data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "The Dingoberry Farm is a 10 mile by 10 mile square, broken up into 1 mile by 1 mile patches. Each patch is farmed either by Farmer Keith or by Farmer Ann. Whenever Ann farms a patch, she also farms all the patches due west of it and all the patches due south of it. Ann puts up ... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 39549,
"solution": "Whenever Ann farms a patch $P$, she also farms all the patches due west of $P$ and due south of $P$. So, the only way she can put a scarecrow on $P$ is if Keith farms the patch immediately north of $P$ and the patch immediately east of $P$, in which case Ann cannot... |
deepscaler | [
{
"role": "user",
"content": "The distance from the origin to the line $3x+2y-13=0$ is $\\sqrt {13}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{13}"
} | {
"split": "train",
"index": 39550,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that for triangle $ABC$, the internal angles $A$ and $B$ satisfy $$\\frac {\\sin B}{\\sin A} = \\cos(A + B),$$ find the maximum value of $\\tan B$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{4}"
} | {
"split": "train",
"index": 39551,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each edge of a regular tetrahedron is divided into three equal parts. Through each point of division, two planes are drawn, each parallel to one of the two faces of the tetrahedron that do not pass through this point. Into how many parts do the constructed planes divide the tet... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 39552,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rationalize the denominator of $\\frac{7}{3+\\sqrt{8}}$. Write the simplified expression in the form $\\frac{A\\sqrt{B}+C}{D}$, where $A$, $B$, $C$, and $D$ are integers, $D$ is positive, and $B$ is not divisible by the square of any prime. Determine $A+B+C+D$ if the greatest c... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 39553,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose a parabola has vertex $\\left(\\frac{3}{2},-\\frac{25}{4}\\right)$ and follows the equation $y = ax^2 + bx + c$, where $a < 0$ and the product $abc$ is an integer. Find the largest possible value of $a$. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 39554,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An icosidodecahedron is a convex polyhedron with 20 triangular faces and 12 pentagonal faces. How many vertices does it have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 39555,
"solution": "Since every edge is shared by exactly two faces, there are $(20 \\cdot 3+12 \\cdot 5) / 2=60$ edges. Using Euler's formula $v-e+f=2$, we see that there are 30 vertices."
} |
deepscaler | [
{
"role": "user",
"content": "A one-meter gas pipe has rusted in two places. Determine the probability that all three resulting parts can be used as connections to gas stoves, if the regulations stipulate that the stove should not be closer than 25 cm to the main gas pipe. Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "9/16"
} | {
"split": "train",
"index": 39556,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( f \\) be the function defined by \\( f(x) = -3 \\sin(\\pi x) \\). How many values of \\( x \\) such that \\(-3 \\le x \\le 3\\) satisfy the equation \\( f(f(f(x))) = f(x) \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "79"
} | {
"split": "train",
"index": 39557,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express \\( 0.3\\overline{45} \\) as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{83}{110}"
} | {
"split": "train",
"index": 39558,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sector $OAB$ is a quarter of a circle with a radius of 6 cm. A circle is inscribed within this sector, tangent to the two radii and the arc at three points. Determine the radius of the inscribed circle, expressed in simplest radical form. Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "6\\sqrt{2} - 6"
} | {
"split": "train",
"index": 39559,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Eliminate every second number in a clockwise direction from numbers $1, 2, 3, \\cdots, 2001$ that have been placed on a circle, starting with the number 2, until only one number remains. What is the last remaining number? Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "1955"
} | {
"split": "train",
"index": 39560,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a rectangular room is 15 feet long and 108 inches wide, calculate the area of the new extended room after adding a 3 feet wide walkway along the entire length of one side, in square yards, where 1 yard equals 3 feet and 1 foot equals 12 inches. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 39561,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The curves $x^{2}+y^{2}=36$ and $y=x^{2}-7$ intersect at four points. Find the sum of the squares of the $x$-coordinates of these points. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 39562,
"solution": "If we use the system of equations to solve for $y$, we get $y^{2}+y-29=0$ (since $x^{2}=y+7$). The sum of the roots of this equation is -1. Combine this with $x^{2}=y+7$ to see that the sum of the square of the possible values of $x$ is $2 \\cdot(-1+7 \\cdot 2)=26$... |
deepscaler | [
{
"role": "user",
"content": "Jia and Yi are playing a guessing game with the following rules: It is known that there are five cards, each with the numbers $1-\\left( \\frac{1}{2} \\right)^n$ ($n\\in \\mathbf{N}^*, 1\\leqslant n\\leqslant 5$) written on them. Now, Jia and Yi each randomly draw one card and ... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{8}"
} | {
"split": "train",
"index": 39563,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given non-zero plane vectors $\\overrightarrow{a}$, $\\overrightarrow{b}$, $\\overrightarrow{c}$ satisfy $|\\overrightarrow{a}|=2$, $|\\overrightarrow{b}-\\overrightarrow{c}|=1$. If the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is $\\frac{π}{3}$, calculate t... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3} - 1"
} | {
"split": "train",
"index": 39564,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a$ and $b$ are integers such that $x^2 - x - 1$ is a factor of $ax^3 + bx^2 + 1,$ find $b.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 39565,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A ball travels on a parabolic path in which the height (in feet) is given by the expression $-16t^2+80t+21$, where $t$ is the time after launch. What is the maximum height of the ball, in feet? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "121"
} | {
"split": "train",
"index": 39566,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given: $$\\frac { A_{ n }^{ 3 }}{6}=n$$ (where $n\\in\\mathbb{N}^{*}$), and $(2-x)^{n}=a_{0}+a_{1}x+a_{2}x^{2}+\\ldots+a_{n}x^{n}$ \nFind the value of $a_{0}-a_{1}+a_{2}-\\ldots+(-1)^{n}a_{n}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 39567,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There were no more than 70 mushrooms in the basket, among which 52% were white. If you throw out the three smallest mushrooms, the white mushrooms will become half of the total.\n\nHow many mushrooms are in the basket? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 39568,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alicia had two containers. The first was $\\frac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\\frac{3}{4}$ full of water. What is the ratio of the volume of ... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{10}"
} | {
"split": "train",
"index": 39569,
"solution": "\n1. **Define Variables:**\n Let the volume of the first container be $A$ and the volume of the second container be $B$.\n\n2. **Set Up the Equation:**\n Since Alicia poured $\\frac{5}{6}$ of the first container into the second container, and this amount filled... |
deepscaler | [
{
"role": "user",
"content": "The perimeter of the polygon shown is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 39570,
"solution": "To solve this problem, we need to determine the perimeter of the polygon. The solution suggests visualizing the polygon as part of a rectangle and then using the properties of the rectangle to find the perimeter of the polygon.\n\n1. **Visualize the Polygon as Part... |
deepscaler | [
{
"role": "user",
"content": "In the right triangle \\(ABC\\), the leg \\(AB = 3\\), and the leg \\(AC = 6\\). The centers of circles with radii 1, 2, and 3 are located at points \\(A\\), \\(B\\), and \\(C\\) respectively. Find the radius of the circle that is externally tangent to each of these three given... | math | {
"style": "rule",
"ground_truth": "\\frac{8 \\sqrt{11} - 19}{7}"
} | {
"split": "train",
"index": 39571,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $\\left(\\frac{7}{8}\\right)^3 \\cdot \\left(\\frac{7}{8}\\right)^{-3}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 39572,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{R}$ be the matrix for rotating about the origin counter-clockwise by an angle of $58^\\circ.$ Find $\\det \\mathbf{R}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 39573,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each number in the list $1,2,3,\\ldots,10$ is either colored red or blue. Numbers are colored independently, and both colors are equally probable. The expected value of the number of positive integers expressible as a sum of a red integer and a blue integer can be written as ... | math | {
"style": "rule",
"ground_truth": "455"
} | {
"split": "train",
"index": 39574,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the triangular pyramid $A-BCD$, where $AB=AC=BD=CD=BC=4$, the plane $\\alpha$ passes through the midpoint $E$ of $AC$ and is perpendicular to $BC$, calculate the maximum value of the area of the section cut by plane $\\alpha$. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 39575,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jake will roll two standard six-sided dice and make a two-digit number from the numbers he rolls. If he rolls a 4 and a 2, he can form either 42 or 24. What is the probability that he will be able to make an integer between 30 and 40, inclusive? Express your answer as a common ... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{36}"
} | {
"split": "train",
"index": 39576,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Lyndy conducted a survey of the 300 students in her school. She found that 60 students own dogs, while only 30 students own cats. What percent of the students own cats? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10\\%"
} | {
"split": "train",
"index": 39577,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagram below shows a $4\\times4$ rectangular array of points, each of which is $1$ unit away from its nearest neighbors.\n[asy] unitsize(0.25inch); defaultpen(linewidth(0.7)); int i, j; for(i = 0; i < 4; ++i) \tfor(j = 0; j < 4; ++j) \t\tdot(((real)i, (real)j)); [/asy]\nD... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 39578,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mena listed the numbers from 1 to 30 one by one. Emily copied these numbers and substituted every digit 2 with digit 1. Both calculated the sum of the numbers they wrote. By how much is the sum that Mena calculated greater than the sum that Emily calculated? Let's think step by... | math | {
"style": "rule",
"ground_truth": "103"
} | {
"split": "train",
"index": 39579,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At 12 o'clock, the angle between the hour hand and the minute hand is 0 degrees. After that, at what time do the hour hand and the minute hand form a 90-degree angle for the 6th time? (12-hour format) Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "3:00"
} | {
"split": "train",
"index": 39580,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least positive integer with exactly $12$ positive factors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 39581,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ be a real number. Find the maximum value of $2^{x(1-x)}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[4]{2}"
} | {
"split": "train",
"index": 39582,
"solution": "Consider the function $2^{y}$. This is monotonically increasing, so to maximize $2^{y}$, you simply want to maximize $y$. Here, $y=x(1-x)=-x^{2}+x$ is a parabola opening downwards. The vertex of the parabola occurs at $x=(-1) /(-2)=1 / 2$, so the maximum value of t... |
deepscaler | [
{
"role": "user",
"content": "If the measure of angle $C$ is double the measure of angle $B$, what is the measure of angle $A$ in triangle $ABC$?\n\n[asy]\n\npair A,B,C;\n\nA=(0,0);\nB=(5,0);\nC=(-0.8,2.5);\n\ndraw(A--B--C--A);\n\nlabel(\"$A$\",A,SW);\nlabel(\"$B$\",B,SE);\nlabel(\"$C$\",C,N);\n\ndraw((4,0.... | math | {
"style": "rule",
"ground_truth": "117^\\circ"
} | {
"split": "train",
"index": 39583,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate: $(2^2)^3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 39584,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convert 2 meters plus 3 centimeters plus 5 millimeters into meters. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2.035 \\text{ m}"
} | {
"split": "train",
"index": 39585,
"solution": "Since there are 100 cm in 1 m, then 1 cm is 0.01 m. Thus, 3 cm equals 0.03 m. Since there are 1000 mm in 1 m, then 1 mm is 0.001 m. Thus, 5 mm equals 0.005 m. Therefore, 2 m plus 3 cm plus 5 mm equals $2+0.03+0.005=2.035 \\mathrm{~m}$."
} |
deepscaler | [
{
"role": "user",
"content": "For what value of $n$ is the five-digit number $\\underline{7n933}$ divisible by 33? (Note: the underlining is meant to indicate that the number should be interpreted as a five-digit number whose ten thousands digit is 7, whose thousands digit is $n$, and so on). Let's think st... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 39586,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A refrigerator is offered at sale at $250.00 less successive discounts of 20% and 15%. The sale price of the refrigerator is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "77\\% of 250.00"
} | {
"split": "train",
"index": 39587,
"solution": "1. **Calculate the price after the first discount:** \n The original price of the refrigerator is $250.00$. The first discount is $20\\%$ of the original price. Therefore, the price after the first discount is calculated as follows:\n \\[\n 250.00 \\times (1... |
deepscaler | [
{
"role": "user",
"content": "In a school there are \\( n \\) students, each with a different student number. Each student number is a positive factor of \\( 60^{60} \\), and the H.C.F. of any two student numbers is not a student number in the school. Find the greatest possible value of \\( n \\). Let's thi... | math | {
"style": "rule",
"ground_truth": "3721"
} | {
"split": "train",
"index": 39588,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jason is trying to remember the five digit combination to his safe. He knows that he only used digits 1 through 5 (possibly repeated), that every even digit was followed by an odd digit, and every odd digit was followed by an even digit. How many possible combinations does Ja... | math | {
"style": "rule",
"ground_truth": "180"
} | {
"split": "train",
"index": 39589,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a dark room drawer, there are 100 red socks, 80 green socks, 60 blue socks, and 40 black socks. A young person picks out one sock at a time without seeing its color. To ensure that at least 10 pairs of socks are obtained, what is the minimum number of socks they must pick ou... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 39590,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A$ and $B$ be real numbers such that $\\frac{A}{x-5}+B(x+1)=\\frac{-3x^2+12x+22}{x-5}$. What is $A+B$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 39591,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many positive integers \\( n \\) less than or equal to 500 is \n$$(\\cos t - i\\sin t)^n = \\cos nt - i\\sin nt$$ \ntrue for all real \\( t \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "500"
} | {
"split": "train",
"index": 39592,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mrs. Delta's language class has 52 students, each with unique initials, and no two students have initials that are alphabetically consecutive (e.g., AB cannot follow AC directly). Assuming Y is considered a consonant, what is the probability of randomly picking a student whose ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{52}"
} | {
"split": "train",
"index": 39593,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function \\( f(x)=\\left(1-x^{3}\\right)^{-1 / 3} \\), find \\( f(f(f \\ldots f(2018) \\ldots)) \\) where the function \\( f \\) is applied 2019 times. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2018"
} | {
"split": "train",
"index": 39594,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In \\(\\triangle ABC\\), \\(AB = 13\\), \\(BC = 14\\), and \\(CA = 15\\). \\(P\\) is a point inside \\(\\triangle ABC\\) such that \\(\\angle PAB = \\angle PBC = \\angle PCA\\). Find \\(\\tan \\angle PAB\\). Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "\\frac{168}{295}"
} | {
"split": "train",
"index": 39595,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\( \\frac{10+11+12}{3} = \\frac{2010+2011+2012+N}{4} \\), then find the value of \\(N\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5989"
} | {
"split": "train",
"index": 39596,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a plane, 6 lines intersect pairwise, but only three pass through the same point. Find the number of non-overlapping line segments intercepted. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 39597,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\\leq m\\leq 2012$ and $5^n<2^m<2^{m+2}<5^{n+1}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "279"
} | {
"split": "train",
"index": 39598,
"solution": "To solve this problem, we need to determine how many pairs of integers $(m, n)$ satisfy the inequalities $5^n < 2^m < 2^{m+2} < 5^{n+1}$ for $1 \\leq m \\leq 2012$.\n\n1. **Understanding the inequalities:**\n - $5^n < 2^m$ implies that $n \\log 5 < m \\log 2$.\n ... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=x^{3}- \\frac {3}{2}x^{2}+ \\frac {3}{4}x+ \\frac {1}{8}$, find the value of $\\sum\\limits_{k=1}^{2016}f( \\frac {k}{2017})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "504"
} | {
"split": "train",
"index": 39599,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A particular $12$-hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a $1$, it mistakenly displays a $9$. For example, when it is 1:16 PM the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show t... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 39600,
"solution": "\n#### Analyzing the Problem:\nThe clock displays the time incorrectly whenever a '1' should appear in the hour or minute display. We need to determine the fraction of the day when the clock shows the correct time.\n\n#### Step 1: Determine Incorrect Hours\nThe clo... |
deepscaler | [
{
"role": "user",
"content": "For how many positive integers $n$ less than or equal to 500 is $$(\\cos t - i\\sin t)^n = \\cos nt - i\\sin nt$$ true for all real $t$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "500"
} | {
"split": "train",
"index": 39601,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that an isosceles trapezoid is circumscribed around a circle, find the ratio of the area of the trapezoid to the area of the circle if the distance between the points where the circle touches the non-parallel sides of the trapezoid is related to the radius of the circle a... | math | {
"style": "rule",
"ground_truth": "\\frac{8\\sqrt{3}}{3\\pi}"
} | {
"split": "train",
"index": 39602,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The projection of $\\begin{pmatrix} 0 \\\\ 1 \\\\ 4 \\end{pmatrix}$ onto a certain vector $\\mathbf{w}$ is $\\begin{pmatrix} 1 \\\\ -1/2 \\\\ 1/2 \\end{pmatrix}.$ Find the projection of $\\begin{pmatrix} 3 \\\\ 3 \\\\ -2 \\end{pmatrix}$ onto $\\mathbf{w}.$ Let's think step by ... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 1/3 \\\\ -1/6 \\\\ 1/6 \\end{pmatrix}"
} | {
"split": "train",
"index": 39603,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "1990-1980+1970-1960+\\cdots -20+10 = Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"split": "train",
"index": 39604,
"solution": "1. **Identify the pattern and the number of terms**: The sequence given is $1990 - 1980 + 1970 - 1960 + \\cdots - 20 + 10$. We observe that the sequence alternates between addition and subtraction, starting with a subtraction. The sequence starts at $1990$ and ends... |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\sqrt{28x} \\cdot \\sqrt{15x} \\cdot \\sqrt{21x}$. Express your answer in simplest radical form in terms of $x$.\n\nNote: When entering a square root with more than one character, you must use parentheses or brackets. For example, you should enter $\\sqrt{14}$ as \"... | math | {
"style": "rule",
"ground_truth": "42x\\sqrt{5x}"
} | {
"split": "train",
"index": 39605,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $(3 + 2x + x^{2})(1 + mx + m^{2}x^{2})$ is expanded and fully simplified, the coefficient of $x^{2}$ is equal to 1. What is the sum of all possible values of $m$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{3}"
} | {
"split": "train",
"index": 39606,
"solution": "When $(3 + 2x + x^{2})(1 + mx + m^{2}x^{2})$ is expanded, the terms that include an $x^{2}$ will come from multiplying a constant with a term that includes $x^{2}$ or multiplying two terms that includes $x$. In other words, the term that includes $x^{2}$ will be $3... |
deepscaler | [
{
"role": "user",
"content": "Let $A B C D$ and $W X Y Z$ be two squares that share the same center such that $W X \\| A B$ and $W X<A B$. Lines $C X$ and $A B$ intersect at $P$, and lines $C Z$ and $A D$ intersect at $Q$. If points $P, W$, and $Q$ are collinear, compute the ratio $A B / W X$. Let's think s... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}+1"
} | {
"split": "train",
"index": 39607,
"solution": "Without loss of generality, let $A B=1$. Let $x=W X$. Then, since $B P W X$ is a parallelogram, we have $B P=x$. Moreover, if $T=X Y \\cap A B$, then we have $B T=\\frac{1-x}{2}$, so $P T=x-\\frac{1-x}{2}=\\frac{3 x-1}{2}$. Then, from $\\triangle P X T \\sim \\tria... |
deepscaler | [
{
"role": "user",
"content": "$J K L M$ is a square. Points $P$ and $Q$ are outside the square such that triangles $J M P$ and $M L Q$ are both equilateral. The size, in degrees, of angle $P Q M$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 39608,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the function $f(x)=\\sin(2x+\\varphi)$ is translated to the right by $\\frac{\\pi}{6}$ units, and the resulting graph is symmetric about the origin. Determine the value of $\\varphi$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 39609,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m$ and $n$ be any two odd numbers, with $n$ less than $m$. The largest integer which divides all possible numbers of the form $m^2-n^2$ is:\n$\\textbf{(A)}\\ 2\\qquad \\textbf{(B)}\\ 4\\qquad \\textbf{(C)}\\ 6\\qquad \\textbf{(D)}\\ 8\\qquad \\textbf{(E)}\\ 16$\n Let's thi... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 39610,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an organization, there are five leaders and a number of regular members. Each year, the leaders are expelled, followed by each regular member recruiting three new members to become regular members. After this, five new leaders are elected from outside the organization. Initi... | math | {
"style": "rule",
"ground_truth": "10895"
} | {
"split": "train",
"index": 39611,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Unconventional dice are to be designed such that the six faces are marked with numbers from $1$ to $6$ with $1$ and $2$ appearing on opposite faces. Further, each face is colored either red or yellow with opposite faces always of the same color. Two dice are considered ... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 39612,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xanthia buys hot dogs that come in packages of six, and she buys hot dog buns that come in packages of eight. What is the smallest number of hot dog packages she can buy in order to be able to buy an equal number of hot dogs and hot dog buns? Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 39613,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of the larger base of a truncated pyramid is $T$, the area of the smaller base is $t$, and the height is $m$. The volume is $V = kTm$, where $\\frac{1}{3} \\leq k \\leq 1$. What is the proportionality ratio $\\lambda(<1)$ between the two bases? (The proportionality rat... | math | {
"style": "rule",
"ground_truth": "\\frac{2 - \\sqrt{3}}{2}"
} | {
"split": "train",
"index": 39614,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$. Point $X$ is chosen on edge $A_{1} D_{1}$, and point $Y$ is chosen on edge $B C$. It is known that $A_{1} X=5$, $B Y=3$, and $B_{1} C_{1}=14$. The plane $C_{1} X Y$ intersects ray $D A$ at point $Z$. Find $D Z$. Let's th... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 39615,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The quadratic $10x^2+100x+1000$ can be written in the form $a(x+b)^2+c$, where $a$, $b$, and $c$ are constants. What is $a+b+c$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "765"
} | {
"split": "train",
"index": 39616,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$A$ and $B$ travel around a circular track at uniform speeds in opposite directions, starting from diametrically opposite points. If they start at the same time, meet first after $B$ has travelled $100$ yards, and meet a second time $60$ yards before $A$ completes one lap, then... | math | {
"style": "rule",
"ground_truth": "480"
} | {
"split": "train",
"index": 39617,
"solution": "1. **Define Variables:**\n Let $x$ be half the circumference of the track. Therefore, the full circumference is $2x$.\n\n2. **Analyze First Meeting:**\n $A$ and $B$ start from diametrically opposite points on the track. When they first meet, $B$ has traveled $1... |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 1980 = 0$ has integral solutions? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "290"
} | {
"split": "train",
"index": 39618,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\frac{x}{ax+b}(a≠0)$, and its graph passes through the point $(-4,4)$, and is symmetric about the line $y=-x$, 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": 39619,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p,q,r$ be the roots of $x^3 - 6x^2 + 8x - 1 = 0$, and let $t = \\sqrt{p} + \\sqrt{q} + \\sqrt{r}$. Find $t^4 - 12t^2 - 8t$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 39620,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tetrahedron $ABCD$ has $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\\frac{12}5\\sqrt2$. What is the volume of the tetrahedron? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{24}{5}"
} | {
"split": "train",
"index": 39621,
"solution": "1. **Identify the Midpoint and Lengths**:\n Let the midpoint of $CD$ be $E$. Since $CD = \\frac{12}{5}\\sqrt{2}$, we have $CE = \\frac{1}{2}CD = \\frac{6}{5}\\sqrt{2}$.\n\n2. **Use the Pythagorean Theorem in $\\triangle ACE$ and $\\triangle BCE$**:\n - For $\\t... |
deepscaler | [
{
"role": "user",
"content": "Below is the graph of $y = a \\sin (bx + c) + d$ for some positive constants $a,$ $b,$ $c,$ and $d.$ Find $b.$\n\n[asy]import TrigMacros;\n\nsize(400);\n\nreal f(real x)\n{\n\treturn 2*sin(3*x + pi) + 1;\n}\n\ndraw(graph(f,-3*pi,3*pi,n=700,join=operator ..),red);\ntrig_axes(-3... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 39622,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The formula for the total surface area of a cylinder is $SA = 2\\pi r^2 + 2\\pi rh,$ where $r$ is the radius and $h$ is the height. A particular solid right cylinder of radius 2 feet has a total surface area of $12\\pi$ square feet. What is the height of this cylinder? Let's th... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 39623,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the center of a circular field, there is a geologists' house. Six straight roads radiate from it, dividing the field into six equal sectors. Two geologists set out on a journey from their house at a speed of 5 km/h along randomly chosen roads. Determine the probability that ... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"split": "train",
"index": 39624,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid $EFGH$, the sides $EF$ and $GH$ are equal. The length of $FG$ is 10 and the length of $EH$ is 20. Each of the non-parallel sides makes right triangles with legs 5 and an unknown base segment. What is the perimeter of $EFGH$?\n[asy]\npen p = linetype(\"4 4\");\ndraw... | math | {
"style": "rule",
"ground_truth": "30 + 10\\sqrt{2}"
} | {
"split": "train",
"index": 39625,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Form a six-digit number using the digits 1, 2, 3, 4, 5, 6 without repetition, where both 5 and 6 are on the same side of 3. How many such six-digit numbers are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "480"
} | {
"split": "train",
"index": 39626,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a Queen and the second card is a $\\diamondsuit$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{52}"
} | {
"split": "train",
"index": 39627,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system, the center of circle $C$ is at $(2,0)$, and its radius is $\\sqrt{2}$. Establish a polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis. The parametric equation of line $l$ is: \n$$\n\\begin{... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"split": "train",
"index": 39628,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the value(s) of $n$ for which $|2n - 7| = 3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 39629,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is given that there exists a unique triple of positive primes $(p,q,r)$ such that $p<q<r$ and \\[\\dfrac{p^3+q^3+r^3}{p+q+r} = 249.\\] Find $r$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 39630,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest prime factor of $15! + 18!$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4897"
} | {
"split": "train",
"index": 39631,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The polynomial $x^3 -ax^2 + bx -2010$ has three positive integer roots. What is 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": "78"
} | {
"split": "train",
"index": 39632,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right trapezoid has an upper base that is 60% of the lower base. If the upper base is increased by 24 meters, it becomes a square. What was the original area of the right trapezoid in square meters? Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "2880"
} | {
"split": "train",
"index": 39633,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a square $ABCD$ . Let $P\\in{AB},\\ Q\\in{BC},\\ R\\in{CD}\\ S\\in{DA}$ and $PR\\Vert BC,\\ SQ\\Vert AB$ and let $Z=PR\\cap SQ$ . If $BP=7,\\ BQ=6,\\ DZ=5$ , then find the side length of the square. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 39634,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Round to the nearest tenth: 36.89753 Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36.9"
} | {
"split": "train",
"index": 39635,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When rolling a certain unfair six-sided die with faces numbered 1, 2, 3, 4, 5, and 6, the probability of obtaining face $F$ is greater than $1/6$, the probability of obtaining the face opposite is less than $1/6$, the probability of obtaining any one of the other four faces is ... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 39636,
"solution": "We have that the cube probabilities to land on its faces are $\\frac{1}{6}$, $\\frac{1}{6}$, $\\frac{1}{6}$, $\\frac{1}{6}$ ,$\\frac{1}{6}+x$ ,$\\frac{1}{6}-x$ we also know that the sum could be 7 only when the faces in each of the two tosses are opposite hence the... |
deepscaler | [
{
"role": "user",
"content": "There are 1000 rooms in a row along a long corridor. Initially, the first room contains 1000 people, and the remaining rooms are empty. Each minute, the following happens: for each room containing more than one person, someone in that room decides it is too crowded and moves to... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"split": "train",
"index": 39637,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\tan 225^\\circ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 39638,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\cos (\\alpha - \\frac{\\pi }{3}) - \\cos \\alpha = \\frac{1}{3}$, find the value of $\\sin (\\alpha - \\frac{\\pi }{6})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 39639,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of \\( k \\) such that, for all real numbers \\( a, b, \\) and \\( c \\),\n\n$$\n(a+b)(b+c)(c+a) = (a+b+c)(ab + bc + ca) + k \\cdot abc\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 39640,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For positive real numbers $a,$ $b,$ and $c,$ compute the maximum value of\n\\[\\frac{abc(a + b + c)}{(a + b)^2 (b + c)^2}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 39641,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(x\\) and \\(y\\) be real numbers such that \\(2(x^3 + y^3) = x + y\\). Find the maximum value of \\(x - y\\). 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": 39642,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each even positive integer $x$, let $g(x)$ denote the greatest power of 2 that divides $x.$ For example, $g(20)=4$ and $g(16)=16.$ For each positive integer $n,$ let $S_n=\\sum_{k=1}^{2^{n-1}}g(2k).$ Find the greatest integer $n$ less than 1000 such that $S_n$ is a perfect ... | math | {
"style": "rule",
"ground_truth": "899"
} | {
"split": "train",
"index": 39643,
"solution": "First note that $g(k)=1$ if $k$ is odd and $2g(k/2)$ if $k$ is even. so $S_n=\\sum_{k=1}^{2^{n-1}}g(2k). = \\sum_{k=1}^{2^{n-1}}2g(k) = 2\\sum_{k=1}^{2^{n-1}}g(k) = 2\\sum_{k=1}^{2^{n-2}} g(2k-1)+g(2k).$ $2k-1$ must be odd so this reduces to $2\\sum_{k=1}^{2^{n-2}}... |
deepscaler | [
{
"role": "user",
"content": "If\n\\[\\mathbf{A} = \\begin{pmatrix} 1 & 3 \\\\ 2 & 1 \\end{pmatrix},\\]then compute $\\det (\\mathbf{A}^2 - 2 \\mathbf{A}).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 39644,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In Mr. Smith's class, the ratio of boys to girls is 3 boys for every 4 girls and there are 42 students in his class, calculate the percentage of students that are boys. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42.857\\%"
} | {
"split": "train",
"index": 39645,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\\{A, B, C\\}$ were there in which $A$ beat $B$, $B$ beat $C$, and $C$ beat $A... | math | {
"style": "rule",
"ground_truth": "385"
} | {
"split": "train",
"index": 39646,
"solution": "1. **Understanding the Tournament Setup**: In this round-robin tournament, each team plays every other team exactly once, winning 10 games and losing 10 games. This implies there are 21 teams (since each team plays 20 games). \n\n2. **Arranging Teams in a Circle**:... |
deepscaler | [
{
"role": "user",
"content": "The numbers $a, b, c, d$ belong to the interval $[-6.5, 6.5]$. Find the maximum value of the expression $a + 2b + c + 2d - ab - bc - cd - da$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "182"
} | {
"split": "train",
"index": 39647,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given in $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is known that $a=b\\cos C-c\\sin B$.\n\n(Ⅰ) Find the size of angle $B$;\n\n(Ⅱ) If $b=5$ and $a=3\\sqrt{2}$, find the area $S$ of $\\triangle ABC$. Let... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 39648,
"solution": ""
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.