data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "In a particular sequence, the first term is $a_1=1007$ and the second term is $a_2=1008$. Furthermore, the values of the remaining terms are chosen so that $a_n + a_{n+1} + a_{n+2} = 2n$ for all $n \\geq 1$. Determine $a_{500}$. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "1339"
} | {
"split": "train",
"index": 26249,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths $3$ and $4$ units. In the corner where those sides meet at a right angle, he leaves a small unplanted square $S$ so that from the air it looks like the right angle symbol. The... | math | {
"style": "rule",
"ground_truth": "\\frac{145}{147}"
} | {
"split": "train",
"index": 26250,
"solution": "1. **Identify the Geometry and Setup Variables**:\n Let the right triangle be $\\triangle ABC$ with the right angle at $A$, and sides $AB = 3$ and $AC = 4$. The hypotenuse $BC$ then, by the Pythagorean theorem, is $5$ units. Let the vertices of the square $S$ be ... |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has $AB = 15, BC = 16$, and $AC = 17$. The points $D, E$, and $F$ are the midpoints of $\\overline{AB}, \\overline{BC}$, and $\\overline{AC}$ respectively. Let $X \\neq E$ be the intersection of the circumcircles of $\\triangle BDE$ and $\\triangle CEF$. Determin... | math | {
"style": "rule",
"ground_truth": "\\frac{960 \\sqrt{39}}{14}"
} | {
"split": "train",
"index": 26251,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function \\( f(x) = ax^2 + 8x + 3 \\) where \\( a < 0 \\). For a given negative \\( a \\), there exists a largest positive real number \\( L(a) \\) such that the inequality \\( |f(x)| \\leq 5 \\) holds for the entire interval \\([0, L(a)]\\). Find the value of \\( a \... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5} + 1}{2}"
} | {
"split": "train",
"index": 26252,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Fibonacci sequence is defined as follows: $F_{0}=0, F_{1}=1$, and $F_{n}=F_{n-1}+F_{n-2}$ for all integers $n \\geq 2$. Find the smallest positive integer $m$ such that $F_{m} \\equiv 0(\\bmod 127)$ and $F_{m+1} \\equiv 1(\\bmod 127)$. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "256"
} | {
"split": "train",
"index": 26253,
"solution": "First, note that 5 is not a quadratic residue modulo 127. We are looking for the period of the Fibonacci numbers $\\bmod 127$. Let $p=127$. We work in $\\mathbb{F}_{p^{2}}$ for the remainder of this proof. Let $\\alpha$ and $\\beta$ be the roots of $x^{2}-x-1$. The... |
deepscaler | [
{
"role": "user",
"content": "Given that the number of rabbits in a farm increases such that the difference between the populations in year $n+2$ and year $n$ is directly proportional to the population in year $n+1$, and the populations in the years $2001$, $2002$, and $2004$ were $50$, $80$, and $170$, res... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 26254,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If you walk for $45$ minutes at a rate of $4 \\text{ mph}$ and then run for $30$ minutes at a rate of $10\\text{ mph}$, how many miles will you have gone at the end of one hour and $15$ minutes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 26255,
"solution": "1. **Convert walking time to hours**: \n Walking time is $45$ minutes. To convert minutes to hours, divide by $60$: \n \\[\n 45 \\text{ minutes} = \\frac{45}{60} \\text{ hours} = \\frac{3}{4} \\text{ hours}\n \\]\n\n2. **Calculate distance walked**: \n ... |
deepscaler | [
{
"role": "user",
"content": "A circle is constructed on the side $BC$ of triangle $ABC$ as its diameter, and it intersects segment $AB$ at point $D$. Find the ratio of the areas of triangles $ABC$ and $BCD$, given that $AC = 15$, $BC = 20$, and $\\angle ABC = \\angle ACD$. Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "25/16"
} | {
"split": "train",
"index": 26256,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer \\(n\\) such that \\(\\frac{n}{n+51}\\) is equal to a terminating decimal? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "74"
} | {
"split": "train",
"index": 26257,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Count the total number of possible scenarios in a table tennis match between two players, where the winner is the first one to win three games and they play until a winner is determined. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 26258,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For an arithmetic sequence $b_1, b_2, b_3, \\dots,$ let\n\\[S_n = b_1 + b_2 + b_3 + \\dots + b_n,\\]and let\n\\[T_n = S_1 + S_2 + S_3 + \\dots + S_n.\\]Given the value of $S_{2023},$ then you can uniquely determine the value of $T_n$ for some integer $n.$ What is this integer $... | math | {
"style": "rule",
"ground_truth": "3034"
} | {
"split": "train",
"index": 26259,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\sqrt{54}\\cdot\\sqrt{32}\\cdot \\sqrt{6}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72\\sqrt{2}"
} | {
"split": "train",
"index": 26260,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From point $A$, Leigh walked 40 yards south, 60 yards west, 10 yards north, and 20 yards east to point $B$. What is the length, in yards, of $\\overline{AB}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 26261,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cylinder with a volume of 9 is inscribed in a cone. The plane of the top base of this cylinder cuts off a frustum from the original cone, with a volume of 63. Find the volume of the original cone. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 26262,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( n = 7^{3} \\times 11^{2} \\times 13^{4} \\), find the number of integers that are divisors of \\( n \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 26263,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An urn contains nine balls, numbered from 1 to 9. José and Maria each simultaneously draw one ball from the urn. They form a two-digit number with the number on José's ball as the tens digit and the number on Maria's ball as the units digit. What is the probability that this nu... | math | {
"style": "rule",
"ground_truth": "4/9"
} | {
"split": "train",
"index": 26264,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a rearrangement of the numbers from 1 to $n$, each pair of consecutive elements $a$ and $b$ of the sequence can be either increasing (if $a<b$ ) or decreasing (if $b<a$ ). How many rearrangements of the numbers from 1 to $n$ have exactly two increasing pairs of consecutiv... | math | {
"style": "rule",
"ground_truth": "3^{n}-(n+1) \\cdot 2^{n}+n(n+1) / 2"
} | {
"split": "train",
"index": 26265,
"solution": "Notice that each such permutation consists of 3 disjoint subsets of $\\{1, \\ldots, n\\}$ whose union is $\\{1, \\ldots, n\\}$, each arranged in decreasing order. For instance, if $n=6$, in the permutation 415326 (which has the two increasing pairs 15 and 26), the ... |
deepscaler | [
{
"role": "user",
"content": "The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with $9$ trapezoids, let $x$ be the angle m... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 26266,
"solution": "1. **Understanding the Geometry of the Keystone Arch**:\n The keystone arch is composed of $9$ congruent isosceles trapezoids. These trapezoids are arranged such that their non-parallel sides (legs) meet at a common point when extended. The bottom sides of the en... |
deepscaler | [
{
"role": "user",
"content": "Given points A (-3, 5) and B (2, 15), find a point P on the line $l: 3x - 4y + 4 = 0$ such that $|PA| + |PB|$ is minimized. The minimum value is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5\\sqrt{13}"
} | {
"split": "train",
"index": 26267,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S-ABC$ be a triangular prism with circumscribed sphere centered at $O$. The midpoints of $SB$ and $AC$ are $N$ and $M$, respectively. The midpoint of line segment $MN$ is $P$, and it is given that $SA^{2} + SB^{2} + SC^{2} = AB^{2} + BC^{2} + AC^{2}$. If $SP = 3\\sqrt{7}$ ... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{21}"
} | {
"split": "train",
"index": 26268,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three boys and two girls are to stand in a row according to the following requirements. How many different arrangements are there? (Answer with numbers) \n(Ⅰ) The two girls stand next to each other; \n(Ⅱ) Girls cannot stand at the ends; \n(Ⅲ) Girls are arranged from left to ... | math | {
"style": "rule",
"ground_truth": "78"
} | {
"split": "train",
"index": 26269,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A food factory processes mushrooms, with a cost of 20 yuan per kilogram of mushrooms, and a processing fee of t yuan per kilogram (t is a constant, and $2 \\leq t \\leq 5$). Assume the factory price per kilogram of mushrooms is x yuan ($25 \\leq x \\leq 40$). According to marke... | math | {
"style": "rule",
"ground_truth": "100e^4"
} | {
"split": "train",
"index": 26270,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A palindrome between $10000$ and $100000$ is chosen at random. What is the probability that it is divisible by $11$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{10}"
} | {
"split": "train",
"index": 26271,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nThere are two docks, $A$ and $B$, on a river, where dock $A$ is upstream and dock $B$ is downstream. There are two boats, Boat 1 and Boat 2. The speed of Boat 1 in still water is twice the speed of Boat 2. Both boats start simultaneously from docks $A$ and $B$, respectively, ... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 26272,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the expression \\((x+y+z)^{2030}+(x-y-z)^{2030}\\), the parentheses were expanded, and like terms were collected. How many monomials of the form \\(x^{a} y^{b} z^{c}\\) have a nonzero coefficient? Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "1032256"
} | {
"split": "train",
"index": 26273,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three not necessarily distinct positive integers between 1 and 99, inclusive, are written in a row on a blackboard. Then, the numbers, without including any leading zeros, are concatenated to form a new integer $N$. For example, if the integers written, in order, are 25, 6, and... | math | {
"style": "rule",
"ground_truth": "825957"
} | {
"split": "train",
"index": 26274,
"solution": "We will divide this into cases based on the number of digits of $N$.\n- Case 1: 6 digits. Then each of the three numbers must have two digits, so we have 90 choices for each. So we have a total of $90^{3}=729000$ possibilities.\n- Case 2: 5 digits. Then, exactly on... |
deepscaler | [
{
"role": "user",
"content": "When two positive integers are multiplied, the result is 24. When these two integers are added, the result is 11. What is the result when the smaller integer is subtracted from the larger integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 26275,
"solution": "The positive divisor pairs of 24 are: 1 and 24, 2 and 12, 3 and 8, 4 and 6. Of these, the pair whose sum is 11 is 3 and 8. The difference between these two integers is $8 - 3 = 5$."
} |
deepscaler | [
{
"role": "user",
"content": "Determine the largest positive integer $n$ for which there exists a set $S$ with exactly $n$ numbers such that \n- each member in $S$ is a positive integer not exceeding $2002$ ,\n- if $a,b\\in S$ (not necessarily different), then $ab\\not\\in S$ .\n Let's think ste... | math | {
"style": "rule",
"ground_truth": "1958"
} | {
"split": "train",
"index": 26276,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Pasha wrote down an equation consisting of integers and arithmetic operation signs in his notebook. He then encoded each digit and operation sign in the expression on the left side of the equation with a letter, replacing identical digits or signs with identical letters, and di... | math | {
"style": "rule",
"ground_truth": "2065 + 5 - 47"
} | {
"split": "train",
"index": 26277,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Professor Ma has formulated n different but equivalent statements A_{1}, A_{2}, \\ldots, A_{n}. Every semester, he advises a student to prove an implication A_{i} \\Rightarrow A_{j}, i \\neq j. This is the dissertation topic of this student. Every semester, he has only one stud... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}(n+2)(n-1)"
} | {
"split": "train",
"index": 26278,
"solution": "We will first construct an answer with \\frac{1}{2}(n+2)(n-1) students. Then, we will show this is the best possible answer. Construction: First, (n-1) students sequentially prove A_{1} \\Rightarrow A_{i} for i=2, \\ldots, n. Then, (n-2) students sequentially prove... |
deepscaler | [
{
"role": "user",
"content": "What is the greatest common divisor of $39$ and $91$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 26279,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From three red balls numbered $1$, $2$, $3$ and two white balls numbered $2$, $3$, find the probability that two balls drawn at random have different numbers and colors. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{2}{5}"
} | {
"split": "train",
"index": 26280,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Evil League of Evil plans to set out from their headquarters at (5,1) to poison two pipes: one along the line \\( y = x \\) and the other along the line \\( x = 7 \\). They wish to determine the shortest distance they can travel to visit both pipes and then return to their ... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{5}"
} | {
"split": "train",
"index": 26281,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the eccentricities of a confocal ellipse and a hyperbola are \\( e_1 \\) and \\( e_2 \\), respectively, and the length of the minor axis of the ellipse is twice the length of the imaginary axis of the hyperbola, find the maximum value of \\( \\frac{1}{e_1} + \\frac{1... | math | {
"style": "rule",
"ground_truth": "5/2"
} | {
"split": "train",
"index": 26282,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $ \\#\\left\\{ (x,y)\\in\\mathbb{N}^2\\bigg| \\frac{1}{\\sqrt{x}} -\\frac{1}{\\sqrt{y}} =\\frac{1}{2016}\\right\\} , $ where $ \\# A $ is the cardinal of $ A . $ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "165"
} | {
"split": "train",
"index": 26283,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A man buys a house for $20,000 and wants to earn a $6\\%$ annual return on his investment. He pays $650 a year in taxes and sets aside $15\\%$ of each month's rent for repairs and upkeep. Determine the required monthly rent (in dollars) to meet his financial goals.\nA) $165.25$... | math | {
"style": "rule",
"ground_truth": "181.38"
} | {
"split": "train",
"index": 26284,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x = (2 + \\sqrt{3})^{1000},$ let $n = \\lfloor x \\rfloor,$ and let $f = x - n.$ Find\n\\[x(1 - f).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 26285,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To be eligible for an algebra class a student must have an average of at least $83\\%$ over all four quarters in his or her pre-algebra class. If Fisher had an $82\\%$, $77\\%$, and $75\\%$ in the first three quarters, what is the minimum score he must earn in the 4th quarter t... | math | {
"style": "rule",
"ground_truth": "98\\%"
} | {
"split": "train",
"index": 26286,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate \\(\\left(d^d - d(d-2)^d\\right)^d\\) when \\(d=4\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1358954496"
} | {
"split": "train",
"index": 26287,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the equations: \n(1) $3x^2 -32x -48=0$ \n(2) $4x^2 +x -3=0$ \n(3) $(3x+1)^2 -4=0$ \n(4) $9(x-2)^2 =4(x+1)^2.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"split": "train",
"index": 26288,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square $ABCD$ has center $O,\\ AB=900,\\ E$ and $F$ are on $AB$ with $AE<BF$ and $E$ between $A$ and $F, m\\angle EOF =45^\\circ,$ and $EF=400.$ Given that $BF=p+q\\sqrt{r},$ where $p,q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p... | math | {
"style": "rule",
"ground_truth": "307"
} | {
"split": "train",
"index": 26289,
"solution": "Draw AO, OB, and extend OB to D. Let $\\angle{FOB} = \\alpha.$ Then, after angle chasing, we find that \\[\\angle{AEB} = 90 + \\alpha\\]. Using this, we draw a line perpendicular to $AB$ at $E$ to meet $BD$ at $M$. Since $\\angle{MEO} = \\alpha$ and $\\angle{EMO} =... |
deepscaler | [
{
"role": "user",
"content": "Determine all real values of $A$ for which there exist distinct complex numbers $x_{1}, x_{2}$ such that the following three equations hold: $$ x_{1}(x_{1}+1) =A $$ x_{2}(x_{2}+1) =A $$ x_{1}^{4}+3 x_{1}^{3}+5 x_{1} =x_{2}^{4}+3 x_{2}^{3}+5 x_{2} $$ Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "A = -7"
} | {
"split": "train",
"index": 26290,
"solution": "Applying polynomial division, $$ x_{1}^{4}+3 x_{1}^{3}+5 x_{1} =\\left(x_{1}^{2}+x_{1}-A\\right)\\left(x_{1}^{2}+2 x_{1}+(A-2)\\right)+(A+7) x_{1}+A(A-2) =(A+7) x_{1}+A(A-2) .$$ Thus, in order for the last equation to hold, we need $(A+7) x_{1}=(A+7) x_{2}$, from w... |
deepscaler | [
{
"role": "user",
"content": "Given that point $C$ is the midpoint between points $A$ and $B$. At 7:00 AM, Car 1 departs from $A$ towards $B$, Car 2 departs from $B$ towards $A$, and Car 3 departs from $C$ towards $A$. When Car 1 and Car 3 meet, Car 2 has traveled exactly $\\frac{3}{8}$ of the total distanc... | math | {
"style": "rule",
"ground_truth": "336"
} | {
"split": "train",
"index": 26291,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest natural number in which all digits are different and each pair of adjacent digits differs by 6 or 7. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60718293"
} | {
"split": "train",
"index": 26292,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 26293,
"solution": "Let the large tetrahedron be $ABCD$, and the small tetrahedron be $WXYZ$, with $W$ on $ABC$, $X$ on $BCD$, $Y$ on $ACD$, and $Z$ on $ABD$. Clearly, the two regular tetrahedrons are similar, so if we can find the ratio of the sides, we can find the ratio of the volu... |
deepscaler | [
{
"role": "user",
"content": "Last year, Michael took 8 math tests and received 8 different scores, each an integer between 85 and 95, inclusive. After each test, he noticed that the average of his test scores was an integer. His score on the eighth test was 90. What was his score on the seventh test? Let's... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 26294,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of times and the positions in which it appears $\\frac12$ in the following sequence of fractions: $$ \\frac11, \\frac21, \\frac12 , \\frac31 , \\frac22 , \\frac13 , \\frac41,\\frac32,\\frac23,\\frac14,..., \\frac{1}{1992} $$ Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "664"
} | {
"split": "train",
"index": 26295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the polynomial $p(x)$ such that\n\\[p(p(x)) = xp(x) + x^2.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-x + 1"
} | {
"split": "train",
"index": 26296,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The chord \\( AB \\) divides the circle into two arcs, with the smaller arc being \\( 130^{\\circ} \\). The larger arc is divided by chord \\( AC \\) in the ratio \\( 31:15 \\) from point \\( A \\). Find the angle \\( BAC \\). Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "37.5"
} | {
"split": "train",
"index": 26297,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two unit squares $S_{1}$ and $S_{2}$ have horizontal and vertical sides. Let $x$ be the minimum distance between a point in $S_{1}$ and a point in $S_{2}$, and let $y$ be the maximum distance between a point in $S_{1}$ and a point in $S_{2}$. Given that $x=5$, the difference be... | math | {
"style": "rule",
"ground_truth": "472"
} | {
"split": "train",
"index": 26298,
"solution": "Consider what must happen in order for the minimum distance to be exactly 5 . Let one square, say $S_{1}$ have vertices of $(0,0),(0,1),(1,0)$, and $(1,1)$. Further, assume WLOG that the center of $S_{2}$ is above the line $y=\\frac{1}{2}$ and to the right of the l... |
deepscaler | [
{
"role": "user",
"content": "In trapezoid $ABCD$ with $\\overline{BC}\\parallel\\overline{AD}$, let $BC = 1000$ and $AD = 2008$. Let $\\angle A = 37^\\circ$, $\\angle D = 53^\\circ$, and $M$ and $N$ be the midpoints of $\\overline{BC}$ and $\\overline{AD}$, respectively. Find the length $MN$.\n Let's think... | math | {
"style": "rule",
"ground_truth": "504"
} | {
"split": "train",
"index": 26299,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two counterfeit coins of equal weight are mixed with $8$ identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the $10$ coins. A second pair ... | math | {
"style": "rule",
"ground_truth": "\\frac{15}{19}"
} | {
"split": "train",
"index": 26300,
"solution": "To solve this problem, we need to calculate the probability that all four selected coins are genuine, given that the combined weight of the first pair is equal to the combined weight of the second pair.\n\n#### Step 1: Define the events\nLet $A$ be the event that a... |
deepscaler | [
{
"role": "user",
"content": "A lucky integer is a positive integer which is divisible by the sum of its digits. What is the least positive multiple of 9 that is not a lucky integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 26301,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of ordered triples $(a,b,c)$ where $a$, $b$, and $c$ are positive integers, $a$ is a factor of $b$, $a$ is a factor of $c$, and $a+b+c=100$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 26302,
"solution": "Denote $x = \\frac{b}{a}$ and $y = \\frac{c}{a}$. The last condition reduces to $a(1 + x + y) = 100$. Therefore, $1 + x + y$ is equal to one of the 9 factors of $100 = 2^25^2$.\n Subtracting the one, we see that $x + y = \\{0,1,3,4,9,19,24,49,99\\}$. There are exac... |
deepscaler | [
{
"role": "user",
"content": "On a table, there are 20 cards numbered from 1 to 20. Each time, Xiao Ming picks out 2 cards such that the number on one card is 2 more than twice the number on the other card. What is the maximum number of cards Xiao Ming can pick? Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 26303,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The letter T is formed by placing two $2\\:\\text{inch}\\!\\times\\!4\\:\\text{inch}$ rectangles next to each other, as shown. What is the perimeter of the T, in inches? [asy]\ndraw((1,0)--(3,0)--(3,4)--(4,4)--(4,6)--(0,6)--(0,4)--(1,4)--cycle);\n[/asy] Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 26304,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a^{2}-4a+3=0$, find the value of $\\frac{9-3a}{2a-4} \\div (a+2-\\frac{5}{a-2})$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{8}"
} | {
"split": "train",
"index": 26305,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of diagonals in a polygon with 120 sides, and calculate the length of one diagonal of this polygon assuming it is regular and each side length is 5 cm. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 26306,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( \\left\\lfloor A \\right\\rfloor \\) denote the greatest integer less than or equal to \\( A \\). Given \\( A = 50 + 19 \\sqrt{7} \\), find the value of \\( A^2 - A \\left\\lfloor A \\right\\rfloor \\). Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 26307,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a circle $C$ with the equation $(x-a)^2 + (y-2)^2 = 4$ ($a > 0$) and a line $l$ with the equation $x - y + 3 = 0$, which intersects the circle at points $A$ and $B$:\n1. When the chord AB cut by line $l$ on circle $C$ has a length of $2\\sqrt{2}$, find the value of $a$.\n... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2} - 1"
} | {
"split": "train",
"index": 26308,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagonal of a square is 10 inches, and the diameter of a circle is also 10 inches. Additionally, an equilateral triangle is inscribed within the square. Find the difference in area between the circle and the combined area of the square and the equilateral triangle. Express ... | math | {
"style": "rule",
"ground_truth": "-14.8"
} | {
"split": "train",
"index": 26309,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "in a right-angled triangle $ABC$ with $\\angle C=90$ , $a,b,c$ are the corresponding sides.Circles $K.L$ have their centers on $a,b$ and are tangent to $b,c$ ; $a,c$ respectively,with radii $r,t$ .find the greatest real number $p$ such that the inequality $\\frac{1... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} + 1"
} | {
"split": "train",
"index": 26310,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Below is a portion of the graph of a function, $y=p(x)$:\n\n[asy]\nimport graph; size(6cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.5,xmax=3.5,ymin=-4.25,ymax=6.25;\n\npen cqcqcq=rgb(0.75,0.75,0.75);\n/*grid*/ pen gs=linewi... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 26311,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Right triangle $ABC$ has one leg of length 6 cm, one leg of length 8 cm and a right angle at $A$. A square has one side on the hypotenuse of triangle $ABC$ and a vertex on each of the two legs of triangle $ABC$. What is the length of one side of the square, in cm? Express your ... | math | {
"style": "rule",
"ground_truth": "\\frac{120}{37}"
} | {
"split": "train",
"index": 26312,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An \"$n$-pointed star\" is formed as follows: the sides of a convex polygon are numbered consecutively $1,2,\\cdots ,k,\\cdots,n,\\text{ }n\\ge 5$; for all $n$ values of $k$, sides $k$ and $k+2$ are non-parallel, sides $n+1$ and $n+2$ being respectively identical with sides $1$... | math | {
"style": "rule",
"ground_truth": "180(n-4)"
} | {
"split": "train",
"index": 26313,
"solution": "To solve this problem, we need to determine the sum of the interior angles at the vertices of the $n$-pointed star formed by extending every second side of an $n$-sided convex polygon.\n\n1. **Understanding the Star Formation**: \n - Each vertex of the star is fo... |
deepscaler | [
{
"role": "user",
"content": "A rectangular board of 8 columns has squares numbered beginning in the upper left corner and moving left to right so row one is numbered 1 through 8, row two is 9 through 16, and so on. A student shades square 1, then skips one square and shades square 3, skips two squares and ... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 26314,
"solution": "1. **Identify the Pattern of Shaded Squares**: The shaded squares follow the sequence of triangular numbers, which are given by the formula $T_n = \\frac{n(n+1)}{2}$ for positive integers $n$. This sequence starts with $T_1 = 1$, $T_2 = 3$, $T_3 = 6$, $T_4 = 10$, a... |
deepscaler | [
{
"role": "user",
"content": "Let $M$ be a point on the ellipse $\\frac{x^2}{25} + \\frac{y^2}{16} = 1$, with $F_1$ and $F_2$ as its foci. If $\\angle F_1MF_2 = \\frac{\\pi}{6}$, calculate the area of $\\triangle MF_1F_2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16(2 - \\sqrt{3})"
} | {
"split": "train",
"index": 26315,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m$ be the smallest positive, three-digit integer congruent to 5 (mod 11). Let $n$ be the smallest positive, four-digit integer congruent to 5 (mod 11). What is $n-m$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "902"
} | {
"split": "train",
"index": 26316,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The vertices of a square are the centers of four circles as shown below. Given each side of the square is 6cm and the radius of each circle is $2\\sqrt{3}$cm, find the area in square centimeters of the shaded region. [asy]\nfill( (-1,-1)-- (1,-1) -- (1,1) -- (-1,1)--cycle, gray... | math | {
"style": "rule",
"ground_truth": "36 - 12\\sqrt{3} - 4\\pi"
} | {
"split": "train",
"index": 26317,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ten distinct natural numbers are such that the product of any 5 of them is even, and the sum of all 10 numbers is odd. What is their smallest possible sum? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 26318,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = x^2-2x$. How many distinct real numbers $c$ satisfy $f(f(f(f(c)))) = 3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 26319,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 5 blue chips and 3 yellow chips in a bag. One chip is drawn from the bag. That chip is placed back into the bag. A second chip is then drawn. What is the probability that the two selected chips are of different colors? Express your answer as a common fraction. Let's t... | math | {
"style": "rule",
"ground_truth": "\\frac{15}{32}"
} | {
"split": "train",
"index": 26320,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $BC = 20 \\sqrt{3}$ and $\\angle C = 30^\\circ$. Let the perpendicular bisector of $BC$ intersect $BC$ and $AC$ at $D$ and $E$, respectively. Find the length of $DE$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 26321,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangle whose side lengths are all positive integers, and only one side length is 5, which is not the shortest side, the number of such triangles is . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 26322,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $AB$ be a segment of length $2$ . The locus of points $P$ such that the $P$ -median of triangle $ABP$ and its reflection over the $P$ -angle bisector of triangle $ABP$ are perpendicular determines some region $R$ . Find the area of $R$ . Let's think step by s... | math | {
"style": "rule",
"ground_truth": "2\\pi"
} | {
"split": "train",
"index": 26323,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the acute angle \\( x \\) that satisfies the equation \\( \\sin^3 x + \\cos^3 x = \\frac{\\sqrt{2}}{2} \\), find \\( x \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 26324,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate how many numbers from 1 to 30030 are not divisible by any of the numbers between 2 and 16. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5760"
} | {
"split": "train",
"index": 26325,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a joint maritime search and rescue exercise between China and foreign countries, the Chinese side participated with 4 ships and 3 airplanes; the foreign side with 5 ships and 2 airplanes. If 2 units are selected from each group (either 1 airplane or 1 ship counts as one unit... | math | {
"style": "rule",
"ground_truth": "180"
} | {
"split": "train",
"index": 26326,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two men at points $R$ and $S$, $76$ miles apart, set out at the same time to walk towards each other.\nThe man at $R$ walks uniformly at the rate of $4\\tfrac{1}{2}$ miles per hour; the man at $S$ walks at the constant\nrate of $3\\tfrac{1}{4}$ miles per hour for the first hour... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 26327,
"solution": "1. **Calculate the distance each person walks**: \n - The man starting from $R$ walks at a rate of $4.5$ miles per hour. Therefore, in $h$ hours, he walks a distance of $4.5h$ miles.\n - The man starting from $S$ walks at a rate that increases by $0.5$ miles pe... |
deepscaler | [
{
"role": "user",
"content": "What is the least five-digit positive integer which is congruent to 7 (mod 21)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10,003"
} | {
"split": "train",
"index": 26328,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(x_1, x_2, \\ldots, x_n\\) be real numbers in arithmetic sequence, which satisfy \\(|x_i| < 1\\) for \\(i = 1, 2, \\dots, n,\\) and \\[|x_1| + |x_2| + \\dots + |x_n| = 25 + |x_1 + x_2 + \\dots + x_n|.\\] What is the smallest possible value of \\(n\\)? Let's think step by ... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 26329,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest sum of two $3$-digit numbers that can be obtained by placing each of the six digits $4,5,6,7,8,9$ in one of the six boxes in this addition problem?\n[asy]\nunitsize(12);\ndraw((0,0)--(10,0));\ndraw((-1.5,1.5)--(-1.5,2.5));\ndraw((-1,2)--(-2,2));\ndraw((1,1)... | math | {
"style": "rule",
"ground_truth": "1047"
} | {
"split": "train",
"index": 26330,
"solution": "1. **Identify the problem**: We need to find the smallest sum of two 3-digit numbers formed by the digits 4, 5, 6, 7, 8, and 9, each used exactly once.\n\n2. **Understand the sum of two numbers**: If the two numbers are $\\overline{abc}$ and $\\overline{def}$, thei... |
deepscaler | [
{
"role": "user",
"content": "Evaluate $3x^y + 4y^x$ when $x=2$ and $y=3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 26331,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line $l$ passing through the focus of the parabola $y=4x^2$ intersects the parabola at points $A(x_1, y_1)$ and $B(x_2, y_2)$. If $y_1+y_2=2$, then the length of segment $AB$ equals \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{8}"
} | {
"split": "train",
"index": 26332,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many square units are in the area of the pentagon shown here with sides of length 15, 20, 27, 24 and 20 units?\n\n[asy]\npair a,b,c,d,e;\na=(0,0);\nb=(24,0);\nc=(24,27);\nd=(5.3,34);\ne=(0,20);\ndraw((0,0)--(24,0)--(24,27)--(5.3,34)--(0,20)--cycle);\ndraw((4.8,32.7)--(6.1... | math | {
"style": "rule",
"ground_truth": "714"
} | {
"split": "train",
"index": 26333,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A box contains 5 white balls and 6 black balls. I draw them out of the box, one at a time. What is the probability that all of my draws alternate in color starting with a black ball? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{462}"
} | {
"split": "train",
"index": 26334,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers from 1 to 2001 have a digit sum that is divisible by 5? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "399"
} | {
"split": "train",
"index": 26335,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The solutions of $x(3x-7)=-3$ may be expressed in the form $\\frac{m+\\sqrt{n}}{p}$ and $\\frac{m-\\sqrt{n}}{p}$, where $m$, $n$, and $p$ have a greatest common divisor of 1. Find $m+n+p$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 26336,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of the expression $\\sqrt{16\\sqrt{8\\sqrt{4}}}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 26337,
"solution": "To solve the expression $\\sqrt{16\\sqrt{8\\sqrt{4}}}$, we will simplify the expression step-by-step:\n\n1. **Simplify the innermost square root:**\n \\[\n \\sqrt{4} = 2\n \\]\n\n2. **Substitute and simplify the next square root:**\n \\[\n \\sqrt{8\\sqrt{... |
deepscaler | [
{
"role": "user",
"content": "Let $x$ and $y$ satisfy the constraints:\n\\[\n\\begin{cases}\nx + 2y - 1 \\geqslant 0, \\\\\nx - y \\geqslant 0, \\\\\n0 \\leqslant x \\leqslant k,\n\\end{cases}\n\\]\nIf the minimum value of $z = x + ky$ is $-2$, then the maximum value of $z$ is ______. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 26338,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $a^2$ varies inversely with $b^3$. If $a=7$ when $b=3$, find the value of $a^2$ when $b=6$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6.125"
} | {
"split": "train",
"index": 26339,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each positive integer $n$, the mean of the first $n$ terms of a sequence is $n$. What is the 2008th term of the sequence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4015"
} | {
"split": "train",
"index": 26340,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $θ$ is an angle in the second quadrant, if $\\tan \\left(θ+ \\frac {π}{4}\\right)= \\frac {1}{2}$, calculate the value of $\\sin θ-\\cos θ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {2 \\sqrt {10}}{5}"
} | {
"split": "train",
"index": 26341,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the remainder when 10002000400080016003200640128025605121024204840968192 is divided by 100020004000800160032. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40968192"
} | {
"split": "train",
"index": 26342,
"solution": "Let $X_{k}$ denote $2^{k}$ except with leading zeroes added to make it four digits long. Let $\\overline{a b c \\cdots}$ denote the number obtained upon concatenating $a, b, c, \\ldots$ We have $$2^{6} \\cdot \\overline{X_{0} X_{1} \\ldots X_{5}}=\\overline{X_{6} X... |
deepscaler | [
{
"role": "user",
"content": "What is $88 \\div (4 \\div 2)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 26343,
"solution": ""
} |
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": "288"
} | {
"split": "train",
"index": 26344,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the volume of the region in space defined by\n\\[ |x + y + 2z| + |x + y - 2z| \\le 12 \\]\nand $x, y, z \\ge 0.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 26345,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive real number $x$ such that\n\\[\\lfloor x^2 \\rfloor - x \\lfloor x \\rfloor = 6.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{55}{7}"
} | {
"split": "train",
"index": 26346,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Arrange for 7 staff members to be on duty from May 1st to May 7th. Each person is on duty for one day, with both members A and B not being scheduled on May 1st and 2nd. The total number of different scheduling methods is $\\_\\_\\_\\_\\_\\_\\_$. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "2400"
} | {
"split": "train",
"index": 26347,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of the first $n$ terms of a geometric sequence ${a_{n}}$ is $S_{n}=k+2( \\frac {1}{3})^{n}$, find the value of the constant $k$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 26348,
"solution": ""
} |
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