data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Alex has 5 pieces of paper, each with a different math problem. In how many ways can he give these problems to his 10 friends (he can give each friend more than one problem)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100,\\!000"
} | {
"split": "train",
"index": 18549,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "2019 students are voting on the distribution of \\(N\\) items. For each item, each student submits a vote on who should receive that item, and the person with the most votes receives the item (in case of a tie, no one gets the item). Suppose that no student votes for the same p... | math | {
"style": "rule",
"ground_truth": "1009"
} | {
"split": "train",
"index": 18550,
"solution": "To get an item, a student must receive at least 2 votes on that item. Since each student receives at most 2019 votes, the number of items one student can receive does not exceed \\(\\frac{2019}{2}=1009.5\\). So, the answer is at most 1009. This occurs when \\(N=201... |
deepscaler | [
{
"role": "user",
"content": "There are 4 boys and 3 girls standing in a row. (You must write down the formula before calculating the result to score points) \n(β
) If the 3 girls must stand together, how many different arrangements are there? \n(β
‘) If no two girls are next to each other, how many differen... | math | {
"style": "rule",
"ground_truth": "720"
} | {
"split": "train",
"index": 18551,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the decomposition rate $v$ of a certain type of garbage approximately satisfies the relationship $v=a\\cdot b^{t}$, where $a$ and $b$ are non-zero constants, and $v=5\\%$ after $6$ months and $v=10\\%$ after $12$ months, determine the time needed for this type of garbage ... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 18552,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As $t$ takes on all real values, the set of points $(x,y)$ defined by\n\\begin{align*}\nx &= t^2 - 2, \\\\\ny &= t^3 - 9t + 5\n\\end{align*}forms a curve that crosses itself. Compute the ordered pair $(x,y)$ where this crossing occurs. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "(7,5)"
} | {
"split": "train",
"index": 18553,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Please fold a long rope in half, then fold it in half again along the middle of the folded rope, and continue to fold it in half 5 times in total. Finally, cut the rope along the middle after it has been folded 5 times. At this point, the rope will be cut into ___ segments. Let... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 18554,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of integers $n$ that satisfy:\n\\[50 < n^2 < 200.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 18555,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three friends are driving to New York City and splitting the gas cost equally. At the last minute, 2 more friends joined the trip. The cost of gas was then evenly redistributed among all of the friends. The cost for each of the original 3 decreased by $\\$$11.00. What was the t... | math | {
"style": "rule",
"ground_truth": "82.50"
} | {
"split": "train",
"index": 18556,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "100 participants came to the international StarCraft championship. The game is played in a knockout format, meaning two players participate in each match, the loser is eliminated from the championship, and the winner remains. Find the maximum possible number of participants who... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 18557,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $tan({ΞΈ+\\frac{Ο}{{12}}})=2$, find $sin({\\frac{Ο}{3}-2ΞΈ})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{5}"
} | {
"split": "train",
"index": 18558,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\frac{x}{\\ln x}$, and $g(x) = f(x) - mx (m \\in \\mathbb{R})$,\n\n(I) Find the interval of monotonic decrease for function $f(x)$.\n\n(II) If function $g(x)$ is monotonically decreasing on the interval $(1, +\\infty)$, find the range of the real num... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2} - \\frac{1}{4e^2}"
} | {
"split": "train",
"index": 18559,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice and Bob each draw one number from 50 slips of paper numbered from $1$ to $50$ placed in a hat. Alice says, \"I can't tell who has the larger number.\" Bob then says, \"I am certain who has the larger number.\" Understanding Bob's certainty, Alice asks Bob if his number is... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"split": "train",
"index": 18560,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least positive integer $n$ such that $4125$ is a factor of $n!$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 18561,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle $JKL$, angle $J$ measures 60 degrees and angle $K$ measures 30 degrees. When drawn, the angle bisectors of angles $J$ and $K$ intersect at a point $M$. What is the measure of obtuse angle $JMK$?\n\n[asy]\nimport geometry;\nimport olympiad;\nunitsize(0.8inch);\... | math | {
"style": "rule",
"ground_truth": "135"
} | {
"split": "train",
"index": 18562,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the polar coordinate system, find the length of the segment cut by the curve $\\rho=1$ from the line $\\rho\\sin\\theta-\\rho\\cos\\theta=1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 18563,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An element is randomly chosen from among the first $20$ rows of Pascalβs Triangle. What is the probability that the value of the element chosen is $1$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{39}{210}"
} | {
"split": "train",
"index": 18564,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An equilateral triangle of side length $12$ is completely filled by non-overlapping equilateral triangles of side length $2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 18565,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area in the first quadrant bounded by the hyperbola $x^{2}-y^{2}=1$, the $x$-axis, and the line $3 x=4 y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\ln 7}{4}"
} | {
"split": "train",
"index": 18566,
"solution": "Convert to polar coordinates: the hyperbola becomes $$1=r^{2}\\left(\\cos ^{2} \\theta-\\sin ^{2} \\theta\\right)=r^{2} \\cos (2 \\theta)$$ so, letting $\\alpha:=\\arctan (3 / 4)$, the area is $$S:=\\int_{0}^{\\alpha} \\frac{r^{2}}{2} d \\theta=\\frac{1}{2} \\int_{... |
deepscaler | [
{
"role": "user",
"content": "The positive integers $A,$ $B,$ $A-B,$ and $A+B$ are all prime numbers. The sum of these four primes is\n\n$\\bullet$ A. even\n\n$\\bullet$ B. divisible by $3$\n\n$\\bullet$ C. divisible by $5$\n\n$\\bullet$ D. divisible by $7$\n\n$\\bullet$ E. prime\n\nExpress your answer usin... | math | {
"style": "rule",
"ground_truth": "\\text{(E)},"
} | {
"split": "train",
"index": 18567,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The smallest of nine consecutive integers is 2012. These nine integers are placed in the circles to the right. The sum of the three integers along each of the four lines is the same. If this sum is as small as possible, what is the value of $u$? Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "2015"
} | {
"split": "train",
"index": 18568,
"solution": "If we have a configuration of the numbers that has the required property, then we can add or subtract the same number from each of the numbers in the circles and maintain the property. (This is because there are the same number of circles in each line.) Therefore, ... |
deepscaler | [
{
"role": "user",
"content": "Convert the binary number $1110011_2$ to its decimal equivalent. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "115"
} | {
"split": "train",
"index": 18569,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the function $f(x)=(x-3)^3 +x-1$. The sequence $\\{a_n\\}$ is an arithmetic sequence with a non-zero common difference. If $f(a_1)+f(a_2) + \\ldots +f(a_7) =14$, then $a_1 +a_2 +\\ldots +a_7 =$ ______. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 18570,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the $x$-coordinate of the point on the $x$-axis that is equidistant from points $A( -2, 0)$ and $B(0,4)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 18571,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be a positive integer. If $a\\equiv (3^{2n}+4)^{-1}\\pmod{9}$, what is the remainder when $a$ is divided by $9$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 18572,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that the graph of a certain function, $y=f(x)$, has the property that if it is shifted $20$ units to the right, then the resulting graph is identical to the original graph of $y=f(x)$.\n\nWhat is the smallest positive $a$ such that if the graph of $y=f\\left(\\frac x5\\... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 18573,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A group consists of 4 male students and 3 female students. From this group, 4 people are selected to complete three different tasks, with the condition that at least two of the selected individuals must be female, and each task must have at least one person assigned to it. The ... | math | {
"style": "rule",
"ground_truth": "792"
} | {
"split": "train",
"index": 18574,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of $y = ax^2 + bx + c$ has a maximum value of 72, and passes through the points $(0, -1)$ and $(6, -1)$. Find $a + b + c$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{356}{9}"
} | {
"split": "train",
"index": 18575,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Regions I, II, and III are bounded by shapes. The perimeter of region I is 16 units and the perimeter of region II is 36 units. Region III is a triangle with a perimeter equal to the average of the perimeters of regions I and II. What is the ratio of the area of region I to the... | math | {
"style": "rule",
"ground_truth": "\\frac{144}{169\\sqrt{3}}"
} | {
"split": "train",
"index": 18576,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{p}$ be the projection of $\\mathbf{v}$ onto $\\mathbf{w}$, let $\\mathbf{q}$ be the projection of $\\mathbf{p}$ onto $\\mathbf{v}$, and let $\\mathbf{r}$ be the projection of $\\mathbf{w}$ onto $\\mathbf{p}$. If $\\frac{\\|\\mathbf{p}\\|}{\\|\\mathbf{v}\\|} = \\fr... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 18577,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are several solid and hollow circles arranged in a certain pattern as follows: ββββββββββββββββββ¦ Among the first 2001 circles, find the number of hollow circles. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "667"
} | {
"split": "train",
"index": 18578,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $k$ be a positive integer. Marco and Vera play a game on an infinite grid of square cells. At the beginning, only one cell is black and the rest are white. \n\tA turn in this game consists of the following. Marco moves first, and for every move he must choose a cell which... | math | {
"style": "rule",
"ground_truth": "203"
} | {
"split": "train",
"index": 18579,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cauldron has the shape of a paraboloid of revolution. The radius of its base is \\( R = 3 \\) meters, and the depth is \\( H = 5 \\) meters. The cauldron is filled with a liquid, the specific weight of which is \\( 0.8 \\Gamma / \\text{cm}^3 \\). Calculate the work required t... | math | {
"style": "rule",
"ground_truth": "294300\\pi"
} | {
"split": "train",
"index": 18580,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(Full score for this problem is 12 points) Given $f(x) = e^x - ax - 1$.\n\n(1) Find the intervals where $f(x)$ is monotonically increasing.\n(2) If $f(x)$ is monotonically increasing on the domain $\\mathbb{R}$, find the range of possible values for $a$.\n(3) Does there exist a... | math | {
"style": "rule",
"ground_truth": "a = 1"
} | {
"split": "train",
"index": 18581,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be a positive integer. Claudio has $n$ cards, each labeled with a different number from 1 to n. He takes a subset of these cards, and multiplies together the numbers on the cards. He remarks that, given any positive integer $m$, it is possible to select some subset of t... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 18582,
"solution": "We require that $n \\geq 15$ so that the product can be divisible by 25 without being even. In addition, for any $n>15$, if we can acquire all residues relatively prime to 100, we may multiply them by some product of $\\{1,2,4,5,15\\}$ to achieve all residues modul... |
deepscaler | [
{
"role": "user",
"content": "Given that an ellipse and a hyperbola share common foci $F_1$, $F_2$, and $P$ is one of their intersection points, $\\angle F_1PF_2=60^\\circ$, find the minimum value of $e_1^2+e_2^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1+\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 18583,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the digits 1, 2, 3, 4, 5, 6 to form a six-digit number (without repeating any digit), requiring that any two adjacent digits have different parity, and 1 and 2 are adjacent, determine the number of such six-digit numbers. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 18584,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest natural number $n$ with the following property: in any $n$-element subset of $\\{1, 2, \\cdots, 60\\}$, there must be three numbers that are pairwise coprime. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 18585,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of ordered triples $(x,y,z)$ of real numbers for which\n\\begin{align*} \\label{eq:1} \\log_{10}(x+y) = z \\\\\n\\log_{10}(x^{2}+y^{2}) = z+1 \\end{align*} There are real numbers $a$ and $b$ such that for all ordered triples $(x,y.z)$ in $S$ we have $x^{3}+y^... | math | {
"style": "rule",
"ground_truth": "\\frac{29}{2}"
} | {
"split": "train",
"index": 18586,
"solution": "1. **Convert the logarithmic equations to exponential form:**\n Given $\\log_{10}(x+y) = z$, we can rewrite this as:\n \\[\n x+y = 10^z\n \\]\n Similarly, from $\\log_{10}(x^2+y^2) = z+1$, we have:\n \\[\n x^2+y^2 = 10^{z+1}\n \\]\n\n2. **Express $(... |
deepscaler | [
{
"role": "user",
"content": "A piece of string fits exactly once around the perimeter of a rectangle with a length of 16 and a width of 10. Rounded to the nearest whole number, what is the area of the largest circle that can be formed from the piece of string? Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "215"
} | {
"split": "train",
"index": 18587,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express $4.\\overline{054}$ as a common fraction in lowest terms. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{150}{37}"
} | {
"split": "train",
"index": 18588,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On circle $O$, points $C$ and $D$ are on the same side of diameter $\\overline{AB}$, $\\angle AOC = 30^\\circ$, and $\\angle DOB = 45^\\circ$. What is the ratio of the area of the smaller sector $COD$ to the area of the circle? Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{24}"
} | {
"split": "train",
"index": 18589,
"solution": "1. **Identify the angles at the center involving points $C$ and $D$:**\n - $\\angle AOC = 30^\\circ$\n - $\\angle DOB = 45^\\circ$\n\n2. **Calculate $\\angle AOB$:**\n Since $AB$ is a diameter, $\\angle AOB$ spans a semicircle, hence $\\angle AOB = 180^\\circ... |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be nonnegative real numbers such that\n\\[\\sin (ax + b) = \\sin 29x\\]for all integers $x.$ Find the smallest possible value of $a.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10 \\pi - 29"
} | {
"split": "train",
"index": 18590,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Trapezoid $ABCD$ has $\\overline{AB}\\parallel\\overline{CD}, BC=CD=43$, and $\\overline{AD}\\perp\\overline{BD}$. Let $O$ be the intersection of the diagonals $\\overline{AC}$ and $\\overline{BD}$, and let $P$ be the midpoint of $\\overline{BD}$. Given that $OP=11$, the length... | math | {
"style": "rule",
"ground_truth": "194"
} | {
"split": "train",
"index": 18591,
"solution": "1. **Identify the properties of the trapezoid**: Given that $ABCD$ is a trapezoid with $\\overline{AB}\\parallel\\overline{CD}$ and $BC=CD=43$. Also, $\\overline{AD}\\perp\\overline{BD}$, which implies that $\\triangle ABD$ is a right triangle.\n\n2. **Intersection... |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the terminal side of angle $\\theta$ with $Ox$ as the initial side passes through the point $\\left( \\frac{3}{5}, \\frac{4}{5} \\right)$, then $\\sin \\theta=$ ______, $\\tan 2\\theta=$ ______. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "-\\frac{24}{7}"
} | {
"split": "train",
"index": 18592,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\\ldots$, $a_n$ satisfying: \\[a_{k+1}=\\frac{a_k^2+1}{a_{k-1}+1}-1\\] for every $k$ with $2\\leq k\\leq n-1$.\n\n[i] Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "n=1,2,3,4"
} | {
"split": "train",
"index": 18593,
"solution": "\nTo solve the given problem, we need to determine all positive integers \\( n \\) such that there exists a sequence of positive integers \\( a_1, a_2, \\ldots, a_n \\) satisfying the recurrence relation:\n\n\\[\na_{k+1} = \\frac{a_k^2 + 1}{a_{k-1} + 1} - 1\n\\]\n\... |
deepscaler | [
{
"role": "user",
"content": "Calculate $6!-5\\cdot5!-5!$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 18594,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a$ and $b$ are rational numbers, a new operation is defined as follows: $a$βΌ$b=a^{3}-2ab+4$. For example, $2$βΌ$5=2^{3}-2\\times 2\\times 5+4=-8$. Find $4$βΌ$\\left(-9\\right)=\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "140"
} | {
"split": "train",
"index": 18595,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all odd integers between $400$ and $600$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50000"
} | {
"split": "train",
"index": 18596,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six distinct integers are picked at random from $\\{1,2,3,\\ldots,10\\}$. What is the probability that, among those selected, the second smallest is $3$?\n$\\textbf{(A)}\\ \\frac{1}{60}\\qquad \\textbf{(B)}\\ \\frac{1}{6}\\qquad \\textbf{(C)}\\ \\frac{1}{3}\\qquad \\textbf{(D)}... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 18597,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 99 children standing in a circle, each initially holding a ball. Every minute, each child with a ball throws their ball to one of their two neighbors. If a child receives two balls, one of the balls is irrevocably lost. What is the minimum amount of time after which o... | math | {
"style": "rule",
"ground_truth": "98"
} | {
"split": "train",
"index": 18598,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given line segments $OA$, $OB$, $OC$ are pairwise perpendicular, with $OA=1$, $OB=1$, $OC=2$. If the projections of line segments $OA$, $OB$, $OC$ on line $OP$ have equal lengths, then the length of these projections is $\\_\\_\\_\\_\\_\\_.$ Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 18599,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system \\( xOy \\), the equation of ellipse \\( C \\) is \\( \\frac{x^2}{9} + \\frac{y^2}{10} = 1 \\). Let \\( F \\) and \\( A \\) be the upper focus and the right vertex of ellipse \\( C \\), respectively. If \\( P \\) is a point on ellipse \\( C ... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{11}}{2}"
} | {
"split": "train",
"index": 18600,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $XY$ in the triangle below.\n\n\n[asy]\nunitsize(1inch);\npair P,Q,R;\nP = (0,0);\nQ= (1,0);\nR = (0,1);\ndraw (P--Q--R--P,linewidth(0.9));\ndraw(rightanglemark(Q,P,R,3));\nlabel(\"$X$\",P,S);\nlabel(\"$Y$\",Q,S);\nlabel(\"$Z$\",R,N);\nlabel(\"$12\\sqrt{2}$\",(Q+R)/2,NE);\... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 18601,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has side lengths $AB=120,BC=220$, and $AC=180$. Lines $\\ell_A,\\ell_B$, and $\\ell_C$ are drawn parallel to $\\overline{BC},\\overline{AC}$, and $\\overline{AB}$, respectively, such that the intersections of $\\ell_A,\\ell_B$, and $\\ell_C$ with the interior of ... | math | {
"style": "rule",
"ground_truth": "715"
} | {
"split": "train",
"index": 18602,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S_n$ be the sum of the first $n$ terms of the sequence $\\{a_n\\}$. If $2a_n + (-1)^n \\cdot a_n = 2^n + (-1)^n \\cdot 2^n$ ($n \\in \\mathbb{N}^*$), then $S_{10}=$ \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{2728}{3}"
} | {
"split": "train",
"index": 18603,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many nonnegative integers can be written in the form\\[a_7\\cdot3^7+a_6\\cdot3^6+a_5\\cdot3^5+a_4\\cdot3^4+a_3\\cdot3^3+a_2\\cdot3^2+a_1\\cdot3^1+a_0\\cdot3^0,\\]where $a_i\\in \\{-1,0,1\\}$ for $0\\le i \\le 7$?\n$\\textbf{(A) } 512 \\qquad \\textbf{(B) } 729 \\qquad \\t... | math | {
"style": "rule",
"ground_truth": "3281"
} | {
"split": "train",
"index": 18604,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $x$ and $y$ given:\n1. $\\frac{4x - 2}{5x - 5} = \\frac{3}{4}$\n2. $x + y = 3$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 18605,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $F$ is the focus of the parabola $C_{1}$: $y^{2}=2Οx (Ο > 0)$, and point $A$ is a common point of one of the asymptotes of the hyperbola $C_{2}$: $\\frac{{{x}^{2}}}{{{a}^{2}}}-\\frac{{{y}^{2}}}{{{b}^{2}}}=1 (a > 0, b > 0)$ and $AF \\perp x$-axis, find the eccentricit... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 18606,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers less than $1000$ are either a perfect cube or a perfect square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 18607,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the real numbers \\( x \\) and \\( y \\) satisfy the equations:\n\\[ 2^x + 4x + 12 = \\log_2{(y-1)^3} + 3y + 12 = 0 \\]\nfind 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"
} | {
"split": "train",
"index": 18608,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four equal circles with diameter $6$ are arranged such that three circles are tangent to one side of a rectangle and the fourth circle is tangent to the opposite side. All circles are tangent to at least one other circle with their centers forming a straight line that is parall... | math | {
"style": "rule",
"ground_truth": "648"
} | {
"split": "train",
"index": 18609,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many three-digit whole numbers have at least one 5 or consecutively have the digit 1 followed by the digit 2? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "270"
} | {
"split": "train",
"index": 18610,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Form five-digit numbers without repeating digits using the numbers \\\\(0\\\\), \\\\(1\\\\), \\\\(2\\\\), \\\\(3\\\\), and \\\\(4\\\\).\n\n\\\\((\\\\)I\\\\()\\\\) How many of these five-digit numbers are even?\n\n\\\\((\\\\)II\\\\()\\\\) How many of these five-digit numbers are... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 18611,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In this diagram, both polygons are regular. What is the value, in degrees, of the sum of the measures of angles $ABC$ and $ABD$?\n\n[asy]\ndraw(10dir(0)--10dir(60)--10dir(120)--10dir(180)--10dir(240)--10dir(300)--10dir(360)--cycle,linewidth(2));\ndraw(10dir(240)--10dir(300)--10... | math | {
"style": "rule",
"ground_truth": "210"
} | {
"split": "train",
"index": 18612,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $x$, $y$, and $r$ with $|x|>|y|$,\n\\[(x+y)^r=x^r+rx^{r-1}y^1+\\frac{r(r-1)}2x^{r-2}y^2+\\frac{r(r-1)(r-2)}{3!}x^{r-3}y^3+\\cdots\\]What are the first three digits to the right of... | math | {
"style": "rule",
"ground_truth": "428"
} | {
"split": "train",
"index": 18613,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the sum of $1001101_2$ and $111000_2$, and then add the decimal equivalent of $1010_2$. Write your final answer in base $10$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "143"
} | {
"split": "train",
"index": 18614,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the terminal side of angle $\\alpha$ passes through the point $(-3, 4)$, then $\\cos\\alpha=$ _______; $\\cos2\\alpha=$ _______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{7}{25}"
} | {
"split": "train",
"index": 18615,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The least positive integer with exactly $2021$ distinct positive divisors can be written in the form $m \\cdot 6^k$, where $m$ and $k$ are integers and $6$ is not a divisor of $m$. What is $m+k?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "58"
} | {
"split": "train",
"index": 18616,
"solution": "To find the least positive integer with exactly $2021$ distinct positive divisors, we start by understanding the divisor function. If a number $n$ has a prime factorization of the form $n = p_1^{a_1} p_2^{a_2} \\cdots p_k^{a_k}$, then the number of divisors of $n$ ... |
deepscaler | [
{
"role": "user",
"content": "How many zeros are there at the end of $\\frac{2018!}{30!\\times 11!}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "493"
} | {
"split": "train",
"index": 18617,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among all pairs of real numbers $(x, y)$ such that $\\cos \\sin x = \\cos \\sin y$ with $-\\frac{15\\pi}{2} \\le x, y \\le \\frac{15\\pi}{2}$, Ana randomly selects a pair $(X, Y)$. Compute the probability that $X = Y$. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 18618,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate using your preferred method: \n(1) $42.67-(12.95-7.33)$ \n(2) $\\left[8.4-8.4\\times(3.12-3.7)\\right]\\div0.42$ \n(3) $5.13\\times0.23+8.7\\times0.513-5.13$ \n(4) $6.66\\times222+3.33\\times556$ Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "3330"
} | {
"split": "train",
"index": 18619,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A random binary string of length 1000 is chosen. Let \\(L\\) be the expected length of its longest (contiguous) palindromic substring. Estimate \\(L\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23.120"
} | {
"split": "train",
"index": 18620,
"solution": "The probability that there exists a palindromic substring of length \\(2n+1\\) is approximately \\(2^{-n} \\cdot 1000\\). Thus, we can expect to often see a length 21 palindrome, and sometimes longer ones. This leads to a guess a bit above 21. \\(L\\) was approxima... |
deepscaler | [
{
"role": "user",
"content": "A herd of 183 elephants could drink the lake in 1 day, and a herd of 37 elephants could do it in 5 days.\n\nIn how many days will one elephant drink the lake? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "365"
} | {
"split": "train",
"index": 18621,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $b$ is a multiple of $570$, find the greatest common divisor of $4b^3 + 2b^2 + 5b + 171$ and $b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "171"
} | {
"split": "train",
"index": 18622,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the right triangular prism $ABC - A_1B_1C_1$, $\\angle ACB = 90^\\circ$, $AC = 2BC$, and $A_1B \\perp B_1C$. Find the sine of the angle between $B_1C$ and the lateral face $A_1ABB_1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{5}"
} | {
"split": "train",
"index": 18623,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $d$ be a real number such that every non-degenerate quadrilateral has at least two interior angles with measure less than $d$ degrees. What is the minimum possible value for $d$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 18624,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle with dimensions 100 cm by 150 cm is tilted so that one corner is 20 cm above a horizontal line, as shown. To the nearest centimetre, the height of vertex $Z$ above the horizontal line is $(100+x) \\mathrm{cm}$. What is the value of $x$? Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 18625,
"solution": "We enclose the given rectangle in a larger rectangle with horizontal and vertical sides so that the vertices of the smaller rectangle lie on the sides of the larger rectangle. We will also remove the units from the problem and deal with dimensionless quantities. Si... |
deepscaler | [
{
"role": "user",
"content": "The quartic (4th-degree) polynomial P(x) satisο¬es $P(1)=0$ and attains its maximum value of $3$ at both $x=2$ and $x=3$ . Compute $P(5)$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-24"
} | {
"split": "train",
"index": 18626,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of all non-zero real numbers. Define a function $f : S \\to S$ such that for all $x, y \\in S$ with $x + y \\neq 0$, the following holds:\n\\[f(x) + f(y) = f\\left(\\frac{xy}{x+y}f(x+y)\\right).\\]\nDetermine the number of possible values of $f(3)$, denoted a... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 18627,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the slope of the line that is tangent to a circle at point (5,5) if the center of the circle is (3,2)? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{3}"
} | {
"split": "train",
"index": 18628,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among all natural numbers not greater than 200, how many numbers are coprime to both 2 and 3 and are not prime numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 18629,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\left\\{a_{n}\\right\\}$ that satisfies $a_{1}=1$ and $S_{n+1}=2 S_{n}-\\frac{n(n+1)}{2}+1$, where $S_{n}=a_{1}+a_{2}+\\cdots+a_{n}$ $(n=1,2, \\cdots)$. If $\\Delta a_{n}=a_{n+1}-a_{n}$, find the number of elements in the set $S=\\left\\{n \\in \\mathbf{N}^... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 18630,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let point $P$ be the intersection point in the first quadrant of the hyperbola $\\frac{x^{2}}{a^{2}}- \\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ and the circle $x^{2}+y^{2}=a^{2}+b^{2}$. Let $F\\_1$ and $F\\_2$ be the left and right foci of the hyperbola, respectively, and $|PF\\_1... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 18631,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a circular cone with vertex $V$, and let $ABC$ be a triangle inscribed in the base of the cone, such that $AB$ is a diameter and $AC=BC$. Let $L$ be a point on $BV$ such that the volume of the cone is 4 times the volume of the tetrahedron $ABCL$. Find the value of $BL/... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4-\\pi}"
} | {
"split": "train",
"index": 18632,
"solution": "Let $R$ be the radius of the base, $H$ the height of the cone, $h$ the height of the pyramid and let $BL/LV=x/y$. Let [.] denote volume. Then [cone] $=\\frac{1}{3} \\pi R^{2} H$ and $[ABCL]=\\frac{1}{3} \\pi R^{2} h$ and $h=\\frac{x}{x+y} H$. We are given that [con... |
deepscaler | [
{
"role": "user",
"content": "Given a rhombus with diagonals of length $12$ and $30$, find the radius of the circle inscribed in this rhombus. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{90\\sqrt{261}}{261}"
} | {
"split": "train",
"index": 18633,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the ratio of the length, width, and height of a rectangular prism is $4: 3: 2$, and that a plane cuts through the prism to form a hexagonal cross-section (as shown in the diagram), with the minimum perimeter of such hexagons being 36, find the surface area of the rec... | math | {
"style": "rule",
"ground_truth": "208"
} | {
"split": "train",
"index": 18634,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\\\(f(x)=a(x-5)^{2}+6\\ln x\\\\), where \\\\(a\\in\\mathbb{R}\\\\), the tangent line of the curve \\\\(y=f(x)\\\\) at point \\\\((1,f(1))\\\\) intersects the \\\\(y\\\\)-axis at point \\\\((0,6)\\\\). \n\\\\((1)\\\\) Determine the value of \\\\(a\\\\); \n\\\\((2)\\\\) Fi... | math | {
"style": "rule",
"ground_truth": "2+6\\ln 3"
} | {
"split": "train",
"index": 18635,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the function $y=a\\sqrt{1-x^2} + \\sqrt{1+x} + \\sqrt{1-x}$ ($a\\in\\mathbb{R}$), and let $t= \\sqrt{1+x} + \\sqrt{1-x}$ ($\\sqrt{2} \\leq t \\leq 2$).\n\n(1) Express $y$ as a function of $t$, denoted as $m(t)$.\n\n(2) Let the maximum value of the function $m(t)$ be $g... | math | {
"style": "rule",
"ground_truth": "a = 1"
} | {
"split": "train",
"index": 18636,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the expression $2^{3}+2^{2}+2^{1}$ equal to? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 18637,
"solution": "Since $2^{1}=2$ and $2^{2}=2 \times 2=4$ and $2^{3}=2 \times 2 \times 2=8$, then $2^{3}+2^{2}+2^{1}=8+4+2=14$."
} |
deepscaler | [
{
"role": "user",
"content": "In triangle ABC below, find the length of side AB.\n\n[asy]\nunitsize(1inch);\npair A,B,C;\nA = (0,0);\nB = (1,0);\nC = (0,1);\ndraw (A--B--C--A,linewidth(0.9));\ndraw(rightanglemark(B,A,C,3));\nlabel(\"$A$\",A,S);\nlabel(\"$B$\",B,S);\nlabel(\"$C$\",C,N);\nlabel(\"$18\\sqrt{2}... | math | {
"style": "rule",
"ground_truth": "18\\sqrt{2}"
} | {
"split": "train",
"index": 18638,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a recent basketball game, Shenille attempted only three-point shots and two-point shots. She was successful on $20\\%$ of her three-point shots and $30\\%$ of her two-point shots. Shenille attempted $30$ shots. How many points did she score? Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 18639,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest value of $c$ such that $1$ is in the range of $f(x)=x^2-5x+c$. 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": 18640,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum possible volume of a cylinder inscribed in a cone with a height of 27 and a base radius of 9. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "324\\pi"
} | {
"split": "train",
"index": 18641,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( S = \\{1, 2, 3, \\ldots, 30\\} \\). Determine the number of vectors \\((x, y, z, w)\\) with \\(x, y, z, w \\in S\\) such that \\(x < w\\) and \\(y < z < w\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90335"
} | {
"split": "train",
"index": 18642,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, if points $ A$ , $ B$ and $ C$ are points of tangency, then $ x$ equals:\n[asy]unitsize(5cm);\ndefaultpen(linewidth(.8pt)+fontsize(8pt));\ndotfactor=3;\n\npair A=(-3*sqrt(3)/32,9/32), B=(3*sqrt(3)/32, 9/32), C=(0,9/16);\npair O=(0,3/8);\n\ndraw((-2/3,9/16... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"split": "train",
"index": 18643,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "During the 2011 Universiade in Shenzhen, a 12-person tour group initially stood in two rows with 4 people in the front row and 8 people in the back row. The photographer plans to keep the order of the front row unchanged, and move 2 people from the back row to the front row, en... | math | {
"style": "rule",
"ground_truth": "560"
} | {
"split": "train",
"index": 18644,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of the region in the first quadrant \\(x>0, y>0\\) bounded above by the graph of \\(y=\\arcsin(x)\\) and below by the graph of \\(y=\\arccos(x)\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 - \\sqrt{2}"
} | {
"split": "train",
"index": 18645,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$ x$ and $ y$ are two distinct positive integers. What is the minimum positive integer value of $ (x + y^2)(x^2 - y)/(xy)$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 18646,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The year 2013 has arrived, and Xiao Ming's older brother sighed and said, \"This is the first year in my life that has no repeated digits.\" It is known that Xiao Ming's older brother was born in a year that is a multiple of 19. How old is the older brother in 2013? Let's think... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 18647,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$ , let $P$ and $R$ be the feet of the perpendiculars from $A$ onto the external and internal bisectors of $\\angle ABC$ , respectively; and let $Q$ and $S$ be the feet of the perpendiculars from $A$ onto the internal and external bisectors of $\\... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 18648,
"solution": ""
} |
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