data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "When $x=1$, the value of the expression $px^3+qx-10$ is 2006; when $x=-1$, find the value of the expression $px^3+qx-10$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2026"
} | {
"split": "train",
"index": 7300,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Country music songs are added to a playlist so that now $40\\%$ of the songs are Country. If the ratio of Hip Hop songs to Pop songs remains the same, what percentage of the total number of songs are now Hip Hop? Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "39\\%"
} | {
"split": "train",
"index": 7301,
"solution": "Since $40\\%$ of the songs on the updated playlist are Country, then the remaining $100\\%-40\\%$ or $60\\%$ must be Hip Hop or Pop songs. Since the ratio of Hip Hop songs to Pop songs does not change, then $65\\%$ of this remaining $60\\%$ must be Hip Hop songs. Ov... |
deepscaler | [
{
"role": "user",
"content": "Find the number of positive integers $n$ such that \n\\[(n - 2)(n - 4)(n - 6) \\dotsm (n - 98) > 0.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 7302,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An 8-by-8 square is divided into 64 unit squares in the usual way. Each unit square is colored black or white. The number of black unit squares is even. We can take two adjacent unit squares (forming a 1-by-2 or 2-by-1 rectangle), and flip their colors: black becomes white a... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 7303,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many 9-digit numbers, divisible by 2, can be formed by permuting the digits of the number 231157152? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3360"
} | {
"split": "train",
"index": 7304,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(a)=a-2$ and $F(a,b)=b^2+a$, then $F(3,f(4))$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 7305,
"solution": "1. **Calculate $f(4)$**: Given the function $f(a) = a - 2$, substitute $a = 4$:\n \\[\n f(4) = 4 - 2 = 2\n \\]\n\n2. **Evaluate $F(3, f(4))$**: With $f(4) = 2$, we need to find $F(3, 2)$. The function $F(a, b) = b^2 + a$ is given, so substitute $a = 3$ and $b ... |
deepscaler | [
{
"role": "user",
"content": "What is the value of $(5 \nabla 1)+(4 \nabla 1)$ if the operation $k \nabla m$ is defined as $k(k-m)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 7306,
"solution": "From the definition, $(5 \nabla 1)+(4 \nabla 1)=5(5-1)+4(4-1)=5(4)+4(3)=20+12=32$."
} |
deepscaler | [
{
"role": "user",
"content": "Given functions \\(f(x)\\) and \\(g(x)\\) are defined on \\(\\mathbb{R}\\) with \\(g(x)\\neq 0\\), and \\(f(x)g'(x) > f'(x)g(x)\\). It is also given that \\(f(x)=a^{x}\\cdot g(x)\\) where \\(a > 0, a\\neq 1\\), and \\(\\frac{f(1)}{g(1)} + \\frac{f(-1)}{g(-1)} = \\frac{5}{2}\\).... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 7307,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A convex quadrilateral $ABCD$ with area $2002$ contains a point $P$ in its interior such that $PA = 24, PB = 32, PC = 28, PD = 45$. Find the perimeter of $ABCD$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4(36 + \\sqrt{113})"
} | {
"split": "train",
"index": 7308,
"solution": "1. **Area and Diagonal Relationship**: \n Given that the area of convex quadrilateral $ABCD$ is $2002$, we can use the inequality for the area of a convex quadrilateral split by its diagonals:\n \\[\n [ABCD] \\leq \\frac{1}{2} (AC \\cdot BD)\n \\]\n This ... |
deepscaler | [
{
"role": "user",
"content": "The following grid represents a mountain range; the number in each cell represents the height of the mountain located there. Moving from a mountain of height \\( a \\) to a mountain of height \\( b \\) takes \\( (b-a)^{2} \\) time. Suppose that you start on the mountain of heig... | math | {
"style": "rule",
"ground_truth": "212"
} | {
"split": "train",
"index": 7309,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many four-digit positive integers exist, all of whose digits are 0's, 2's, and/or 5's, and the number does not start with 0? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 7310,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$S$ is a set of complex numbers such that if $u, v \\in S$, then $u v \\in S$ and $u^{2}+v^{2} \\in S$. Suppose that the number $N$ of elements of $S$ with absolute value at most 1 is finite. What is the largest possible value of $N$ ? Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 7311,
"solution": "First, if $S$ contained some $u \\neq 0$ with absolute value $<1$, then (by the first condition) every power of $u$ would be in $S$, and $S$ would contain infinitely many different numbers of absolute value $<1$. This is a contradiction. Now suppose $S$ contains som... |
deepscaler | [
{
"role": "user",
"content": "Suppose $ n$ is a product of four distinct primes $ a,b,c,d$ such that:\r $ (i)$ $ a\\plus{}c\\equal{}d;$ \r $ (ii)$ $ a(a\\plus{}b\\plus{}c\\plus{}d)\\equal{}c(d\\minus{}b);$ \r $ (iii)$ $ 1\\plus{}bc\\plus{}d\\equal{}bd$ .\r\nDetermine $ n$ . Let's think step by st... | math | {
"style": "rule",
"ground_truth": "2002"
} | {
"split": "train",
"index": 7312,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $BC = 1$ unit and $\\measuredangle BAC = 40^\\circ$, $\\measuredangle ABC = 90^\\circ$, hence $\\measuredangle ACB = 50^\\circ$. Point $D$ is midway on side $AB$, and point $E$ is the midpoint of side $AC$. If $\\measuredangle CDE = 50^\\circ$, compute the ar... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{16}"
} | {
"split": "train",
"index": 7313,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest integer $n \\geq 5$ for which there exists a set of $n$ distinct pairs $\\left(x_{1}, y_{1}\\right), \\ldots,\\left(x_{n}, y_{n}\\right)$ of positive integers with $1 \\leq x_{i}, y_{i} \\leq 4$ for $i=1,2, \\ldots, n$, such that for any indices $r, s \\in\\{1... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 7314,
"solution": "In other words, we have a set $S$ of $n$ pairs in $(\\mathbb{Z} / 4 \\mathbb{Z})^{2}$ closed under addition. Since $1+1+1+1 \\equiv 0(\\bmod 4)$ and $1+1+1 \\equiv-1(\\bmod 4),(0,0) \\in S$ and $S$ is closed under (additive) inverses. Thus $S$ forms a group under ad... |
deepscaler | [
{
"role": "user",
"content": "Given $\\sin \\alpha + \\cos \\beta = \\frac{1}{3}$ and $\\sin \\beta - \\cos \\alpha = \\frac{1}{2}$, find $\\sin (\\alpha-\\beta)=$ ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{59}{72}"
} | {
"split": "train",
"index": 7315,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 7 balls of each of the three colors: red, blue, and yellow. When randomly selecting 3 balls with different numbers, determine the total number of ways to pick such that the 3 balls are of different colors and their numbers are not consecutive. Let's think step by step... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 7316,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, side $a = \\sqrt{3}$, side $b = \\sqrt{3}$, and side $c > 3$. Let $x$ be the largest number such that the magnitude, in degrees, of the angle opposite side $c$ exceeds $x$. Then $x$ equals: Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "120^{\\circ}"
} | {
"split": "train",
"index": 7317,
"solution": "1. **Identify the given information**: In $\\triangle ABC$, we have $a = \\sqrt{3}$, $b = \\sqrt{3}$, and $c > 3$. We need to find the largest $x$ such that the angle opposite to side $c$, denoted as $\\angle C$, exceeds $x$ degrees.\n\n2. **Apply the Law of Cosines... |
deepscaler | [
{
"role": "user",
"content": "In a certain colony of bacteria, the number of bacteria doubles every day. The colony starts with 3 bacteria, and has 6 at the end of day 1, 12 at the end of day 2, and so on. What is the number of the first day which ends with the colony having more than 100 bacteria? Let's th... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 7318,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line with slope $2$ passes through the focus $F$ of the parabola $y^2 = 2px$ $(p > 0)$ and intersects the parabola at points $A$ and $B$. The projections of $A$ and $B$ on the $y$-axis are $D$ and $C$ respectively. If the area of trapezoid $\\triangle BCD$ is $6\\sqrt{5}$, th... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 7319,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An author writes a book containing 60,000 words in 100 hours. For the first 20% of the time, the author writes with 50% increased productivity. How many words per hour did the author write on average? Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "600"
} | {
"split": "train",
"index": 7320,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given 8 people are sitting around a circular table for a meeting, including one leader, one vice leader, and one recorder, and the recorder is seated between the leader and vice leader, determine the number of different seating arrangements possible, considering that arrangemen... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 7321,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that\n\\[\\begin{pmatrix} \\cos 170^\\circ & -\\sin 170^\\circ \\\\ \\sin 170^\\circ & \\cos 170^\\circ \\end{pmatrix}^n = \\mathbf{I}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 7322,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $3+x=5$ and $-3+y=5$, what is the value of $x+y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 7323,
"solution": "Since $3+x=5$, then $x=2$. Since $-3+y=5$, then $y=8$. Thus, $x+y=10$. Alternatively, we could have added the original two equations to obtain $(3+x)+(-3+y)=5+5$ which simplifies to $x+y=10$."
} |
deepscaler | [
{
"role": "user",
"content": "What is the constant term of the expansion of $\\left(6x+\\dfrac{1}{3x}\\right)^6$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "160"
} | {
"split": "train",
"index": 7324,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $A=30^{\\circ}$, $AB=\\sqrt {3}$, $BC=1$, find the area of $\\triangle ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {\\sqrt {3}}{4}"
} | {
"split": "train",
"index": 7325,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of ways to arrange the letters of the word ALABAMA. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "210"
} | {
"split": "train",
"index": 7326,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = (m^2 - m - 1)x^{-5m-3}$ is a power function, and it is increasing on the interval $(0, +\\infty)$, determine the value of $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 7327,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle A B C$ be a triangle with $A B=7, B C=1$, and $C A=4 \\sqrt{3}$. The angle trisectors of $C$ intersect $\\overline{A B}$ at $D$ and $E$, and lines $\\overline{A C}$ and $\\overline{B C}$ intersect the circumcircle of $\\triangle C D E$ again at $X$ and $Y$, resp... | math | {
"style": "rule",
"ground_truth": "\\frac{112}{65}"
} | {
"split": "train",
"index": 7328,
"solution": "Let $O$ be the cirumcenter of $\\triangle C D E$. Observe that $\\triangle A B C \\sim \\triangle X Y C$. Moreover, $\\triangle A B C$ is a right triangle because $1^{2}+(4 \\sqrt{3})^{2}=7^{2}$, so the length $X Y$ is just equal to $2 r$, where $r$ is the radius of... |
deepscaler | [
{
"role": "user",
"content": "The greatest integer function of $x$ is expressed as $[x]$ and is defined to be the greatest integer less than or equal to $x$. Find $[\\pi - 4]$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 7329,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Pauline Bunyan can shovel snow at the rate of $20$ cubic yards for the first hour, $19$ cubic yards for the second, $18$ for the third, etc., always shoveling one cubic yard less per hour than the previous hour. If her driveway is $4$ yards wide, $10$ yards long, and covered w... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 7330,
"solution": "1. **Calculate the total volume of snow**: The driveway is $4$ yards wide, $10$ yards long, and covered with snow $3$ yards deep. Therefore, the total volume of snow is:\n \\[\n 4 \\text{ yards} \\times 10 \\text{ yards} \\times 3 \\text{ yards} = 120 \\text{ cu... |
deepscaler | [
{
"role": "user",
"content": "There are 200 candies. What is the minimum number of schoolchildren that these candies can be distributed to so that, no matter how the candies are distributed, there are always at least two schoolchildren who receive the same number of candies (possibly none)? Let's think step... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 7331,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $c>0$ be a given positive real and $\\mathbb{R}_{>0}$ be the set of all positive reals. Find all functions $f \\colon \\mathbb{R}_{>0} \\to \\mathbb{R}_{>0}$ such that \\[f((c+1)x+f(y))=f(x+2y)+2cx \\quad \\textrm{for all } x,y \\in \\mathbb{R}_{>0}.\\] Let's think step by ... | math | {
"style": "rule",
"ground_truth": "f(x) = 2x"
} | {
"split": "train",
"index": 7332,
"solution": "\nTo solve the functional equation \n\n\\[\nf((c+1)x + f(y)) = f(x + 2y) + 2cx\n\\]\n\nfor all \\( x, y \\in \\mathbb{R}_{>0} \\), we aim to find all functions \\( f \\colon \\mathbb{R}_{>0} \\to \\mathbb{R}_{>0} \\) that satisfy this condition.\n\n### Step 1: Analy... |
deepscaler | [
{
"role": "user",
"content": "The equation $z^6+z^3+1=0$ has complex roots with argument $\\theta$ between $90^\\circ$ and $180^\\circ$ in the complex plane. Determine the degree measure of $\\theta$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "160"
} | {
"split": "train",
"index": 7333,
"solution": "The substitution $y=z^3$ simplifies the equation to $y^2+y+1 = 0$. Applying the quadratic formula gives roots $y=-\\frac{1}{2}\\pm \\frac{\\sqrt{3}i}{2}$, which have arguments of $120$ and $240,$ respectively. We can write them as $z^3 = \\cos 240^\\circ + i\\sin 24... |
deepscaler | [
{
"role": "user",
"content": "Let $R_n$ and $U_n$ be the respective sums of the first $n$ terms of two arithmetic sequences. If the ratio $R_n:U_n = (3n+5):(2n+13)$ for all $n$, find the ratio of the seventh term of the first sequence to the seventh term of the second sequence.\nA) $7:6$\nB) $4:3$\nC) $3:2$... | math | {
"style": "rule",
"ground_truth": "4:3"
} | {
"split": "train",
"index": 7334,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a\\in\\{0,1,2\\}$ and $b\\in\\{-1,1,3,5\\}$, find the probability that the function $f(x)=ax^{2}-2bx$ is increasing on the interval $(1,+\\infty)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{12}"
} | {
"split": "train",
"index": 7335,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $R$ be the set of points $(x, y)$ such that $\\lfloor x^2 \\rfloor = \\lfloor y \\rfloor$ and $\\lfloor y^2 \\rfloor = \\lfloor x \\rfloor$ . Compute the area of region $R$ . Recall that $\\lfloor z \\rfloor$ is the greatest integer that is less than or equal to... | math | {
"style": "rule",
"ground_truth": "4 - 2\\sqrt{2}"
} | {
"split": "train",
"index": 7336,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many unordered pairs of edges of a given octahedron determine a plane? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "66"
} | {
"split": "train",
"index": 7337,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the polar coordinate equation of curve C is ρ - 4cosθ = 0, establish a rectangular coordinate system with the pole as the origin and the polar axis as the positive semi-axis. Line l passes through point M(3, 0) with a slope angle of $\\frac{\\pi}{6}$.\n(I) Find the r... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{15}}{3}"
} | {
"split": "train",
"index": 7338,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are five students, A, B, C, D, and E, arranged to participate in the volunteer services for the Shanghai World Expo. Each student is assigned one of four jobs: translator, guide, etiquette, or driver. Each job must be filled by at least one person. Students A and B cannot... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 7339,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Some mice live in three neighboring houses. Last night, every mouse left its house and moved to one of the other two houses, always taking the shortest route. The numbers in the diagram show the number of mice per house, yesterday and today. How many mice used the path at the b... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 7340,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In isosceles $\\triangle ABC$, $|AB|=|AC|$, vertex $A$ is the intersection point of line $l: x-y+1=0$ with the y-axis, and $l$ bisects $\\angle A$. If $B(1,3)$, find: \n(I) The equation of line $BC$; \n(II) The area of $\\triangle ABC$. Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "\\frac {3}{2}"
} | {
"split": "train",
"index": 7341,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $α \\in (0, \\frac{π}{2})$, and $\\sin (\\frac{π}{6} - α) = -\\frac{1}{3}$, find the value of $\\cos α$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{6} - 1}{6}"
} | {
"split": "train",
"index": 7342,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose you have two bank cards for making purchases: a debit card and a credit card. Today you decided to buy airline tickets worth 20,000 rubles. If you pay for the purchase with the credit card (the credit limit allows it), you will have to repay the bank within $\\mathrm{N}... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 7343,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the curve $C_{1}$ is defined by $\\begin{cases} x=-2+\\cos \\alpha \\\\ y=-1+\\sin \\alpha \\end{cases}$ (where $\\alpha$ is a parameter). In the polar coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ ... | math | {
"style": "rule",
"ground_truth": "\\frac {1}{2}"
} | {
"split": "train",
"index": 7344,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\cos(α-β)= \\frac{3}{5}$, $\\sin β= -\\frac{5}{13}$, and $α∈(0, \\frac{π}{2})$, $β∈(-\\frac{π}{2},0)$, find $\\sin α$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{33}{65}"
} | {
"split": "train",
"index": 7345,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest positive integer $n$ such that for each prime $p$ with $2<p<n$ the difference $n-p$ is also prime. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 7346,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose \n\\[\n3 + \\frac{1}{1 + \\frac{1}{3 + \\frac{3}{4+y}}} = \\frac{169}{53}.\n\\]\nSolve for the value of $y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-605}{119}"
} | {
"split": "train",
"index": 7347,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the complex number $z$ satisfies $z(1-i)=|1-i|+i$, then the imaginary part of $\\overline{z}$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\dfrac{\\sqrt{2}+1}{2}"
} | {
"split": "train",
"index": 7348,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $(5^7+3^6)(1^5-(-1)^4)^{10}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 7349,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a regular tetrahedron embedded in 3-dimensional space, the centers of the four faces are the vertices of a smaller tetrahedron. If the vertices of the larger tetrahedron are located on the surface of a sphere of radius \\(r\\), find the ratio of the volume of the smaller tet... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{27}"
} | {
"split": "train",
"index": 7350,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers $1,2, \\ldots, 20$ are put into a hat. Claire draws two numbers from the hat uniformly at random, $a<b$, and then puts them back into the hat. Then, William draws two numbers from the hat uniformly at random, $c<d$. Let $N$ denote the number of integers $n$ that sat... | math | {
"style": "rule",
"ground_truth": "\\frac{181}{361}"
} | {
"split": "train",
"index": 7351,
"solution": "The number of integers that satisfy exactly one of the two inequalities is equal to the number of integers that satisfy the first one, plus the number of integers that satisfy the second one, minus twice the number of integers that satisfy both. Parity-wise, this is... |
deepscaler | [
{
"role": "user",
"content": "A right triangle has legs of lengths 3 and 4. Find the volume of the solid formed by revolving the triangle about its hypotenuse. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{48\\pi}{5}"
} | {
"split": "train",
"index": 7352,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Certain integers, when divided by $\\frac{3}{5}, \\frac{5}{7}, \\frac{7}{9}, \\frac{9}{11}$, result in a mixed number where the fractional part is $\\frac{2}{3}, \\frac{2}{5}, \\frac{2}{7}, \\frac{2}{9}$, respectively. Find the smallest integer greater than 1 that satisfies the... | math | {
"style": "rule",
"ground_truth": "316"
} | {
"split": "train",
"index": 7353,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $s(n)$ denote the number of 1's in the binary representation of $n$. Compute $$\\frac{1}{255} \\sum_{0 \\leq n<16} 2^{n}(-1)^{s(n)}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 7354,
"solution": "Notice that if $n<8,(-1)^{s(n)}=(-1) \\cdot(-1)^{s(n+8)}$ so the sum becomes $\\frac{1}{255}\\left(1-2^{8}\\right) \\sum_{0 \\leq n<8} 2^{n}(-1)^{s(n)}=$ 45 ."
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sin\\alpha + \\cos\\alpha = \\frac{\\sqrt{2}}{3}$ and $0 < \\alpha < \\pi$, find the value of $\\tan(\\alpha - \\frac{\\pi}{4})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 7355,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Add $1_3 + 12_3 + 212_3 + 2121_3.$ Express your answer in base 3. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10200_3"
} | {
"split": "train",
"index": 7356,
"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 $m+n$.\n Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 7357,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate plane \\(xOy\\), the circle \\(\\Omega\\) and the parabola \\(\\Gamma: y^2 = 4x\\) have exactly one common point, and the circle \\(\\Omega\\) is tangent to the \\(x\\)-axis at the focus \\(F\\) of the parabola \\(\\Gamma\\). Find the radius of the c... | math | {
"style": "rule",
"ground_truth": "\\frac{4 \\sqrt{3}}{9}"
} | {
"split": "train",
"index": 7358,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The simplest form of $1 - \\frac{1}{1 + \\frac{a}{1 - a}}$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "a"
} | {
"split": "train",
"index": 7359,
"solution": "We start by simplifying the expression inside the fraction:\n\\[ 1 - \\frac{1}{1 + \\frac{a}{1 - a}}. \\]\n\n1. **Simplify the innermost fraction**:\n \\[ \\frac{a}{1 - a}. \\]\n\n2. **Add this fraction to 1**:\n \\[ 1 + \\frac{a}{1 - a} = \\frac{1 - a + a}{1 - ... |
deepscaler | [
{
"role": "user",
"content": "A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are $189$, $320$, $287$, $234$, $x$, and $y$. Find the greatest possible value of $x+y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "791"
} | {
"split": "train",
"index": 7360,
"solution": "Let the four numbers be $a$, $b$, $c$, and $d$, in no particular order. Adding the pairwise sums, we have $3a+3b+3c+3d=1030+x+y$, so $x+y=3(a+b+c+d)-1030$. Since we want to maximize $x+y$, we must maximize $a+b+c+d$.\n Of the four sums whose values we know, there mu... |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a triangle with $\\angle A = 90^o, \\angle B = 60^o$ and $BC = 1$ cm. Draw outside of $\\vartriangle ABC$ three equilateral triangles $ABD,ACE$ and $BCF$ . Determine the area of $\\vartriangle DEF$ . Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "\\frac{9\\sqrt{3}}{16}"
} | {
"split": "train",
"index": 7361,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), it is known that \\( AC=3 \\) and the three interior angles \\( A \\), \\( B \\), and \\( C \\) form an arithmetic sequence. \n(1) If \\( \\cos C= \\frac {\\sqrt{6}}{3} \\), find \\( AB \\); \n(2) Find the maximum value of the area of \\( \... | math | {
"style": "rule",
"ground_truth": "\\frac {9 \\sqrt {3}}{4}"
} | {
"split": "train",
"index": 7362,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $\\frac{(2112-2021)^2}{169}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 7363,
"solution": "1. **Calculate the difference in the numerator**: \n \\[\n 2112 - 2021 = 91\n \\]\n This is the exact value, not an approximation.\n\n2. **Square the difference**:\n \\[\n (2112 - 2021)^2 = 91^2 = 8281\n \\]\n\n3. **Divide by the denominator**:\n \\[... |
deepscaler | [
{
"role": "user",
"content": "How many four-digit positive integers are multiples of 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1286"
} | {
"split": "train",
"index": 7364,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A collection of 8 cubes consists of one cube with edge-length $k$ for each integer $k, 1 \\le k \\le 8.$ A tower is to be built using all 8 cubes according to the rules:\nAny cube may be the bottom cube in the tower.\nThe cube immediately on top of a cube with edge-length $k$ m... | math | {
"style": "rule",
"ground_truth": "458"
} | {
"split": "train",
"index": 7365,
"solution": "We proceed recursively. Suppose we can build $T_m$ towers using blocks of size $1, 2, \\ldots, m$. How many towers can we build using blocks of size $1, 2, \\ldots, m, m + 1$? If we remove the block of size $m + 1$ from such a tower (keeping all other blocks in orde... |
deepscaler | [
{
"role": "user",
"content": "Find one-fourth of 6 times 8. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 7366,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For what value of the parameter \\( p \\) will the sum of the squares of the roots of the equation\n\n\\[ p x^{2}+(p^{2}+p) x-3 p^{2}+2 p=0 \\]\n\nbe the smallest? What is this smallest value? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.10"
} | {
"split": "train",
"index": 7367,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The difference between a $6.5\\%$ sales tax and a $6\\%$ sales tax on an item priced at $\\$20$ before tax is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$0.10"
} | {
"split": "train",
"index": 7368,
"solution": "1. **Identify the problem**: We need to find the difference between a $6.5\\%$ sales tax and a $6\\%$ sales tax on an item priced at $20.\n\n2. **Convert percentages to decimal form**: \n - $6.5\\%$ as a decimal is $0.065$.\n - $6\\%$ as a decimal is $0.06$.\n\n... |
deepscaler | [
{
"role": "user",
"content": "A chunk fell out of a dictionary. The first page of the chunk has the number 213, and the number of the last page is written using the same digits in a different order. How many pages are in the chunk that fell out? Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 7369,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $G$ be the centroid of triangle $ABC,$ and let $P$ be an arbitrary point. Then there exists a constant $k$ so that\n\\[PA^2 + PB^2 + PC^2 = k \\cdot PG^2 + GA^2 + GB^2 + GC^2.\\]Find $k.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 7370,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence \\(\\left\\{a_{n}\\right\\}\\) satisfies \\(a_{1} = 1\\) and \\(a_{n} = \\frac{1}{4 a_{n-1}} + \\frac{1}{n}\\) for \\(n \\geq 2\\). Find the value of \\(a_{100}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{101}{200}"
} | {
"split": "train",
"index": 7371,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The slope angle of the tangent line to the curve $y=x\\cos x$ at $x=0$ is what angle? 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": 7372,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The polynomial \\( f(x)=x^{2007}+17 x^{2006}+1 \\) has distinct zeroes \\( r_{1}, \\ldots, r_{2007} \\). A polynomial \\( P \\) of degree 2007 has the property that \\( P\\left(r_{j}+\\frac{1}{r_{j}}\\right)=0 \\) for \\( j=1, \\ldots, 2007 \\). Determine the value of \\( P(1) ... | math | {
"style": "rule",
"ground_truth": "289/259"
} | {
"split": "train",
"index": 7373,
"solution": "For some constant \\( k \\), we have \\( P(z)=k \\prod_{j=1}^{2007}\\left(z-\\left(r_{j}+\\frac{1}{r_{j}}\\right)\\right) \\). Now writing \\( \\omega^{3}=1 \\) with \\( \\omega \\neq 1 \\), we have \\( \\omega^{2}+\\omega=-1 \\). Then \\( P(1) / P(-1)=\\frac{k \\pr... |
deepscaler | [
{
"role": "user",
"content": "Let $m$ and $n$ be integers such that $m + n$ and $m - n$ are prime numbers less than $100$ . Find the maximal possible value of $mn$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2350"
} | {
"split": "train",
"index": 7374,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An equilateral triangle is subdivided into 4 smaller equilateral triangles. Using red and yellow to paint the vertices of the triangles, each vertex must be colored and only one color can be used per vertex. If two colorings are considered the same when they can be made identic... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 7375,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\frac{8^x}{4^{x+y}}=16$ and $\\frac{16^{x+y}}{4^{7y}}=1024$, find $x+y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 7376,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As shown in the diagram, square ABCD and square EFGH have their corresponding sides parallel to each other. Line CG is extended to intersect with line BD at point I. Given that BD = 10, the area of triangle BFC is 3, and the area of triangle CHD is 5, what is the length of BI? ... | math | {
"style": "rule",
"ground_truth": "15/4"
} | {
"split": "train",
"index": 7377,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Shelby drives her scooter at a speed of $30$ miles per hour if it is not raining, and $20$ miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of $16$ miles in $40$ minutes. How many minutes did she drive in the... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 7378,
"solution": "1. **Define Variables:**\n Let $x$ be the time (in minutes) Shelby drove in the rain. Therefore, the time she drove when it was not raining is $40 - x$ minutes.\n\n2. **Convert Speeds to Miles per Minute:**\n - Shelby's speed in non-rainy conditions is $30$ mile... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = x^3 + ax^2 + bx + a^2$ has an extremum at $x = 1$ with the value of 10, find the values of $a$ and $b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-11"
} | {
"split": "train",
"index": 7379,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A dormitory is installing a shower room for 100 students. How many shower heads are economical if the boiler preheating takes 3 minutes per shower head, and it also needs to be heated during the shower? Each group is allocated 12 minutes for showering. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 7380,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A spider is on the edge of a ceiling of a circular room with a radius of 65 feet. The spider moves straight across the ceiling to the opposite edge, passing through the circle's center. It then moves directly to another point on the edge of the circle, not passing through the c... | math | {
"style": "rule",
"ground_truth": "220 + 20\\sqrt{22}"
} | {
"split": "train",
"index": 7381,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\cos ( \\frac {π}{6}+α) \\cdot \\cos ( \\frac {π}{3}-α)=- \\frac {1}{4}$, where $α \\in ( \\frac {π}{3}, \\frac {π}{2})$, find the value of $\\sin 2α$ and the value of $\\tan α - \\frac {1}{\\tan α}$. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 7382,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"split": "train",
"index": 7383,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a triangle, the lengths of the three sides are $4$, $x$, and $12-x$.<br/>$(1)$ The range of values for $x$ is ______;<br/>$(2)$ If this is an isosceles triangle, then the perimeter of the isosceles triangle is ______. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 7384,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $P$ is a moving point on the parabola $y^{2}=4x$. The minimum value of the sum of the distances from point $P$ to point $A(0,-1)$ and from point $P$ to the line $x=-1$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 7385,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Arrange the sequence $\\{2n+1\\} (n\\in\\mathbb{N}^{*})$ sequentially in brackets with one number in the first bracket, two numbers in the second bracket, three numbers in the third bracket, four numbers in the fourth bracket, one number in the fifth bracket, and so on in a cyc... | math | {
"style": "rule",
"ground_truth": "2392"
} | {
"split": "train",
"index": 7386,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rectangle ABCD and right triangle AEF share side AD and have the same area. Side AD = 8, and side AB = 7. If EF, which is perpendicular to AD, is denoted as x, determine the length of hypotenuse AF. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{65}"
} | {
"split": "train",
"index": 7387,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a windless day, a polar bear found itself on a small ice floe that broke off from an iceberg, floating in still water. Rescuers from a helicopter hovering above the floe noted that the animal was walking in a circle with a diameter of 9.5 meters. They were surprised when lat... | math | {
"style": "rule",
"ground_truth": "11400"
} | {
"split": "train",
"index": 7388,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the absolute value of the difference between the squares of 103 and 97? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1200"
} | {
"split": "train",
"index": 7389,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The lines $l_1$ and $l_2$ are two tangents to the circle $x^2+y^2=2$. If the intersection point of $l_1$ and $l_2$ is $(1,3)$, then the tangent of the angle between $l_1$ and $l_2$ equals \\_\\_\\_\\_\\_\\_. 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": 7390,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a given positive integer $n$, we define $\\varphi(n)$ to be the number of positive integers less than or equal to $n$ which share no common prime factors with $n$. Find all positive integers $n$ for which $\\varphi(2019 n)=\\varphi\\left(n^{2}\\right)$. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "1346, 2016, 2019"
} | {
"split": "train",
"index": 7391,
"solution": "Let $p_{1}, p_{2}, \\ldots, p_{k}$ be the prime divisors of $n$. Then it is known that $\\varphi(n)=n \\cdot \\frac{p_{1}-1}{p_{1}} \\ldots \\frac{p_{k}-1}{p_{k}}$. As $n^{2}$ and $n$ has the same set of prime divisors, it also holds that $\\varphi\\left(n^{2}\\righ... |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{a},$ $\\mathbf{b},$ $\\mathbf{c}$ be unit vectors such that\n\\[\\mathbf{a} \\times (\\mathbf{b} \\times \\mathbf{c}) = \\frac{\\mathbf{b} + \\mathbf{c}}{\\sqrt{2}},\\]and such that $\\{\\mathbf{a}, \\mathbf{b}, \\mathbf{c}\\}$ is a linearly independent set.\n\nFi... | math | {
"style": "rule",
"ground_truth": "135^\\circ"
} | {
"split": "train",
"index": 7392,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}=(2\\cos x,1)$, $\\overrightarrow{b}=(\\sqrt{3}\\sin x+\\cos x,-1)$, and the function $f(x)=\\overrightarrow{a}\\cdot\\overrightarrow{b}$.\n\n1. Find the maximum and minimum values of $f(x)$ in the interval $[0,\\frac{\\pi}{4}]$.\n2. If $f(x_{0... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 7393,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A Vandal and a Moderator are editing a Wikipedia article. The article originally is error-free. Each day, the Vandal introduces one new error into the Wikipedia article. At the end of the day, the moderator checks the article and has a $2 / 3$ chance of catching each individual... | math | {
"style": "rule",
"ground_truth": "\\frac{416}{729}"
} | {
"split": "train",
"index": 7394,
"solution": "Consider the error that was introduced on day 1. The probability that the Moderator misses this error on all three checks is $1 / 3^{3}$, so the probability that this error gets removed is $1-\\frac{1}{3^{3}}$. Similarly, the probability that the moderator misses th... |
deepscaler | [
{
"role": "user",
"content": "Assuming that the clock hands move without jumps, determine how many minutes after the clock shows 8:00 will the minute hand catch up with the hour hand. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "43 \\frac{7}{11}"
} | {
"split": "train",
"index": 7395,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), sides \\(AB\\) and \\(BC\\) are equal, \\(AC = 2\\), and \\(\\angle ACB = 30^\\circ\\). From vertex \\(A\\), the angle bisector \\(AE\\) and the median \\(AD\\) are drawn to the side \\(BC\\). Find the area of triangle \\(ADE\\). Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\sqrt{3} - 3}{6}"
} | {
"split": "train",
"index": 7396,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), angle \\(B\\) is \\(120^\\circ\\), and \\(AB = 2BC\\). The perpendicular bisector of side \\(AB\\) intersects \\(AC\\) at point \\(D\\). Find the ratio \\(AD:DC\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3/2"
} | {
"split": "train",
"index": 7397,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $K$ be the product of all factors $(b-a)$ (not necessarily distinct) where $a$ and $b$ are integers satisfying $1\\le a < b \\le 20$. Find the greatest positive integer $n$ such that $2^n$ divides $K$. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 7398,
"solution": "In general, there are $20-n$ pairs of integers $(a, b)$ that differ by $n$ because we can let $b$ be any integer from $n+1$ to $20$ and set $a$ equal to $b-n$. Thus, the product is $(1^{19})(2^{18})\\cdots(19^1)$ (or alternatively, $19! \\cdot 18! \\cdots 1!$.)\n Wh... |
deepscaler | [
{
"role": "user",
"content": "If $N$, written in base $2$, is $11000$, the integer immediately preceding $N$, written in base $2$, is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10111"
} | {
"split": "train",
"index": 7399,
"solution": "To find the integer immediately preceding $N$ in binary, we need to subtract $1$ from $N$.\n\n1. **Convert $N$ from binary to decimal**: \n The binary number $11000_2$ can be converted to decimal as follows:\n \\[\n 1 \\cdot 2^4 + 1 \\cdot 2^3 + 0 \\cdot 2^2 +... |
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