data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Let $b_n$ be the number obtained by writing the integers 1 to $n$ from left to right, where each integer is squared. For example, $b_3 = 149$ (since $1^2 = 1$, $2^2 = 4$, $3^2 = 9$), and $b_5 = 1491625$. For $1 \\le k \\le 100$, determine how many $b_k$ are divisible by 4. Let'... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 200,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Andrew and John are both Beatles fans. Their respective collections share nine of the same albums. Andrew has seventeen albums in his collection. Six albums are in John's collection, but not Andrew's. How many albums are in either Andrew's or John's collection, but not both... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 201,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the values of $x$ such that $\\cos^3 3x+ \\cos^3 5x = 8 \\cos^3 4x \\cos^3 x$, where $x$ is measured in degrees and $100< x< 200.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "906"
} | {
"split": "train",
"index": 202,
"solution": "Observe that $2\\cos 4x\\cos x = \\cos 5x + \\cos 3x$ by the sum-to-product formulas. Defining $a = \\cos 3x$ and $b = \\cos 5x$, we have $a^3 + b^3 = (a+b)^3 \\rightarrow ab(a+b) = 0$. But $a+b = 2\\cos 4x\\cos x$, so we require $\\cos x = 0$, $\\cos 3x = 0$, $\\cos... |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{a},$ $\\mathbf{b},$ and $\\mathbf{c}$ be vectors such that $\\|\\mathbf{a}\\| = 2,$ $\\|\\mathbf{b}\\| = 3,$ and $\\|\\mathbf{c}\\| = 6,$ and\n\\[2\\mathbf{a} + 3\\mathbf{b} + 4\\mathbf{c} = \\mathbf{0}.\\]Compute $2(\\mathbf{a} \\cdot \\mathbf{b}) + 3(\\mathbf{a}... | math | {
"style": "rule",
"ground_truth": "-673/4"
} | {
"split": "train",
"index": 203,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The volume of a cube in cubic meters and its surface area in square meters is numerically equal to four-thirds of the sum of the lengths of its edges in meters. What is the total volume in cubic meters of twenty-seven such cubes? Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "216"
} | {
"split": "train",
"index": 204,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the matrix that corresponds to reflecting over the vector $\\begin{pmatrix} 3 \\\\ 2 \\end{pmatrix}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 5/13 & 12/13 \\\\ 12/13 & -5/13 \\end{pmatrix}"
} | {
"split": "train",
"index": 205,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the triangular prism $P-ABC$, the three edges $PA$, $PB$, and $PC$ are mutually perpendicular, with $PA=1$, $PB=2$, and $PC=2$. If $Q$ is any point on the circumsphere of the triangular prism $P-ABC$, what is the maximum distance from $Q$ to the plane $ABC$? Let's think step... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2} + \\frac{\\sqrt{6}}{6}"
} | {
"split": "train",
"index": 206,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\\frac{\\cos B}{b} + \\frac{\\cos C}{c} = \\frac{2\\sqrt{3}\\sin A}{3\\sin C}$.\n\n(1) Find the value of $b$;\n\n(2) If $B = \\frac{\\pi}{3}$, find the maximum area of tri... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{16}"
} | {
"split": "train",
"index": 207,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A jar contains $2$ yellow candies, $4$ red candies, and $6$ blue candies. Candies are randomly drawn out of the jar one-by-one and eaten. The probability that the $2$ yellow candies will be eaten before any of the red candies are eaten is given by the fraction $\\frac{... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 208,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "2500 chess kings have to be placed on a $100 \\times 100$ chessboard so that\n\n[b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex);\n[b](ii)[/b] each row and each column contains exactly 25 kings.\n\nFind the numbe... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 209,
"solution": "\nLet us consider a \\(100 \\times 100\\) chessboard and the placement of 2500 kings such that:\n\n1. No king can capture another king, meaning no two kings can be placed on squares that share a common vertex.\n2. Each row and each column contains exactly 25 kings.\n... |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a triangle with $AB=13$ , $BC=14$ , and $CA=15$ . Points $P$ , $Q$ , and $R$ are chosen on segments $BC$ , $CA$ , and $AB$ , respectively, such that triangles $AQR$ , $BPR$ , $CPQ$ have the same perimeter, which is $\\frac{4}{5}$ of the perimeter ... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 210,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For all positive numbers $a,b \\in \\mathbb{R}$ such that $a+b=1$, find the supremum of the expression $-\\frac{1}{2a}-\\frac{2}{b}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{9}{2}"
} | {
"split": "train",
"index": 211,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The lottery now consists of two drawings. First, a PowerBall is picked from among 30 numbered balls. Second, six LuckyBalls are picked from among 49 numbered balls. To win the lottery, you must pick the PowerBall number correctly and also correctly pick the numbers on all six L... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{419,512,480}"
} | {
"split": "train",
"index": 212,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A ball is dropped from a height of $128$ meters, and each time it hits the ground, it bounces back to half of its original height. When it hits the ground for the $9$th time, the total distance it has traveled is ______ meters. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "383"
} | {
"split": "train",
"index": 213,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the definite integral:\n\n$$\n\\int_{0}^{\\frac{\\pi}{2}}\\left(x^{2}-5x+6\\right) \\sin 3x \\, dx\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{67 - 3\\pi}{27}"
} | {
"split": "train",
"index": 214,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum positive integer $n\\ge 3$, such that there exist $n$ points $A_1,A_2,\\cdots, A_n$ satisfying no three points are collinear and for any $1\\le i\\le n$, there exist $1\\le j \\le n (j\\neq i)$, segment $A_jA_{j+1}$ pass through the midpoint of segment $A_iA_{i... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 215,
"solution": "\nTo find the minimum positive integer \\( n \\geq 3 \\) such that there exist \\( n \\) points \\( A_1, A_2, \\ldots, A_n \\) satisfying no three points are collinear and for any \\( 1 \\leq i \\leq n \\), there exists \\( 1 \\leq j \\leq n \\) (with \\( j \\neq i \... |
deepscaler | [
{
"role": "user",
"content": "If $wxyz$ is a four-digit positive integer with $w \\neq 0$, the layer sum of this integer equals $wxyz + xyz + yz + z$. If the layer sum of $wxyz$ equals 2014, what is the value of $w + x + y + z$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 216,
"solution": "The layer sum of $wxyz$ equals 2014. This means that the sum of the integer with digits $wxyz$, the integer with digits $xyz$, the integer with digits $yz$, and the integer $z$ is 2014. Note that the integer with digits $wxyz$ equals $1000w + 100x + 10y + z$, the int... |
deepscaler | [
{
"role": "user",
"content": "How many different $4\\times 4$ arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row is 0 and the sum of the entries in each column is 0?\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 217,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sufficiently large positive integer \\( n \\), which can be divided by all the integers from 1 to 250 except for two consecutive integers \\( k \\) and \\( k+1 \\), find \\( k \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "127"
} | {
"split": "train",
"index": 218,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Seven cards numbered $1$ through $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed leaving the remaining six cards in either ascending or descending order. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 219,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cryptographic code is designed as follows. The first time a letter appears in a given message it is replaced by the letter that is $1$ place to its right in the alphabet (asumming that the letter $A$ is one place to the right of the letter $Z$). The second time this same lett... | math | {
"style": "rule",
"ground_truth": "s"
} | {
"split": "train",
"index": 220,
"solution": "1. **Identify the pattern of replacement:** The problem states that each time a letter appears, it is replaced by the letter that is $1, 1+2, 1+2+3, \\ldots$ places to the right in the alphabet. This sequence represents the sum of the first $n$ natural numbers, where... |
deepscaler | [
{
"role": "user",
"content": "If $f(x)$ is a function defined on $R$, and $f(x) - x^2$ is an odd function, and $f(x) + 2^x$ is an even function, then the minimum value of $f(x)$ on the interval $\\left[-2,-1\\right]$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{4}"
} | {
"split": "train",
"index": 221,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( S = \\{(x, y) \\mid x, y \\in \\mathbb{Z}, 0 \\leq x, y \\leq 2016\\} \\). Given points \\( A = (x_1, y_1), B = (x_2, y_2) \\) in \\( S \\), define\n\\[ d_{2017}(A, B) = (x_1 - x_2)^2 + (y_1 - y_2)^2 \\pmod{2017} \\]\n\nThe points \\( A = (5, 5) \\), \\( B = (2, 6) \\),... | math | {
"style": "rule",
"ground_truth": "1021"
} | {
"split": "train",
"index": 222,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Bus stop \\(B\\) is located on a straight highway between stops \\(A\\) and \\(C\\). After some time driving from \\(A\\), the bus finds itself at a point on the highway where the distance to one of the three stops is equal to the sum of the distances to the other two stops. Af... | math | {
"style": "rule",
"ground_truth": "180"
} | {
"split": "train",
"index": 223,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At what value of $a$ do the graphs of $y=ax^2+3x+1$ and $y=-x-1$ intersect at exactly one point? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 224,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be a subset of $\\{1,2,3,...,100\\}$ such that no pair of distinct elements in $S$ has a product divisible by $5$. What is the maximum number of elements in $S$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 225,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $3 \\times 3$ table starts with every entry equal to 0 and is modified using the following steps: (i) adding 1 to all three numbers in any row; (ii) adding 2 to all three numbers in any column. After step (i) has been used a total of $a$ times and step (ii) has been used a to... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 226,
"solution": "Since the second column includes the number 1, then step (ii) was never used on the second column, otherwise each entry would be at least 2. To generate the 1,3 and 2 in the second column, we thus need to have used step (i) 1 time on row 1,3 times on row 2, and 2 tim... |
deepscaler | [
{
"role": "user",
"content": "Compute $i^{-100}+i^{-99}+i^{-98}+\\cdots+i^{-1}+i^0+i^1+\\cdots+i^{99}+i^{100}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 227,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point P is any point on the graph of the function $f(x) = 2\\sqrt{2x}$, and a tangent line is drawn from point P to circle D: $x^2 + y^2 - 4x + 3 = 0$, with the points of tangency being A and B, find the minimum value of the area of quadrilateral PADB. Let's think st... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 228,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "13. Given that $a$, $b$, $c$, are the lengths of the sides opposite to angles $A$, $B$, $C$ in $\\triangle ABC$ respectively, with $a=2$, and $(2+b)(\\sin A-\\sin B)=(c-b)\\sin C$, find the maximum area of $\\triangle ABC$. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 229,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a positive integer \\( n \\), let \\( s(n) \\) denote the sum of its digits, and let \\( p(n) \\) denote the product of its digits. If the equation \\( s(n) + p(n) = n \\) holds true, then \\( n \\) is called a coincidence number. What is the sum of all coincidence numbers?... | math | {
"style": "rule",
"ground_truth": "531"
} | {
"split": "train",
"index": 230,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(6?3) + 4 - (2 - 1) = 5. To make this statement true, the question mark between the 6 and the 3 should be replaced by Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\div"
} | {
"split": "train",
"index": 231,
"solution": "We start by simplifying the expression on the right side of the equation:\n\\[ (6?3) + 4 - (2 - 1) = 5. \\]\n\nFirst, simplify the parentheses on the right:\n\\[ 2 - 1 = 1. \\]\n\nNow, substitute this back into the equation:\n\\[ (6?3) + 4 - 1 = 5. \\]\n\nSimplify fu... |
deepscaler | [
{
"role": "user",
"content": "Given that the general term of the sequence $\\{a_{n}\\}$ is ${a}_{n}=97-3n(n∈{N}^{*})$, find the value of $n$ for which the sum of the first $n$ terms of the sequence $\\{{a}_{n}{a}_{n+1}{a}_{n+2}\\}(n∈{N}^{*})$ reaches its maximum value. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 232,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In three independent repeated trials, the probability of event $A$ occurring in each trial is the same. If the probability of event $A$ occurring at least once is $\\frac{63}{64}$, then the probability of event $A$ occurring exactly once is $\\_\\_\\_\\_\\_\\_$. Let's think ste... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{64}"
} | {
"split": "train",
"index": 233,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be the number of functions $f$ from $\\{1,2, \\ldots, 101\\} \\rightarrow\\{1,2, \\ldots, 101\\}$ such that $f^{101}(1)=2$. Find the remainder when $N$ is divided by 103. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"split": "train",
"index": 234,
"solution": "For convenience, let $n=101$. Compute the number of functions such that $f^{n}(1)=1$. Since $n$ is a prime, there are 2 cases: the order of 1 is either 1 or $n$. The first case gives $n^{n-1}$ functions, and the second case gives $(n-1)$ ! functions. By symmetry, the... |
deepscaler | [
{
"role": "user",
"content": "What is the value of \\(6102 - 2016\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4086"
} | {
"split": "train",
"index": 235,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Acute triangle $A B C$ has circumcenter $O$. The bisector of $\\angle A B C$ and the altitude from $C$ to side $A B$ intersect at $X$. Suppose that there is a circle passing through $B, O, X$, and $C$. If $\\angle B A C=n^{\\circ}$, where $n$ is a positive integer, compute the ... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 236,
"solution": "We have $\\angle X B C=B / 2$ and $\\angle X C B=90^{\\circ}-B$. Thus, $\\angle B X C=90^{\\circ}+B / 2$. We have $\\angle B O C=2 A$, so $$90^{\\circ}+B / 2=2 A$$ This gives $B=4 A-180^{\\circ}$, which gives $C=360^{\\circ}-5 A$. In order for $0^{\\circ}<B<90^{\\cir... |
deepscaler | [
{
"role": "user",
"content": "Find the least positive integer $n$ , such that there is a polynomial \\[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\\dots+a_1x+a_0 \\] with real coefficients that satisfies both of the following properties:\n - For $i=0,1,\\dots,2n$ it is $2014 \\leq a_i \\leq 2015$ .\n ... | math | {
"style": "rule",
"ground_truth": "2014"
} | {
"split": "train",
"index": 237,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Select two distinct numbers simultaneously and at random from the set $\\{1, 2, 3, 4, 5, 6\\}$. What is the probability that the smaller one divides the larger one and both numbers are either both even or both odd? Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{15}"
} | {
"split": "train",
"index": 238,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function \\( y = \\sqrt{a x^2 + b x + c} \\) (where \\(a, b, c \\in \\mathbb{R}\\) and \\(a < 0\\)), the domain is \\( D \\). If the points \\( (s, f(t)) \\) (where \\( s, t \\in D \\)) form a square, then the real number \\( a \\) equals ______. Let's think step by s... | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 239,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different real numbers $x$ satisfy the equation $(x^{2}-5)^{2}=16$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 240,
"solution": "1. **Rewrite the given equation**: We start with the equation \\[(x^2-5)^2 = 16.\\]\n\n2. **Simplify the equation**: We can simplify this equation by taking the square root of both sides, remembering to consider both the positive and negative roots:\n \\[x^2 - 5 = ... |
deepscaler | [
{
"role": "user",
"content": "\nCalculate the limit of the function:\n\n$$\n\\lim _{x \\rightarrow 0}\\left(3-\\frac{2}{\\cos x}\\right)^{\\operatorname{cosec}^{2} x}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "e^{-1}"
} | {
"split": "train",
"index": 241,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum positive integer \\( n \\) such that\n\\[ \nn^{2} \\leq 160 \\times 170 \\times 180 \\times 190\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30499"
} | {
"split": "train",
"index": 242,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For the nonzero numbers $a$, $b$, and $c$, define $$\n\\text{{J}}(a,b,c) = \\frac{a}{b} + \\frac{b}{c} + \\frac{c}{a}.\n$$Find $\\text{{J}}(2,12, 9)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 243,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Paint both sides of a small wooden board. It takes 1 minute to paint one side, but you must wait 5 minutes for the paint to dry before painting the other side. How many minutes will it take to paint 6 wooden boards in total? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 244,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all the ways in which the number 1987 can be written in another base as a three-digit number where the sum of the digits is 25. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 245,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Fifteen distinct points are designated on $\\triangle ABC$: the 3 vertices $A$, $B$, and $C$; $3$ other points on side $\\overline{AB}$; $4$ other points on side $\\overline{BC}$; and $5$ other points on side $\\overline{CA}$. Find the number of triangles with positive area who... | math | {
"style": "rule",
"ground_truth": "390"
} | {
"split": "train",
"index": 246,
"solution": "Every triangle is uniquely determined by 3 points. There are $\\binom{15}{3}=455$ ways to choose 3 points, but that counts the degenerate triangles formed by choosing three points on a line. There are $\\binom{5}{3}$ invalid cases on segment $AB$, $\\binom{6}{3}$ inv... |
deepscaler | [
{
"role": "user",
"content": "For environmental protection, Wuyang Mineral Water recycles empty bottles. Consumers can exchange 4 empty bottles for 1 bottle of mineral water (if there are fewer than 4 empty bottles, they cannot be exchanged). Huacheng Middle School bought 1999 bottles of Wuyang brand minera... | math | {
"style": "rule",
"ground_truth": "2345"
} | {
"split": "train",
"index": 247,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all the positive two-digit integers divisible by both the sum and product of their digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 248,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is known that exactly one of the three (distinguishable) musketeers stole the truffles. Each musketeer makes one statement, in which he either claims that one of the three is guilty, or claims that one of the three is innocent. It is possible for two or more of the musketeer... | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 249,
"solution": "We divide into cases, based on the number of distinct people that statements are made about. - The statements are made about 3 distinct people. Then, since exactly one person is guilty, and because exactly one of the three lied, there are either zero statements of gu... |
deepscaler | [
{
"role": "user",
"content": "Consider the parametric equations for a curve given by\n\\begin{align*}\nx &= \\cos t + \\frac{t}{3}, \\\\\ny &= \\sin t.\n\\end{align*}\nDetermine how many times the graph intersects itself between $x = 0$ and $x = 60$. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 250,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A$ be the vertex of the graph of the equation $y=x^2 - 2x + 3 $. Let $B$ be the vertex of the graph of the equation $y=x^2 + 4x + 10 $. What is the distance between $A$ and $B$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 251,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each positive integer \\(1 \\leqq k \\leqq 100\\), let \\(a_{k}\\) denote the sum \\(\\frac{1}{k}+\\frac{1}{k+1}+\\ldots+\\frac{1}{100}\\). Calculate the value of\n\n\\[ a_{1} + a_{1}^{2} + a_{2}^{2} + \\ldots + a_{100}^{2}. \\] Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 252,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given positive real numbers $a$ and $b$ satisfying $a+b=1$, find the maximum value of $\\dfrac {2a}{a^{2}+b}+ \\dfrac {b}{a+b^{2}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac {2 \\sqrt {3}+3}{3}"
} | {
"split": "train",
"index": 253,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, the rectangular wire grid contains 15 identical squares. The length of the rectangular grid is 10. What is the length of wire needed to construct the grid? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "76"
} | {
"split": "train",
"index": 254,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a triangle with circumcenter $O$, incenter $I, \\angle B=45^{\\circ}$, and $OI \\parallel BC$. Find $\\cos \\angle C$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1-\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 255,
"solution": "Let $M$ be the midpoint of $BC$, and $D$ the foot of the perpendicular of $I$ with $BC$. Because $OI \\parallel BC$, we have $OM=ID$. Since $\\angle BOC=2 \\angle A$, the length of $OM$ is $OA \\cos \\angle BOM=OA \\cos A=R \\cos A$, and the length of $ID$ is $r$, wh... |
deepscaler | [
{
"role": "user",
"content": "Find $\\begin{pmatrix} 3 & 0 \\\\ 1 & 2 \\end{pmatrix} + \\begin{pmatrix} -5 & -7 \\\\ 4 & -9 \\end{pmatrix}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -2 & -7 \\\\ 5 & -7 \\end{pmatrix}"
} | {
"split": "train",
"index": 256,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( S = \\{1, 2, \\ldots, 1963\\} \\). What is the maximum number of elements that can be chosen from \\( S \\) such that the sum of any two chosen numbers is not divisible by their difference? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "655"
} | {
"split": "train",
"index": 257,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the graphs of $y=h(x)$ and $y=j(x)$ intersect at $(3,3),$ $(5,5),$ $(7,7),$ and $(9,9),$ determine whether there is a point where the graphs of $y=h(3x)$ and $y=3j(x)$ intersect, and find the sum of the coordinates of that point if it exists. Let's think step by step... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 258,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Which number has the property that if it is multiplied by $1, 2, 3, 4, 5$, or $6$, the resulting product contains only the digits that appear in the original number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "142857"
} | {
"split": "train",
"index": 259,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given there are ten steps from the first floor to the second floor, calculate the total number of ways Xiao Ming can go from the first floor to the second floor. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "89"
} | {
"split": "train",
"index": 260,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the sum of two numbers is $1$ and their product is $1$, then the sum of their cubes is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 261,
"solution": "Let the two numbers be $a$ and $b$. Given that the sum of the two numbers is $1$ and their product is $1$, we have:\n\\[ a + b = 1 \\]\n\\[ ab = 1 \\]\n\nWe need to find the sum of their cubes, $a^3 + b^3$. We can use the identity for the sum of cubes:\n\\[ a^3 + b^3... |
deepscaler | [
{
"role": "user",
"content": "Let \\\\(\\{a_n\\}\\\\) be a geometric sequence where each term is positive, and let \\\\(S_n\\\\) be the sum of the first \\\\(n\\\\) terms with \\\\(S_{10}=10\\\\) and \\\\(S_{30}=70\\\\). Find \\\\(S_{40}=\\\\)_______. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 262,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different positive three-digit integers can be formed using only the digits in the set $\\{2, 3, 5, 5, 5, 6, 6\\}$ if no digit may be used more times than it appears in the given set of available digits? Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "43"
} | {
"split": "train",
"index": 263,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Notice that in the fraction $\\frac{16}{64}$ we can perform a simplification as $\\cancel{\\frac{16}{64}}=\\frac 14$ obtaining a correct equality. Find all fractions whose numerators and denominators are two-digit positive integers for which such a simplification is correct. Le... | math | {
"style": "rule",
"ground_truth": "\\tfrac{19}{95}, \\tfrac{16}{64}, \\tfrac{11}{11}, \\tfrac{26}{65}, \\tfrac{22}{22}, \\tfrac{33}{33} , \\tfrac{49}{98}, \\tfrac{44}{44}, \\tfrac{55}{55}, \\tfrac{66}{66}, \\tfrac{77}{77}, \\tfrac{88}{88} , \\tfrac{99}{99}"
} | {
"split": "train",
"index": 264,
"solution": "\nThe problem requires us to find all fractions with two-digit numerators and denominators where the simplification by \"cancelling\" a common digit in the numerator and denominator incorrectly leads to a correct fraction. \n\nTo solve this problem, let's consider a ... |
deepscaler | [
{
"role": "user",
"content": "A deck of forty cards consists of four $1$'s, four $2$'s,..., and four $10$'s. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let $m/n$ be the probability that two randomly selected cards also form... | math | {
"style": "rule",
"ground_truth": "758"
} | {
"split": "train",
"index": 265,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $C$ be a circle with two diameters intersecting at an angle of 30 degrees. A circle $S$ is tangent to both diameters and to $C$, and has radius 1. Find the largest possible radius of $C$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1+\\sqrt{2}+\\sqrt{6}"
} | {
"split": "train",
"index": 266,
"solution": "For $C$ to be as large as possible we want $S$ to be as small as possible. It is not hard to see that this happens in the situation shown below. Then the radius of $C$ is $1+\\csc 15=\\mathbf{1}+\\sqrt{\\mathbf{2}}+\\sqrt{\\mathbf{6}}$. The computation of $\\sin 15$ ... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sin \\alpha = 2 \\cos \\alpha$, find the value of $\\cos ( \\frac {2015\\pi}{2}-2\\alpha)$. 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": 267,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Gerard cuts a large rectangle into four smaller rectangles. The perimeters of three of these smaller rectangles are 16, 18, and 24. What is the perimeter of the fourth small rectangle?\nA) 8\nB) 10\nC) 12\nD) 14\nE) 16 Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 268,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The average score of 60 students is 72. After disqualifying two students whose scores are 85 and 90, calculate the new average score for the remaining class. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "71.47"
} | {
"split": "train",
"index": 269,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In Mrs. Warner's class, there are 30 students. Strangely, 15 of the students have a height of 1.60 m and 15 of the students have a height of 1.22 m. Mrs. Warner lines up \\(n\\) students so that the average height of any four consecutive students is greater than 1.50 m and the ... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 270,
"solution": "We refer to the students with height 1.60 m as 'taller' students and to those with height 1.22 m as 'shorter' students. For the average of four consecutive heights to be greater than 1.50 m, the sum of these four heights must be greater than \\(4 \\times 1.50 \\mathr... |
deepscaler | [
{
"role": "user",
"content": "A polygon \\(\\mathcal{P}\\) is drawn on the 2D coordinate plane. Each side of \\(\\mathcal{P}\\) is either parallel to the \\(x\\) axis or the \\(y\\) axis (the vertices of \\(\\mathcal{P}\\) do not have to be lattice points). Given that the interior of \\(\\mathcal{P}\\) incl... | math | {
"style": "rule",
"ground_truth": "8 \\sqrt{2022}"
} | {
"split": "train",
"index": 271,
"solution": "The minimum possible perimeter is achieved by an axis-aligned square with all four sides tangent to the circle, which has area \\(8 \\sqrt{2022}\\). To see why this is true, notice that there must be at least \\(2 \\sqrt{2022}\\) length of total perimeter facing left... |
deepscaler | [
{
"role": "user",
"content": "This pattern is made from toothpicks. If the pattern is continued by adding two toothpicks to the previous stage, how many toothpicks are used to create the figure for the $15^{th}$ stage?\n\n[asy]draw((0,0)--(7.5,13)--(-7.5,13)--cycle);\ndraw((0,0)--(-15,0)--(-7.5,13)--cycle);... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 272,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let be the set $ \\mathcal{C} =\\left\\{ f:[0,1]\\longrightarrow\\mathbb{R}\\left| \\exists f''\\bigg|_{[0,1]} \\right.\\quad\\exists x_1,x_2\\in [0,1]\\quad x_1\\neq x_2\\wedge \\left( f\\left( \nx_1 \\right) = f\\left( x_2 \\right) =0\\vee f\\left( \nx_1 \\right) = f'\\left... | math | {
"style": "rule",
"ground_truth": "1/12"
} | {
"split": "train",
"index": 273,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The water storage volume of two reservoirs, A and B, changes with time. The relationship between the water storage volume of reservoir A, $f(t)$ (in hundreds of tons), and time, $t$ (in hours), is given by $f(t)=2+\\sin t$, where $t\\in[0,12]$. The relationship between the wate... | math | {
"style": "rule",
"ground_truth": "6.721"
} | {
"split": "train",
"index": 274,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A three-meter gas pipe has rusted in two places. Determine the probability that all three resulting parts can be used as connections to gas stoves, given that regulations require a stove to be no closer than 75 cm from the main gas pipeline. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 275,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $OAB$ has $O=(0,0)$, $B=(5,0)$, and $A$ in the first quadrant. In addition, $\\angle ABO=90^\\circ$ and $\\angle AOB=30^\\circ$. Suppose that $OA$ is rotated $90^\\circ$ counterclockwise about $O$. What are the coordinates of the image of $A$? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "\\left( - \\frac {5}{3}\\sqrt {3},5\\right)"
} | {
"split": "train",
"index": 276,
"solution": "1. **Identify the Coordinates of Point $A$:**\n Given that $\\triangle OAB$ has $O = (0,0)$, $B = (5,0)$, and $\\angle ABO = 90^\\circ$, point $A$ must lie on the line perpendicular to $OB$ at $B$. Since $\\angle AOB = 30^\\circ$, we can use trigonometric relations... |
deepscaler | [
{
"role": "user",
"content": "Betty used a calculator to find the product $0.075 \\times 2.56$. She forgot to enter the decimal points. The calculator showed $19200$. If Betty had entered the decimal points correctly, the answer would have been Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": ".192"
} | {
"split": "train",
"index": 277,
"solution": "1. **Identify the error in decimal placement**: Betty entered the numbers without the decimal points, so she effectively multiplied $75$ by $256$ instead of $0.075$ by $2.56$.\n\n2. **Calculate the product without decimal error**: \n \\[\n 75 \\times 256 = 19200\... |
deepscaler | [
{
"role": "user",
"content": "The function \\( f(x) = \\frac{x^{2}}{8} + x \\cos x + \\cos (2x) \\) (for \\( x \\in \\mathbf{R} \\)) has a minimum value of ___ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 278,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rabbit escapes and runs 100 steps ahead before a dog starts chasing it. The rabbit can cover 8 steps in the same distance that the dog can cover in 3 steps. Additionally, the dog can run 4 steps in the same time that the rabbit can run 9 steps. How many steps must the dog run... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 279,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six trees are equally spaced along one side of a straight road. The distance from the first tree to the fourth is 60 feet. What is the distance in feet between the first and last trees? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 280,
"solution": "1. **Identify the number of spaces between trees**: \n - The problem states that there are six trees equally spaced along one side of a straight road. \n - The trees are numbered from 1 to 6. \n - The distance from the first tree to the fourth tree is given as ... |
deepscaler | [
{
"role": "user",
"content": "The angle bisector of angle \\(ABC\\) forms an angle with its sides that is three times smaller than the adjacent angle to \\(ABC\\). Find the measure of angle \\(ABC\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 281,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum number of natural numbers $x_1,x_2, ... , x_m$ satisfying the conditions:\na) No $x_i - x_j , 1 \\le i < j \\le m$ is divisible by $11$ , and\nb) The sum $x_2x_3 ...x_m + x_1x_3 ... x_m + \\cdot \\cdot \\cdot + x_1x_2... x_{m-1}$ is divisible by $11$... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 282,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $f(x), g(x), h(x), p(x)$ are linear equations where $f(x) = x + 1$, $g(x) = -x + 5$, $h(x) = 4$, and $p(x) = 1$. Define new functions $j(x)$ and $k(x)$ as follows:\n$$j(x) = \\max\\{f(x), g(x), h(x), p(x)\\},$$\n$$k(x)= \\min\\{f(x), g(x), h(x), p(x)\\}.$$\nFind the len... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 283,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ''roundness'' of an integer greater than 1 is the sum of the exponents of the prime factorization of the number. For example, $20 = 2^2 \\times 5^1$, so 20 has a roundness of 3. What is the roundness of 1,000,000? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 284,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let set $M=\\{-1, 0, 1\\}$, and set $N=\\{a, a^2\\}$. Find the real number $a$ such that $M \\cap N = N$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 285,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evgeny is laying tiles on the floor of his living room, which measures 12 by 16 meters. He plans to place square tiles of size 1 m × 1 m along the border of the room and to lay the rest of the floor with square tiles of size 2 m × 2 m. How many tiles will he need in total? Let'... | math | {
"style": "rule",
"ground_truth": "87"
} | {
"split": "train",
"index": 286,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( S = \\{1, 2, \\cdots, 2005\\} \\). Find the minimum value of \\( n \\) such that any set of \\( n \\) pairwise coprime elements from \\( S \\) contains at least one prime number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 287,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of a quadrilateral is 3 cm², and the lengths of its diagonals are 6 cm and 2 cm. Find the angle between the diagonals. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 288,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Acme Corporation has released an alphabet soup in which each of the vowels (A, E, I, O, U) of the English alphabet appears five times (and the consonants do not appear at all). How many five-letter words can be formed from a bowl of Acme Vowel Soup? (Note: The words do not have... | math | {
"style": "rule",
"ground_truth": "3125"
} | {
"split": "train",
"index": 289,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A large rectangle has side lengths of $(x+7)$ and $(x+5)$. In the large rectangle, there is a rectangular hole with side lengths of $(2x-3)$ and $(x-2)$. What is the area of the large rectangle (not including the area of the hole)? Express your answer as a polynomial in $x$. Le... | math | {
"style": "rule",
"ground_truth": "-x^2+19x+29"
} | {
"split": "train",
"index": 290,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "John has saved up $5555_8$ dollars for a new laptop. A laptop he desires costs $1500_{10}$ dollars. How many dollars will he have left after purchasing the laptop if all his money is in base eight and the laptop price is in base ten? Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "1425"
} | {
"split": "train",
"index": 291,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two sides of a right triangle have the lengths 4 and 5. What is the product of the possible lengths of the third side? Express the product as a decimal rounded to the nearest tenth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19.2"
} | {
"split": "train",
"index": 292,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $S_{n}$ is the sum of the first $n$ terms of the sequence $\\{a_{n}\\}$, if ${a_1}=\\frac{5}{2}$, and ${a_{n+1}}({2-{a_n}})=2$ for $n\\in\\mathbb{N}^*$, then $S_{22}=$____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{3}"
} | {
"split": "train",
"index": 293,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a+b=1$, find the supremum of $$- \\frac {1}{2a}- \\frac {2}{b}.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac {9}{2}"
} | {
"split": "train",
"index": 294,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the probability of player A winning a single game is $\\frac{2}{3}$, calculate the probability that A wins the match with a score of 3:1 in a best of five games format. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{27}"
} | {
"split": "train",
"index": 295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to $\\angle A$, $\\angle B$, and $\\angle C$ are $a$, $b$, and $c$ respectively. $M$ is the midpoint of $BC$ with $BM=MC=2$, and $AM=b-c$. Find the maximum area of $\\triangle ABC$. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 296,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\lfloor{\\sqrt{12}}\\rfloor^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 297,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "**p1.** The Evergreen School booked buses for a field trip. Altogether, $138$ people went to West Lake, while $115$ people went to East Lake. The buses all had the same number of seats and every bus has more than one seat. All seats were occupied and everybody had a seat. H... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 298,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $w$ and $z$ are complex numbers such that $|w+z|=1$ and $|w^2+z^2|=14,$ find the smallest possible value of $|w^3+z^3|.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\tfrac{41}2"
} | {
"split": "train",
"index": 299,
"solution": ""
} |
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