x-y+1\\geqslant 0 \\\\\\\\ & x\\leqslant 3 \\end{cases}\\\\). If the minimum value of \\\\(z=mx+y\\\\) is \\\\(-3\\\\), then the value of \\\\(m\\\\) is . Let's think ste... | math | {
"style": "rule",
"ground_truth": "- \\dfrac{2}{3}"
} | {
"split": "train",
"index": 2503,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=- \\sqrt {2}\\sin (2x+ \\frac {\\pi}{4})+6\\sin x\\cos x-2\\cos ^{2}x+1$, where $x\\in\\mathbb{R}$. \n(I) Find the smallest positive period of $f(x)$. \n(II) Find the maximum and minimum values of $f(x)$ in the interval $\\left[0, \\frac {\\pi}{2}\\ri... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 2504,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each value of $x$, $g(x)$ is defined to be the minimum value of the three numbers $3x + 3$, $\\frac{1}{3}x + 1$, and $-\\frac{2}{3}x + 8$. Find the maximum value of $g(x)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10}{3}"
} | {
"split": "train",
"index": 2505,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For the polynomial\n\\[ p(x) = 985 x^{2021} + 211 x^{2020} - 211, \\]\n\nlet its 2021 complex roots be \\( x_1, x_2, \\cdots, x_{2021} \\). Calculate\n\\[ \\sum_{k=1}^{2021} \\frac{1}{x_{k}^{2} + 1} = \\]\n Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "2021"
} | {
"split": "train",
"index": 2506,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sides of a non-degenerate isosceles triangle are \\(x\\), \\(x\\), and \\(24\\) units. How many integer values of \\(x\\) are possible? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 2507,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T_i$ be the set of all integers $n$ such that $50i \\leq n < 50(i + 1)$. For example, $T_4$ is the set $\\{200, 201, 202, \\ldots, 249\\}$. How many of the sets $T_0, T_1, T_2, \\ldots, T_{1999}$ do not contain a perfect square? Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "1733"
} | {
"split": "train",
"index": 2508,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The lines $y=5x+3$, $y=-2x-25$ and $y=3x+k$ intersect at the same point. What is the value of $k$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 2509,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, $2,5,8,11,14$ is an arithmetic sequence with five terms, in which the first term is $2$ and the constant added is $3$. Each row and each c... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 2510,
"solution": "\n#### Step 1: Understanding the Problem\nThe problem states that each row and column in a $5 \\times 5$ array forms an arithmetic sequence. The center square is labeled $X$. We need to determine the value of $X$.\n\n#### Step 2: Analyzing Given Information\nWe are ... |
deepscaler | [
{
"role": "user",
"content": "How many of the smallest 2401 positive integers written in base 7 include the digits 4, 5, or 6? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2146"
} | {
"split": "train",
"index": 2511,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Vasya has 9 different books by Arkady and Boris Strugatsky, each containing a single work by the authors. Vasya wants to arrange these books on a shelf in such a way that:\n(a) The novels \"Beetle in the Anthill\" and \"Waves Extinguish the Wind\" are next to each other (in any... | math | {
"style": "rule",
"ground_truth": "4 \\cdot 7!"
} | {
"split": "train",
"index": 2512,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Source: 2018 Canadian Open Math Challenge Part B Problem 2\n-----\n\nLet ABCD be a square with side length 1. Points $X$ and $Y$ are on sides $BC$ and $CD$ respectively such that the areas of triangels $ABX$ , $XCY$ , and $YDA$ are equal. Find the ratio of the area ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 2513,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( a_{n} = 1 + 2 + \\cdots + n \\), where \\( n \\in \\mathbf{N}_{+} \\), and \\( S_{m} = a_{1} + a_{2} + \\cdots + a_{m} \\), \\( m = 1, 2, \\cdots, m \\). Find the number of values among \\( S_{1}, S_{2}, \\cdots, S_{2017} \\) that are divisible by 2 but not by 4. Let's ... | math | {
"style": "rule",
"ground_truth": "252"
} | {
"split": "train",
"index": 2514,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mr. Wang drives from his home to location $A$. On the way there, he drives the first $\\frac{1}{2}$ of the distance at a speed of 50 km/h and increases his speed by $20\\%$ for the remaining distance. On the way back, he drives the first $\\frac{1}{3}$ of the distance at a spee... | math | {
"style": "rule",
"ground_truth": "330"
} | {
"split": "train",
"index": 2515,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many multiples of 4 are there between 200 and 500? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "74"
} | {
"split": "train",
"index": 2516,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the following image, there is a hexagon $ABEFGD$. Quadrilaterals $ABCD$ and $EFGC$ are congruent rectangles, and quadrilateral $BEGD$ is also a rectangle. Determine the ratio of the areas of the white and shaded parts of the hexagon, given that $|AB| = 5 \\text{ cm}$ and tri... | math | {
"style": "rule",
"ground_truth": "2:1"
} | {
"split": "train",
"index": 2517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the value of the expression \\((5(5(5(5+1)+1)+1)+1)\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "781"
} | {
"split": "train",
"index": 2518,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rationalize the denominator of $\\frac{\\sqrt{8}+\\sqrt{3}}{\\sqrt{2}+\\sqrt{3}}$. Express your answer in simplest form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}-1"
} | {
"split": "train",
"index": 2519,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rhombus $ABCD$ is inscribed in rectangle $WXYZ$ such that vertices $A$, $B$, $C$, and $D$ are on sides $\\overline{WX}$, $\\overline{XY}$, $\\overline{YZ}$, and $\\overline{ZW}$, respectively. It is given that $WA=12$, $XB=9$, $BD=15$, and the diagonal $AC$ of rhombus equals si... | math | {
"style": "rule",
"ground_truth": "66"
} | {
"split": "train",
"index": 2520,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x,$ $y,$ $z$ be real numbers such that $x + y + z = 1,$ and $x \\ge -\\frac{1}{3},$ $y \\ge -1,$ and $z \\ge -\\frac{5}{3}.$ Find the maximum value of\n\\[\\sqrt{3x + 1} + \\sqrt{3y + 3} + \\sqrt{3z + 5}.\\] Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 2521,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the distance from the point $(1,-1,2)$ to the line passing through $(-2,2,1)$ and $(-1,-1,3).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 2522,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Samson writes down the number 123456789 on a piece of paper. He can insert multiplication signs between any two adjacent digits, any number of times at different places, or none at all. By reading the digits between the multiplication signs as individual numbers, he creates an ... | math | {
"style": "rule",
"ground_truth": "123456789"
} | {
"split": "train",
"index": 2523,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "During the Double 11 shopping festival, a certain online store purchased two types of toys, $A$ and $B$, directly from the factory. The purchase price and selling price are as shown in the table: (Note: Profit = Selling Price - Purchase Price)\n\n| | $A$ toy | $B$ toy ... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 2524,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose there is an octahedral die with the numbers 1, 2, 3, 4, 5, 6, 7, and 8 written on its eight faces. Each time the die is rolled, the chance of any of these numbers appearing is the same. If the die is rolled three times, and the numbers appearing on the top face are reco... | math | {
"style": "rule",
"ground_truth": "\\frac{45}{256}"
} | {
"split": "train",
"index": 2525,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum value of the expression \\( (\\sin 3x + \\sin 2y + \\sin z)(\\cos 3x + \\cos 2y + \\cos z) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"split": "train",
"index": 2526,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given:\n$$ \\frac{ \\left( \\frac{1}{3} \\right)^2 + \\left( \\frac{1}{4} \\right)^2 }{ \\left( \\frac{1}{5} \\right)^2 + \\left( \\frac{1}{6} \\right)^2} = \\frac{37x}{73y} $$\nExpress $\\sqrt{x} \\div \\sqrt{y}$ as a common fraction. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "\\frac{75 \\sqrt{73}}{6 \\sqrt{61} \\sqrt{37}}"
} | {
"split": "train",
"index": 2527,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $h(x) = \\sqrt[3]{\\frac{2x+5}{5}}$. For what value of $x$ will $h(3x) = 3(h(x))$? Express your answer in simplest form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{65}{24}"
} | {
"split": "train",
"index": 2528,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer ... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 2529,
"solution": "1. Let $x$ be the number of acorns that both the chipmunk and the squirrel hid. According to the problem, the chipmunk hides 3 acorns per hole and the squirrel hides 4 acorns per hole.\n\n2. Let $h_c$ be the number of holes dug by the chipmunk and $h_s$ be the numbe... |
deepscaler | [
{
"role": "user",
"content": "Given that there are 5 advertisements (3 commercial advertisements and 2 Olympic promotional advertisements), the last advertisement is an Olympic promotional advertisement, and the two Olympic promotional advertisements cannot be broadcast consecutively, determine the number o... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 2530,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f$ is defined on positive integers as follows:\n\\[f(n) = \\left\\{\n\\begin{array}{cl}\nn + 15 & \\text{if $n < 15$}, \\\\\nf(n - 7) & \\text{if $n \\ge 15$}.\n\\end{array}\n\\right.\\]\nFind the maximum value of the function. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 2531,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two adjacent faces of a tetrahedron, which are equilateral triangles with side length 3, form a dihedral angle of 30 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron on the plane conta... | math | {
"style": "rule",
"ground_truth": "\\frac{9\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 2532,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a quadratic function $f(x)$ with a second-degree coefficient $a$, and the inequality $f(x) > -2x$ has the solution set $(1,3)$:\n(1) If the function $y = f(x) + 6a$ has exactly one zero, find the explicit form of $f(x)$.\n(2) Let $h(a)$ be the maximum value of $f(x)$, fin... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 2533,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a moving point \\( P \\) on the \\( x \\)-axis. Given the fixed points \\( A(0,2) \\) and \\( B(0,4) \\), what is the maximum value of \\( \\sin \\angle APB \\) when \\( P \\) moves along the entire \\( x \\)-axis? Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 2534,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $G_{1} G_{2} G_{3}$ be a triangle with $G_{1} G_{2}=7, G_{2} G_{3}=13$, and $G_{3} G_{1}=15$. Let $G_{4}$ be a point outside triangle $G_{1} G_{2} G_{3}$ so that ray $\\overrightarrow{G_{1} G_{4}}$ cuts through the interior of the triangle, $G_{3} G_{4}=G_{4} G_{2}$, and $\... | math | {
"style": "rule",
"ground_truth": "\\frac{169}{23}"
} | {
"split": "train",
"index": 2535,
"solution": "We first show that quadrilateral $G_{1} G_{2} G_{4} G_{3}$ is cyclic. Note that by the law of cosines, $$\\cos \\angle G_{2} G_{1} G_{3}=\\frac{7^{2}+15^{2}-13^{2}}{2 \\cdot 7 \\cdot 15}=\\frac{1}{2}$$ so $\\angle G_{2} G_{1} G_{3}=60^{\\circ}$. However, we know tha... |
deepscaler | [
{
"role": "user",
"content": "One digit of the decimal representation of $\\frac{5}{7}$ is randomly selected. What is the probability that the digit is a 4? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 2536,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a cube of unit side. Let $A$ and $B$ be two opposite vertex. Determine the radius of the sphere, with center inside the cube, tangent to the three faces of the cube with common point $A$ and tangent to the three sides with common point $B$ . Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 2537,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The increasing sequence $3, 15, 24, 48, \\ldots\\,$ consists of those positive multiples of 3 that are one less than a perfect square. What is the remainder when the 1994th term of the sequence is divided by 1000? Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 2538,
"solution": "One less than a perfect square can be represented by $n^2 - 1 = (n+1)(n-1)$. Either $n+1$ or $n-1$ must be divisible by 3. This is true when $n \\equiv -1,\\ 1 \\equiv 2,\\ 1 \\pmod{3}$. Since 1994 is even, $n$ must $\\equiv 1 \\pmod{3}$. It will be the $\\frac{1994... |
deepscaler | [
{
"role": "user",
"content": "Located inside equilateral triangle $ABC$ is a point $P$ such that $PA=8$, $PB=6$, and $PC=10$. To the nearest integer the area of triangle $ABC$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "79"
} | {
"split": "train",
"index": 2539,
"solution": "1. **Identify the relationship between the distances from $P$ to the vertices**:\n Given $PA = 8$, $PB = 6$, and $PC = 10$, we observe that:\n \\[\n PA^2 + PB^2 = 8^2 + 6^2 = 64 + 36 = 100 = 10^2 = PC^2.\n \\]\n This implies that $\\triangle PAB$ is a righ... |
deepscaler | [
{
"role": "user",
"content": "If $x$ men working $x$ hours a day for $x$ days produce $x$ articles, then the number of articles \n(not necessarily an integer) produced by $y$ men working $y$ hours a day for $y$ days is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{y^3}{x^2}"
} | {
"split": "train",
"index": 2540,
"solution": "1. **Identify the production rate**: Let $k$ be the number of articles produced per hour per person. The equation for the production of articles by $x$ men working $x$ hours a day for $x$ days is given by:\n \\[\n x \\text{ men} \\times x \\text{ hours/day} \\ti... |
deepscaler | [
{
"role": "user",
"content": "Four people (A, B, C, D) are practicing passing a ball. The ball is initially passed by A, and each person who receives the ball has an equal probability of passing it to one of the other three people. Let \\( p_{n} \\) represent the probability that the ball returns to A after... | math | {
"style": "rule",
"ground_truth": "\\frac{61}{243}"
} | {
"split": "train",
"index": 2541,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $(a+i)i=b+ai$, solve for $|a+bi|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 2542,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangle $\\triangle ABC$ with angles $A$, $B$, $C$ and their corresponding opposite sides $a$, $b$, $c$. It is known that $2a\\sin (C+ \\frac{\\pi}{6})=b+c$.\n\n1. Find the value of angle $A$.\n2. If $B= \\frac{\\pi}{4}$ and $b-a= \\sqrt{2}- \\sqrt{3}$, find the area o... | math | {
"style": "rule",
"ground_truth": "\\frac{3 + \\sqrt{3}}{4}"
} | {
"split": "train",
"index": 2543,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the equation: $2\\left(x-1\\right)^{2}=x-1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 2544,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a function $g(x),$ for positive integer values of $x,$ by \\[g(x) = \\left\\{\\begin{aligned} \\log_3 x & \\quad \\text{ if } \\log_3 x \\text{ is an integer} \\\\ 1 + g(x + 2) & \\quad \\text{ otherwise}. \\end{aligned} \\right.\\] Compute $g(50).$ Let's think step by s... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 2545,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the number 105,000, express it in scientific notation. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.05\\times 10^{5}"
} | {
"split": "train",
"index": 2546,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many hits does \"3.1415\" get on Google? Quotes are for clarity only, and not part of the search phrase. Also note that Google does not search substrings, so a webpage with 3.14159 on it will not match 3.1415. If $A$ is your answer, and $S$ is the correct answer, then you w... | math | {
"style": "rule",
"ground_truth": "422000"
} | {
"split": "train",
"index": 2547,
"solution": "The answer is 422000."
} |
deepscaler | [
{
"role": "user",
"content": "Dr. Math's house number $WXYZ$ is a four-digit number where each digit $W$, $X$, $Y$, and $Z$ is non-zero, and the two portions of the house number, $WX$ and $YZ$, form two-digit primes. Every prime number selected must be less than 50. Additionally, the sum of the digits in $Y... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 2548,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five blue beads, three green beads, and one red bead are placed in line in random order. Calculate the probability that no two blue beads are immediately adjacent to each other. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{126}"
} | {
"split": "train",
"index": 2549,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a four-digit natural number $M$, if the digit in the thousands place is $6$ more than the digit in the units place, and the digit in the hundreds place is $2$ more than the digit in the tens place, then $M$ is called a \"naive number.\" For example, the four-digit number $7... | math | {
"style": "rule",
"ground_truth": "9313"
} | {
"split": "train",
"index": 2550,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Specify a six-digit number $N$ consisting of distinct digits such that the numbers $2N$, $3N$, $4N$, $5N$, and $6N$ are permutations of its digits. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "142857"
} | {
"split": "train",
"index": 2551,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$2(81+83+85+87+89+91+93+95+97+99)= $ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1800"
} | {
"split": "train",
"index": 2552,
"solution": "1. **Identify the sequence and its properties**: The sequence given in the problem is $81, 83, 85, 87, 89, 91, 93, 95, 97, 99$. This is an arithmetic sequence where each term increases by $2$.\n\n2. **Sum the sequence**: The sum of an arithmetic sequence can be calc... |
deepscaler | [
{
"role": "user",
"content": "When $0.73\\overline{864}$ is expressed as a fraction in the form $\\frac{y}{999900}$, what is the value of $y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "737910"
} | {
"split": "train",
"index": 2553,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of the closed shape formed by the graph of the function $y=2\\sin x$, where $x \\in \\left[\\frac{\\pi}{2}, \\frac{5\\pi}{2}\\right]$, and the lines $y=\\pm2$ is what? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"split": "train",
"index": 2554,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine, with proof, the smallest positive integer \\( n \\) with the following property: For every choice of \\( n \\) integers, there exist at least two whose sum or difference is divisible by 2009. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "1006"
} | {
"split": "train",
"index": 2555,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $x$ such that $\\lceil x \\rceil \\cdot x = 156$. Express $x$ as a decimal. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 2556,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $q(x)$ be a quadratic polynomial such that $[q(x)]^2 - x^2$ is divisible by $(x - 2)(x + 2)(x - 5)$. Find $q(10)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{110}{7}"
} | {
"split": "train",
"index": 2557,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $[x]$, where $x = -3.7 + 1.5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 2558,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nIn a right triangle $ABC$ with legs $AB = 3$ and $BC = 4$, a circle is drawn through the midpoints of sides $AB$ and $AC$, touching the leg $BC$. Find the length of the segment of the hypotenuse $AC$ that lies inside this circle. Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "11/10"
} | {
"split": "train",
"index": 2559,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of the region bounded by the graph of \\[x^2+y^2 = 3|x-y| + 3|x+y|\\] is $m+n\\pi$, where $m$ and $n$ are integers. What is $m + n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 2560,
"solution": "We are given the equation \\(x^2+y^2 = 3|x-y| + 3|x+y|\\) and need to find the area of the region it describes. We will consider different cases based on the absolute values \\(|x-y|\\) and \\(|x+y|\\).\n\n#### Case 1: \\(|x-y|=x-y, |x+y|=x+y\\)\nSubstituting these ... |
deepscaler | [
{
"role": "user",
"content": "Splicing two identical small cubes into a cuboid, the surface area is reduced by approximately \\_\\_\\_\\_\\_\\_%. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16.7"
} | {
"split": "train",
"index": 2561,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Arnaldo claimed that one billion is the same as one million millions. Professor Piraldo corrected him and said, correctly, that one billion is the same as one thousand millions. What is the difference between the correct value of one billion and Arnaldo's assertion?\n(a) 1000\n... | math | {
"style": "rule",
"ground_truth": "999000000000"
} | {
"split": "train",
"index": 2562,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At the end of a professional bowling tournament, the top 5 bowlers have a playoff. First #5 bowls #4. The loser receives $5$th prize and the winner bowls #3 in another game. The loser of this game receives $4$th prize and the winner bowls #2. The loser of this game receives $3$... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 2563,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ordered pairs $(a, b)$ of positive integers satisfy the equation \n\\[a \\cdot b + 63 = 20 \\cdot \\text{lcm}(a, b) + 12 \\cdot \\text{gcd}(a,b),\\] where $\\text{gcd}(a,b)$ denotes the greatest common divisor of $a$ and $b$, and $\\text{lcm}(a,b)$ denotes their least ... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 2564,
"solution": "1. **Expressing $a$ and $b$ in terms of $\\text{lcm}$ and $\\text{gcd}$**: \n Let $x = \\text{lcm}(a, b)$, and $y = \\text{gcd}(a, b)$. We know that $a \\cdot b = \\text{lcm}(a, b) \\cdot \\text{gcd}(a, b) = x \\cdot y$. Substituting this into the given equation,... |
deepscaler | [
{
"role": "user",
"content": "A malfunctioning thermometer shows a temperature of $+1^{\\circ}$ in freezing water and $+105^{\\circ}$ in the steam of boiling water. Currently, this thermometer shows $+17^{\\circ}$; what is the true temperature? Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "15.38"
} | {
"split": "train",
"index": 2565,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two students, A and B, are preparing to have a table tennis match during their physical education class. Assuming that the probability of A winning against B in each game is $\\frac{1}{3}$, the match follows a best-of-three format (the first player to win two games wins the mat... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{27}"
} | {
"split": "train",
"index": 2566,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a hyperbola with the equation $\\frac{y^2}{4} - \\frac{x^2}{a} = 1$ and asymptote equations $y = \\pm\\frac{2\\sqrt{3}}{3}x$, find the eccentricity of this hyperbola. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{7}}{2}"
} | {
"split": "train",
"index": 2567,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "During the New Year, Xiaoming's family bought many bottles of juice. On New Year's Eve, they drank half of the total amount minus 1 bottle. On the first day of the New Year, they drank half of the remaining amount again. On the second day of the New Year, they drank half of the... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 2568,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), point \\(O\\) is the center of the circumcircle, and point \\(L\\) is the midpoint of side \\(AB\\). The circumcircle of triangle \\(ALO\\) intersects the line \\(AC\\) at point \\(K\\). Find the area of triangle \\(ABC\\) if \\(\\angle LOA = 45^\\circ\\)... | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 2569,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The bases of a trapezoid are 3 cm and 5 cm. One of the diagonals of the trapezoid is 8 cm, and the angle between the diagonals is $60^{\\circ}$. Find the perimeter of the trapezoid. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 2570,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The taxi fare in Gotham City is $2.40 for the first $\\frac{1}{2}$ mile and additional mileage charged at the rate $0.20 for each additional 0.1 mile. You plan to give the driver a $2 tip. How many miles can you ride for $10? Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "3.3"
} | {
"split": "train",
"index": 2571,
"solution": "1. **Understanding the Fare Structure**: The initial fare is $2.40 for the first $\\frac{1}{2}$ mile. After that, each additional $0.1$ mile costs $0.20.\n\n2. **Total Fare and Tip**: The total amount you are willing to spend is $10, which includes a $2 tip. Therefo... |
deepscaler | [
{
"role": "user",
"content": "Given non-zero vectors $a=(-x, x)$ and $b=(2x+3, 1)$, where $x \\in \\mathbb{R}$.\n\n$(1)$ If $a \\perp b$, find the value of $x$;\n\n$(2)$ If $a \\nparallel b$, find $|a - b|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"split": "train",
"index": 2572,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using two red, three blue, and four green small cubes, calculate the number of different towers that can be built using eight of these cubes. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1260"
} | {
"split": "train",
"index": 2573,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The volume of the solid of revolution generated by rotating the region bounded by the curve $y= \\sqrt{2x}$, the line $y=x-4$, and the x-axis around the x-axis is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{128\\pi}{3}"
} | {
"split": "train",
"index": 2574,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a = \\log 8$ and $b = \\log 25,$ compute\n\\[5^{a/b} + 2^{b/a}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{2} + 5^{2/3}"
} | {
"split": "train",
"index": 2575,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Petya and his three classmates started a 100-meter race simultaneously, and Petya finished first. Twelve seconds after the race began, no one had finished yet, and all four participants had collectively run a total of 288 meters. When Petya finished the race, the other three pa... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 2576,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $r$ be the speed in miles per hour at which a wheel, $13$ feet in circumference, travels. If the time for a complete rotation of the wheel is shortened by $\\frac{1}{3}$ of a second, the speed $r$ is increased by $6$ miles per hour. Find $r$.\nA) 10\nB) 11\nC) 12\nD) 13\nE)... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 2577,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain store sells a type of student backpack. It is known that the cost price of this backpack is $30$ yuan each. Market research shows that the daily sales quantity $y$ (in units) of this backpack is related to the selling price $x$ (in yuan) as follows: $y=-x+60$ ($30\\le... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 2578,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all the functions $f: \\mathbb{R} \\to\\mathbb{R}$ such that\n\\[f(x-f(y))=f(f(y))+xf(y)+f(x)-1\\]\nfor all $x,y \\in \\mathbb{R} $. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "f(x)=1-\\dfrac{x^2}{2}"
} | {
"split": "train",
"index": 2579,
"solution": "\nWe are given a functional equation for functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\):\n\n\\[\nf(x - f(y)) = f(f(y)) + x f(y) + f(x) - 1\n\\]\n\nfor all \\( x, y \\in \\mathbb{R} \\). We seek to find all possible functions \\( f \\) that satisfy this equation.\... |
deepscaler | [
{
"role": "user",
"content": "If $4^6=8^n$, what is $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 2580,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( f(x) = \\sum_{k=1}^{2017} \\frac{\\cos k x}{\\cos^k x} \\), find \\( f\\left(\\frac{\\pi}{2018}\\right) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 2581,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\sigma(n)$ be the number of positive divisors of $n$ , and let $\\operatorname{rad} n$ be the product of the distinct prime divisors of $n$ . By convention, $\\operatorname{rad} 1 = 1$ . Find the greatest integer not exceeding \\[ 100\\left(\\sum_{n=1}^{\\infty}\\... | math | {
"style": "rule",
"ground_truth": "164"
} | {
"split": "train",
"index": 2582,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We define \\( a @ b = a \\times (a + 1) \\times \\ldots \\times (a + b - 1) \\). Given \\( x @ y @ 2 = 420 \\), then \\( y @ x = \\) ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 2583,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ordered triples $(a, b, c)$ of non-zero real numbers have the property that each number is the product of the other two? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 2584,
"solution": "1. **Setting up the equations:** Given the conditions $ab = c$, $bc = a$, and $ca = b$, we need to find the ordered triples $(a, b, c)$ of non-zero real numbers that satisfy these equations.\n\n2. **Multiplying the equations:** Multiply the three given equations:\n ... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\frac{x}{2} = y^2$ and $\\frac{x}{5} = 3y$, solve for $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "112.5"
} | {
"split": "train",
"index": 2585,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "20 points were marked inside a square and connected with non-intersecting segments to each other and to the vertices of the square, such that the square was divided into triangles. How many triangles were formed? Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 2586,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many total strikes do the clocks make in a day if they strike once every half an hour and strike $1,2,3 \\ldots 12$ times each hour? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "180"
} | {
"split": "train",
"index": 2587,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rotate an equilateral triangle with side length $2$ around one of its sides to form a solid of revolution. The surface area of this solid is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}\\pi"
} | {
"split": "train",
"index": 2588,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There exists a positive real number $x$ such that $ \\cos (\\arctan (x)) = x $. Find the value of $x^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-1 + \\sqrt{5}}{2}"
} | {
"split": "train",
"index": 2589,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways can 6 people stand in a row such that persons A and B are not next to each other and neither is at either end of the row? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 2590,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "5 people are standing in a row for a photo, among them one person must stand in the middle. There are ways to arrange them. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 2591,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the sequence defined by $a_k =\\dfrac{1}{k^2+k}$ for $k\\geq 1$. Given that $a_m+a_{m+1}+\\cdots+a_{n-1}=\\dfrac{1}{29}$, for positive integers $m$ and $n$ with $m<n$, find $m+n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "840"
} | {
"split": "train",
"index": 2592,
"solution": "Note that $a_1 + a_2 + \\cdots + a_i = \\dfrac{i}{i+1}$. This can be proven by induction. Thus, $\\sum\\limits_{i=m}^{n-1} a_i = \\sum\\limits_{i=1}^{n-1} a_i - \\sum\\limits_{i=1}^{m-1} a_i = \\dfrac{n-1}{n} - \\dfrac{m-1}{m} = \\dfrac{n-m}{mn} = 1/29$. Cross-multi... |
deepscaler | [
{
"role": "user",
"content": "Given a triangle \\( ABC \\) with sides \\( AB=13 \\), \\( BC=20 \\), and \\( AC=21 \\). Point \\( K \\) is on side \\( AB \\), point \\( L \\) is on side \\( AC \\), and point \\( N \\) is on side \\( BC \\). It is known that \\( AK=4 \\), \\( CN=1 \\), and \\( CL=\\frac{20}{2... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 2593,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the smallest base-10 positive integer greater than 7 that is a palindrome when written in both base 3 and 5. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 2594,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a cube $ABCD$-$A\\_1B\\_1C\\_1D\\_1$ with edge length $1$, point $M$ is the midpoint of $BC\\_1$, and $P$ is a moving point on edge $BB\\_1$. Determine the minimum value of $AP + MP$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{2}"
} | {
"split": "train",
"index": 2595,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a section of the map, three roads form a right triangle. When motorcyclists were asked about the distance between $A$ and $B$, one of them responded that after traveling from $A$ to $B$, then to $C$, and back to $A$, his odometer showed 60 km. The second motorcyclist added t... | math | {
"style": "rule",
"ground_truth": "22.5"
} | {
"split": "train",
"index": 2596,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sequence $\\{a_n\\}$ forms a geometric sequence, and $a_n > 0$.\n\n(1) If $a_2 - a_1 = 8$, $a_3 = m$. ① When $m = 48$, find the general formula for the sequence $\\{a_n\\}$. ② If the sequence $\\{a_n\\}$ is unique, find the value of $m$.\n\n(2) If $a_{2k} + a_{2k... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 2597,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A set of $n$ people participate in an online video basketball tournament. Each person may be a member of any number of $5$-player teams, but no two teams may have exactly the same $5$ members. The site statistics show a curious fact: The average, over all subsets of size $9$ of... | math | {
"style": "rule",
"ground_truth": "557"
} | {
"split": "train",
"index": 2598,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In this addition problem, each letter stands for a different digit.\n$\\begin{array}{cccc}&T & W & O\\\\ +&T & W & O\\\\ \\hline F& O & U & R\\end{array}$\nIf T = 7 and the letter O represents an even number, what is the only possible value for W? Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 2599,
"solution": "1. **Identify the value of T**: Given that $T = 7$, we substitute $T$ in the addition problem:\n \\[\n \\begin{array}{cccc}\n & 7 & W & O \\\\\n + & 7 & W & O \\\\\n \\hline\n F & O & U & R \\\\\n \\end{array}\n \\]\n\n2. **Determine the value of O... |
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