sungyub/dapo-math-17k-verl huggingface dataset:
user: Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB'=3 \cdot AB$. Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC'=3 \cdot BC$, and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA'=3 \cdot CA$. Find the ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$. The original answer is in the format m:n, please give the value of m + n.

Answer: 38
user: Let $N=123456789101112\dots4344$ be the $79$-digit number that is formed by writing the integers from $1$ to $44$ in order, one after the other. What is the remainder when $N$ is divided by $45$?

Answer: 9
user: For any positive integer $M$, the notation $M!$ denotes the product of the integers $1$ through $M$. What is the largest integer $n$ for which $5^n$ is a factor of the sum $98!+99!+100!$? Find the value of $n$.

Answer: 26
user: The number $21! = 51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd? Express your answer in the form of \frac{k}{m} and find the value of k + m.

Answer: 20
user: How many ordered pairs $(a,b)$ such that $a$ is a positive real number and $b$ is an integer between $2$ and $200$, inclusive, satisfy the equation $(\log_b a)^{2017}=\log_b(a^{2017})?$

Answer: 597
user: Real numbers \(x\) and \(y\) are chosen independently and uniformly at random from the interval \((0,1)\). Find the probability that \(\lfloor\log_2x\rfloor=\lfloor\log_2y\rfloor\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).

Answer: 4
user: An integer between $1000$ and $9999$, inclusive, is chosen at random. Find the probability that it is an odd integer whose digits are all distinct. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 281
user: A square with side length \(x\) is inscribed in a right triangle with sides of length \(3\), \(4\), and \(5\) so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length \(y\) is inscribed in another right triangle with sides of length \(3\), \(4\), and \(5\) so that one side of the square lies on the hypotenuse of the triangle. Find the value of \(\dfrac{x}{y}\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).

Answer: 72
user: In \(\triangle ABC\), \(AB=6\), \(AC=8\), \(BC=10\), and \(D\) is the midpoint of \(\overline{BC}\). Find the sum of the radii of the circles inscribed in \(\triangle ADB\) and \(\triangle ADC\). The original answer is in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please give the value of \(k + m\).

Answer: 23
user: Last year, Isabella took 7 math tests and received 7 different scores, each an integer between 91 and 100, inclusive. After each test, she noticed that the average of her test scores was an integer. Her score on the seventh test was 95. What was her score on the sixth test?

Answer: 100
user: Sides \(\overline{AB}\) and \(\overline{AC}\) of equilateral triangle \(ABC\) are tangent to a circle at points \(B\) and \(C\) respectively. Find the fraction of the area of \(\triangle ABC\) that lies outside the circle. The original answer is in the form \(\frac{k}{m}-\frac{n\sqrt{3}\pi}{p}\). Please give the value of \(k + m + n + p\).

Answer: 38
user: The diameter $\overline{AB}$ of a circle of radius $2$ is extended to a point $D$ outside the circle so that $BD=3$. Point $E$ is chosen so that $ED=5$ and line $ED$ is perpendicular to line $AD$. Segment $\overline{AE}$ intersects the circle at a point $C$ between $A$ and $E$. Find the area of $\triangle ABC$. The original answer is in the format \frac{k}{m}, please give the value of k + m.

Answer: 177
user: A square is drawn in the Cartesian coordinate plane with vertices at $(2, 2)$, $(-2, 2)$, $(-2, -2)$, $(2, -2)$. A particle starts at $(0,0)$. Every second it moves with equal probability to one of the eight lattice points (points with integer coordinates) closest to its current position, independently of its previous moves. In other words, the probability is $\frac{1}{8}$ that the particle will move from $(x, y)$ to each of $(x, y + 1)$, $(x + 1, y + 1)$, $(x + 1, y)$, $(x + 1, y - 1)$, $(x, y - 1)$, $(x - 1, y - 1)$, $(x - 1, y)$, or $(x - 1, y + 1)$. The particle will eventually hit the square for the first time, either at one of the $4$ corners of the square or at one of the $12$ lattice points in the interior of one of the sides of the square. The probability that it will hit at a corner rather than at an interior point of a side is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?

Answer: 39
user: How many triangles with positive area have all their vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $5$, inclusive?

Answer: 2148
user: Let $N=123456789101112\dots4344$ be the $79$-digit number that is formed by writing the integers from $1$ to $44$ in order, one after the other. Find the remainder when $N$ is divided by $45$.

Answer: 9
user: The graph of \(y=f(x)\), where \(f(x)\) is a polynomial of degree \(3\), contains points \(A(2,4)\), \(B(3,9)\), and \(C(4,16)\). Lines \(AB\), \(AC\), and \(BC\) intersect the graph again at points \(D\), \(E\), and \(F\), respectively, and the sum of the \(x\)-coordinates of \(D\), \(E\), and \(F\) is \(24\). The original answer is in \(\frac{k}{m}\) format. Please give the value of \(k + m\) for \(f(0)\).

Answer: 29
user: Each day for four days, Linda traveled for one hour at a speed that resulted in her traveling one mile in an integer number of minutes. Each day after the first, her speed decreased so that the number of minutes to travel one mile increased by $5$ minutes over the preceding day. Each of the four days, her distance traveled was also an integer number of miles. What was the total number of miles for the four trips?

Answer: 25
user: For certain real numbers \(a\), \(b\), and \(c\), the polynomial \[g(x) = x^3 + ax^2 + x + 10\] has three distinct roots, and each root of \(g(x)\) is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 100x + c.\] What is the value of \(f(1)\)?

Answer: -7007
user: Quadrilateral $ABCD$ has right angles at $B$ and $C$, $\triangle ABC \sim \triangle BCD$, and $AB > BC$. There is a point $E$ in the interior of $ABCD$ such that $\triangle ABC \sim \triangle CEB$ and the area of $\triangle AED$ is $17$ times the area of $\triangle CEB$. The original answer for \(\tfrac{AB}{BC}\) is in the form \(k + \sqrt{m}\). Please find the value of \(k + m\).

Answer: 7
user: Mrs. Sanders has three grandchildren, who call her regularly. One calls her every three days, one calls her every four days, and one calls her every five days. All three called her on December 31, 2016. Find the number of days during the next year when she did not receive a phone call from any of her grandchildren.

Answer: 146
user: How many integers between $100$ and $999$, inclusive, have the property that some permutation of its digits is a multiple of $11$ between $100$ and $999$? For example, both $121$ and $211$ have this property. Provide your answer as an integer.

Answer: 226
user: Last year Isabella took $7$ math tests and received $7$ different scores, each an integer between $91$ and $100$, inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was $95$. What was her score on the sixth test?

Answer: 100
user: The vertices \(V\) of a centrally symmetric hexagon in the complex plane are given by 
\[ V=\left\{ \sqrt{2}i,-\sqrt{2}i, \frac{1}{\sqrt{8}}(1+i),\frac{1}{\sqrt{8}}(-1+i),\frac{1}{\sqrt{8}}(1-i),\frac{1}{\sqrt{8}}(-1-i) \right\}. \] 
For each \(j\), \(1\leq j\leq 12\), an element \(z_j\) is chosen from \(V\) at random, independently of the other choices. Let \(P={\prod}_{j=1}^{12}z_j\) be the product of the \(12\) numbers selected. Find the probability that \(P=-1\). The original answer is in the form \(\frac{k}{m^n}\), please provide the value of \(k + m + n\).

Answer: 233
user: What is the value of

\(\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1?\)

Provide your answer in the form of \(\frac{k}{m}\), where the fraction is in the simplest form. Calculate the value of \(k + m\).

Answer: 18
user: While exploring a cave, Carl comes across a collection of $5$-pound rocks worth $14 each, $4$-pound rocks worth $11 each, and $1$-pound rocks worth $2 each. There are at least $20$ of each size. He can carry at most $18$ pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?

Answer: 50
user: A unit of blood expires after $10!=10\cdot 9 \cdot 8 \cdots 1$ seconds. Yasin donates a unit of blood at noon of January $1$. On what day does his unit of blood expire? Provide the answer by converting the month to its corresponding number (e.g., January is 1, February is 2, etc.) and the day to a number, then give the sum of these two numbers.

Answer: 14
user: How many ways can a student schedule $3$ mathematics courses -- algebra, geometry, and number theory -- in a $6$-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other $3$ periods is of no concern here.) Find the total number of ways.

Answer: 24
user: A three-dimensional rectangular box with dimensions $X$, $Y$, and $Z$ has faces whose surface areas are $24$, $24$, $48$, $48$, $72$, and $72$ square units. What is $X + Y + Z$?

Answer: 22
user: A box contains $5$ chips, numbered $1$, $2$, $3$, $4$, and $5$. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds $4$. Find the probability that $3$ draws are required. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 6
user: For positive integers $m$ and $n$ such that $m+10<n+1$, both the mean and the median of the set $\{m, m+4, m+10, n+1, n+2, 2n\}$ are equal to $n$. What is the value of $m+n$?

Answer: 21
user: For how many (not necessarily positive) integer values of $n$ is the value of $4000\cdot \left(\tfrac{2}{5}\right)^n$ an integer?

Answer: 9
user: In the figure below, $N$ congruent semicircles lie on the diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let $A$ be the combined area of the small semicircles and $B$ be the area of the region inside the large semicircle but outside the semicircles. The ratio $A:B$ is $1:18$. What is the value of $N$?

Answer: 19
user: What is the value of 
$$\log_3 7 \cdot \log_5 9 \cdot \log_7 11 \cdot \log_9 13 \cdots \log_{21} 25 \cdot \log_{23} 27?$$

Answer: 6
user: Line segment $\overline{AB}$ is a diameter of a circle with $AB = 24$. Point $C$, not equal to $A$ or $B$, lies on the circle. As point $C$ moves around the circle, the centroid (center of mass) of $\triangle ABC$ traces out a closed curve missing two points. To the nearest positive integer, find the area of the region bounded by this curve.

Answer: 50
user: The faces of each of $7$ standard dice are labeled with the integers from $1$ to $6$. Let $p$ be the probability that when all $7$ dice are rolled, the sum of the numbers on the top faces is $10$. What other sum occurs with the same probability as $p$?

Answer: 39
user: Monica is tiling the floor of her $12$-foot by $16$-foot living room. She plans to place one-foot by one-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will she use in total?

Answer: 87
user: In the rectangular parallelepiped shown, $AB = 3$, $BC = 1$, and $CG = 2$. Point $M$ is the midpoint of $\overline{FG}$. Find the volume of the rectangular pyramid with base $BCHE$ and apex $M$.

Answer: 2
user: How many ordered pairs of real numbers $(x,y)$ satisfy the following system of equations? 
$$\begin{align*} x+3y&=3 \\ \big||x|-|y|\big|&=1 \end{align*}$$ 
Fill in the number of such ordered pairs.

Answer: 3
user: A list of $2018$ positive integers has a unique mode, which occurs exactly $10$ times. What is the least number of distinct values that can occur in the list? Provide your answer as an integer.

Answer: 225
user: A paper triangle with sides of lengths \(3, 4,\) and \(5\) inches, as shown, is folded so that point \(A\) falls on point \(B\). The original answer for the length of the crease is in the form \(\frac{k}{m}\). Please find the value of \(k + m\).

Answer: 23
user: Abby, Bridget, and four of their classmates will be seated in two rows of three for a group picture, as shown. 
$$\begin{array}{cccc}\mathrm{X}&&\mathrm{X}&&\mathrm{X}\\\mathrm{X}&&\mathrm{X}&&\mathrm{X}\end{array}$$ 
If the seating positions are assigned randomly, find the probability that Abby and Bridget are adjacent to each other in the same row or the same column. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 22
user: How many ordered pairs of real numbers $(x,y)$ satisfy the following system of equations?
\[ x+3y=3 \]
\[ \big||x|-|y|\big|=1 \]

Answer: 3
user: 令 $a=\sin 14^{\circ}+\cos 14^{\circ}, b=\sin 16^{\circ}+\cos 16^{\circ}, c=\frac{1}{2}\left(a^{2}+b^{2}\right)$，请找出 $a, b, c$ 的大小顺序，并用字母x, y, z分别表示a, b, c的大小顺序（x表示最小，z表示最大）。请计算x + y + z的值，其中x, y, z分别对应大小顺序的数值：x=1, y=2, z=3。

Answer: 6
user: 在直三棱柱 $A_{1} B_{1} C_{1}-A B C$ 中, $\angle B A C=\frac{\pi}{2}, A B=A C=A A_{1}=1$. 已知 $G$ 与 $E$ 分别为 $A_{1} B_{1}$ 和 $C C_{1}$ 的中点, $D$ 与 $F$ 分别为线段 $A C$ 和 $A B$ 上的动点(不包括端点). 若 $G D \perp E F$, 求线段 $D F$ 的长度的最小值和最大值（不包括最大值），并分别用$x$和$y$表示。答案的形式为区间$[ \frac{k}{\sqrt{m}}, n )$，请给出$k + m + n$的值。

Answer: 7
user: Side $\overline{AB}$ of $\triangle ABC$ has length $10$. The bisector of angle $A$ meets $\overline{BC}$ at $D$, and $CD = 3$. The set of all possible values of $AC$ is an open interval $(m,n)$. What is the value of $m+n$?

Answer: 18
user: The clock in Sri's car, which is not accurate, gains time at a constant rate. One day as he begins shopping, he notes that his car clock and his watch (which is accurate) both say $12:00$ noon. When he is done shopping, his watch says $12:30$ and his car clock says $12:35$. Later that day, Sri loses his watch. He looks at his car clock and it says $7:00$. What is the actual time in hours and minutes? If the actual time is $a$ hours and $b$ minutes, what is the value of $a + b$?

Answer: 6
user: A paper triangle with sides of lengths \(3, 4,\) and \(5\) inches, as shown, is folded so that point \(A\) falls on point \(B\). The original answer for the length of the crease is in \(\frac{m}{n}\) format, where \(m\) and \(n\) are integers in simplest form. Please find the value of \(m + n\).

Answer: 23
user: How many of the first $2018$ numbers in the sequence $101, 1001, 10001, 100001, \ldots$ are divisible by $101$? Fill in the blank with the correct number.

Answer: 505
user: How many nonnegative integers can be written in the form $$a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2+a_1\cdot3^1+a_0\cdot3^0,$$
where $a_i\in \{-1,0,1\}$ for $0\le i \le 7$? Find the total number of such integers.

Answer: 3281
user: Square $ABCD$ has side length $30$. Point $P$ lies inside the square so that $AP = 12$ and $BP = 26$. The centroids of $\triangle{ABP}$, $\triangle{BCP}$, $\triangle{CDP}$, and $\triangle{DAP}$ are the vertices of a convex quadrilateral. What is the area of that quadrilateral?

Answer: 200
user: Lailai took five math tests, each worth a maximum of 100 points. Lailai's score on each test was an integer between 0 and 100, inclusive. Lailai received the same score on the first four tests, and she received a higher score on the last test. Her average score on the five tests was 82. How many values are possible for Lailai's score on the last test?

Answer: 4
user: What is the greatest integer less than or equal to 
$$\frac{3^{100}+2^{100}}{3^{96}+2^{96}}?$$ 
Find this integer.

Answer: 80
user: The solution to the equation $\log_{3x} 4 = \log_{2x} 8$, where $x$ is a positive real number other than $\frac{1}{3}$ or $\frac{1}{2}$, can be written as $\frac {p}{q}$ where $p$ and $q$ are relatively prime positive integers. What is $p + q$?

Answer: 31
user: Joey and Chloe and their daughter Zoe all have the same birthday. Joey is $1$ year older than Chloe, and Zoe is exactly $1$ year old today. Today is the first of the $9$ birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?

Answer: 11
user: Let $N$ be the greatest five-digit number whose digits have a product of $120$. What is the sum of the digits of $N$?

Answer: 18
user: Two circles of radius $5$ are externally tangent to each other and are internally tangent to a circle of radius $13$ at points $A$ and $B$, as shown in the diagram. The distance $AB$ can be written in the form $\frac{k}{l}$, where $k$ and $l$ are relatively prime positive integers. Find the value of $k+l$.

Answer: 69
user: A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point \(A\) in the figure on the right. The box has base length \(w\) and height \(h\). Find the area of the sheet of wrapping paper in terms of \(w\) and \(h\). The answer should be in the form \(a(w+h)^b\). Please find the value of \(a + b\).

Answer: 4
user: A scanning code consists of a $7 \times 7$ grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of $49$ squares. A scanning code is called $\textit{symmetric}$ if its look does not change when the entire square is rotated by a multiple of $90 ^{\circ}$ counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?

Answer: 1022
user: How many odd positive $3$-digit integers are divisible by $3$ but do not contain the digit $3$? Provide your answer as an integer.

Answer: 96
user: Let $a_1,a_2,\dots,a_{2018}$ be a strictly increasing sequence of positive integers such that \[a_1+a_2+\cdots+a_{2018}=2018^{2018}.\]
Find the remainder when $a_1^3+a_2^3+\cdots+a_{2018}^3$ is divided by $6$.

Answer: 4
user: The solutions to the equation \((z+6)^8=81\) are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled \(A,B,\) and \(C\). Find the least possible area of \(\triangle ABC\). The original answer is in the form \(\frac{k}{m}\sqrt{n} - \frac{k}{m}\), where k, m, and n are integers. Please provide the sum of k, m, and n.

Answer: 7
user: Professor Chang has nine different language books lined up on a bookshelf: two Arabic, three German, and four Spanish. Find the number of ways to arrange the nine books on the shelf keeping the Arabic books together and keeping the Spanish books together.

Answer: 5760
user: In rectangle $PQRS$, $PQ=8$ and $QR=6$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, points $E$ and $F$ lie on $\overline{RS}$, and points $G$ and $H$ lie on $\overline{SP}$ so that $AP=BQ<4$ and the convex octagon $ABCDEFGH$ is equilateral. The length of a side of this octagon can be expressed in the form $k+m\sqrt{n}$, where $k$, $m$, and $n$ are integers and $n$ is not divisible by the square of any prime. What is the value of $k+m+n$?

Answer: 7
user: Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths \(3\) and \(4\) units. In the corner where those sides meet at a right angle, he leaves a small unplanted square \(S\) so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from \(S\) to the hypotenuse is \(2\) units. What fraction of the field is planted? The original answer is in \(\frac{m}{n}\) format, please give the value of \(m + n\).

Answer: 292
user: Let $p$ and $q$ be positive integers such that \[\frac{5}{9} < \frac{p}{q} < \frac{4}{7}\] and $q$ is as small as possible. Find the value of $q-p$.

Answer: 7
user: How many nonnegative integers can be written in the form 
$$a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2+a_1\cdot3^1+a_0\cdot3^0,$$
where $a_i\in \{-1,0,1\}$ for $0\le i \le 7$?

Answer: 3281
user: Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. Find the area of quadrilateral $FDBG$.

Answer: 75
user: A function $f$ is defined recursively by $f(1)=f(2)=1$ and \[f(n)=f(n-1)-f(n-2)+n\] for all integers $n \geq 3$. Find the value of $f(2018)$.

Answer: 2017
user: A number $m$ is randomly selected from the set $\{11,13,15,17,19\}$, and a number $n$ is randomly selected from $\{1999,2000,2001,\cdots,2018\}$. Find the probability that $m^n$ has a units digit of $1$. Express your answer in the form of a fraction \frac{k}{l} in simplest form, and provide the value of k + l.

Answer: 7
user: Let $A$ be the set of positive integers that have no prime factors other than $2$, $3$, or $5$. The infinite sum \[\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{12} + \frac{1}{15} + \frac{1}{16} + \frac{1}{18} + \frac{1}{20} + \cdots\] of the reciprocals of the elements of $A$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the value of $m+n$.

Answer: 19
user: Mary chose an even $4$-digit number $n$. She wrote down all the divisors of $n$ in increasing order from left to right: $1,2,\ldots,\dfrac{n}{2},n$. At some moment Mary wrote $323$ as a divisor of $n$. What is the smallest possible value of the next divisor written to the right of $323$?

Answer: 340
user: In a sign pyramid, a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. Find the number of possible ways to fill the four cells in the bottom row to produce a "+" at the top of the pyramid.

Answer: 8
user: A scanning code consists of a $7 \times 7$ grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of $49$ squares. A scanning code is called symmetric if its look does not change when the entire square is rotated by a multiple of $90 ^{\circ}$ counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?

Answer: 1022
user: Triangle $ABC$ is an isosceles right triangle with $AB=AC=3$. Let $M$ be the midpoint of hypotenuse $\overline{BC}$. Points $I$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively, so that $AI>AE$ and $AIME$ is a cyclic quadrilateral. Given that triangle $EMI$ has area $2$, the length $CI$ can be written as $\frac{a-\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers and $b$ is not divisible by the square of any prime. What is the value of $a+b+c$?

Answer: 12
user: Let \(ABCDEF\) be a regular hexagon with side length \(1\). Denote by \(X\), \(Y\), and \(Z\) the midpoints of sides \(\overline{AB}\), \(\overline{CD}\), and \(\overline{EF}\), respectively. Find the area of the convex hexagon whose interior is the intersection of the interiors of \(\triangle ACE\) and \(\triangle XYZ\). The original answer is in the form \(\frac{k}{m}\sqrt{n}\), where \(\frac{k}{m}\) is in its simplest form. Please give the value of \(k + m + n\).

Answer: 50