data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "A room is 19 feet long and 11 feet wide. Find the ratio of the length of the room to its perimeter. Express your answer in the form $a:b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19:60"
} | {
"split": "train",
"index": 17649,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point \\( M \\) belongs to the edge \\( CD \\) of the parallelepiped \\( ABCDA_1B_1C_1D_1 \\), where \\( CM: MD = 1:2 \\). Construct the section of the parallelepiped with a plane passing through point \\( M \\) parallel to the lines \\( DB \\) and \\( AC_1 \\). In what ratio d... | math | {
"style": "rule",
"ground_truth": "1 : 11"
} | {
"split": "train",
"index": 17650,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer $n$ which cannot be written in any of the following forms? - $n=1+2+\\cdots+k$ for a positive integer $k$. - $n=p^{k}$ for a prime number $p$ and integer $k$ - $n=p+1$ for a prime number $p$. - $n=p q$ for some distinct prime numbers $p$ an... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 17651,
"solution": "The first numbers which are neither of the form $p^{k}$ nor $p q$ are 12, 18, 20, 24, 28, 30, 36, 40, ... Of these $12,18,20,24,30$ are of the form $p+1$ and 28,36 are triangular. Hence the answer is 40."
} |
deepscaler | [
{
"role": "user",
"content": "There are 5 weights. Their masses are 1000 g, 1001 g, 1002 g, 1004 g, and 1007 g, but they have no markings and are visually indistinguishable. There is a scale with a pointer that shows the mass in grams. How can you determine the 1000 g weight using three weighings? Let's thi... | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"split": "train",
"index": 17652,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a speech competition, judges will score participants based on the content, delivery, and effectiveness of the speech, with weights of $4:4:2$ respectively. If a student receives scores of $91$, $94$, and $90$ in these three aspects, then the student's total score is ______ p... | math | {
"style": "rule",
"ground_truth": "92"
} | {
"split": "train",
"index": 17653,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define $E(a,b,c) = a \\cdot b^2 + c$. What value of $a$ is the solution to the equation $E(a,4,5) = E(a,6,7)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{10}"
} | {
"split": "train",
"index": 17654,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the area of the region of the circle defined by $x^2 + y^2 - 8x + 16 = 0$ that lies below the $x$-axis and to the left of the line $y = x - 4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"split": "train",
"index": 17655,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Denote by $P(n)$ the greatest prime divisor of $n$. Find all integers $n\\geq 2$ for which \\[P(n)+\\lfloor\\sqrt{n}\\rfloor=P(n+1)+\\lfloor\\sqrt{n+1}\\rfloor\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 17656,
"solution": "\nWe seek all integers \\( n \\geq 2 \\) such that the greatest prime divisor of \\( n \\), denoted \\( P(n) \\), together with the integer part of the square root of \\( n \\), satisfies the equation:\n\n\\[\nP(n) + \\lfloor \\sqrt{n} \\rfloor = P(n+1) + \\lfloor ... |
deepscaler | [
{
"role": "user",
"content": "The cells of a $8 \\times 8$ table are initially white. Alice and Bob play a game. First Alice paints $n$ of the fields in red. Then Bob chooses $4$ rows and $4$ columns from the table and paints all fields in them in black. Alice wins if there is at least one red field left. F... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 17657,
"solution": "\nConsider a \\( 8 \\times 8 \\) table where Alice and Bob play a game. Initially, all cells in this table are white. Alice begins by painting \\( n \\) of the cells red. After that, Bob selects 4 rows and 4 columns and paints all cells in these rows and columns bl... |
deepscaler | [
{
"role": "user",
"content": "Evaluate the infinite geometric series: $$\\frac{3}{2}-\\frac{2}{3}+\\frac{8}{27}-\\frac{32}{243}+\\dots$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{27}{26}"
} | {
"split": "train",
"index": 17658,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Use \\((a, b)\\) to represent the greatest common divisor of \\(a\\) and \\(b\\). Let \\(n\\) be an integer greater than 2021, and \\((63, n+120) = 21\\) and \\((n+63, 120) = 60\\). What is the sum of the digits of the smallest \\(n\\) that satisfies the above conditions? Let's... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 17659,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A large supermarket purchased a popular disinfectant laundry detergent. Due to the rise in raw material prices, the cost price per bottle of detergent this year increased by $4$ compared to last year. The quantity of detergent purchased for $1440$ yuan this year is the same as ... | math | {
"style": "rule",
"ground_truth": "8100"
} | {
"split": "train",
"index": 17660,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute: $104 \\times 96$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9984"
} | {
"split": "train",
"index": 17661,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an isosceles triangle \\(ABC\\) with base \\(AC\\) equal to 37, the exterior angle at vertex \\(B\\) is \\(60^\\circ\\). Find the distance from vertex \\(C\\) to line \\(AB\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18.5"
} | {
"split": "train",
"index": 17662,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum value of the area of triangle $\\triangle ABC$ that satisfies $AB=4$ and $AC=2BC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{16}{3}"
} | {
"split": "train",
"index": 17663,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\log_8{3}=p$ and $\\log_3{5}=q$, then, in terms of $p$ and $q$, $\\log_{10}{5}$ equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3pq}{1+3pq}"
} | {
"split": "train",
"index": 17664,
"solution": "1. **Given Information**: We are given $\\log_8{3}=p$ and $\\log_3{5}=q$. We need to find $\\log_{10}{5}$ in terms of $p$ and $q$.\n\n2. **Change of Base Formula**: Recall the change of base formula for logarithms:\n \\[\n \\log_b{a} = \\frac{\\log_c{a}}{\\log_... |
deepscaler | [
{
"role": "user",
"content": "Find $x^2+y^2$ if $x$ and $y$ are positive integers such that \\[\\begin{aligned} xy+x+y&=71 \\\\ x^2y+xy^2 &= 880.\\end{aligned}\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "146"
} | {
"split": "train",
"index": 17665,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the area of rectangle $ABCD$ is $8$, when the perimeter of the rectangle is minimized, fold $\\triangle ACD$ along the diagonal $AC$, then the surface area of the circumscribed sphere of the pyramid $D-ABC$ is ______. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "16\\pi"
} | {
"split": "train",
"index": 17666,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( x, y, z \\in \\mathbb{R}^{+} \\) and \\( x + 2y + 3z = 1 \\), find the minimum value of \\( \\frac{16}{x^{3}}+\\frac{81}{8y^{3}}+\\frac{1}{27z^{3}} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1296"
} | {
"split": "train",
"index": 17667,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an arithmetic sequence \\(\\{a_{n}\\}\\), where \\(a_{1} > 0\\) and \\(5a_{8} = 8a_{13}\\), what is the value of \\(n\\) when the sum of the first \\(n\\) terms \\(S_{n}\\) is maximized? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 17668,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An iterative process is used to find an average of the numbers -1, 0, 5, 10, and 15. Arrange the five numbers in a certain sequence. Find the average of the first two numbers, then the average of the result with the third number, and so on until the fifth number is included. Wh... | math | {
"style": "rule",
"ground_truth": "8.875"
} | {
"split": "train",
"index": 17669,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Take 3 segments randomly, each shorter than a unit. What is the probability that these 3 segments can form a triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 17670,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $2\\cos(2\\alpha)=\\sin\\left(\\alpha-\\frac{\\pi}{4}\\right)$ and $\\alpha\\in\\left(\\frac{\\pi}{2},\\pi\\right)$, calculate the value of $\\cos 2\\alpha$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{15}}{8}"
} | {
"split": "train",
"index": 17671,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mr. Ambulando is at the intersection of $5^{\\text{th}}$ and $\\text{A St}$ , and needs to walk to the intersection of $1^{\\text{st}}$ and $\\text{F St}$ . There's an accident at the intersection of $4^{\\text{th}}$ and $\\text{B St}$ , which he'd like to avoid.\n\n[c... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 17672,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 5 people standing in a row, where A and B must stand next to each other, and C and D cannot stand next to each other. How many different arrangements are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 17673,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be integers such that $ab = 100.$ Find the minimum value of $a + b.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-101"
} | {
"split": "train",
"index": 17674,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Yellers are coached by Coach Loud. The Yellers have 15 players, but three of them, Max, Rex, and Tex, refuse to play together in any combination. How many starting lineups (of 5 players) can Coach Loud make, if the starting lineup can't contain any two of Max, Rex, and Tex ... | math | {
"style": "rule",
"ground_truth": "2277"
} | {
"split": "train",
"index": 17675,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse C: $\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1(a>b>0)$ with its right focus F, upper vertex B, the line BF intersects C at another point A, and the projection of point A on the x-axis is A1. O is the origin, and if $\\overrightarrow{BO}=2\\overrightarrow{A_{1}A}$, det... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 17676,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = x^3 - ax^2 + 3x$, and $x=3$ is an extremum of $f(x)$.\n(Ⅰ) Determine the value of the real number $a$;\n(Ⅱ) Find the equation of the tangent line $l$ to the graph of $y=f(x)$ at point $P(1, f(1))$;\n(Ⅲ) Find the minimum and maximum values of $f(x)$ on... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 17677,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In parallelogram \\( A B C D \\), the height drawn from vertex \\( B \\) of the obtuse angle to side \\( DA \\) divides it in a ratio of 5:3, starting from vertex \\( D \\). Find the ratio \\( AC:BD \\) if \\( AD:AB=2 \\). Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "2:1"
} | {
"split": "train",
"index": 17678,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $\\sqrt[4]{5^9 \\cdot 7^2}$ is fully simplified, the result is $a\\sqrt[4]{b}$, where $a$ and $b$ are positive integers. What is $a+b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "270"
} | {
"split": "train",
"index": 17679,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $F\\_1$ and $F\\_2$ are the left and right foci of the ellipse $C: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$, $D$ and $E$ are the upper and right vertices of the ellipse $C$, and $S_{\\triangle DEF_2} = \\frac{\\sqrt{3}}{2}$, eccentricity $e = \\frac{1}{2}... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 17680,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\log 2 = 0.3010$ and $\\log 5 = 0.6990$, calculate the value of $x$ for the equation $2^{x+2} = 200$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5.64"
} | {
"split": "train",
"index": 17681,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the game of rock-paper-scissors-lizard-Spock, rock defeats scissors and lizard, paper defeats rock and Spock, scissors defeats paper and lizard, lizard defeats paper and Spock, and Spock defeats rock and scissors. If three people each play a game of rock-paper-scissors-lizar... | math | {
"style": "rule",
"ground_truth": "\\frac{12}{25}"
} | {
"split": "train",
"index": 17682,
"solution": "Let the three players be $A, B, C$. Our answer will simply be the sum of the probability that $A$ beats both $B$ and $C$, the probability that $B$ beats both $C$ and $A$, and the probability that $C$ beats $A$ and $B$, because these events are all mutually exclusiv... |
deepscaler | [
{
"role": "user",
"content": "Let $S = \\{1, 22, 333, \\dots , 999999999\\}$ . For how many pairs of integers $(a, b)$ where $a, b \\in S$ and $a < b$ is it the case that $a$ divides $b$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 17683,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $\\tan B= \\sqrt {3}$, $AB=3$, and the area of triangle $ABC$ is $\\frac {3 \\sqrt {3}}{2}$. Find the length of $AC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt {7}"
} | {
"split": "train",
"index": 17684,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the function machine shown, the input is 10. What is the output?\n\n[asy]\nsize(200); currentpen = fontsize(10pt); picture a,b,c,d,e,f;\nreal height = 3, width1 = 10, width2 = 11, width3 = 10, width4 = 10;\nreal widthC = 20,heightC = 6;\nreal widthE = 10, lengthE = 4.5,angle... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 17685,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a race, all runners must start at point $A$, touch any part of a 1500-meter wall, and then stop at point $B$. Given that the distance from $A$ directly to the wall is 400 meters and from the wall directly to $B$ is 600 meters, calculate the minimum distance a participant mus... | math | {
"style": "rule",
"ground_truth": "1803"
} | {
"split": "train",
"index": 17686,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For positive integers $n$, let $f(n)$ return the smallest positive integer $k$ such that $\\frac{1}{k}$ has exactly $n$ digits after the decimal point. How many positive integer divisors does $f(2010)$ have? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "2011"
} | {
"split": "train",
"index": 17687,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Chuck the llama is tied to the corner of a $2\\text{ m}$ by $3\\text{ m}$ shed on a $3\\text{ m}$ leash. How much area (in square meters) does Chuck have in which to play if he can go only around the outside of the shed? [asy]\ndraw((0,0)--(15,0)--(15,10)--(0,10)--cycle,black+l... | math | {
"style": "rule",
"ground_truth": "7\\pi"
} | {
"split": "train",
"index": 17688,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate: $$\\frac {1}{2}\\log_{2}3 \\cdot \\frac {1}{2}\\log_{9}8 = \\_\\_\\_\\_\\_\\_ ．$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {3}{8}"
} | {
"split": "train",
"index": 17689,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive $3$-digit numbers are multiples of $20$, but not of $55$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 17690,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers and $b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the floor. The unpainted part of the floor forms a border of width $1$ foot around the painte... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 17691,
"solution": "1. **Identify the area of the painted and unpainted regions**: Given that the unpainted border is 1 foot wide and surrounds the entire painted rectangle, the dimensions of the painted rectangle are $(a-2)$ by $(b-2)$. The total area of the floor is $ab$, and the ar... |
deepscaler | [
{
"role": "user",
"content": "The sum of the digits of the year 2004 is 6. What is the first year after 2000 for which the sum of the digits is 12? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2019"
} | {
"split": "train",
"index": 17692,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An $\\textit{abundant number}$ is a positive integer, the sum of whose distinct proper factors is greater than the number. (The $\\textit{proper factors}$ of a number are all of its factors except the number itself.) How many numbers less than 25 are abundant numbers? Let's thi... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 17693,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the imaginary unit $i$, let $z=1+i+i^{2}+i^{3}+\\ldots+i^{9}$, then $|z|=$______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt {2}"
} | {
"split": "train",
"index": 17694,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( x = 1 + \\frac{1}{\\sqrt{2}} + \\frac{1}{\\sqrt{3}} + \\cdots + \\frac{1}{\\sqrt{10^{6}}} \\), calculate the value of \\([x]\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1998"
} | {
"split": "train",
"index": 17695,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Margo walks to her friend's house in 10 minutes. Using the same route, it takes Margo 20 minutes to get back home. If her average walking rate for the entire trip is 4 miles per hour, how many total miles did she walk? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 17696,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Bag $A$ contains two 10 yuan bills and three 1 yuan bills, and Bag $B$ contains four 5 yuan bills and three 1 yuan bills. Two bills are randomly drawn from each bag. What is the probability that the total value of the bills remaining in Bag $A$ is greater than the total value o... | math | {
"style": "rule",
"ground_truth": "9/35"
} | {
"split": "train",
"index": 17697,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find\n\\[\\min_{y \\in \\mathbb{R}} \\max_{0 \\le x \\le 1} |x^2 - xy|.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 - 2 \\sqrt{2}"
} | {
"split": "train",
"index": 17698,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using the Horner's method, calculate the value of the polynomial $f(x)=2x^{4}-x^{3}+3x^{2}+7$ at $x=3$, and find the corresponding value of $v_{3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 17699,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\\cos A= \\frac {c}{a}\\cos C$, $b+c=2+ \\sqrt {2}$, and $\\cos B= \\frac {3}{4}$, find the area of $\\triangle ABC$. Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {7}}{2}"
} | {
"split": "train",
"index": 17700,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are two rows of seats, with 4 seats in the front row and 5 seats in the back row. Now, we need to arrange seating for 2 people, and these 2 people cannot sit next to each other (sitting one in front and one behind is also considered as not adjacent). How many different se... | math | {
"style": "rule",
"ground_truth": "58"
} | {
"split": "train",
"index": 17701,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $\\angle ACB=60^{\\circ}$, $BC > 1$, and $AC=AB+\\frac{1}{2}$. When the perimeter of $\\triangle ABC$ is at its minimum, the length of $BC$ is $\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1 + \\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 17702,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A graph $G(V,E)$ is triangle-free, but adding any edges to the graph will form a triangle. It's given that $|V|=2019$, $|E|>2018$, find the minimum of $|E|$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4033"
} | {
"split": "train",
"index": 17703,
"solution": "\nGiven a graph \\( G(V, E) \\) that is triangle-free, but adding any edges to the graph will form a triangle, and with \\( |V| = 2019 \\) and \\( |E| > 2018 \\), we need to find the minimum number of edges \\( |E| \\).\n\nWe claim that the minimum number of edges ... |
deepscaler | [
{
"role": "user",
"content": "It takes four painters working at the same rate $1.25$ work-days to finish a job. If only three painters are available, how many work-days will it take them to finish the job, working at the same rate? Express your answer as a mixed number. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "1\\frac{2}{3}"
} | {
"split": "train",
"index": 17704,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are four points that are $7$ units from the line $y = 20$ and $10$ units from the point $(10, 20)$. What is the sum of the $x$- and $y$-coordinates of all four of these points? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 17705,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $x$: $0.04x + 0.05(25 + x) = 13.5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "136.\\overline{1}"
} | {
"split": "train",
"index": 17706,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of several distinct positive integers is divisible by ${2006}^{2}$ . Determine the minimum value the sum of such numbers can take. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "228"
} | {
"split": "train",
"index": 17707,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose the polar coordinate equation of circle $C$ is $ρ=2\\cos θ$, the parametric equation of line $l$ is $\\begin{cases}x=\\frac{1}{2}+\\frac{\\sqrt{3}}{2}t\\\\y=\\frac{1}{2}+\\frac{1}{2}t\\end{cases}$ ($t$ is the parameter), and the polar coordinates of point $A$ are $(\\fr... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 17708,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid EFGH, sides EF and GH are equal. It is known that EF = 12 units and GH = 10 units. Additionally, each of the non-parallel sides forms a right-angled triangle with half of the difference in lengths of EF and GH and a given leg of 6 units. Determine the perimeter of ... | math | {
"style": "rule",
"ground_truth": "22 + 2\\sqrt{37}"
} | {
"split": "train",
"index": 17709,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, if the area of $\\triangle ABC$ is 36 where $A(3, 15)$, $B(15, 0)$, and $C(0, q)$ lie on a Cartesian plane. Determine the value of $q$.\n[asy]\nsize(5cm);defaultpen(fontsize(9));\npair a = (3, 15); pair b = (15, 0); pair c = (0, 12);pair d= (3, 0);\ndraw(a--b--c... | math | {
"style": "rule",
"ground_truth": "12.75"
} | {
"split": "train",
"index": 17710,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right-angled triangle has sides of lengths 6, 8, and 10. A circle is drawn so that the area inside the circle but outside this triangle equals the area inside the triangle but outside the circle. The radius of the circle is closest to: Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "2.8"
} | {
"split": "train",
"index": 17711,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sequence \\( a_1, a_2, \\cdots, a_n, \\cdots \\) satisfies \\( a_1 = a_2 = 1 \\) and \\( a_3 = 2 \\), and for any \\( n \\in \\mathbf{N}^{*} \\), it holds that \\( a_n \\cdot a_{n+1} \\cdot a_{n+2} \\cdot a_{n+3} = a_n + a_{n+1} + a_{n+2} + a_{n+3} \\). Find the ... | math | {
"style": "rule",
"ground_truth": "4044"
} | {
"split": "train",
"index": 17712,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\cos (x- \\frac {π}{4})-\\sin (x- \\frac {π}{4}).$\n(I) Determine the evenness or oddness of the function $f(x)$ and provide a proof;\n(II) If $θ$ is an angle in the first quadrant and $f(θ+ \\frac {π}{3})= \\frac { \\sqrt {2}}{3}$, find the value of $... | math | {
"style": "rule",
"ground_truth": "\\frac {4 \\sqrt {2}}{9}"
} | {
"split": "train",
"index": 17713,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nCalculate the limit of the numerical sequence:\n\n$$\\lim _{n \\rightarrow \\infty} \\frac{\\sqrt[4]{2+n^{5}}-\\sqrt{2 n^{3}+3}}{(n+\\sin n) \\sqrt{7 n}}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\sqrt{\\frac{2}{7}}"
} | {
"split": "train",
"index": 17714,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bag contains nine blue marbles, ten ugly marbles, and one special marble. Ryan picks marbles randomly from this bag with replacement until he draws the special marble. He notices that none of the marbles he drew were ugly. Given this information, what is the expected value of... | math | {
"style": "rule",
"ground_truth": "\\frac{20}{11}"
} | {
"split": "train",
"index": 17715,
"solution": "The probability of drawing $k$ marbles is the probability of drawing $k-1$ blue marbles and then the special marble, which is $p_{k}=\\left(\\frac{9}{20}\\right)^{k-1} \\times \\frac{1}{20}$. The probability of drawing no ugly marbles is therefore $\\sum_{k=1}^{\\i... |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\{a\\_n\\}$ satisfies $a\\_1=2$, $a_{n+1}-2a_{n}=2$, and the sequence $b_{n}=\\log _{2}(a_{n}+2)$. If $S_{n}$ is the sum of the first $n$ terms of the sequence $\\{b_{n}\\}$, then the minimum value of $\\{\\frac{S_{n}+4}{n}\\}$ is ___. Let's think step by s... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"split": "train",
"index": 17716,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C D$ be a square of side length 10 . Point $E$ is on ray $\\overrightarrow{A B}$ such that $A E=17$, and point $F$ is on ray $\\overrightarrow{A D}$ such that $A F=14$. The line through $B$ parallel to $C E$ and the line through $D$ parallel to $C F$ meet at $P$. Compu... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 17717,
"solution": "From $B P \\| C E$, we get that $[B P E]=[B P C]$. From $D P \\| C F$, we get that $[D P F]=[D P C]$. Thus, $$\\begin{aligned} {[A E P F] } & =[B A C P]+[B P E]+[D P F] \\\\ & =[B A C P]+[B P C]+[D P C] \\\\ & =[A B C D] \\\\ & =10^{2}=100 \\end{aligned}$$"
} |
deepscaler | [
{
"role": "user",
"content": "At 1:00 PM, two identical recreational boats set off in opposite directions from a pier on a river. At the same time, a raft also departed from the pier. An hour later, one of the boats turned around and started moving back. The other boat did the same at 3:00 PM. What is the s... | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"split": "train",
"index": 17718,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "20 balls numbered 1 through 20 are placed in a bin. In how many ways can 4 balls be drawn, in order, from the bin, if each ball remains outside the bin after it is drawn and each drawn ball has a consecutive number as the previous one? Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 17719,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each side of a cube has a stripe drawn diagonally from one vertex to the opposite vertex. The stripes can either go from the top-right to bottom-left or from top-left to bottom-right, chosen at random for each face. What is the probability that there is at least one continuous ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{64}"
} | {
"split": "train",
"index": 17720,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The members of a club are arranged in a rectangular formation. When they are arranged in 10 rows, there are 4 positions unoccupied in the formation. When they are arranged in 11 rows, there are 5 positions unoccupied. How many members are in the club if the membership is betwee... | math | {
"style": "rule",
"ground_truth": "226"
} | {
"split": "train",
"index": 17721,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many solutions does the equation \n\\[\n\\frac{(x-1)(x-2)(x-3)\\dotsm(x-200)}{(x-1^2)(x-2^2)(x-3^2)\\dotsm(x-10^2)(x-11^3)(x-12^3)\\dotsm(x-13^3)}\n\\]\nhave for \\(x\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "190"
} | {
"split": "train",
"index": 17722,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A$ and $B$ be two points on the parabola $y = x^2,$ such that when the tangents at $A$ and $B$ drawn, they are perpendicular. Then for any such pair of tangents, the $y$-coordinate of their point of intersection $P$ is always the same. Find this $y$-coordinate.\n\n[asy]\... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{4}"
} | {
"split": "train",
"index": 17723,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( f(n) = \\sum_{k=2}^{\\infty} \\frac{1}{k^n \\cdot k!} \\). Calculate \\( \\sum_{n=2}^{\\infty} f(n) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 - e"
} | {
"split": "train",
"index": 17724,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The writer Arthur has $n \\ge1$ co-authors who write books with him. Each book has a list of authors including Arthur himself. No two books have the same set of authors. At a party with all his co-author, each co-author writes on a note how many books they remember having writt... | math | {
"style": "rule",
"ground_truth": "n\\le6"
} | {
"split": "train",
"index": 17725,
"solution": "\nTo solve the problem, we need to determine the values of \\( n \\) for which it is possible that each co-author accurately remembers the number of books written with Arthur, and these numbers correspond to the first \\( n \\) Fibonacci numbers. \n\nThe Fibonacci ... |
deepscaler | [
{
"role": "user",
"content": "A triangular box is to be cut from an equilateral triangle of length 30 cm. Find the largest possible volume of the box (in cm³). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "500"
} | {
"split": "train",
"index": 17726,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the case when all numbers are equal. $\\frac{5}{4} n + \\frac{5}{4} = n$. If the first number is -5, then all numbers will be equal to -5. The same applies to all cases where the first number is equal to $-5 + 1024n$, $n \\in \\mathbb{Z}$. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 17727,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of the expression $5^3 - 3 \\times 5^2 + 3 \\times 5 - 1$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 17728,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest prime whose digits sum to $19$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "199"
} | {
"split": "train",
"index": 17729,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $E: \\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1(a>b>0)$ with an eccentricity of $\\frac{\\sqrt{2}}{2}$ and upper vertex at B. Point P is on E, point D is at (0, -2b), and the maximum area of △PBD is $\\frac{3\\sqrt{2}}{2}$.\n(I) Find the equation of E;\n(II) I... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 17730,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $x$ is a multiple of $18720$, what is the greatest common divisor of $f(x)=(5x+3)(8x+2)(12x+7)(3x+11)$ and $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "462"
} | {
"split": "train",
"index": 17731,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "[help me] Let m and n denote the number of digits in $2^{2007}$ and $5^{2007}$ when expressed in base 10. What is the sum m + n? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2008"
} | {
"split": "train",
"index": 17732,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be a positive integer and $a$ be an integer such that $a$ is its own inverse modulo $n$. What is the remainder when $a^2$ is divided by $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 17733,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sample has a capacity of $80$. After grouping, the frequency of the second group is $0.15$. Then, the frequency of the second group is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 17734,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence ${a_n}$, $a_1=1$ and $a_n a_{n+1} + \\sqrt{3}(a_n - a_{n+1}) + 1 = 0$. Determine the value of $a_{2016}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 - \\sqrt{3}"
} | {
"split": "train",
"index": 17735,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The left and right foci of the hyperbola $E$: $\\dfrac{x^2}{a^2} - \\dfrac{y^2}{b^2} = 1$ ($a > 0, b > 0$) are $F_1$ and $F_2$, respectively. Point $M$ is a point on the asymptote of hyperbola $E$, and $MF_1 \\perpendicular MF_2$. If $\\sin \\angle MF_1F_2 = \\dfrac{1}{3}$, the... | math | {
"style": "rule",
"ground_truth": "\\dfrac{9}{7}"
} | {
"split": "train",
"index": 17736,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x+7$ is a factor of $cx^3 + 19x^2 - 3cx + 35$, find the constant $c$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 17737,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Eight congruent disks are arranged around a square with side length 2 in such a manner that each disk is tangent to two others, with four disks centered on the midpoints of the square’s sides and four at the corners. The disks completely cover the boundary of the square. Calcul... | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"split": "train",
"index": 17738,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the probability of a contestant winning the quiz given that they answer at least 3 out of 4 questions correctly, assuming random guesses for each question with 3 possible answer options. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"split": "train",
"index": 17739,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the point F(0,1) is the focus of the parabola $x^2=2py$,\n\n(1) Find the equation of the parabola C;\n(2) Points A, B, and C are three points on the parabola such that $\\overrightarrow{FA} + \\overrightarrow{FB} + \\overrightarrow{FC} = \\overrightarrow{0}$, find th... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{6}}{2}"
} | {
"split": "train",
"index": 17740,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $a$, $b$, $c \\in \\mathbb{R}$, and $2a+2b+c=8$, find the minimum value of $(a-1)^2+(b+2)^2+(c-3)^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{49}{9}"
} | {
"split": "train",
"index": 17741,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given positive integers $x$ and $y$ such that $x\\neq y$ and $\\frac{1}{x} + \\frac{1}{y} = \\frac{1}{12}$, what is the smallest possible value for $x + y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 17742,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest three-digit integer that is divisible by each of its distinct, non-zero digits. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "936"
} | {
"split": "train",
"index": 17743,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find an ordered pair $(u,v)$ that solves the system: \\begin{align*} 5u &= -7 - 2v,\\\\ 3u &= 4v - 25 \\end{align*} Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-3,4)"
} | {
"split": "train",
"index": 17744,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $$E: \\frac{x^{2}}{a^{2}} - \\frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$$ with left and right vertices A and B, respectively. Let M be a point on the hyperbola such that ∆ABM is an isosceles triangle, and the area of its circumcircle is 4πa², then the eccentrici... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 17745,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two machine tools, A and B, produce the same product. The products are divided into first-class and second-class according to quality. In order to compare the quality of the products produced by the two machine tools, each machine tool produced 200 products. The quality of the ... | math | {
"style": "rule",
"ground_truth": "99\\%"
} | {
"split": "train",
"index": 17746,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x,$ $y,$ and $z$ be nonnegative real numbers such that $x + y + z = 5.$ Find the maximum value of\n\\[\\sqrt{2x + 1} + \\sqrt{2y + 1} + \\sqrt{2z + 1}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{39}"
} | {
"split": "train",
"index": 17747,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a board on $2013 \\times 2013$ squares, what is the maximum number of chess knights that can be placed so that no $2$ attack each other? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2026085"
} | {
"split": "train",
"index": 17748,
"solution": ""
} |
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