data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "In an equilateral triangle $\\triangle PRS$, if $QS=QT$ and $\\angle QTS=40^\\circ$, what is the value of $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 36749,
"solution": "Since $\\triangle PRS$ is equilateral, then all three of its angles equal $60^\\circ$. In particular, $\\angle RSP=60^\\circ$. Since $QS=QT$, then $\\triangle QST$ is isosceles and so $\\angle TSQ=\\angle STQ=40^\\circ$. Since $RST$ is a straight line segment, then... |
deepscaler | [
{
"role": "user",
"content": "The diagram shows a regular dodecagon and a square, whose vertices are also vertices of the dodecagon. What is the value of the ratio of the area of the square to the area of the dodecagon? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2:3"
} | {
"split": "train",
"index": 36750,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all the three-digit positive integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "494550"
} | {
"split": "train",
"index": 36751,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $x_{1}$ is a root of the one-variable quadratic equation about $x$, $\\frac{1}{2}m{x^2}+\\sqrt{2}x+{m^2}=0$, and ${x_1}=\\sqrt{a+2}-\\sqrt{8-a}+\\sqrt{-{a^2}}$ (where $a$ is a real number), find the values of $m$ and the other root of the equation. Let's think step b... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 36752,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A parallelogram has side lengths of 10, 12, $10y-2$, and $4x+6$. Determine the value of $x+y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2.7"
} | {
"split": "train",
"index": 36753,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, if $|\\overrightarrow{AB}|=2$, $|\\overrightarrow{AC}|=3$, $|\\overrightarrow{BC}|=4$, and $O$ is the incenter of $\\triangle ABC$, and $\\overrightarrow{AO}=\\lambda \\overrightarrow{AB}+\\mu \\overrightarrow{BC}$, what is $\\lambda+\\mu$? Let's think step... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{9}"
} | {
"split": "train",
"index": 36754,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a function $f$ for which \\[f(x) = f(398 - x) = f(2158 - x) = f(3214 - x)\\] holds for all real $x,$ what is the largest number of different values that can appear in the list $f(0),f(1),f(2),\\ldots,f(999)?$ Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "177"
} | {
"split": "train",
"index": 36755,
"solution": "\\begin{align*}f(2158 - x) = f(x) &= f(3214 - (2158 - x)) &= f(1056 + x)\\\\ f(398 - x) = f(x) &= f(2158 - (398 - x)) &= f(1760 + x)\\end{align*}\n Since $\\mathrm{gcd}(1056, 1760) = 352$ we can conclude that (by the Euclidean algorithm)\n\\[f(x) = f(352 + x)\\]\n ... |
deepscaler | [
{
"role": "user",
"content": "Human beings have discovered a habitable planet and soon after, they find 10 more habitable planets. Of these 11, only 5 are deemed ``Earth-like'' in their resources and the rest are deemed ``Mars-like'' since they lack many important resources. Assume that planets like Earth t... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 36756,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The perimeter of a rectangle is 30 inches. One side of the rectangle is fixed at 7 inches. What is the number of square inches in the maximum possible area for this rectangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 36757,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, the numbers 1 to 10 are placed around a circle. Sandy crosses out 1, then 4, and then 7. Continuing in a clockwise direction, she crosses out every third number of those remaining, until only two numbers are left. The sum of these two numbers is: Let's think ste... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 36758,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the integer $n$, $0 \\le n \\le 11$, such that \\[n \\equiv 10389 \\pmod{12}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 36759,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a_{n}\\}$ that satisfies ${a}_{1}+3{a}_{2}+9{a}_{3}+⋯+{3}^{n-1}{a}_{n}=\\frac{n+1}{3}$, where the sum of the first $n$ terms of the sequence $\\{a_{n}\\}$ is denoted as $S_{n}$, find the minimum value of the real number $k$ such that $S_{n} \\lt k$ holds fo... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"split": "train",
"index": 36760,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "15 balls numbered 1 through 15 are placed in a bin. Joe produces a list of four numbers by performing the following sequence four times: he chooses a ball, records the number, and places the ball back in the bin. Finally, Joe chooses to make a unique list by selecting 3 numbers... | math | {
"style": "rule",
"ground_truth": "202500"
} | {
"split": "train",
"index": 36761,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Amina and Bert alternate turns tossing a fair coin. Amina goes first and each player takes three turns. The first player to toss a tail wins. If neither Amina nor Bert tosses a tail, then neither wins. What is the probability that Amina wins? Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "\\frac{21}{32}"
} | {
"split": "train",
"index": 36762,
"solution": "If Amina wins, she can win on her first turn, on her second turn, or on her third turn. If she wins on her first turn, then she went first and tossed tails. This occurs with probability $\\frac{1}{2}$. If she wins on her second turn, then she tossed heads, then Ber... |
deepscaler | [
{
"role": "user",
"content": "Given that \\( x \\) and \\( y \\) are positive numbers, determine the minimum value of \\(\\left(x+\\frac{1}{y}\\right)^{2}+\\left(y+\\frac{1}{2x}\\right)^{2}\\). 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": 36763,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parametric equation of circle $C$ as $\\begin{cases} x=1+3\\cos \\theta \\\\ y=3\\sin \\theta \\end{cases}$ (where $\\theta$ is the parameter), and establishing a polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis, ... | math | {
"style": "rule",
"ground_truth": "\\frac {3 \\sqrt {3}}{2}"
} | {
"split": "train",
"index": 36764,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many even integers between 4000 and 7000 have four different digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "728"
} | {
"split": "train",
"index": 36765,
"solution": "The thousands digit is $\\in \\{4,5,6\\}$.\n Case $1$: Thousands digit is even\n$4, 6$, two possibilities, then there are only $\\frac{10}{2} - 1 = 4$ possibilities for the units digit. This leaves $8$ possible digits for the hundreds and $7$ for the tens places, y... |
deepscaler | [
{
"role": "user",
"content": "How many numbers with less than four digits (from 0 to 9999) are neither divisible by 3, nor by 5, nor by 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4571"
} | {
"split": "train",
"index": 36766,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a rectangular table of size \\( x \\) cm \\(\\times 80\\) cm, identical sheets of paper of size 5 cm \\(\\times 8\\) cm are placed. The first sheet is placed in the bottom left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right ... | math | {
"style": "rule",
"ground_truth": "77"
} | {
"split": "train",
"index": 36767,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the range of $$f(A)=\\frac{(\\sin A)\\left(3 \\cos ^{2} A+\\cos ^{4} A+3 \\sin ^{2} A+\\left(\\sin ^{2} A\\right)\\left(\\cos ^{2} A\\right)\\right)}{(\\tan A)(\\sec A-(\\sin A)(\\tan A))}$$ if $A \\neq \\frac{n \\pi}{2}$. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "(3,4)"
} | {
"split": "train",
"index": 36768,
"solution": "We factor the numerator and write the denominator in terms of fractions to get \\(\\frac{(\\sin A)\\left(3+\\cos ^{2} A\\right)\\left(\\sin ^{2} A+\\cos ^{2} A\\right)}{\\left(\\frac{\\sin A}{\\cos A}\\right)\\left(\\frac{1}{\\cos A}-\\frac{\\sin ^{2} A}{\\cos A}\\... |
deepscaler | [
{
"role": "user",
"content": "Let $x<0.1$ be a positive real number. Let the foury series be $4+4 x+4 x^{2}+4 x^{3}+\\ldots$, and let the fourier series be $4+44 x+444 x^{2}+4444 x^{3}+\\ldots$ Suppose that the sum of the fourier series is four times the sum of the foury series. Compute $x$. Let's think ste... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{40}"
} | {
"split": "train",
"index": 36769,
"solution": "The sum of the foury series can be expressed as \\(\\frac{4}{1-x}\\) by geometric series. The fourier series can be expressed as $$ \\begin{aligned} & \\frac{4}{9}\\left((10-1)+(100-1) x+(1000-1) x^{2}+\\ldots\\right) \\\\ & =\\frac{4}{9}\\left(\\left(10+100 x+1000... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sum_{k=1}^{35}\\sin 5k=\\tan \\frac mn,$ where angles are measured in degrees, and $m_{}$ and $n_{}$ are relatively prime positive integers that satisfy $\\frac mn<90,$ find $m+n.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "177"
} | {
"split": "train",
"index": 36770,
"solution": "We note that $\\sin x = \\mbox{Im } e^{ix}$. We thus have that \\begin{align*} \\sum_{k = 1}^{35} \\sin 5k &= \\sum_{k = 1}^{35} \\mbox{Im } e^{5ki}\\\\ &= \\mbox{Im } \\sum_{k = 1}^{35} e^{5ki}\\\\ &= \\mbox{Im } \\frac{e^{5i}(1 - e^{180i})}{1 - e^{5i}}\\\\ &= \\m... |
deepscaler | [
{
"role": "user",
"content": "$(1)$ When throwing two uniformly weighted dice, let $A=$\"the first time an odd number appears,\" $B=$\"the sum of the two dice is a multiple of $3$.\" Determine whether events $A$ and $B$ are independent, and explain the reason;<br/>$(2)$ Athletes A and B are participating in... | math | {
"style": "rule",
"ground_truth": "0.2212"
} | {
"split": "train",
"index": 36771,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of distinct numbers in the list\n\\[\\left\\lfloor \\frac{1^2}{1000} \\right\\rfloor, \\ \\left\\lfloor \\frac{2^2}{1000} \\right\\rfloor, \\ \\left\\lfloor \\frac{3^2}{1000} \\right\\rfloor, \\ \\dots, \\ \\left\\lfloor \\frac{1000^2}{1000} \\right\\rfloor.\\] ... | math | {
"style": "rule",
"ground_truth": "751"
} | {
"split": "train",
"index": 36772,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{v}$ and $\\mathbf{w}$ be vectors such that\n\\[\\operatorname{proj}_{\\mathbf{w}} \\mathbf{v} = \\begin{pmatrix} 1 \\\\ 0 \\\\ -3 \\end{pmatrix}.\\]Compute $\\operatorname{proj}_{\\mathbf{w}} (-2 \\mathbf{v}).$ Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -2 \\\\ 0 \\\\ 6 \\end{pmatrix}"
} | {
"split": "train",
"index": 36773,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $α \\in (0, \\frac{π}{2})$ and $\\sin (\\frac{π}{4} - α) = \\frac{\\sqrt{10}}{10}$,\n(1) find the value of $\\tan 2α$;\n(2) find the value of $\\frac{\\sin (α + \\frac{π}{4})}{\\sin 2α + \\cos 2α + 1}$. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{8}"
} | {
"split": "train",
"index": 36774,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence of 11 positive real numbers, $a_{1}, a_{2}, a_{3}, \\ldots, a_{11}$, satisfies $a_{1}=4$ and $a_{11}=1024$ and $a_{n}+a_{n-1}=\\frac{5}{2} \\sqrt{a_{n} \\cdot a_{n-1}}$ for every integer $n$ with $2 \\leq n \\leq 11$. For example when $n=7, a_{7}+a_{6}=\\frac{5}{2} \... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 36775,
"solution": "Suppose that, for some integer $n \\geq 2$, we have $a_{n}=x$ and $a_{n-1}=y$. The equation $a_{n}+a_{n-1}=\\frac{5}{2} \\sqrt{a_{n} \\cdot a_{n-1}}$ can be re-written as $x+y=\\frac{5}{2} \\sqrt{x y}$. Since $x>0$ and $y>0$, squaring both sides of the equation giv... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = e^{\\sin x + \\cos x} - \\frac{1}{2}\\sin 2x$ ($x \\in \\mathbb{R}$), find the difference between the maximum and minimum values of the function $f(x)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "e^{\\sqrt{2}} - e^{-\\sqrt{2}}"
} | {
"split": "train",
"index": 36776,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the island, there are 2001 inhabitants including liars and knights. Knights always tell the truth, and liars always lie. Each inhabitant of the island declared, \"Among the remaining inhabitants of the island, more than half are liars\". How many liars are there on the islan... | math | {
"style": "rule",
"ground_truth": "1001"
} | {
"split": "train",
"index": 36777,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $$\\alpha \\in \\left( \\frac{5}{4}\\pi, \\frac{3}{2}\\pi \\right)$$ and it satisfies $$\\tan\\alpha + \\frac{1}{\\tan\\alpha} = 8$$, then $\\sin\\alpha\\cos\\alpha = \\_\\_\\_\\_\\_\\_$; $\\sin\\alpha - \\cos\\alpha = \\_\\_\\_\\_\\_\\_$. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 36778,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is ... | math | {
"style": "rule",
"ground_truth": "132"
} | {
"split": "train",
"index": 36779,
"solution": "\nWe are given that Amy, Beth, and Jo listen to four different songs and discuss which ones they like, with the conditions that no song is liked by all three and for each pair of the girls, there is at least one song liked by those two but disliked by the third. We... |
deepscaler | [
{
"role": "user",
"content": "In multiplying two positive integers $a$ and $b$, Ron reversed the digits of the two-digit number $a$. His erroneous product was $161$. What is the correct value of the product of $a$ and $b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "224"
} | {
"split": "train",
"index": 36780,
"solution": "1. **Identify the Error in Multiplication**: Ron reversed the digits of $a$ when multiplying. Let's denote the reversed number as $a'$. The erroneous product given is $161$, so we have $a' \\cdot b = 161$.\n\n2. **Factorize 161**: To find possible values of $a'$ an... |
deepscaler | [
{
"role": "user",
"content": "An $11 \\times 11 \\times 11$ wooden cube is formed by gluing together $11^3$ unit cubes. What is the greatest number of unit cubes that can be seen from a single point? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "331"
} | {
"split": "train",
"index": 36781,
"solution": "To find the greatest number of unit cubes that can be seen from a single point, we consider viewing the cube from one of its corners. From this vantage point, three faces of the cube are visible: the top face, the front face, and the side face.\n\n1. **Counting the... |
deepscaler | [
{
"role": "user",
"content": "Find the minimum possible value of the largest of $x y, 1-x-y+x y$, and $x+y-2 x y$ if $0 \\leq x \\leq y \\leq 1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"split": "train",
"index": 36782,
"solution": "I claim the answer is $4 / 9$. Let $s=x+y, p=x y$, so $x$ and $y$ are $\\frac{s \\pm \\sqrt{s^{2}-4 p}}{2}$. Since $x$ and $y$ are real, $s^{2}-4 p \\geq 0$. If one of the three quantities is less than or equal to $1 / 9$, then at least one of the others is at leas... |
deepscaler | [
{
"role": "user",
"content": "Xiao Ming attempts to remove all 24 bottles of beer from a box, with each attempt allowing him to remove either three or four bottles at a time. How many different methods are there for Xiao Ming to remove all the beer bottles? Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 36783,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are five positive integers that are common divisors of each number in the list $$36, 72, -24, 120, 96.$$ Find the sum of these five positive integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 36784,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A teacher tells the class,\n\n\"Think of a number, add 1 to it, and double the result. Give the answer to your partner. Partner, subtract 1 from the number you are given and double the result to get your answer.\"\n\nBen thinks of $6$, and gives his answer to Sue. What shoul... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 36785,
"solution": "1. **Ben's Calculation:**\n - Ben thinks of a number, which is $6$.\n - He adds $1$ to this number: $6 + 1 = 7$.\n - He then doubles the result: $7 \\times 2 = 14$.\n - Ben gives the number $14$ to Sue.\n\n2. **Sue's Calculation:**\n - Sue receives the nu... |
deepscaler | [
{
"role": "user",
"content": "A class has a group of 7 people, and now 3 of them are chosen to swap seats with each other, while the remaining 4 people's seats remain unchanged. Calculate the number of different rearrangement plans. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 36786,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Every pair of communities in a county are linked directly by one mode of transportation; bus, train, or airplane. All three methods of transportation are used in the county with no community being serviced by all three modes and no three communities being linked pairwise by the... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 36787,
"solution": "\nLet us consider a set of communities, denoted as vertices in a graph, where each edge between a pair of communities is labeled with one of the following modes of transportation: bus, train, or airplane. The problem imposes the following conditions:\n\n1. All thre... |
deepscaler | [
{
"role": "user",
"content": "Some marbles in a bag are red and the rest are blue. If one red marble is removed, then one-seventh of the remaining marbles are red. If two blue marbles are removed instead of one red, then one-fifth of the remaining marbles are red. How many marbles were in the bag originally... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 36788,
"solution": "Let $r$ and $b$ be the number of red and blue marbles originally in the bag, respectively.\n\n1. **Removing one red marble:**\n - After removing one red marble, the number of red marbles becomes $r-1$.\n - The total number of marbles left is $r+b-1$.\n - Give... |
deepscaler | [
{
"role": "user",
"content": "Don has a bottle of milk with $\\frac{1}{5}$ of a gallon of milk in it. Don's friend Rachel drinks $\\frac{2}{3}$ of the milk he has. How many gallons of milk did Rachel drink? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{15}"
} | {
"split": "train",
"index": 36789,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $|x|=4$, $|y|=2$, and $x<y$, then the value of $x\\div y$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 36790,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $d$ be a randomly chosen divisor of 2016. Find the expected value of $\\frac{d^{2}}{d^{2}+2016}$. 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": 36791,
"solution": "Let $ab=2016$. Then $$\\frac{a^{2}}{a^{2}+2016}+\\frac{b^{2}}{b^{2}+2016}=\\frac{a^{2}}{a^{2}+2016}+\\frac{\\left(\\frac{2016}{a}\\right)^{2}}{\\left(\\frac{2016}{a}\\right)^{2}+2016}=\\frac{a^{2}}{a^{2}+2016}+\\frac{2016}{a^{2}+2016}=1$$ Thus, every divisor $d$ pa... |
deepscaler | [
{
"role": "user",
"content": "A three-meter gas pipe has rusted in two places. Determine the probability that all three resulting segments can be used as connections to gas stoves, given that regulations require the stove to be no closer than 75 cm to the main gas pipe. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "1/4"
} | {
"split": "train",
"index": 36792,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle with vertices \\(A = (4, 3)\\), \\(B = (6, -2)\\), and \\(C = (7, 1)\\) is reflected about the line \\(x = 6\\) to create a second triangle. Determine the area of the union of the two triangles. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 36793,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The perimeter of triangle $APM$ is $152$, and the angle $PAM$ is a right angle. A circle of radius $19$ with center $O$ on $\\overline{AP}$ is drawn so that it is tangent to $\\overline{AM}$ and $\\overline{PM}$. Given that $OP=m/n$ where $m$ and $n$ are relatively prime positi... | math | {
"style": "rule",
"ground_truth": "98"
} | {
"split": "train",
"index": 36794,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ whose sum of the first $n$ terms is $S_n$, and the point $(n, \\frac{S_n}{n})$ lies on the line $y= \\frac{1}{2}x+ \\frac{11}{2}$. The sequence $\\{b_n\\}$ satisfies $b_{n+2}-2b_{n+1}+b_n=0$ $(n\\in{N}^*)$, and $b_3=11$, with the sum of the first $9... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 36795,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\\( p(x, y, z) \\) is a polynomial with real coefficients such that:\n1. \\( p(tx, ty, tz) = t^2 f(y - x, z - x) \\) for all real \\( x, y, z, t \\) (and some function \\( f \\));\n2. \\( p(1, 0, 0) = 4 \\), \\( p(0, 1, 0) = 5 \\), and \\( p(0, 0, 1) = 6 \\);\n3. \\( p(\\alpha,... | math | {
"style": "rule",
"ground_truth": "\\frac{5 \\sqrt{30}}{3}"
} | {
"split": "train",
"index": 36796,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a Cartesian coordinate system, the \"rectangular distance\" between points $P\\left(x_{1}, y_{1}\\right)$ and $Q\\left(x_{2}, y_{2}\\right)$ is defined as $d(P, Q) = \\left|x_{1}-x_{2}\\right| + \\left|y_{1}-y_{2}\\right|$. If the \"rectangular distance\" from point $C(x, y)... | math | {
"style": "rule",
"ground_truth": "5(\\sqrt{2} + 1)"
} | {
"split": "train",
"index": 36797,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Place 5 balls numbered 1, 2, 3, 4, 5 into 5 boxes also numbered 1, 2, 3, 4, 5. \n(1) How many ways are there to do this? \n(2) If each box can hold at most one ball, how many ways are there? \n(3) If exactly one box is to remain empty, how many ways are there? \n(4) If each... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 36798,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\left( 1 + \\cos \\frac {\\pi}{8} \\right) \\left( 1 + \\cos \\frac {3 \\pi}{8} \\right) \\left( 1 + \\cos \\frac {5 \\pi}{8} \\right) \\left( 1 + \\cos \\frac {7 \\pi}{8} \\right).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 36799,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Chris and Paul each rent a different room of a hotel from rooms $1-60$. However, the hotel manager mistakes them for one person and gives \"Chris Paul\" a room with Chris's and Paul's room concatenated. For example, if Chris had 15 and Paul had 9, \"Chris Paul\" has 159. If the... | math | {
"style": "rule",
"ground_truth": "\\frac{153}{1180}"
} | {
"split": "train",
"index": 36800,
"solution": "There are $60 \\cdot 59=3540$ total possible outcomes, and we need to count the number of these which concatenate into a number at most 60. Of these, $9 \\cdot 8$ result from both Chris and Paul getting one-digit room numbers. If Chris gets a two-digit number, then... |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $AB = 8$, $BC = 8$, and $CA = 6$.\n\nPoint $P$ is randomly selected inside triangle $ABC$. What is the probability that $P$ is closer to vertex $C$ than it is to either vertex $A$ or $B$? 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": 36801,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $x$ is the median of the data set $1$, $2$, $3$, $x$, $5$, $6$, $7$, and the average of the data set $1$, $2$, $x^{2}$, $-y$ is $1$, find the minimum value of $y- \\frac {1}{x}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {23}{3}"
} | {
"split": "train",
"index": 36802,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A hotel has three types of rooms available: a triple room, a double room, and a single room (each type of room can only accommodate the corresponding number of people). There are 4 adult men and 2 little boys looking for accommodation. The little boys should not stay in a room ... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 36803,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\triangle A_1A_2A_3$ is equilateral and $A_{n+3}$ is the midpoint of line segment $A_nA_{n+1}$ for all positive integers $n$, then the measure of $\\measuredangle A_{44}A_{45}A_{43}$ equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60^{\\circ}"
} | {
"split": "train",
"index": 36804,
"solution": "To solve this problem, we need to understand the geometric sequence generated by the points $A_n$ and how the angles between these points evolve.\n\n1. **Initial Setup**: Given that $\\triangle A_1A_2A_3$ is equilateral, each angle in this triangle is $60^\\circ$.\... |
deepscaler | [
{
"role": "user",
"content": "Square $ABCD$ has sides of length 4. Set $T$ is the set of all line segments that have length 4 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $T$ enclose a region whose area to the nearest hundredth is $m$. Find $100m$. Let's... | math | {
"style": "rule",
"ground_truth": "343"
} | {
"split": "train",
"index": 36805,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $$x \\in \\left(- \\frac{\\pi}{2}, \\frac{\\pi}{2}\\right)$$ and $$\\sin x + \\cos x = \\frac{1}{5}$$, calculate the value of $\\tan 2x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{24}{7}"
} | {
"split": "train",
"index": 36806,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $E$: $x^2 + \\frac{y^2}{b^2} = 1 (0 < b < 1)$, and the line $l$ passing through $F_1$ intersects $E$ at points $A$ and $B$. If the sequence $|AF_2|, |AB|, |BF_2|$ forms an arithmetic progression, then:\n\n(1)... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 36807,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $z$ is a complex number such that $z+\\frac{1}{z}=2\\cos 5^\\circ$, find $z^{1500}+\\frac{1}{z^{1500}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\sqrt{3}"
} | {
"split": "train",
"index": 36808,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that in $\\triangle ABC$, $BD:DC = 3:2$ and $AE:EC = 3:4$, and the area of $\\triangle ABC$ is 1, find the area of $\\triangle BMD$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{15}"
} | {
"split": "train",
"index": 36809,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For arbitrary real numbers \\(a\\) and \\(b\\) (\\(a \\neq 0\\)), find the minimum value of the expression \\(\\frac{1}{a^{2}} + 2a^{2} + 3b^{2} + 4ab\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\frac{8}{3}}"
} | {
"split": "train",
"index": 36810,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P(z)=z^3+az^2+bz+c$, where a, b, and c are real. There exists a complex number $w$ such that the three roots of $P(z)$ are $w+3i$, $w+9i$, and $2w-4$, where $i^2=-1$. Find $|a+b+c|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "136"
} | {
"split": "train",
"index": 36811,
"solution": "Set $w=x+yi$, so $x_1 = x+(y+3)i$, $x_2 = x+(y+9)i$, $x_3 = 2x-4+2yi$.\n Since $a,b,c\\in{R}$, the imaginary part of $a,b,c$ must be $0$.\n Start with a, since it's the easiest one to do: $y+3+y+9+2y=0, y=-3$,\n and therefore: $x_1 = x$, $x_2 = x+6i$, $x_3 = 2x-4-6... |
deepscaler | [
{
"role": "user",
"content": "In the tetrahedron $ABCD$, $\\triangle ABC$ is an equilateral triangle, $AD = BD = 2$, $AD \\perp BD$, and $AD \\perp CD$. Find the distance from point $D$ to the plane $ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 36812,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum real number \\( k \\) such that for any positive numbers \\( a \\) and \\( b \\), the following inequality holds:\n$$\n(a+b)(ab+1)(b+1) \\geqslant k \\, ab^2.\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27/4"
} | {
"split": "train",
"index": 36813,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f(x) = (m^2 - m - 1)x^m$ is a power function, and it is a decreasing function on $x \\in (0, +\\infty)$. The value of the real number $m$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 36814,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Construct a square on one side of an equilateral triangle. On one non-adjacent side of the square, construct a regular pentagon, as shown. On a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 36815,
"solution": "1. **Identify the shapes and their sides**: We are given an equilateral triangle (3 sides), a square (4 sides), a regular pentagon (5 sides), a regular hexagon (6 sides), a regular heptagon (7 sides), and a regular octagon (8 sides).\n\n2. **Determine the adjacency... |
deepscaler | [
{
"role": "user",
"content": "A frequency distribution of the scores for Mr. Sampson's algebra class is shown. What percent of the class received a score in the $60\\%$-$69\\%$ range? \\begin{tabular}{|c|c|}\nTest Scores & Frequencies\\\\\n\\hline\n$90\\% - 100\\%$& IIII\\\\\n$80\\% - 89\\%$& IIII IIII\\\\\... | math | {
"style": "rule",
"ground_truth": "20\\%"
} | {
"split": "train",
"index": 36816,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The eccentricity of the ellipse $\\frac {x^{2}}{9}+ \\frac {y^{2}}{4+k}=1$ is $\\frac {4}{5}$. Find the value of $k$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 36817,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Place the numbers 1, 2, 3, 4, 2, 6, 7, and 8 at the eight vertices of a cube such that the sum of the numbers on any given face is at least 10. Determine the minimum possible sum of the numbers on each face. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 36818,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a box contains $5$ shiny pennies and $6$ dull pennies, determine the sum of the numerator and denominator of the probability that it will take exactly six draws to get the fourth shiny penny. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "236"
} | {
"split": "train",
"index": 36819,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of right-angled triangles with integer side lengths $a$ and $b$ (where $b < 2011$), and hypotenuse length $b + 1$ is . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 36820,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bag contains 70 balls that differ only in color: 20 red, 20 blue, 20 yellow, and the rest are black and white.\n\nWhat is the minimum number of balls that must be drawn from the bag, without looking, to ensure that among them there are at least 10 balls of a single color? Let... | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 36821,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least prime factor of $7^4 - 7^3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 36822,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Use the five digits $0$, $1$, $2$, $3$, $4$ to form integers that satisfy the following conditions:\n(I) All four-digit integers;\n(II) Five-digit integers without repetition that are greater than $21000$. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "66"
} | {
"split": "train",
"index": 36823,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of $y=\\frac{5x^2-9}{3x^2+5x+2}$ has a horizontal asymptote at $y=a$. What is $a$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac53"
} | {
"split": "train",
"index": 36824,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a certain math exam, $10\\%$ of the students got $70$ points, $25\\%$ got $80$ points, $20\\%$ got $85$ points, $15\\%$ got $90$ points, and the rest got $95$ points. What is the difference between the mean and the median score on this exam? Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 36825,
"solution": "1. **Calculate the percentage of students scoring 95 points**: \n Given that $10\\%$ scored 70 points, $25\\%$ scored 80 points, $20\\%$ scored 85 points, and $15\\%$ scored 90 points, the remaining percentage of students who scored 95 points is:\n \\[\n 100\... |
deepscaler | [
{
"role": "user",
"content": "The number $947$ can be written as $23q + r$ where $q$ and $r$ are positive integers. What is the greatest possible value of $q - r$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 36826,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Positive real numbers \\( a, b, c \\) are in a geometric progression \\((q \\neq 1)\\), and \\( \\log _{a} b, \\log _{b} c, \\log _{c} a \\) are in an arithmetic progression. Find the common difference \\( d \\). Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"split": "train",
"index": 36827,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many positive integer values of $n$ are both $\\frac{n}{3}$ and $3n$ three-digit whole numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 36828,
"solution": "1. **Identify the conditions for $\\frac{n}{3}$ and $3n$:**\n - $\\frac{n}{3}$ must be a three-digit whole number. This implies $\\frac{n}{3} \\geq 100$ and $\\frac{n}{3} \\leq 999$.\n - $3n$ must also be a three-digit whole number. This implies $3n \\geq 100$ ... |
deepscaler | [
{
"role": "user",
"content": "There are three piles containing 22, 14, and 12 nuts. It is necessary to equalize the number of nuts in all piles by making three moves, while adhering to the following condition: it is only allowed to move as many nuts from one pile to another as there are in the pile to which... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 36829,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form $a + b \\sqrt{2} + c \\sqrt{3} + d \\sqrt{6},$ where $a^{}_{}$, $b^{}_{}$, $c^{}_{}$, and $d^{}_{}$ are positive integers. Find $a ... | math | {
"style": "rule",
"ground_truth": "720"
} | {
"split": "train",
"index": 36830,
"solution": "Begin as in solution 2, drawing a triangle connecting the center of the 12-gon to two vertices of the 12-gon. Apply law of cosines on $\\theta = {30^{\\circ}, 60^{\\circ}, 90^{\\circ}, 120^{\\circ}, 150^{\\circ}, 180^{\\circ}}$ to get $d^2 = 288 - 288 \\cos \\theta... |
deepscaler | [
{
"role": "user",
"content": "An isosceles right triangle $A B C$ has area 1. Points $D, E, F$ are chosen on $B C, C A, A B$ respectively such that $D E F$ is also an isosceles right triangle. Find the smallest possible area of $D E F$. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 36831,
"solution": "Without loss of generality, suppose that $A B$ is the hypotenuse. If $F$ is the right angle, then $F$ must be the midpoint of $A B$. To prove this, let $X$ and $Y$ be the feet from $F$ to $B C$ and $A C$. Since $\\angle X F Y=\\angle D F E=90^{\\circ}$, we have $\\... |
deepscaler | [
{
"role": "user",
"content": "Three people, A, B, and C, visit three tourist spots, with each person visiting only one spot. Let event $A$ be \"the three people visit different spots,\" and event $B$ be \"person A visits a spot alone.\" Then, the probability $P(A|B)=$ ______. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{2}"
} | {
"split": "train",
"index": 36832,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for \\(x\\): \\[\\frac{x-60}{3} = \\frac{5-3x}{4}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{255}{13}"
} | {
"split": "train",
"index": 36833,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The base of a triangle is 20; the medians drawn to the lateral sides are 18 and 24. Find the area of the triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "288"
} | {
"split": "train",
"index": 36834,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)$ be a function defined on $\\mathbb{R}$ with a period of 2. On the interval $[-1,1)$, $f(x)$ is given by\n$$\nf(x) = \\begin{cases}\nx+a & \\text{for } -1 \\leq x < 0,\\\\\n\\left| \\frac{2}{5} - x \\right| & \\text{for } 0 \\leq x < 1,\n\\end{cases}\n$$\nwhere $a \\i... | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{5}"
} | {
"split": "train",
"index": 36835,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Assume the adjoining chart shows the $1980$ U.S. population, in millions, for each region by ethnic group. To the nearest percent, what percent of the U.S. Black population lived in the South?\n\\begin{tabular}{|c|cccc|} \\hline & NE & MW & South & West \\\\ \\hline White & 4... | math | {
"style": "rule",
"ground_truth": "56\\%"
} | {
"split": "train",
"index": 36836,
"solution": "1. **Calculate the total Black population in the U.S.**:\n From the table, the Black population in each region is given as:\n - NE: 5 million\n - MW: 5 million\n - South: 15 million\n - West: 2 million\n\n Summing these values gives the total Black popu... |
deepscaler | [
{
"role": "user",
"content": "I'm going to dinner at a large restaurant which my friend recommended, unaware that I am vegan and have both gluten and dairy allergies. Initially, there are 6 dishes that are vegan, which constitutes one-sixth of the entire menu. Unfortunately, 4 of those vegan dishes contain ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{18}"
} | {
"split": "train",
"index": 36837,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer \\( n \\) for which there are exactly 2323 positive integers less than or equal to \\( n \\) that are divisible by 2 or 23, but not both. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4644"
} | {
"split": "train",
"index": 36838,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular prism has edges $a=b=8$ units and $c=27$ units. Divide the prism into four parts from which a cube can be assembled. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 36839,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given triangle \\( \\triangle ABC \\) with \\( AB = 1 \\), \\( AC = 2 \\), and \\( \\cos B + \\sin C = 1 \\), find the length of side \\( BC \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3 + 2 \\sqrt{21}}{5}"
} | {
"split": "train",
"index": 36840,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six men and their wives are sitting at a round table with 12 seats. These men and women are very jealous - no man will allow his wife to sit next to any man except for himself, and no woman will allow her husband to sit next to any woman except for herself. In how many distinct... | math | {
"style": "rule",
"ground_truth": "288000"
} | {
"split": "train",
"index": 36841,
"solution": "Think of this problem in terms of \"blocks\" of men and women, that is, groups of men and women sitting together. Each block must contain at least two people; otherwise you have a man sitting next to two women (or vice-versa). We will define the notation $[a_{1}, b... |
deepscaler | [
{
"role": "user",
"content": "Given a plane $\\alpha$ and two non-coincident straight lines $m$ and $n$, consider the following four propositions:\n(1) If $m \\parallel \\alpha$ and $n \\subseteq \\alpha$, then $m \\parallel n$.\n(2) If $m \\parallel \\alpha$ and $n \\parallel \\alpha$, then $m \\parallel n... | math | {
"style": "rule",
"ground_truth": "(4)"
} | {
"split": "train",
"index": 36842,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the infinite series $1+2\\left(\\dfrac{1}{1998}\\right)+3\\left(\\dfrac{1}{1998}\\right)^2+4\\left(\\dfrac{1}{1998}\\right)^3+\\cdots$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3992004}{3988009}"
} | {
"split": "train",
"index": 36843,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are two ribbons, one is 28 cm long and the other is 16 cm long. Now, we want to cut them into shorter ribbons of the same length without any leftovers. What is the maximum length of each short ribbon in centimeters? And how many such short ribbons can be cut? Let's think ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 36844,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle with a radius of 6 is inscribed around the trapezoid \\(ABCD\\). The center of this circle lies on the base \\(AD\\), and \\(BC = 4\\). Find the area of the trapezoid. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24\\sqrt{2}"
} | {
"split": "train",
"index": 36845,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ is inscribed in a circle, and $\\angle B = \\angle C = 4\\angle A$. If $B$ and $C$ are adjacent vertices of a regular polygon of $n$ sides inscribed in this circle, then $n=$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 36846,
"solution": "1. **Identify the relationship between angles**: Given that $\\angle B = \\angle C = 4\\angle A$, we can use the fact that the sum of angles in a triangle is $180^\\circ$. Therefore, we write:\n \\[\n \\angle B + \\angle C + \\angle A = 180^\\circ\n \\]\n S... |
deepscaler | [
{
"role": "user",
"content": "Triangle $GHI$ has sides of length 7, 24, and 25 units, and triangle $JKL$ has sides of length 9, 40, and 41 units. Both triangles have an altitude to the hypotenuse such that for $GHI$, the altitude splits the triangle into two triangles whose areas have a ratio of 2:3. For $J... | math | {
"style": "rule",
"ground_truth": "\\dfrac{7}{15}"
} | {
"split": "train",
"index": 36847,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\( k \\) is the smallest positive integer such that \\(\\left(2^{k}\\right)\\left(5^{300}\\right)\\) has 303 digits when expanded, then the sum of the digits of the expanded number is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 36848,
"solution": ""
} |
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