data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "A contest has six problems worth seven points each. On any given problem, a contestant can score either 0,1 , or 7 points. How many possible total scores can a contestant achieve over all six problems? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 4100,
"solution": "For $0 \\leq k \\leq 6$, to obtain a score that is $k(\\bmod 6)$ exactly $k$ problems must get a score of 1 . The remaining $6-k$ problems can generate any multiple of 7 from 0 to $7(6-k)$, of which there are $7-k$. So the total number of possible scores is $\\sum_{... |
deepscaler | [
{
"role": "user",
"content": "From a bottle containing 1 liter of alcohol, $\\frac{1}{3}$ liter of alcohol is poured out, an equal amount of water is added and mixed thoroughly. Then, $\\frac{1}{3}$ liter of the mixture is poured out, an equal amount of water is added and mixed thoroughly. Finally, 1 liter ... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{27}"
} | {
"split": "train",
"index": 4101,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A three-digit number is formed by the digits $0$, $1$, $2$, $3$, $4$, $5$, with exactly two digits being the same. There are a total of \\_\\_\\_\\_\\_ such numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 4102,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Regular octagonal pyramid $\\allowbreak PABCDEFGH$ has the octagon $ABCDEFGH$ as its base. Each side of the octagon has length 5. Pyramid $PABCDEFGH$ has an additional feature where triangle $PAD$ is an equilateral triangle with side length 10. Calculate the volume of the pyram... | math | {
"style": "rule",
"ground_truth": "\\frac{250\\sqrt{3}(1 + \\sqrt{2})}{3}"
} | {
"split": "train",
"index": 4103,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of $202_4 + 330_4 + 1000_4$. Express your answer first in base 4, then convert that sum to base 10. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "158"
} | {
"split": "train",
"index": 4104,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $cos\\left( \\frac{\\pi}{6}-\\alpha\\right) = \\frac{\\sqrt{3}}{2}$, find the value of $cos\\left( \\frac{5\\pi}{6}+\\alpha\\right) - sin^2\\left(-\\alpha+\\frac{7\\pi}{6}\\right)$.\nIf $cos\\alpha = \\frac{2}{3}$ and $\\alpha$ is an angle in the fourth quadrant, fin... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 4105,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( S = \\{1, 2, \\ldots, 280\\} \\). Find the smallest natural number \\( n \\) such that every \\( n \\)-element subset of \\( S \\) contains 5 pairwise coprime numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "217"
} | {
"split": "train",
"index": 4106,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is known that $\\sin y = 2 \\cos x + \\frac{5}{2} \\sin x$ and $\\cos y = 2 \\sin x + \\frac{5}{2} \\cos x$. Find $\\sin 2x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{37}{20}"
} | {
"split": "train",
"index": 4107,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are four complex numbers $z$ such that\n\\[z \\overline{z}^3 + \\overline{z} z^3 = 350,\\]and both the real and imaginary parts of $z$ are integers. These four complex numbers are plotted in the complex plane. Find the area of the quadrilateral formed by the four comple... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 4108,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As shown below, there is a $40\\times30$ paper with a filled $10\\times5$ rectangle inside of it. We want to cut out the filled rectangle from the paper using four straight cuts. Each straight cut is a straight line that divides the paper into two pieces, and we keep the piece ... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 4109,
"solution": "\nWe are given a \\( 40 \\times 30 \\) rectangle (the paper) with a filled \\( 10 \\times 5 \\) rectangle inside it. The objective is to cut out the filled rectangle using four straight cuts with the aim of minimizing the total length of the cuts. Each cut divides t... |
deepscaler | [
{
"role": "user",
"content": "In a certain competition, two players of equal level meet in the final. The final is played in a best-of-five format, with the winner receiving all the prize money. When the score was $2:1$ after the first three games, the competition was terminated due to unforeseen circumstan... | math | {
"style": "rule",
"ground_truth": "9000"
} | {
"split": "train",
"index": 4110,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Pascal High School organized three different trips. Fifty percent of the students went on the first trip, $80 \\%$ went on the second trip, and $90 \\%$ went on the third trip. A total of 160 students went on all three trips, and all of the other students went on exactly two tr... | math | {
"style": "rule",
"ground_truth": "800"
} | {
"split": "train",
"index": 4111,
"solution": "Let $x$ be the total number of students at Pascal H.S. Let $a$ be the total number of students who went on both the first trip and the second trip, but did not go on the third trip. Let $b$ be the total number of students who went on both the first trip and the thir... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC, AB = 8, BC = 7, CA = 6$ and side $BC$ is extended, as shown in the figure, to a point $P$ so that $\\triangle PAB$ is similar to $\\triangle PCA$. The length of $PC$ is\n[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, P=(1.5,5), B=(8,0), C=P+2.... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 4112,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular prism with integer edge lengths is painted red on its entire surface and then cut into smaller cubes with edge length 1. Among these smaller cubes, 40 cubes have two red faces and 66 cubes have one red face. What is the volume of this rectangular prism? Let's thin... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 4113,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $a=2b$. Also, $\\sin A$, $\\sin C$, $\\sin B$ form an arithmetic sequence.\n\n$(I)$ Find the value of $\\cos (B+C)$;\n\n$(II)$ If the area of $\\triangle ABC$ is $\\frac{8\\sq... | math | {
"style": "rule",
"ground_truth": "4 \\sqrt {2}"
} | {
"split": "train",
"index": 4114,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $F_{1}$ and $F_{2}$ be the foci of the ellipse $C_{1}$: $\\frac{x^{2}}{6}+\\frac{y^{2}}{2}=1$, and let $P$ be a point of intersection between the ellipse $C_{1}$ and the hyperbola $C_{2}$: $\\frac{x^{2}}{3}-y^{2}=1$. Find the value of $\\cos \\angle F_{1}PF_{2}$. Let's thin... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 4115,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For what value of $k$ does the equation $x^2+10x+y^2+6y-k=0$ represent a circle of radius 6? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 4116,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let triangle $ABC$ be equilateral with each side measuring 18 inches. Let $O$ be the intersection point of medians $AP$ and $CQ$ of triangle $ABC$. If $OQ$ is 6 inches, then $AQ$, in inches, is:\nA) $6$ inches\nB) $12$ inches\nC) $15$ inches\nD) $18$ inches\nE) $21$ inches Let'... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 4117,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the point $(9,7)$ is on the graph of $y=f(x)$, there is one point that must be on the graph of $2y=\\frac{f(2x)}2+2$. What is the sum of coordinates of that point? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{29}4"
} | {
"split": "train",
"index": 4118,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Parallelogram $PQRS$ has vertices $P(4,4)$, $Q(-2,-2)$, $R(-8,-2)$, and $S(-2,4)$. If a point is chosen at random from the region defined by the parallelogram, what is the probability that the point lies below or on the line $y = -1$? Express your answer as a common fraction. L... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 4119,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the complex numbers \\( z_{1} \\) and \\( z_{2} \\) such that \\( \\left| z_{2} \\right| = 4 \\) and \\( 4z_{1}^{2} - 2z_{1}z_{2} + z_{2}^{2} = 0 \\), find the maximum value of \\( \\left| \\left( z_{1} + 1 \\right)^{2} \\left( z_{1} - 2 \\right) \\right| \\). Let's think... | math | {
"style": "rule",
"ground_truth": "6\\sqrt{6}"
} | {
"split": "train",
"index": 4120,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convert $5214_8$ to a base 10 integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2700"
} | {
"split": "train",
"index": 4121,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The polynomial \\( x^{2n} + 1 + (x+1)^{2n} \\) cannot be divided by \\( x^2 + x + 1 \\) under the condition that \\( n \\) is equal to: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 4122,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice and Carol each have a rectangular sheet of paper. Alice has a sheet of paper measuring 10 inches by 12 inches and rolls it into a tube by taping the two 10-inch sides together. Carol rolls her sheet, which measures 8 inches by 15 inches, by taping the two 15-inch sides to... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 4123,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle touches the extensions of two sides \\( AB \\) and \\( AD \\) of a square \\( ABCD \\) with a side length of 4 cm. From point \\( C \\), two tangents are drawn to this circle. Find the radius of the circle if the angle between the tangents is \\( 60^{\\circ} \\). Let's... | math | {
"style": "rule",
"ground_truth": "4 (\\sqrt{2} + 1)"
} | {
"split": "train",
"index": 4124,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f : \\mathbb{R} \\to \\mathbb{R}$ be a function such that $f(1) = 1$ and\n\\[f(xy + f(x)) = xf(y) + f(x)\\]for all real numbers $x$ and $y.$\n\nLet $n$ be the number of possible values of $f \\left( \\frac{1}{2} \\right),$ and let $s$ be the sum of all possible values of $... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 4125,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be the greatest integer multiple of 8, no two of whose digits are the same. What is the remainder when $N$ is divided by 1000? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 4126,
"solution": "We want a number with no digits repeating, so we can only use the digits $0-9$ once in constructing our number. To make the greatest number, we want the greatest digit to occupy the leftmost side and the least digit to occupy the rightmost side. Therefore, the last ... |
deepscaler | [
{
"role": "user",
"content": "Given that the ratio of $3x - 4$ to $y + 15$ is constant, and $y = 3$ when $x = 2$, then, when $y = 12$, $x$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{3}"
} | {
"split": "train",
"index": 4127,
"solution": "1. **Identify the constant ratio**: Given that the ratio of $3x - 4$ to $y + 15$ is constant, we can express this relationship as:\n \\[\n \\frac{3x - 4}{y + 15} = k\n \\]\n where $k$ is the constant ratio.\n\n2. **Use the given condition to find $k$**: We k... |
deepscaler | [
{
"role": "user",
"content": "The Nanjing Youth Olympic Games are about to open, and a clothing store owner, Mr. Chen, spent 3600 yuan to purchase two types of sportswear, A and B, and sold them out quickly. When Mr. Chen went to purchase the same types and quantities of clothing again, he found that the pu... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 4128,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\\( 427 \\div 2.68 \\times 16 \\times 26.8 \\div 42.7 \\times 16 \\) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25600"
} | {
"split": "train",
"index": 4129,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Factor $t^2-49$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(t-7)(t+7)"
} | {
"split": "train",
"index": 4130,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, $c$, $\\left(a+c\\right)\\sin A=\\sin A+\\sin C$, $c^{2}+c=b^{2}-1$. Find:<br/>\n$(1)$ $B$;<br/>\n$(2)$ Given $D$ is the midpoint of $AC$, $BD=\\frac{\\sqrt{3}}{2}$, find the area of $\\tria... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 4131,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convert the binary number $101111011_{(2)}$ to its decimal equivalent. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "379"
} | {
"split": "train",
"index": 4132,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If 2035 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 4133,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different positive three-digit integers can be formed using only the digits in the set $\\{2, 3, 5, 5, 7, 7, 7\\}$ if no digit may be used more times than it appears in the given set of available digits? Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "43"
} | {
"split": "train",
"index": 4134,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The nine horizontal and nine vertical lines on an $8\\times8$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$\n Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 4135,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the game of Galactic Dominion, players compete to amass cards, each of which is worth a certain number of points. Say you are playing a version of this game with only two kinds of cards, planet cards and hegemon cards. Each planet card is worth 2010 points, and each hegemon ... | math | {
"style": "rule",
"ground_truth": "503"
} | {
"split": "train",
"index": 4136,
"solution": "If you have $P$ planets and $H$ hegemons, buying a planet gives you $2010+4 H$ points while buying a hegemon gives you $4 P$ points. Thus you buy a hegemon whenever $P-H \\geq 502.5$, and you buy a planet whenever $P-H \\leq 502.5$. Therefore $a_{i}=1$ for $1 \\leq ... |
deepscaler | [
{
"role": "user",
"content": "The New Year's gala has a total of 8 programs, 3 of which are non-singing programs. When arranging the program list, it is stipulated that the non-singing programs are not adjacent, and the first and last programs are singing programs. How many different ways are there to arran... | math | {
"style": "rule",
"ground_truth": "720"
} | {
"split": "train",
"index": 4137,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A \"progressive number\" refers to a positive integer in which each digit is greater than the digit to its left (such as 1 458). If the four-digit \"progressive numbers\" are arranged in ascending order, then the 30th number is \\_\\_\\_\\_\\_\\_\\_\\_. Let's think step by step... | math | {
"style": "rule",
"ground_truth": "1359"
} | {
"split": "train",
"index": 4138,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Pirate Pete shares his treasure with Pirate Paul in an interesting way. Pete first says, ``One for me, one for you,'' giving himself one coin and starting Paul's pile with one coin. Then Pete says, ``Two for me, and two for you,'' giving himself two more coins but making Paul's... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 4139,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice is making a batch of cookies and needs $2\\frac{1}{2}$ cups of sugar. Unfortunately, her measuring cup holds only $\\frac{1}{4}$ cup of sugar. How many times must she fill that cup to get the correct amount of sugar? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 4140,
"solution": "1. **Convert the mixed number to an improper fraction**: Alice needs $2\\frac{1}{2}$ cups of sugar. This can be converted to an improper fraction:\n \\[\n 2\\frac{1}{2} = 2 + \\frac{1}{2} = \\frac{4}{2} + \\frac{1}{2} = \\frac{5}{2}\n \\]\n\n2. **Set up the di... |
deepscaler | [
{
"role": "user",
"content": "If the line $l_1: x + ay + 6 = 0$ is parallel to the line $l_2: (a-2)x + 3y + 2a = 0$, calculate the distance between lines $l_1$ and $l_2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8\\sqrt{2}}{3}"
} | {
"split": "train",
"index": 4141,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many bases between two and nine inclusive does the representation of $576_{10}$ have a final digit of 1? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 4142,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fair six-sided die has faces numbered $1, 2, 3, 4, 5, 6$. The die is rolled four times, and the results are $a, b, c, d$. What is the probability that one of the numbers in the set $\\{a, a+b, a+b+c, a+b+c+d\\}$ is equal to 4? Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "\\frac{343}{1296}"
} | {
"split": "train",
"index": 4143,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P= \\left( 1 - \\frac{1}{2} \\right) \\left( 1 - \\frac{1}{3} \\right) \\left( 1 - \\frac{1}{4} \\right) \\dotsm \\left( 1 - \\frac{1}{n} \\right)$. What is the value of $P$ if $n=2007$? Express your answer as a common fraction. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2007}"
} | {
"split": "train",
"index": 4144,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Here are two functions: $$\\begin{array}{ccc}\nf(x) & = & 3x^2-2x+ 4\\\\\ng(x) & = & x^2-kx-6\n\\end{array}$$ If $f(10) - g(10) = 10,$ 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": "-18"
} | {
"split": "train",
"index": 4145,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $\\angle A = 60^{\\circ}$ and $AB > AC$. Point $O$ is the circumcenter. The two altitudes $BE$ and $CF$ intersect at point $H$. Points $M$ and $N$ are on segments $BH$ and $HF$ respectively, and satisfy $BM = CN$. Find the value of $\\frac{MH + NH}{OH}$. Le... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 4146,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ellipse \\(\\frac{x^{2}}{2002^{2}}+\\frac{y^{2}}{1949^{2}}=1\\) intersects with two lines passing through the origin at points \\(A, B, C,\\) and \\(D\\). The slope of line \\(AB\\) is \\(k\\), and the slope of line \\(CD\\) is \\(-\\frac{1949^{2}}{2002^{2} \\cdot k}\\). Gi... | math | {
"style": "rule",
"ground_truth": "53"
} | {
"split": "train",
"index": 4147,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $XYZ$, the sides are in the ratio $3:4:5$. If segment $XM$ bisects the largest angle at $X$ and divides side $YZ$ into two segments, find the length of the shorter segment given that the length of side $YZ$ is $12$ inches. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"split": "train",
"index": 4148,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In Rad's garden there are exactly 30 red roses, exactly 19 yellow roses, and no other roses. How many of the yellow roses does Rad need to remove so that $\\frac{2}{7}$ of the roses in the garden are yellow? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 4149,
"solution": "If $\\frac{2}{7}$ of the roses are to be yellow, then the remaining $\\frac{5}{7}$ of the roses are to be red. Since there are 30 red roses and these are to be $\\frac{5}{7}$ of the roses, then $\\frac{1}{7}$ of the total number of roses would be $30 \\div 5 = 6$, w... |
deepscaler | [
{
"role": "user",
"content": "Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player... | math | {
"style": "rule",
"ground_truth": "5/11"
} | {
"split": "train",
"index": 4150,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fair coin is flipped 8 times. What is the probability that exactly 6 of the flips come up heads? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{64}"
} | {
"split": "train",
"index": 4151,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all positive divisors of $50$ that are also divisors of $15$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 4152,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The line passing through the points (3, 9) and (-1, 1) intersects the x-axis at a point whose x-coordinate is $\\frac{9-1}{3-(-1)}$ 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": 4153,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A teen age boy wrote his own age after his father's. From this new four place number, he subtracted the absolute value of the difference of their ages to get $4,289$ . The sum of their ages was Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 4154,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) Given a sequence $\\{a_n\\}$ that satisfies $a_1a_2…a_n=n+1$, find $a_3=$\n\n(2) Let $\\overrightarrow{e_1}, \\overrightarrow{e_2}$ be unit vectors, where $\\overrightarrow{a}=2\\overrightarrow{e_1}+\\overrightarrow{e_2}, \\overrightarrow{b}=\\overrightarrow{e_2}$, and $\\o... | math | {
"style": "rule",
"ground_truth": "6:5:4"
} | {
"split": "train",
"index": 4155,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Shown below are rows 1, 2, and 3 of Pascal's triangle.\n\n\\[\n\\begin{array}{ccccccc}\n& & 1 & & 1 & & \\\\\n& 1 & & 2 & & 1 & \\\\\n1 & & 3 & & 3 & & 1\n\\end{array}\n\\]Let $(a_i),$ $(b_i),$ $(c_i)$ be the sequence, from left to right, of elements in the 2005th, 2006th, and ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 4156,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle has vertices \\( P=(-7,4) \\), \\( Q=(-14,-20) \\), and \\( R=(2,-8) \\). The equation of the bisector of \\( \\angle P \\) can be written in the form \\( ax+2y+c=0 \\). Find \\( a+c \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 4157,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each half of this figure is composed of 3 red triangles, 5 blue triangles and 8 white triangles. When the upper half is folded down over the centerline, 2 pairs of red triangles coincide, as do 3 pairs of blue triangles. There are 2 red-white pairs. How many white pairs coincid... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 4158,
"solution": "1. **Identify the total number of triangles in each half**: Each half of the figure contains 3 red triangles, 5 blue triangles, and 8 white triangles.\n\n2. **Analyze the red triangles**: When the upper half is folded down, 2 pairs of red triangles coincide. This me... |
deepscaler | [
{
"role": "user",
"content": "Find all solutions $x$ (real and otherwise) to the equation\n\\[x^4+64=0.\\]Enter all the solutions, separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2+2i,\\,-2-2i,\\,-2+2i,\\,2-2i"
} | {
"split": "train",
"index": 4159,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a convex 13-gon, all diagonals are drawn. They divide it into polygons. Consider the polygon with the largest number of sides among them. What is the greatest number of sides that it can have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 4160,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A shopping mall's main staircase from the 1st floor to the 2nd floor consists of 15 steps. Each step has a height of 16 centimeters and a depth of 26 centimeters. The width of the staircase is 3 meters. If the cost of carpeting is 80 yuan per square meter, how much will it cost... | math | {
"style": "rule",
"ground_truth": "1512"
} | {
"split": "train",
"index": 4161,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The three roots of the equation \\[ax^3+bx^2+cx+d=0\\]are $1,$ $2,$ and $3.$ Compute $\\frac{c}{d}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{11}{6}"
} | {
"split": "train",
"index": 4162,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any $x \\in \\mathbb{R}$, the function $f(x)$ represents the minimum value among the three function values $y_{1}=4x+1$, $y_{2}=x+2$, $y_{3}=-2x+4$. The maximum value of $f(x)$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}"
} | {
"split": "train",
"index": 4163,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the equations \\( x^{2} - xy + x = 2018 \\) and \\( y^{2} - xy - y = 52 \\), find the sum of all possible values of \\( x - y \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 4164,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $XYZ$ where $XY=60$ and $XZ=15$, the area of the triangle is given as $225$. Let $W$ be the midpoint of $\\overline{XY}$, and $V$ be the midpoint of $\\overline{XZ}$. The angle bisector of $\\angle YXZ$ intersects $\\overline{WV}$ and $\\overline{YZ}$ at points $P$ ... | math | {
"style": "rule",
"ground_truth": "123.75"
} | {
"split": "train",
"index": 4165,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\begin{vmatrix} x & y \\\\ z & w \\end{vmatrix} = 3,$ find $\\begin{vmatrix} 3x & 3y \\\\ 3z & 3w \\end{vmatrix}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 4166,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of pairs \\((a, b)\\) of integers with \\(1 \\leq b < a \\leq 200\\) such that the sum \\((a+b) + (a-b) + ab + \\frac{a}{b}\\) is a square of a number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 4167,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two numbers are independently selected from the set of positive integers less than or equal to 6. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"split": "train",
"index": 4168,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at $1\\!:\\!00$ PM and finishes the second task at $2\\!:\\!40$ PM. When does she finish the third task? Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "3:30 PM"
} | {
"split": "train",
"index": 4169,
"solution": "1. **Identify the total time taken for two tasks**: Marie finishes the second task at $2\\!:\\!40$ PM and she started the first task at $1\\!:\\!00$ PM. The total time taken for two tasks is from $1\\!:\\!00$ PM to $2\\!:\\!40$ PM.\n - Convert this time span into ... |
deepscaler | [
{
"role": "user",
"content": "The sum to infinity of the terms of an infinite geometric progression is 10. The sum of the first two terms is 7. Compute the first term of the progression. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10\\left(1 + \\sqrt{\\frac{3}{10}}\\right)"
} | {
"split": "train",
"index": 4170,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{281}{13}"
} | {
"split": "train",
"index": 4171,
"solution": "1. **Find the intercepts of the line with the axes:**\n - Set $y = 0$ in the equation $12x + 5y = 60$:\n \\[\n 12x = 60 \\implies x = \\frac{60}{12} = 5\n \\]\n - Set $x = 0$ in the equation $12x + 5y = 60$:\n \\[\n 5y = 60 \\implies y = \\fr... |
deepscaler | [
{
"role": "user",
"content": "Determine the maximum possible value of the expression\n\n$$\n27abc + a\\sqrt{a^2 + 2bc} + b\\sqrt{b^2 + 2ca} + c\\sqrt{c^2 + 2ab}\n$$\n\nwhere \\(a, b, c\\) are positive real numbers such that \\(a + b + c = \\frac{1}{\\sqrt{3}}\\). Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3 \\sqrt{3}}"
} | {
"split": "train",
"index": 4172,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For real numbers $a, b$ , and $c$ the polynomial $p(x) = 3x^7 - 291x^6 + ax^5 + bx^4 + cx^2 + 134x - 2$ has $7$ real roots whose sum is $97$ . Find the sum of the reciprocals of those $7$ roots. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 4173,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$ , $AB=13$ , $BC=14$ and $CA=15$ . Segment $BC$ is split into $n+1$ congruent segments by $n$ points. Among these points are the feet of the altitude, median, and angle bisector from $A$ . Find the smallest possible value of $n$ .\n\n*Proposed by... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 4174,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $A B C$ is given in the plane. Let $A D$ be the angle bisector of $\\angle B A C$; let $B E$ be the altitude from $B$ to $A D$, and let $F$ be the midpoint of $A B$. Given that $A B=28, B C=33, C A=37$, what is the length of $E F$ ? Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 4175,
"solution": "$14 \\triangle A B E$ is a right triangle, and $F$ is the midpoint of the hypotenuse (and therefore the circumcenter), so $E F=B F=A F=14$."
} |
deepscaler | [
{
"role": "user",
"content": "When the base-16 number $66666_{16}$ is written in base 2, how many base-2 digits (bits) does it have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 4176,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a function $y=f(x)$ defined on the domain $I$, if there exists an interval $[m,n] \\subseteq I$ that simultaneously satisfies the following conditions: $①f(x)$ is a monotonic function on $[m,n]$; $②$when the domain is $[m,n]$, the range of $f(x)$ is also $[m,n]$, then we ... | math | {
"style": "rule",
"ground_truth": "\\frac {2 \\sqrt {3}}{3}"
} | {
"split": "train",
"index": 4177,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = (2 - a)(x - 1) - 2 \\ln x (a \\in \\mathbb{R})$.\n\n(1) If the tangent line of the curve $g(x) = f(x) + x$ at the point $(1, g(1))$ passes through the point $(0, 2)$, find the monotonically decreasing interval of the function $g(x)$;\n\n(2) If the fun... | math | {
"style": "rule",
"ground_truth": "2 - 4 \\ln 2"
} | {
"split": "train",
"index": 4178,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the least possible value of $f(1998),$ where $f:\\Bbb{N}\\to \\Bbb{N}$ is a function such that for all $m,n\\in {\\Bbb N}$, \n\n\\[f\\left( n^{2}f(m)\\right) =m\\left( f(n)\\right) ^{2}. \\] Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 4179,
"solution": "\nTo find the least possible value of \\( f(1998) \\), where \\( f: \\mathbb{N} \\to \\mathbb{N} \\) satisfies the functional equation\n\n\\[\nf\\left( n^{2}f(m)\\right) = m\\left( f(n)\\right) ^{2}\n\\]\n\nfor all \\( m, n \\in \\mathbb{N} \\), we begin by analyzin... |
deepscaler | [
{
"role": "user",
"content": "You are given the numbers 1, 2, 3, 4, 5, 6, 7, 8 to be placed at the eight vertices of a cube, such that the sum of any three numbers on each face of the cube is at least 10. Find the minimum possible sum of the four numbers on any face. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 4180,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbb{R}^+$ be the set of positive real numbers. Find all functions $f \\colon \\mathbb{R}^+ \\to \\mathbb{R}^+$ such that, for all $x,y \\in \\mathbb{R}^+$,\n$$f(xy+f(x))=xf(y)+2.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x + 1"
} | {
"split": "train",
"index": 4181,
"solution": "\nLet \\( f: \\mathbb{R}^+ \\to \\mathbb{R}^+ \\) be a function satisfying the functional equation for all \\( x, y \\in \\mathbb{R}^+ \\):\n\n\\[\nf(xy + f(x)) = x f(y) + 2.\n\\]\n\nTo find \\( f \\), consider substituting specific values for \\( x \\) and \\( y \\... |
deepscaler | [
{
"role": "user",
"content": "For positive integers $i = 2, 3, \\ldots, 2020$ , let \\[\na_i = \\frac{\\sqrt{3i^2+2i-1}}{i^3-i}.\n\\]Let $x_2$ , $\\ldots$ , $x_{2020}$ be positive reals such that $x_2^4 + x_3^4 + \\cdots + x_{2020}^4 = 1-\\frac{1}{1010\\cdot 2020\\cdot 2021}$ . Let $S$ be the maximu... | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 4182,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the maximum number of queens that can be placed on the black squares of an $8 \\times 8$ chessboard so that each queen is attacked by at least one of the others? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 4183,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two riders simultaneously departed from points \\( A \\) and \\( C \\) towards point \\( B \\). Despite the fact that \\( C \\) was 20 km farther from \\( B \\) than \\( A \\) was from \\( B \\), both riders arrived at \\( B \\) at the same time. Find the distance from \\( C \\... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 4184,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest natural number consisting of distinct digits such that the product of its digits equals 2016. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "876321"
} | {
"split": "train",
"index": 4185,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The lengths of the diagonals of a rhombus and the length of its side form a geometric progression. Find the sine of the angle between the side of the rhombus and its longer diagonal, given that it is greater than \\( \\frac{1}{2} \\). Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\frac{\\sqrt{17}-1}{8}}"
} | {
"split": "train",
"index": 4186,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a $10\\times10$ checkerboard with alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 6 black squares, can be drawn on the checkerboard? Let's think step by s... | math | {
"style": "rule",
"ground_truth": "140"
} | {
"split": "train",
"index": 4187,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the expression:\n\\[4(1+4(1+4(1+4(1+4(1+4(1+4(1+4(1))))))))\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "87380"
} | {
"split": "train",
"index": 4188,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all divisors \\(d=2^a \\cdot 3^b\\) (where \\(a, b > 0\\)) of \\(N=19^{88}-1\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "744"
} | {
"split": "train",
"index": 4189,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the function $f(x)$ is monotonic on $(-1, +\\infty)$, and the graph of the function $y = f(x - 2)$ is symmetrical about the line $x = 1$, if the sequence $\\{a_n\\}$ is an arithmetic sequence with a nonzero common difference and $f(a_{50}) = f(a_{51})$, determine the... | math | {
"style": "rule",
"ground_truth": "-100"
} | {
"split": "train",
"index": 4190,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $10$ people standing equally spaced around a circle. Each person knows exactly $3$ of the other $9$ people: the $2$ people standing next to her or him, as well as the person directly across the circle. How many ways are there for the $10$ people to split up into $5$ p... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 4191,
"solution": "To solve this problem, we need to consider the relationships each person has with others in the circle. Each person knows exactly three others: the two adjacent people and the one directly across the circle. We need to find how many ways we can pair up these 10 peop... |
deepscaler | [
{
"role": "user",
"content": "Jar C initially contains 6 red buttons and 12 green buttons. Michelle removes the same number of red buttons as green buttons from Jar C and places them into an empty Jar D. After the removal, Jar C is left with $\\frac{3}{4}$ of its initial button count. If Michelle were to ra... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{14}"
} | {
"split": "train",
"index": 4192,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is $1$ foot wide on all four sides. What is the length in feet of the ... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 4193,
"solution": "1. **Define the dimensions and areas:**\n - Let the length of the inner rectangle be $x$ feet. Since the width is given as 1 foot, the area of the inner rectangle is $x \\cdot 1 = x$ square feet.\n - The second largest rectangle, which includes the inner rectang... |
deepscaler | [
{
"role": "user",
"content": "Let $a$, $b$, and $c$ be the $3$ roots of $x^3-x+1=0$. Find $\\frac{1}{a+1}+\\frac{1}{b+1}+\\frac{1}{c+1}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 4194,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of an infinite geometric series is $64$ times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 4195,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a paralleogram $ABCD$ , a point $P$ on the segment $AB$ is taken such that $\\frac{AP}{AB}=\\frac{61}{2022}$ and a point $Q$ on the segment $AD$ is taken such that $\\frac{AQ}{AD}=\\frac{61}{2065}$ .If $PQ$ intersects $AC$ at $T$ , find $\\frac{AC}{AT}$ to... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 4196,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four vehicles were traveling on the highway at constant speeds: a car, a motorcycle, a scooter, and a bicycle. The car passed the scooter at 12:00, encountered the bicyclist at 14:00, and met the motorcyclist at 16:00. The motorcyclist met the scooter at 17:00 and caught up wit... | math | {
"style": "rule",
"ground_truth": "15:20"
} | {
"split": "train",
"index": 4197,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In Ms. Smith's class, each student averages one day absent out of thirty. What is the probability that out of any two students chosen at random, one student will be absent while the other is present? Express your answer as a percent rounded to the nearest tenth. Let's think ste... | math | {
"style": "rule",
"ground_truth": "6.4"
} | {
"split": "train",
"index": 4198,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A satellite is launched vertically from the Earth's pole with the first cosmic velocity. What is the maximum distance the satellite will reach from the Earth's surface? (The gravitational acceleration at the Earth's surface is $g = 10 \\, \\mathrm{m/s^2}$, and the Earth's radiu... | math | {
"style": "rule",
"ground_truth": "6400"
} | {
"split": "train",
"index": 4199,
"solution": ""
} |
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