data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "What is the product of the coordinates of the midpoint of a line segment with endpoints at $(2,3)$ and $(-6,5)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"split": "train",
"index": 1700,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be nonnegative real numbers such that\n\\[\\sin (ax + b) = \\sin 17x\\]for all integers $x.$ Find the smallest possible value of $a.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 1701,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle with a radius of 3 units has its center at $(0, 0)$. A circle with a radius of 5 units has its center at $(12, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. What is the value of $x$? Express your answer as a common ... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"split": "train",
"index": 1702,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be a fixed integer, $n \\geqslant 2$.\n1. Determine the smallest constant $c$ such that the inequality\n$$\n\\sum_{1 \\leqslant i<j \\leqslant n} x_i x_j (x_i^2 + x_j^2) \\leqslant c \\left(\\sum_{i=1}^n x_i \\right)^4\n$$\nholds for all non-negative real numbers $x_1, ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 1703,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\\frac{n}{180}$ terminate? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 1704,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are three identical red balls, three identical yellow balls, and three identical green balls. In how many different ways can they be split into three groups of three balls each? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 1705,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many numbers from the set $\\{1, 2, 3, \\ldots, 100\\}$ have a perfect square factor greater than one? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 1706,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the largest integer that can be expressed in the form $3^{x(3-x)}$ for some real number $x$ .\n\n[i]Proposed by James Lin Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 1707,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The increasing sequence of positive integers $b_1,$ $b_2,$ $b_3,$ $\\dots$ follows the rule\n\\[b_{n + 2} = b_{n + 1} + b_n\\]for all $n \\ge 1.$ If $b_9 = 544,$ then find $b_{10}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "883"
} | {
"split": "train",
"index": 1708,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the numbers $1, 2, \\cdots, 10$, a number $a$ is randomly selected, and from the numbers $-1, -2, \\cdots, -10$, a number $b$ is randomly selected. What is the probability that $a^{2} + b$ is divisible by 3? Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "0.3"
} | {
"split": "train",
"index": 1709,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A natural number greater than 1 is called \"good\" if it is exactly equal to the product of its distinct proper divisors (excluding 1 and the number itself). Find the sum of the first ten \"good\" natural numbers. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "182"
} | {
"split": "train",
"index": 1710,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If one side of a triangle is $12$ inches and the opposite angle is $30^{\\circ}$, then the diameter of the circumscribed circle is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 1711,
"solution": "1. **Identify the Known Values:**\n - One side of the triangle (let's call it $a$) is given as $12$ inches.\n - The angle opposite to this side ($\\angle A$) is $30^\\circ$.\n\n2. **Apply the Extended Law of Sines:**\n - The Extended Law of Sines states that f... |
deepscaler | [
{
"role": "user",
"content": "The function $f(x)$ satisfies\n\\[f(x - y) = f(x) f(y)\\]for all real numbers $x$ and $y,$ and $f(x) \\neq 0$ for all real numbers $x.$ Find $f(3).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 1712,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A snake of length $k$ is an animal which occupies an ordered $k$-tuple $\\left(s_{1}, \\ldots, s_{k}\\right)$ of cells in a $n \\times n$ grid of square unit cells. These cells must be pairwise distinct, and $s_{i}$ and $s_{i+1}$ must share a side for $i=1, \\ldots, k-1$. If th... | math | {
"style": "rule",
"ground_truth": "4571.8706930"
} | {
"split": "train",
"index": 1713,
"solution": "Let $n=30$. The snake can get stuck in only 8 positions, while the total number of positions is about $n^{2} \\times 4 \\times 3 \\times 3=36 n^{2}$. We can estimate the answer as $\\frac{36 n^{2}}{8}=4050$, which is good enough for 13 points. Let's try to compute t... |
deepscaler | [
{
"role": "user",
"content": "In chess, there are two types of minor pieces, the bishop and the knight. A bishop may move along a diagonal, as long as there are no pieces obstructing its path. A knight may jump to any lattice square $\\sqrt{5}$ away as long as it isn't occupied.\n\nOne day, a bishop and a... | math | {
"style": "rule",
"ground_truth": "102"
} | {
"split": "train",
"index": 1714,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangular pyramid $S-ABC$ with the base being an isosceles right triangle with $AB$ as the hypotenuse, and $SA = SB = SC = 2$, $AB = 2$, let points $S$, $A$, $B$, and $C$ all lie on the surface of a sphere centered at $O$. What is the distance from point $O$ to the pla... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 1715,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $180^\\circ$ rotation around the origin in the counter-clockwise direction is applied to $-6 - 3i.$ What is the resulting complex number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6 + 3i"
} | {
"split": "train",
"index": 1716,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a computer game, a player can choose to play as one of three factions: \\( T \\), \\( Z \\), or \\( P \\). There is an online mode where 8 players are divided into two teams of 4 players each. How many total different matches are possible, considering the sets of factions? T... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 1717,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the expression $\\frac{\\sqrt{10} + \\sqrt{15}}{\\sqrt{3} + \\sqrt{5} - \\sqrt{2}}$.\nA) $\\frac{2\\sqrt{30} + 5\\sqrt{2} + 11\\sqrt{5} + 5\\sqrt{3}}{6}$ \nB) $\\sqrt{3} + \\sqrt{5} + \\sqrt{2}$ \nC) $\\frac{\\sqrt{10} + \\sqrt{15}}{6}$ \nD) $\\sqrt{3} + \\sqrt{5} -... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{30} + 5\\sqrt{2} + 11\\sqrt{5} + 5\\sqrt{3}}{6}"
} | {
"split": "train",
"index": 1718,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $PQR$ has positive integer side lengths with $PQ = PR$. Let $J$ be the intersection of the bisectors of $\\angle Q$ and $\\angle R$. Suppose $QJ = 10$. Find the smallest possible perimeter of $\\triangle PQR$. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "1818"
} | {
"split": "train",
"index": 1719,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $4535_6 + 23243_6$. Express your answer in base $6$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32222_6"
} | {
"split": "train",
"index": 1720,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A wholesaler gives different discounts based on the size of the customer's order. The steps to calculate the amount payable by the customer are as follows: \nS1 Input the order amount x (unit: pieces); input the unit price A (unit: yuan); \nS2 If x < 250, then the discount ra... | math | {
"style": "rule",
"ground_truth": "980"
} | {
"split": "train",
"index": 1721,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\(\\frac{x+y}{x-y}+\\frac{x-y}{x+y}=3\\). Find the value of the expression \\(\\frac{x^{2}+y^{2}}{x^{2}-y^{2}}+\\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13/6"
} | {
"split": "train",
"index": 1722,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\pi$ be a permutation of the numbers from 1 through 2012. What is the maximum possible number of integers $n$ with $1 \\leq n \\leq 2011$ such that $\\pi(n)$ divides $\\pi(n+1)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1006"
} | {
"split": "train",
"index": 1723,
"solution": "Since any proper divisor of $n$ must be less than or equal to $n / 2$, none of the numbers greater than 1006 can divide any other number less than or equal to 2012. Since there are at most 1006 values of $n$ for which $\\pi(n) \\leq 1006$, this means that there can ... |
deepscaler | [
{
"role": "user",
"content": "A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are $R_1 = 100$ inches, $R_2 = 60$ inches, and $R_3 = 80$ inches, respectively. The ball always remains in contact with the track and does no... | math | {
"style": "rule",
"ground_truth": "238\\pi"
} | {
"split": "train",
"index": 1724,
"solution": "1. **Identify the radius of the ball and adjust the radii of the arcs:**\n The diameter of the ball is given as 4 inches, so the radius of the ball is half of that, which is 2 inches. When the ball rolls along the track, the center of the ball follows a path that ... |
deepscaler | [
{
"role": "user",
"content": "Two identical rectangular crates are packed with cylindrical pipes, using different methods. Each pipe has diameter $10\\text{ cm}.$ A side view of the first four rows of each of the two different methods of packing is shown below.\n\n[asy]\ndraw(circle((1,1),1),black+linewidth... | math | {
"style": "rule",
"ground_truth": "10 + 5\\sqrt{3}"
} | {
"split": "train",
"index": 1725,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a list of five positive integers, none greater than 100, the mean is 1.5 times the mode. If 31, 58, 98, $x$ and $x$ are the five integers, what is the value of $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 1726,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=3\\sin x+4\\cos x$, if for any $x\\in R$ we have $f(x)\\geqslant f(α)$, then the value of $\\tan α$ is equal to ___. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {3}{4}"
} | {
"split": "train",
"index": 1727,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the probability of bus No. 3 arriving at the bus stop within 5 minutes is 0.20 and the probability of bus No. 6 arriving within 5 minutes is 0.60, calculate the probability that the passenger can catch the bus he needs within 5 minutes. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "0.80"
} | {
"split": "train",
"index": 1728,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a class of 35 students, everyone participates in physics and chemistry. Nine students received an A in chemistry and 15 received an A in physics, including five that received an A in both subjects. How many students did not receive an A in either of these two subjects? Let's... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 1729,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the constant term in the expansion of $\\left( \\sqrt {x}+ \\dfrac {1}{2 \\sqrt {x}}\\right)^{8}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac {35}{8}"
} | {
"split": "train",
"index": 1730,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the total number of triples of integers $(x,y,n)$ satisfying the equation $\\tfrac 1x+\\tfrac 1y=\\tfrac1{n^2}$ , where $n$ is either $2012$ or $2013$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "338"
} | {
"split": "train",
"index": 1731,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the distance between the points $(-5,-2)$ and $(7,3)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 1732,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Construct a square such that two adjacent vertices lie on a circle with a unit radius, and the side connecting the other two vertices is tangent to the circle. Calculate the sides of the square! Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8/5"
} | {
"split": "train",
"index": 1733,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $XYZ$, $\\angle Z=90^\\circ$, $XZ=3$ and $YZ=4$. Points $W$ and $V$ are on $\\overline{XY}$ and $\\overline{YZ}$, respectively, and $\\angle WVZ=90^\\circ$. If $WV=2$, then what is the length of $WY$? Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "\\frac{10}{3}"
} | {
"split": "train",
"index": 1734,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Thirty-nine students from seven classes invented 60 problems, with the students from each class inventing the same number of problems (which is not zero), and the students from different classes inventing different numbers of problems. How many students invented one problem eac... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 1735,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers less than 100 are relatively prime to 200? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 1736,
"solution": "$401 \\leq n<100$ is relatively prime to 200 if and only if it's relatively prime to 100 (200, 100 have the same prime factors). Thus our answer is $\\phi(100)=100 \\frac{1}{2} \\frac{4}{5}=40$."
} |
deepscaler | [
{
"role": "user",
"content": "Given \\\\(\\alpha \\in (0, \\frac{\\pi}{2})\\\\) and \\\\(\\beta \\in (\\frac{\\pi}{2}, \\pi)\\\\) with \\\\(\\sin(\\alpha + \\beta) = \\frac{3}{5}\\\\) and \\\\(\\cos \\beta = -\\frac{5}{13}\\\\), find the value of \\\\(\\sin \\alpha\\\\). Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "\\frac{33}{65}"
} | {
"split": "train",
"index": 1737,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence of figures is formed using tiles. Each tile is an equilateral triangle with side length 7 cm. The first figure consists of 1 tile. Each figure after the first is formed by adding 1 tile to the previous figure. How many tiles are used to form the figure in the sequenc... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 1738,
"solution": "The first figure consists of one tile with perimeter $3 \\times 7 \\mathrm{~cm} = 21 \\mathrm{~cm}$. Each time an additional tile is added, the perimeter of the figure increases by 7 cm (one side length of a tile), because one side length of the previous figure is '... |
deepscaler | [
{
"role": "user",
"content": "Three numbers, $a_1\\,$, $a_2\\,$, $a_3\\,$, are drawn randomly and without replacement from the set $\\{1, 2, 3, \\dots, 1000\\}\\,$. Three other numbers, $b_1\\,$, $b_2\\,$, $b_3\\,$, are then drawn randomly and without replacement from the remaining set of 997 numbers. Let $... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 1739,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the coefficient of the $x^{3}$ term in the expansion of $(2x+1)(x-1)^{5}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-10"
} | {
"split": "train",
"index": 1740,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC,$ $a = 7,$ $b = 9,$ and $c = 4.$ Let $I$ be the incenter.\n\n[asy]\nunitsize(0.8 cm);\n\npair A, B, C, D, E, F, I;\n\nB = (0,0);\nC = (7,0);\nA = intersectionpoint(arc(B,4,0,180),arc(C,9,0,180));\nI = incenter(A,B,C);\n\ndraw(A--B--C--cycle);\ndraw(incircle(A,... | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{7}{20}, \\frac{9}{20}, \\frac{1}{5} \\right)"
} | {
"split": "train",
"index": 1741,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parabola $y^2 = 2px (0 < p < 4)$, with a focus at point $F$, and a point $P$ moving along $C$. Let $A(4,0)$ and $B(p, \\sqrt{2}p)$ with the minimum value of $|PA|$ being $\\sqrt{15}$, find the value of $|BF|$. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "\\dfrac{9}{2}"
} | {
"split": "train",
"index": 1742,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The points $(2, 5), (10, 9)$, and $(6, m)$, where $m$ is an integer, are vertices of a triangle. What is the sum of the values of $m$ for which the area of the triangle is a minimum? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 1743,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define $\\displaystyle{f(x) = x + \\sqrt{x + \\sqrt{x + \\sqrt{x + \\sqrt{x + \\ldots}}}}}$ . Find the smallest integer $x$ such that $f(x)\\ge50\\sqrt{x}$ .\n\n(Edit: The official question asked for the \"smallest integer\"; the intended question was the \"smallest positi... | math | {
"style": "rule",
"ground_truth": "2500"
} | {
"split": "train",
"index": 1744,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A straight concrete sidewalk is to be $3$ feet wide, $60$ feet long, and $3$ inches thick. How many cubic yards of concrete must a contractor order for the sidewalk if concrete must be ordered in a whole number of cubic yards? Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 1745,
"solution": "To find the volume of the concrete needed for the sidewalk, we first convert all dimensions to yards (since 1 yard = 3 feet and 1 yard = 36 inches).\n\n1. **Convert dimensions to yards:**\n - Width: $3$ feet = $\\frac{3}{3} = 1$ yard.\n - Length: $60$ feet = $\\... |
deepscaler | [
{
"role": "user",
"content": "Find \\(n\\) such that \\(2^6 \\cdot 3^3 \\cdot n = 10!\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2100"
} | {
"split": "train",
"index": 1746,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that triangle XYZ is a right triangle with two altitudes of lengths 6 and 18, determine the largest possible integer length for the third altitude. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 1747,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the function $f(x)=\\frac{x}{x+a}$ is symmetric about the point $(1,1)$, and the function $g(x)=\\log_{10}(10^x+1)+bx$ is even. Find the value of $a+b$. 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": 1748,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the squares of the solutions of $x^2-13x+4=0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "161"
} | {
"split": "train",
"index": 1749,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $a$ and $b$ are positive integers such that $a$ has $4$ factors and $b$ has $a$ factors. If $b$ is divisible by $a$, then 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": "24"
} | {
"split": "train",
"index": 1750,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $c$ and $d$ be constants. Suppose that the equation \\[\\frac{(x+c)(x+d)(x-7)}{(x+4)^2} = 0\\] has exactly $3$ distinct roots, while the equation \\[\\frac{(x+2c)(x+5)(x+8)}{(x+d)(x-7)} = 0\\] has exactly $1$ distinct root. Compute $100c + d.$ Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "408"
} | {
"split": "train",
"index": 1751,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mackenzie bought 142 feet of fencing with which to enclose her rectangular garden. If the numbers of feet in the lengths of the garden's sides are natural numbers, what is the maximum number of square feet that can be enclosed by the fencing? Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "1260"
} | {
"split": "train",
"index": 1752,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ and $c$ be nonzero real numbers, and let\n\\[x = \\frac{b}{c} + \\frac{c}{b}, \\quad y = \\frac{a}{c} + \\frac{c}{a}, \\quad z = \\frac{a}{b} + \\frac{b}{a}.\\]Simplify $x^2 + y^2 + z^2 - xyz.$ Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 1753,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many sequences of 5 positive integers $(a, b, c, d, e)$ satisfy $a b c d e \\leq a+b+c+d+e \\leq 10$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "116"
} | {
"split": "train",
"index": 1754,
"solution": "We count based on how many 1's the sequence contains. If $a=b=c=d=e=1$ then this gives us 1 possibility. If $a=b=c=d=1$ and $e \\neq 1$, $e$ can be $2,3,4,5,6$. Each such sequence $(1,1,1,1, e)$ can be arranged in 5 different ways, for a total of $5 \\cdot 5=25$ way... |
deepscaler | [
{
"role": "user",
"content": "Ranu starts with one standard die on a table. At each step, she rolls all the dice on the table: if all of them show a 6 on top, then she places one more die on the table; otherwise, she does nothing more on this step. After 2013 such steps, let $D$ be the number of dice on... | math | {
"style": "rule",
"ground_truth": "10071"
} | {
"split": "train",
"index": 1755,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A box contains $2$ pennies, $4$ nickels, and $6$ dimes. Six coins are drawn without replacement, with each coin having an equal probability of being chosen. What is the probability that the value of coins drawn is at least $50$ cents? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "\\frac{127}{924}"
} | {
"split": "train",
"index": 1756,
"solution": "To solve this problem, we need to calculate the probability that the value of the coins drawn is at least 50 cents. We will first determine the total number of ways to draw 6 coins from the 12 available, and then find the number of successful outcomes where the tota... |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a rectangle with $AB=10$ and $BC=26$ . Let $\\omega_1$ be the circle with diameter $\\overline{AB}$ and $\\omega_2$ be the circle with diameter $\\overline{CD}$ . Suppose $\\ell$ is a common internal tangent to $\\omega_1$ and $\\omega_2$ and tha... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 1757,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the isosceles trapezoid \\(ABCD\\) with bases \\(AD\\) and \\(BC\\), perpendiculars \\(BH\\) and \\(DK\\) are drawn from vertices \\(B\\) and \\(D\\) to the diagonal \\(AC\\). It is known that the feet of the perpendiculars lie on the segment \\(AC\\) and that \\(AC = 20\\),... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 1758,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tamara has three rows of two $6$-feet by $2$-feet flower beds in her garden. The beds are separated and also surrounded by $1$-foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet? Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "78"
} | {
"split": "train",
"index": 1759,
"solution": "1. **Calculate the dimensions of the garden including walkways:**\n - Each flower bed measures $6$ feet by $2$ feet.\n - There are $1$-foot-wide walkways around and between the beds.\n\n For the width:\n - There are two beds in each row, so the total width o... |
deepscaler | [
{
"role": "user",
"content": "Find all values of $x$ that satisfy\n\\[5x - 1 < (x + 1)^2 < 7x - 3.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(2,4)"
} | {
"split": "train",
"index": 1760,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $\\frac {x^{2}}{9} + \\frac {y^{2}}{4} = 1$, and the line $L: x + 2y - 10 = 0$.\n\n(1) Does there exist a point $M$ on the ellipse for which the distance to line $L$ is minimal? If so, find the coordinates of point $M$ and the minimum distance.\n\n(2) Does the... | math | {
"style": "rule",
"ground_truth": "3\\sqrt {5}"
} | {
"split": "train",
"index": 1761,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest real number \\(\\lambda\\) such that for the real-coefficient polynomial \\(f(x) = x^3 + ax^2 + bx + c\\) with all roots non-negative real numbers, we have \\(f(x) \\geqslant \\lambda(x - a)^3\\) for all \\(x \\geqslant 0\\). Also, determine when the equality h... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{27}"
} | {
"split": "train",
"index": 1762,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the real values of $x$ satisfying the equality $|x+2|=2|x-2|$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6\\frac{2}{3}"
} | {
"split": "train",
"index": 1763,
"solution": "To solve the equation $|x+2| = 2|x-2|$, we need to consider the different cases for the absolute values based on the sign of the expressions inside them.\n\n#### Case 1: $x + 2 \\geq 0$ and $x - 2 \\geq 0$\nThis implies $x \\geq 2$. The equation becomes:\n\\[ x+2 = ... |
deepscaler | [
{
"role": "user",
"content": "Let \\( x, y, z, w \\) be four consecutive vertices of a regular \\( A \\)-gon. If the length of the line segment \\( xy \\) is 2 and the area of the quadrilateral \\( xyzw \\) is \\( a + \\sqrt{b} \\), find the value of \\( B = 2^a \\cdot 3^b \\). Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 1764,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Person A and person B independently attempt to decrypt a password. Their probabilities of successfully decrypting the password are $\\dfrac{1}{3}$ and $\\dfrac{1}{4}$, respectively. Calculate:\n\n$(1)$ The probability that exactly one of them decrypts the password.\n\n$(2)$ If ... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 1765,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Create a five-digit number without repeated digits using the numbers 0, 3, 4, 5, and 6, with the condition that even numbers must be adjacent to each other, as well as odd numbers. How many such five-digit numbers exist? Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 1766,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $n$, if $9^n\\cdot9^n\\cdot9^n\\cdot9^n=81^4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 1767,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\cos(\\omega x - \\frac{\\pi}{3}) - \\cos(\\omega x)$ $(x \\in \\mathbb{R}, \\omega$ is a constant, and $1 < \\omega < 2)$, the graph of function $f(x)$ is symmetric about the line $x = \\pi$. \n(Ⅰ) Find the smallest positive period of the function ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 1768,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square $ABCD$ is inscribed in the region bound by the parabola $y = x^2 - 8x + 12$ and the $x$-axis, as shown below. Find the area of square $ABCD.$\n\n[asy]\nunitsize(0.8 cm);\n\nreal parab (real x) {\n return(x^2 - 8*x + 12);\n}\n\npair A, B, C, D;\nreal x = -1 + sqrt(5);\n... | math | {
"style": "rule",
"ground_truth": "24 - 8 \\sqrt{5}"
} | {
"split": "train",
"index": 1769,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Perform the calculations. \n36×17+129 \n320×(300-294) \n25×5×4 \n18.45-25.6-24.4. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-31.55"
} | {
"split": "train",
"index": 1770,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest integral value of $x$ which solves: $\\frac{1}{3}<{\\frac{x}{5}}<{\\frac{5}{8}}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 1771,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the convex quadrilateral \\(ABCD\\), the intersection point of its diagonals is \\(O\\). What is the minimum area of the quadrilateral if the area of triangle \\(AOB\\) is \\(4 \\mathrm{~cm}^2\\) and the area of triangle \\(COD\\) is \\(9 \\mathrm{~cm}^2\\)? Let's think step... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 1772,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The side surface of a cylinder unfolds into a rectangle with side lengths of $6\\pi$ and $4\\pi$. The surface area of the cylinder is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24\\pi^2 + 8\\pi"
} | {
"split": "train",
"index": 1773,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Egor wrote a number on the board and encoded it according to the rules of letter puzzles (different letters correspond to different digits, and identical letters correspond to identical digits). The word \"GUATEMALA\" was the result. How many different numbers could Egor have o... | math | {
"style": "rule",
"ground_truth": "20160"
} | {
"split": "train",
"index": 1774,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate\n\n$$\n\\operatorname{tg} \\frac{\\pi}{43} \\cdot \\operatorname{tg} \\frac{2 \\pi}{43}+\\operatorname{tg} \\frac{2 \\pi}{43} \\cdot \\operatorname{tg} \\frac{3 \\pi}{43}+\\ldots+\\operatorname{tg} \\frac{k \\pi}{43} \\cdot \\operatorname{tg} \\frac{(k+1) \\pi}{43}+\\... | math | {
"style": "rule",
"ground_truth": "-2021"
} | {
"split": "train",
"index": 1775,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In △ABC, the sides opposite to angles A, B, C are a, b, c respectively. If acosB - bcosA = $$\\frac {c}{3}$$, then the minimum value of $$\\frac {acosA + bcosB}{acosB}$$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt {2}"
} | {
"split": "train",
"index": 1776,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an inverted cone with a base radius of $15 \\mathrm{cm}$ and a height of $15 \\mathrm{cm}$, and a cylinder with a horizontal base radius of $18 \\mathrm{cm}$, determine the height in centimeters of the water in the cylinder after $10\\%$ of the water is lost from the cone... | math | {
"style": "rule",
"ground_truth": "3.125"
} | {
"split": "train",
"index": 1777,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are two schools, A and B, each sending 5 students to participate in a long-distance race. The rule is: the student who finishes in the \\( K \\)-th place receives \\( K \\) points (no two students finish at the same time). The school with the lower total score wins. How m... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 1778,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A number in the set $\\{50, 51, 52, 53, ... , 500\\}$ is randomly selected. What is the probability that it is a two-digit number divisible by 3? 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{17}{451}"
} | {
"split": "train",
"index": 1779,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given non-negative numbers $x$, $y$, and $z$ such that $x+y+z=2$, determine the minimum value of $\\frac{1}{3}x^3+y^2+z$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{12}"
} | {
"split": "train",
"index": 1780,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively in $\\triangle ABC$, and $(\\sin A + \\sin B)(a-b) = (\\sin C - \\sin B)c$.\n\n1. Find the measure of angle $A$.\n2. If $a=4$, find the maximum area of $\\triangle ABC$. Let's think ste... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"split": "train",
"index": 1781,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the required distance between the pins and the length of the string in order to draw an ellipse with a length of 12 cm and a width of 8 cm. Find a simple rule that allows for constructing an ellipse of predetermined dimensions. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 1782,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of $\\sin 68^{\\circ} \\sin 67^{\\circ} - \\sin 23^{\\circ} \\cos 68^{\\circ}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 1783,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of all real numbers $x$ such that $0 \\le x \\le 2016 \\pi$ and $\\sin x < 3 \\sin(x/3)$ . The set $S$ is the union of a finite number of disjoint intervals. Compute the total length of all these intervals. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "1008\\pi"
} | {
"split": "train",
"index": 1784,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find a six-digit number where the first digit is 6 times less than the sum of all the digits to its right, and the second digit is 6 times less than the sum of all the digits to its right. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "769999"
} | {
"split": "train",
"index": 1785,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $RSP$ is a straight line and $\\angle QSP = 80^\\circ$. What is the measure of $\\angle PQR$, in degrees?\n\n[asy]\ndraw((.48,-.05)--(.48,.05)); draw((.52,-.05)--(.52,.05)); draw((1.48,-.05)--(1.48,.05)); draw((1.52,-.05)--(1.52,.05));\n\ndraw((1.04,.51)--(1.14... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 1786,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $(-64)\\div (-32)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 1787,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The total investment of the Xiangshan Port Sea-Crossing Bridge, about 7.7 billion yuan, should be expressed in scientific notation. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7.7 \\times 10^9"
} | {
"split": "train",
"index": 1788,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A plane has no vertex of a regular dodecahedron on it,try to find out how many edges at most may the plane intersect the regular dodecahedron? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 1789,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the line $l: x-y+4=0$ and the circle $C: \\begin{cases}x=1+2\\cos \\theta \\\\ y=1+2\\sin \\theta\\end{cases} (\\theta$ is a parameter), find the distance from each point on $C$ to $l$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{2}-2"
} | {
"split": "train",
"index": 1790,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the equation $x^2 + 14x = 32$. Find the values of $a$ and $b$ such that the positive solution of the equation has the form $\\sqrt{a}-b$, where $a$ and $b$ are positive natural numbers. Calculate $a+b$. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "88"
} | {
"split": "train",
"index": 1791,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given points $A(-2,0)$ and $P(1, \\frac{3}{2})$ on the ellipse $M: \\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a>b>0)$, and two lines with slopes $k$ and $-k (k>0)$ passing through point $P$ intersect ellipse $M$ at points $B$ and $C$.\n(I) Find the equation of ellipse $M$ an... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 1792,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of\n\\[(12 - x)(10 - x)(12 + x)(10 + x).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-484"
} | {
"split": "train",
"index": 1793,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all functions $f: (0, \\infty) \\to (0, \\infty)$ such that\n\\begin{align*}\nf(y(f(x))^3 + x) = x^3f(y) + f(x) \n\\end{align*}\nfor all $x, y>0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x"
} | {
"split": "train",
"index": 1794,
"solution": "\nWe are given the functional equation for functions \\( f: (0, \\infty) \\to (0, \\infty) \\) defined by:\n\n\\[\nf(y(f(x))^3 + x) = x^3f(y) + f(x) \n\\]\n\nfor all \\( x, y > 0 \\). We need to find all such functions \\( f \\).\n\n1. **Initial Observation:**\n W... |
deepscaler | [
{
"role": "user",
"content": "Square $ABCD$ has side length $30$. Point $P$ lies inside the square so that $AP = 12$ and $BP = 26$. The centroids of $\\triangle{ABP}$, $\\triangle{BCP}$, $\\triangle{CDP}$, and $\\triangle{DAP}$ are the vertices of a convex quadrilateral. What is the area of that quadrilater... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 1795,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $p$ and $q$ are positive numbers for which $\\log_{9}(p) = \\log_{12}(q) = \\log_{16}(p+q)$. What is the value of $\\frac{q}{p}$? 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": 1796,
"solution": "1. **Rewrite the given logarithmic equations using properties of logarithms**:\n Given that $\\log_9(p) = \\log_{12}(q) = \\log_{16}(p+q)$, we can use the change of base formula to rewrite these equations as:\n \\[\n \\frac{\\log p}{\\log 9} = \\frac{\\log q}{... |
deepscaler | [
{
"role": "user",
"content": "Given that the angle between vectors $\\overrightarrow {a}$ and $\\overrightarrow {b}$ is 120°, and the magnitude of $\\overrightarrow {a}$ is 2. If $(\\overrightarrow {a} + \\overrightarrow {b}) \\cdot (\\overrightarrow {a} - 2\\overrightarrow {b}) = 0$, find the projection of... | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{33} + 1}{8}"
} | {
"split": "train",
"index": 1797,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( f(x) = x^2 + 3x + 2 \\) and \\( S = \\{0, 1, 2, 3, \\cdots, 100\\} \\), if \\( a \\in S \\) and \\( f(a) \\) is divisible by 6, how many such \\( a \\) exist? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 1798,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Add $A85_{12}$ and $2B4_{12}$. Express your answer in base $12$, using $A$ for $10$ and $B$ for $11$ if necessary. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1179_{12}"
} | {
"split": "train",
"index": 1799,
"solution": ""
} |
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