data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "How many three-digit whole numbers contain at least one digit 6 or at least one digit 8? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "452"
} | {
"split": "train",
"index": 5100,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "20 different villages are located along the coast of a circular island. Each of these villages has 20 fighters, with all 400 fighters having different strengths.\n\nTwo neighboring villages $A$ and $B$ now have a competition in which each of the 20 fighters from village $A$ com... | math | {
"style": "rule",
"ground_truth": "290"
} | {
"split": "train",
"index": 5101,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nA circle passing through the vertex \\( P \\) of triangle \\( PQR \\) touches side \\( QR \\) at point \\( F \\) and intersects sides \\( PQ \\) and \\( PR \\) at points \\( M \\) and \\( N \\), respectively, different from vertex \\( P \\). Find the ratio \\( QF : FR \\) if ... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 5102,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the absolute value of the difference of single-digit integers $A$ and $B$ such that $$ \\begin{array}{c@{}c@{\\;}c@{}c@{}c@{}c} & & & B& B & A_6\\\\ & & & \\mathbf{4} & \\mathbf{1} & B_6\\\\& & + & A & \\mathbf{1} & \\mathbf{5_6}\\\\ \\cline{2-6} & & A & \\mathbf{1} & \\m... | math | {
"style": "rule",
"ground_truth": "1_6"
} | {
"split": "train",
"index": 5103,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $g$ be a function defined on the positive integers, such that\n\\[g(xy) = g(x) + g(y)\\] for all positive integers $x$ and $y.$ Given $g(12) = 18$ and $g(48) = 26,$ find $g(600).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 5104,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given any point $P$ on the ellipse $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1\\; \\; (a > b > 0)$ with foci $F\\_{1}$ and $F\\_{2}$, if $\\angle PF\\_1F\\_2=\\alpha$, $\\angle PF\\_2F\\_1=\\beta$, $\\cos \\alpha= \\frac{ \\sqrt{5}}{5}$, and $\\sin (\\alpha+\\beta)= \\frac{3}... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{7}"
} | {
"split": "train",
"index": 5105,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers between 10000 and 100000 include the block of digits 178? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "280"
} | {
"split": "train",
"index": 5106,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A quadratic polynomial with real coefficients and leading coefficient $1$ is called $\\emph{disrespectful}$ if the equation $p(p(x))=0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\\tilde{p}(x... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{16}"
} | {
"split": "train",
"index": 5107,
"solution": "1. **Form of the Polynomial:**\n The disrespectful function $p(x)$ has leading coefficient $1$, so it can be written in factored form as $p(x) = (x-r)(x-s)$.\n\n2. **Condition for $p(p(x)) = 0$:**\n Plugging our form in, we get:\n \\[\n p(p(x)) = ((x-r)(x-s)... |
deepscaler | [
{
"role": "user",
"content": "If $f(x)=x^{2}+bx+c$, and $f(1)=0$, $f(3)=0$, find\n(1) the value of $f(-1)$;\n(2) the maximum and minimum values of $f(x)$ on the interval $[2,4]$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 5108,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fruit wholesaler sells apples at a cost price of 40 yuan per box. The price department stipulates that the selling price per box cannot exceed 55 yuan. Market research has found that if each box is sold for 50 yuan, an average of 90 boxes are sold per day, and for every 1 yua... | math | {
"style": "rule",
"ground_truth": "1125"
} | {
"split": "train",
"index": 5109,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the difference between the sum of the first $2003$ even counting numbers and the sum of the first $2003$ odd counting numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2003"
} | {
"split": "train",
"index": 5110,
"solution": "1. **Identify the sequences**: \n - The first $2003$ odd counting numbers form the sequence $O = 1, 3, 5, \\ldots, 4005$.\n - The first $2003$ even counting numbers can be considered in two cases:\n - Including $0$: $E_1 = 0, 2, 4, \\ldots, 4004$\n - Exc... |
deepscaler | [
{
"role": "user",
"content": "Given triangle $ ABC$ . Point $ O$ is the center of the excircle touching the side $ BC$ . Point $ O_1$ is the reflection of $ O$ in $ BC$ . Determine angle $ A$ if $ O_1$ lies on the circumcircle of $ ABC$ . Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 5111,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The endpoints of a line segment AB, which has a fixed length of 3, move on the parabola $y^2=x$. If M is the midpoint of the line segment AB, then the minimum distance from M to the y-axis is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"split": "train",
"index": 5112,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among the five-digit numbers formed using the digits 0, 1, 2, 3, 4, how many have the first and last digits the same, and the three middle digits all different? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 5113,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider real numbers $A$ , $B$ , \\dots, $Z$ such that \\[\n EVIL = \\frac{5}{31}, \\;\n LOVE = \\frac{6}{29}, \\text{ and }\n IMO = \\frac{7}{3}.\n\\] If $OMO = \\tfrac mn$ for relatively prime positive integers $m$ and $n$ , find the value of $m+n$ .\n\n*Propos... | math | {
"style": "rule",
"ground_truth": "579"
} | {
"split": "train",
"index": 5114,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be an acute triangle with incenter $I$ and circumcenter $O$. Assume that $\\angle OIA=90^{\\circ}$. Given that $AI=97$ and $BC=144$, compute the area of $\\triangle ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14040"
} | {
"split": "train",
"index": 5115,
"solution": "We present five different solutions and outline a sixth and seventh one. In what follows, let $a=BC$, $b=CA$, $c=AB$ as usual, and denote by $r$ and $R$ the inradius and circumradius. Let $s=\\frac{1}{2}(a+b+c)$. In the first five solutions we will only prove that $... |
deepscaler | [
{
"role": "user",
"content": "A sequence of positive integers is given by $a_{1}=1$ and $a_{n}=\\operatorname{gcd}\\left(a_{n-1}, n\\right)+1$ for $n>1$. Calculate $a_{2002}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 5116,
"solution": "3. It is readily seen by induction that $a_{n} \\leq n$ for all $n$. On the other hand, $a_{1999}$ is one greater than a divisor of 1999. Since 1999 is prime, we have $a_{1999}=2$ or 2000; the latter is not possible since $2000>1999$, so we have $a_{1999}=2$. Now we... |
deepscaler | [
{
"role": "user",
"content": "The numbers $1,2, \\ldots, 10$ are randomly arranged in a circle. Let $p$ be the probability that for every positive integer $k<10$, there exists an integer $k^{\\prime}>k$ such that there is at most one number between $k$ and $k^{\\prime}$ in the circle. If $p$ can be expresse... | math | {
"style": "rule",
"ground_truth": "1390"
} | {
"split": "train",
"index": 5117,
"solution": "Let $n=10$ and call two numbers close if there is at most one number between them and an circular permutation focused if only $n$ is greater than all numbers close to it. Let $A_{n}$ be the number of focused circular permutations of $\\{1,2, \\ldots, n\\}$. If $n \\... |
deepscaler | [
{
"role": "user",
"content": "If the arithmetic mean of $a$ and $b$ is double their geometric mean, with $a>b>0$, then a possible value for the ratio $a/b$, to the nearest integer, is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 5118,
"solution": "1. **Given Condition**: The arithmetic mean of $a$ and $b$ is double their geometric mean. This can be expressed as:\n \\[\n \\frac{a+b}{2} = 2 \\sqrt{ab}\n \\]\n Squaring both sides to eliminate the square root, we get:\n \\[\n \\left(\\frac{a+b}{2}\\ri... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are respectively $a$, $b$, $c$. Given that $\\cos B = \\frac{1}{3}$, $ac = 6$, and $b = 3$.\n\n$(1)$ Find the value of $\\cos C$ for side $a$;\n\n$(2)$ Find the value of $\\cos (2C+\\frac{\\pi }{3})$. Let's think s... | math | {
"style": "rule",
"ground_truth": "\\frac{17-56 \\sqrt{6}}{162}"
} | {
"split": "train",
"index": 5119,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express $0.4\\overline5$ as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{41}{90}"
} | {
"split": "train",
"index": 5120,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pawn moves on a $6 \\times 6$ chessboard. It starts from the bottom-left square. At each step, it can either jump to the square directly to its right or the square directly above it. It must reach the top-right square, such that it never lies strictly above the diagonal conne... | math | {
"style": "rule",
"ground_truth": "132"
} | {
"split": "train",
"index": 5121,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among 100 young men, if at least one of the height or weight of person A is greater than that of person B, then A is considered not inferior to B. Determine the maximum possible number of top young men among these 100 young men. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 5122,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five six-sided dice are rolled. It is known that after the roll, there are two pairs of dice showing the same number, and one odd die. The odd die is rerolled. What is the probability that after rerolling the odd die, the five dice show a full house? Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 5123,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\overrightarrow{OA}$, $\\overrightarrow{OB}$, and $\\overrightarrow{OC}$ are all unit vectors, and satisfy $\\frac{1}{2}\\overrightarrow{OA}+\\overrightarrow{OB}+\\overrightarrow{OC}=\\overrightarrow{0}$, calculate the value of $\\overrightarrow{AB} \\cdot \\overri... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{8}"
} | {
"split": "train",
"index": 5124,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $0.42\\overline{153}$ is expressed as a fraction in the form $\\frac{x}{99900}$, 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": "42111"
} | {
"split": "train",
"index": 5125,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a party, Ted's age is 15 years less than twice Sally's age. The sum of their ages is 54. How old is Ted? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 5126,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse \\(\\frac{x^{2}}{4}+\\frac{y^{2}}{3}=1\\) with the left and right foci \\(F_{1}\\) and \\(F_{2}\\) respectively, a line \\(l\\) passes through the right focus and intersects the ellipse at points \\(P\\) and \\(Q\\). Find the maximum area of the inscribed circ... | math | {
"style": "rule",
"ground_truth": "\\frac{9\\pi}{16}"
} | {
"split": "train",
"index": 5127,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle \\(ABCD\\), \\(AB = 2\\) and \\(AD = 1\\). Point \\(P\\) is a moving point on side \\(DC\\) (including \\(D\\) and \\(C\\)), and point \\(Q\\) is a moving point on the extension of side \\(CB\\) (including point \\(B\\)), such that \\(|\\overrightarrow{DP}| = |\\ov... | math | {
"style": "rule",
"ground_truth": "3/4"
} | {
"split": "train",
"index": 5128,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate\n\\[\\begin{vmatrix} \\cos \\alpha \\cos \\beta & \\cos \\alpha \\sin \\beta & -\\sin \\alpha \\\\ -\\sin \\beta & \\cos \\beta & 0 \\\\ \\sin \\alpha \\cos \\beta & \\sin \\alpha \\sin \\beta & \\cos \\alpha \\end{vmatrix}.\\] Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 5129,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the positive integer $n$ such that\n\\[\\sin \\left( \\frac{\\pi}{2n} \\right) + \\cos \\left (\\frac{\\pi}{2n} \\right) = \\frac{\\sqrt{n}}{2}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 5130,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Chords $A B$ and $C D$ of a circle are perpendicular and intersect at a point $P$. If $A P=6, B P=12$, and $C D=22$, find the area of the circle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "130 \\pi"
} | {
"split": "train",
"index": 5131,
"solution": "Let $O$ be the center of the circle and let $M$ be the midpoint of segment $A B$ and let $N$ be the midpoint of segment $C D$. Since quadrilateral $O M P N$ is a rectangle we have that $O N=M P=A M-A P=3$ so $$O C=\\sqrt{O N^{2}+N C^{2}}=\\sqrt{9+121}=\\sqrt{130}$$ ... |
deepscaler | [
{
"role": "user",
"content": "Find all real numbers $k$ such that $r^{4}+k r^{3}+r^{2}+4 k r+16=0$ is true for exactly one real number $r$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\pm \\frac{9}{4}"
} | {
"split": "train",
"index": 5132,
"solution": "Any real quartic has an even number of real roots with multiplicity, so there exists real $r$ such that $x^{4}+k x^{3}+x^{2}+4 k x+16$ either takes the form $(x+r)^{4}$ (clearly impossible) or $(x+r)^{2}\\left(x^{2}+a x+b\\right)$ for some real $a, b$ with $a^{2}<4 ... |
deepscaler | [
{
"role": "user",
"content": "There are two equilateral triangles with a vertex at $(0, 1)$ , with another vertex on the line $y = x + 1$ and with the final vertex on the parabola $y = x^2 + 1$ . Find the area of the larger of the two triangles. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "26\\sqrt{3} + 45"
} | {
"split": "train",
"index": 5133,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $M=123456789101112\\dots5253$ be the $96$-digit number that is formed by writing integers from $1$ to $53$ in order. What is the remainder when $M$ is divided by $55$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "53"
} | {
"split": "train",
"index": 5134,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three distinct vertices are chosen at random from the vertices of a given regular polygon of $(2n+1)$ sides. If all such choices are equally likely, what is the probability that the center of the given polygon lies in the interior of the triangle determined by the three chosen ... | math | {
"style": "rule",
"ground_truth": "\\frac{n+1}{4n-2}"
} | {
"split": "train",
"index": 5135,
"solution": "There are $\\binom{2n+1}{3}$ ways how to pick the three vertices. We will now count the ways where the interior does NOT contain the center. These are obviously exactly the ways where all three picked vertices lie among some $n+1$ consecutive vertices of the polygon... |
deepscaler | [
{
"role": "user",
"content": "In the quadrilateral $MARE$ inscribed in a unit circle $\\omega,$ $AM$ is a diameter of $\\omega,$ and $E$ lies on the angle bisector of $\\angle RAM.$ Given that triangles $RAM$ and $REM$ have the same area, find the area of quadrilateral $MARE.$ Let's think ... | math | {
"style": "rule",
"ground_truth": "\\frac{8\\sqrt{2}}{9}"
} | {
"split": "train",
"index": 5136,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For positive integer $n_{}^{}$, define $S_n^{}$ to be the minimum value of the sum $\\sum_{k=1}^n \\sqrt{(2k-1)^2+a_k^2},$ where $a_1,a_2,\\ldots,a_n^{}$ are positive real numbers whose sum is 17. There is a unique positive integer $n^{}_{}$ for which $S_n^{}$ is also an intege... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 5137,
"solution": "Consider $n$ right triangles joined at their vertices, with bases $a_1,a_2,\\ldots,a_n$ and heights $1,3,\\ldots, 2n - 1$. The sum of their hypotenuses is the value of $S_n$. The minimum value of $S_n$, then, is the length of the straight line connecting the bottom ... |
deepscaler | [
{
"role": "user",
"content": "Mário completed 30 hours in an extra math course. On days when he had class, it lasted only 1 hour, and it took place exclusively in the morning or in the afternoon. Additionally, there were 20 afternoons and 18 mornings without classes during the course period.\n\na) On how ma... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 5138,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T$ be the set of all positive real numbers. Let $g : T \\to T$ be a function such that\n\\[ g(x) + g(y) = x \\cdot y \\cdot g(g(x) + g(y)) \\]\nfor all $x, y \\in T$ where $x + y \\neq 1$.\n\nLet $m$ be the number of possible values of $g(3),$ and let $t$ be the sum of all... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 5139,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "John draws a regular five pointed star in the sand, and at each of the 5 outward-pointing points and 5 inward-pointing points he places one of ten different sea shells. How many ways can he place the shells, if reflections and rotations of an arrangement are considered equivale... | math | {
"style": "rule",
"ground_truth": "362880"
} | {
"split": "train",
"index": 5140,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jia, Yi, Bing, Ding, and Wu sit around a circular table to play cards. Jia has a fixed seat. If Yi and Ding cannot sit next to each other, how many different seating arrangements are possible? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 5141,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The quotient of two positive integers is $\\frac{5}{2}$ and their product is 160. What is the value of the larger of the two integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 5142,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The matrix\n\\[\\begin{pmatrix} 3 & -1 \\\\ c & d \\end{pmatrix}\\]is its own inverse. Enter the ordered pair $(c,d).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(8,-3)"
} | {
"split": "train",
"index": 5143,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of $(3,1) \\nabla (4,2)$ using the operation ' $\\nabla$ ' defined by $(a, b) \\nabla (c, d)=ac+bd$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 5144,
"solution": "From the definition, $(3,1) \\nabla (4,2)=(3)(4)+(1)(2)=12+2=14$."
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $\\sum_{n=1}^{2001} n^{3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4012013006001"
} | {
"split": "train",
"index": 5145,
"solution": "$\\sum_{n=1}^{2001} n^{3}=\\left(\\sum_{n=1}^{2001} n\\right)^{2}=\\left(\\frac{2001 \\cdot 2002}{2}\\right)^{2}=4012013006001$."
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $D$ and $E$ are the midpoints of $\\overline{AB}$ and $\\overline{BC}$ respectively. Determine the sum of the $x$ and $y$ coordinates of $F$, the point of intersection of $\\overline{AE}$ and $\\overline{CD}$. [asy]\nsize(180); defaultpen(linewidth(.7pt)+fontsi... | math | {
"style": "rule",
"ground_truth": "\\frac{14}{3}"
} | {
"split": "train",
"index": 5146,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two numbers are independently selected from the set of positive integers less than or equal to 6. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 5147,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $\\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1$, given that $a^{2} + b^{2} - 3c^{2} = 0$, where $c$ is the semi-latus rectum, find the value of $\\frac{a + c}{a - c}$. 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": 5148,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), \\( AB = \\sqrt{2} \\), \\( AC = \\sqrt{3} \\), and \\( \\angle BAC = 30^\\circ \\). Let \\( P \\) be an arbitrary point in the plane containing \\( \\triangle ABC \\). Find the minimum value of \\( \\mu = \\overrightarrow{PA} \\cdot \\overri... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2} - \\frac{5}{3}"
} | {
"split": "train",
"index": 5149,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given fifty distinct natural numbers, twenty-five of which do not exceed 50, and the remaining are greater than 50 but do not exceed 100. Additionally, no two of these numbers differ by exactly 50. Find the sum of these numbers. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "2525"
} | {
"split": "train",
"index": 5150,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be twice as old as Bill will be then. What is the difference in their current ages? Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 5151,
"solution": "1. **Define the ages**: Let Jack's age be represented as $\\overline{ab} = 10a + b$ and Bill's age as $\\overline{ba} = 10b + a$, where $a$ and $b$ are the tens and units digits of their ages, respectively.\n\n2. **Age in five years**: In five years, Jack's age will... |
deepscaler | [
{
"role": "user",
"content": "Given that $|\\overrightarrow{a}|=5$, $|\\overrightarrow{b}|=3$, and $\\overrightarrow{a} \\cdot \\overrightarrow{b} = -12$, find the projection of vector $\\overrightarrow{a}$ on vector $\\overrightarrow{b}$. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 5152,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If point A $(3,1)$ lies on the line $mx+ny+1=0$, where $mn>0$, then the maximum value of $\\frac {3}{m}+ \\frac {1}{n}$ is \\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-16"
} | {
"split": "train",
"index": 5153,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An unfair coin lands on heads with probability $\\frac34$ and tails with probability $\\frac14$. A heads flip gains $\\$3$, but a tails flip loses $\\$8$. What is the expected worth of a coin flip? Express your answer as a decimal rounded to the nearest hundredth. Let's think... | math | {
"style": "rule",
"ground_truth": "\\$0.25"
} | {
"split": "train",
"index": 5154,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle $DEF$, where $DE=15$, $DF=9$, and $EF=12$ units. What is the distance from $F$ to the midpoint of segment $DE$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7.5"
} | {
"split": "train",
"index": 5155,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all functions $f: \\mathbb{R}^{+} \\rightarrow \\mathbb{R}^{+}$, such that $f(x+f(x)+f(y))=2f(x)+y$ for all positive reals $x,y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x"
} | {
"split": "train",
"index": 5156,
"solution": "\nTo find all functions \\( f: \\mathbb{R}^{+} \\rightarrow \\mathbb{R}^{+} \\) such that \n\n\\[\nf(x+f(x)+f(y)) = 2f(x) + y \n\\]\n\nfor all positive reals \\(x, y\\), we proceed with the following steps:\n\n**Step 1: Analyze the functional equation**\n\nLet's fix... |
deepscaler | [
{
"role": "user",
"content": "In the prime factorization of $24!$, what is the exponent of $3$? (Reminder: The number $n!$ is the product of the integers from 1 to $n$. For example, $5!=5\\cdot 4\\cdot3\\cdot2\\cdot 1= 120$.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 5157,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The terms $x, x + 2, x + 4, \\dots, x + 2n$ form an arithmetic sequence, with $x$ an integer. If each term of the sequence is cubed, the sum of the cubes is $-1197$. What is the value of $n$ if $n > 3$? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 5158,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A target consisting of five zones is hanging on the wall: a central circle (bullseye) and four colored rings. The width of each ring equals the radius of the bullseye. It is known that the number of points for hitting each zone is inversely proportional to the probability of hi... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 5159,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers $a_{1}, a_{2}, \\ldots, a_{100}$ are a permutation of the numbers $1,2, \\ldots, 100$. Let $S_{1}=a_{1}$, $S_{2}=a_{1}+a_{2}, \\ldots, S_{100}=a_{1}+a_{2}+\\ldots+a_{100}$. What maximum number of perfect squares can be among the numbers $S_{1}, S_{2}, \\ldots, S_{10... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 5160,
"solution": "We add initial term \\(S_{0}=0\\) to the sequence \\(S_{1}, S_{2}, \\ldots, S_{100}\\) and consider all the terms \\(S_{n_{0}}<S_{n_{1}}<\\ldots\\) that are perfect squares: \\(S_{n_{k}}=m_{k}^{2}\\) (in particular, \\(n_{0}=m_{0}=0\\)). Since \\(S_{100}=5050<72^{2}... |
deepscaler | [
{
"role": "user",
"content": "The two spinners shown are spun once and each lands on one of the numbered sectors. What is the probability that the sum of the numbers in the two sectors is prime? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{9}"
} | {
"split": "train",
"index": 5161,
"solution": "To solve this problem, we first need to understand the possible outcomes when spinning the two spinners. Each spinner lands on one of the numbered sectors, and we are interested in the sums of these numbers.\n\n#### Step 1: Identify the numbers on each spinner\n- Sp... |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer with exactly 12 positive integer divisors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 5162,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nTwo people are flipping a coin: one flipped it 10 times, and the other 11 times.\n\nWhat is the probability that the second person gets more heads than the first person? 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": 5163,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Selected Exercise $(4-5)$: Inequality Lecture\n\nGiven the function $f(x)=|2x-a|+|x-1|$, where $a\\in R$\n\n(1) Find the range of values for the real number $a$ if the inequality $f(x)\\leqslant 2-|x-1|$ has a solution;\n\n(2) When $a < 2$, the minimum value of the function $f(... | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 5164,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define the sequence $a_{1}, a_{2} \\ldots$ as follows: $a_{1}=1$ and for every $n \\geq 2$, $a_{n}= \\begin{cases}n-2 & \\text { if } a_{n-1}=0 \\\\ a_{n-1}-1 & \\text { if } a_{n-1} \\neq 0\\end{cases}$. A non-negative integer $d$ is said to be jet-lagged if there are non-nega... | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 5165,
"solution": "Let $N=n+r$, and $M=n$. Then $r=N-M$, and $s=a_{N}-a_{M}$, and $d=r+s=\\left(a_{N}+N\\right)-\\left(a_{M}+M\\right)$. So we are trying to find the number of possible values of $\\left(a_{N}+N\\right)-\\left(a_{M}+M\\right)$, subject to $N \\geq M$ and $a_{N} \\geq a... |
deepscaler | [
{
"role": "user",
"content": "Mady now has boxes each capable of holding up to 5 balls instead of 4. Under the same process as described, Mady adds balls and resets boxes. Determine the total number of balls in the boxes after her $2010$th step. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 5166,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest three-digit multiple of 8 whose digits' sum is 24? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "888"
} | {
"split": "train",
"index": 5167,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Carrie sends five text messages to her brother each Saturday and Sunday, and two messages on other days. Over four weeks, how many text messages does Carrie send? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 5168,
"solution": "Each week, Carrie sends 5 messages to her brother on each of 2 days, for a total of 10 messages. Each week, Carrie sends 2 messages to her brother on each of the remaining 5 days, for a total of 10 messages. Therefore, Carrie sends $10+10=20$ messages per week. In f... |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be real numbers such that $a + 4i$ and $b + 5i$ are the roots of\n\\[z^2 - (10 + 9i) z + (4 + 46i) = 0.\\]Enter the ordered pair $(a,b).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(6,4)"
} | {
"split": "train",
"index": 5169,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(a\\), \\(b\\), and \\(c\\) be positive real numbers. Find the minimum value of\n\\[\n\\frac{5c}{a+b} + \\frac{5a}{b+c} + \\frac{3b}{a+c} + 1.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7.25"
} | {
"split": "train",
"index": 5170,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For $-1<r<1$, let $T(r)$ denote the sum of the geometric series \\[20 + 10r + 10r^2 + 10r^3 + \\cdots.\\] Let $b$ between $-1$ and $1$ satisfy $T(b)T(-b)=5040$. Find $T(b)+T(-b)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "504"
} | {
"split": "train",
"index": 5171,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 23 socks in a drawer: 8 white and 15 black. Every minute, Marina goes to the drawer and pulls out a sock. If at any moment Marina has pulled out more black socks than white ones, she exclaims, \"Finally!\" and stops the process.\n\nWhat is the maximum number of socks ... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 5172,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=4\\sin x\\sin \\left(x+ \\frac {\\pi}{3}\\right)-1$. \n$(1)$ Calculate the value of $f\\left( \\frac {5\\pi}{6}\\right)$: \n$(2)$ Let $A$ be the smallest angle in $\\triangle ABC$, and $f(A)= \\frac {8}{5}$, find the value of $f\\left(A+ \\frac {\\pi}... | math | {
"style": "rule",
"ground_truth": "\\frac {6}{5}"
} | {
"split": "train",
"index": 5173,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using the digits 1, 2, and 3 to form four-digit numbers, where each digit must appear, and identical digits cannot be adjacent, how many such four-digit numbers are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 5174,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the value of $x$ such that\n\n$\\left(1+\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{8}\\cdots\\right)\\left(1-\\frac{1}{2}+\\frac{1}{4}-\\frac{1}{8}+\\cdots\\right)=1+\\frac{1}{x}+\\frac{1}{x^2}+\\frac{1}{x^3}+\\cdots$. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 5175,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among the numbers $1, 2, 3, \\cdots, 50$, if 10 consecutive numbers are selected, what is the probability that exactly 3 of them are prime numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22/41"
} | {
"split": "train",
"index": 5176,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Carina has three pins, labeled $A, B$ , and $C$ , respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance $1$ away. What is the least number of moves that Carina can make in order for triangle $ABC$ ... | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 5177,
"solution": "The answer is $128$ , achievable by $A=(10,0), B=(0,-63), C=(-54,1)$ . We now show the bound.\nWe first do the following optimizations:\n-if you have a point goes both left and right, we may obviously delete both of these moves and decrease the number of moves by $2... |
deepscaler | [
{
"role": "user",
"content": "A supermarket has $128$ crates of apples. Each crate contains at least $120$ apples and at most $144$ apples.\nWhat is the largest integer $n$ such that there must be at least $n$ crates containing the same number of apples? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 5178,
"solution": "1. **Identify the range of possible apple counts per crate**: Each crate contains between $120$ and $144$ apples, inclusive. Therefore, the number of different possible apple counts per crate is calculated as:\n \\[\n 144 - 120 + 1 = 25\n \\]\n This means th... |
deepscaler | [
{
"role": "user",
"content": "If\n\\[\n(1 + \\tan 1^\\circ)(1 + \\tan 2^\\circ)(1 + \\tan 3^\\circ) \\dotsm (1 + \\tan 89^\\circ) = 2^m,\n\\]\nthen find $m.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 5179,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x) = 3x^2-5$, what is the value of $f(f(1))$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 5180,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $a$, $b$, $c$, $d$, $e$ are 5 elements taken from the set $\\{1, 2, 3, 4, 5\\}$ without repetition, calculate the probability that $abc+de$ is an odd number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"split": "train",
"index": 5181,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle{ABC}, AB=10, \\angle{A}=30^\\circ$ , and $\\angle{C=45^\\circ}$. Let $H, D,$ and $M$ be points on the line $BC$ such that $AH\\perp{BC}$, $\\angle{BAD}=\\angle{CAD}$, and $BM=CM$. Point $N$ is the midpoint of the segment $HM$, and point $P$ is on ray $AD$ such th... | math | {
"style": "rule",
"ground_truth": "77"
} | {
"split": "train",
"index": 5182,
"solution": "Break our diagram into 2 special right triangle by dropping an altitude from $B$ to $AC$ we then get that \\[AC=5+5\\sqrt{3}, BC=5\\sqrt{2}.\\] Since $\\triangle{HCA}$ is a 45-45-90,\n\\[HC=\\frac{5\\sqrt2+5\\sqrt6}{2}\\] $MC=\\frac{BM}{2},$ \\[HM=\\frac{5\\sqrt6}{2... |
deepscaler | [
{
"role": "user",
"content": "Find all functions $f:\\mathbb{R}\\rightarrow\\mathbb{R}$ that satisfy \\[f(x^2-y)+2yf(x)=f(f(x))+f(y)\\] for all $x,y\\in\\mathbb{R}$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = -x^2, \\quad f(x) = 0, \\quad f(x) = x^2"
} | {
"split": "train",
"index": 5183,
"solution": "Plugging in $y$ as $0:$ \\begin{equation}\nf(x^2)=f(f(x))+f(0) \\text{ } (1)\n\\end{equation}\nPlugging in $x, y$ as $0:$ \\[f(0)=f(f(0))+f(0)\\] or \\[f(f(0))=0\\] Plugging in $x$ as $0:$ \\[f(-y)+2yf(0)=f(f(0))+f(y),\\] but since $f(f(0))=0,$ \\begin{equation}\n... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the length of the side opposite to angle $A$ is $2$, and the vectors $\\overrightarrow{m} = (2, 2\\cos^2\\frac{B+C}{2} - 1)$ and $\\overrightarrow{n} = (\\sin\\frac{A}{2}, -1)$.\n\n1. Find the value of angle $A$ when the dot product $\\overrightarrow{m} \\c... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 5184,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the squares of the lengths of the $\\textbf{medians}$ of a triangle whose side lengths are $10,$ $10,$ and $12$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "258"
} | {
"split": "train",
"index": 5185,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In \\(\\triangle ABC\\), \\(AC = AB = 25\\) and \\(BC = 40\\). From \\(D\\), perpendiculars are drawn to meet \\(AC\\) at \\(E\\) and \\(AB\\) at \\(F\\), calculate the value of \\(DE + DF\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 5186,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A storm in Sydney, Australia, caused $\\$$30 million in damage. That estimate was in Australian dollars. At that time, 1.5 Australian dollars were worth 1 American dollar. Determine the number of American dollars of damage the storm caused. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "20,\\!000,\\!000"
} | {
"split": "train",
"index": 5187,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $sin(\\alpha-\\frac{\\pi}{6})=\\frac{3}{5}$, calculate $cos(\\frac{2\\pi}{3}-\\alpha)$. 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": 5188,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Democratic Republic of Irun, 5 people are voting in an election among 5 candidates. If each person votes for a single candidate at random, what is the expected number of candidates that will be voted for? Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "\\frac{2101}{625}"
} | {
"split": "train",
"index": 5189,
"solution": "The probability that a chosen candidate will receive no votes at all is $\\left(\\frac{4}{5}\\right)^{5}$, or the probability that every person will vote for someone other than that one candidate. Then the probability that a chosen candidate will receive at least on... |
deepscaler | [
{
"role": "user",
"content": "Let $S=\\{1,2,3,...,12\\}$ . How many subsets of $S$ , excluding the empty set, have an even sum but not an even product?\n\n*Proposed by Gabriel Wu* Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 5190,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rhombus \\(ABCD\\), point \\(Q\\) divides side \\(BC\\) in the ratio \\(1:3\\) starting from vertex \\(B\\), and point \\(E\\) is the midpoint of side \\(AB\\). It is known that the median \\(CF\\) of triangle \\(CEQ\\) is equal to \\(2\\sqrt{2}\\), and \\(EQ = \\sqrt{2}... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{7}}{2}"
} | {
"split": "train",
"index": 5191,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An integer is called a \"good number\" if it has 8 positive divisors and the sum of these 8 positive divisors is 3240. For example, 2006 is a good number because the sum of its divisors 1, 2, 17, 34, 59, 118, 1003, and 2006 is 3240. Find the smallest good number. Let's think st... | math | {
"style": "rule",
"ground_truth": "1614"
} | {
"split": "train",
"index": 5192,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ryan has 3 red lava lamps and 3 blue lava lamps. He arranges them in a row on a shelf randomly, and then randomly turns 3 of them on. What is the probability that the leftmost lamp is blue and off, and the rightmost lamp is red and on? Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{9}{100}"
} | {
"split": "train",
"index": 5193,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $P$ and $Q$ are midpoints of two sides of the square. What fraction of the interior of the square is shaded? Express your answer as a common fraction.\n\n[asy]\nfilldraw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)--cycle,gray,linewidth(1));\nfilldraw((0,1)--(1,2)--(2,2)--(0,1)--cy... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{8}"
} | {
"split": "train",
"index": 5194,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sets of consecutive integers $\\{1\\}$,$\\{2, 3\\}$,$\\{4,5,6\\}$,$\\{7,8,9,10\\}$,$\\; \\cdots \\;$, where each set contains one more element than the preceding one, and where the first element of each set is one more than the last element of the preceding set. Let ... | math | {
"style": "rule",
"ground_truth": "4641"
} | {
"split": "train",
"index": 5195,
"solution": "To solve for $S_{21}$, the sum of the elements in the 21st set, we first need to determine the first and last elements of this set.\n\n1. **Determine the first element of the 21st set:**\n Each set starts where the previous one ended, plus one. The $n$-th set star... |
deepscaler | [
{
"role": "user",
"content": "The line $y = \\frac{1}{2} x + 4$ is paramterized by the form\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} -7 \\\\ s \\end{pmatrix} + t \\begin{pmatrix} l \\\\ -5 \\end{pmatrix}.\\]Enter the ordered pair $(s,l).$ Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{1}{2}, -10 \\right)"
} | {
"split": "train",
"index": 5196,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of the region enclosed by the graph of \\( |x-75| + |y| = \\left|\\frac{x}{3}\\right| \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "703.125"
} | {
"split": "train",
"index": 5197,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "330"
} | {
"split": "train",
"index": 5198,
"solution": "1. **Find the prime factorizations** of 180 and 594:\n - $180 = 2^2 \\cdot 3^2 \\cdot 5$\n - $594 = 2 \\cdot 3^3 \\cdot 11$\n\n2. **Calculate the greatest common factor (GCF)**:\n - The GCF is found by taking the lowest power of all prime factors common to bot... |
deepscaler | [
{
"role": "user",
"content": "Determine all positive integers $n$ with the property that the third root of $n$ is obtained by removing its last three decimal digits. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32768"
} | {
"split": "train",
"index": 5199,
"solution": ""
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.