meetforever/GRPO_CL huggingface dataset:
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
user: In \(\triangle ABC\), a point \(E\) is on \(\overline{AB}\) with \(AE=1\) and \(EB=2.\) Point \(D\) is on \(\overline{AC}\) so that \(\overline{DE} \parallel \overline{BC}\) and point \(F\) is on \(\overline{BC}\) so that \(\overline{EF} \parallel \overline{AC}.\) The ratio of the area of \(CDEF\) to the area of \(\triangle ABC\) is in the form \(\frac{k}{m}\). Find the value of \(k + m\).
Answer: 13
user: In \(\triangle{ABC}\) with side lengths \(AB = 13\), \(AC = 12\), and \(BC = 5\), let \(O\) and \(I\) denote the circumcenter and incenter, respectively. A circle with center \(M\) is tangent to the legs \(AC\) and \(BC\) and to the circumcircle of \(\triangle{ABC}\). Find the area of \(\triangle{MOI}\). The answer is in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is in simplest form. Please find the value of \(k + m\).
Answer: 9
user: How many positive three-digit integers have a remainder of $2$ when divided by $6$, a remainder of $5$ when divided by $9$, and a remainder of $7$ when divided by $11$?
Answer: 5
user: Consider polynomials $P(x)$ of degree at most $3$, each of whose coefficients is an element of $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. How many such polynomials satisfy $P(-1) = -9$? Provide your answer as an integer.
Answer: 220
user: How many ordered pairs $(a, b)$ of positive integers satisfy the equation 
\[a\cdot b + 63 = 20\cdot \text{lcm}(a, b) + 12\cdot\text{gcd}(a,b),\]
where $\text{gcd}(a,b)$ denotes the greatest common divisor of $a$ and $b$, and $\text{lcm}(a,b)$ denotes their least common multiple?
Answer: 2
user: In $\triangle PAT,$ $\angle P=36^{\circ},$ $\angle A=56^{\circ},$ and $PA=10.$ Points $U$ and $G$ lie on sides $\overline{TP}$ and $\overline{TA},$ respectively, so that $PU=AG=1.$ Let $M$ and $N$ be the midpoints of segments $\overline{PA}$ and $\overline{UG},$ respectively. If the answer is $x^\circ$, what is the value of $x$ for the degree measure of the acute angle formed by lines $MN$ and $PA?$
Answer: 80
user: Ajay is standing at point $A$ near Pontianak, Indonesia, $0^\circ$ latitude and $110^\circ \text{ E}$ longitude. Billy is standing at point $B$ near Big Baldy Mountain, Idaho, USA, $45^\circ \text{ N}$ latitude and $115^\circ \text{ W}$ longitude. Assume that Earth is a perfect sphere with center $C.$ What is the degree measure of $\angle ACB$? If the answer is $x^\circ$, what is the value of $x$?
Answer: 120
user: Alice, Bob, and Carol play a game in which each of them chooses a real number between \(0\) and \(1.\) The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between \(0\) and \(1,\) and Bob announces that he will choose his number uniformly at random from all the numbers between \(\frac{1}{2}\) and \(\frac{2}{3}.\) Armed with this information, what number should Carol choose to maximize her chance of winning? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 37
user: Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$. Find the number of real numbers $x$ that satisfy the equation $x^2 + 10,000\lfloor x \rfloor = 10,000x$.
Answer: 199
user: In the cube \(ABCDEFGH\) with opposite vertices \(C\) and \(E,\) \(J\) and \(I\) are the midpoints of segments \(\overline{FB}\) and \(\overline{HD},\) respectively. Let \(R\) be the ratio of the area of the cross-section \(EJCI\) to the area of one of the faces of the cube. The original answer is in \(\frac{k}{m}\) format. Please find the value of \(k + m\).
Answer: 5
user: For a positive integer $n$ and nonzero digits $a$, $b$, and $c$, let $A_n$ be the $n$-digit integer each of whose digits is equal to $a$; let $B_n$ be the $n$-digit integer each of whose digits is equal to $b$, and let $C_n$ be the $2n$-digit (not $n$-digit) integer each of whose digits is equal to $c$. Find the greatest possible value of $a + b + c$ for which there are at least two values of $n$ such that $C_n - B_n = A_n^2$.
Answer: 18
user: Circles $\omega_1$, $\omega_2$, and $\omega_3$ each have radius $4$ and are placed in the plane so that each circle is externally tangent to the other two. Points $P_1$, $P_2$, and $P_3$ lie on $\omega_1$, $\omega_2$, and $\omega_3$ respectively such that $P_1P_2=P_2P_3=P_3P_1$ and line $P_iP_{i+1}$ is tangent to $\omega_i$ for each $i=1,2,3$, where $P_4 = P_1$. See the figure below. The area of $\triangle P_1P_2P_3$ can be written in the form $\sqrt{a}+\sqrt{b}$ for positive integers $a$ and $b$. Find the value of $a+b$.
Answer: 552
user: How many perfect cubes lie between $2^8+1$ and $2^{18}+1$, inclusive?
Answer: 58
user: What is the hundreds digit of $(20! - 15!)$?
Answer: 0
user: Three identical rectangles are put together to form rectangle $ABCD$, as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is $5$ feet, what is the area of rectangle $ABCD$ in square feet?
Answer: 150
user: A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls, and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $15$ balls of a single color will be drawn? Provide your answer as an integer.
Answer: 76
user: All lines with equation \(ax+by=c\) such that \(a, b, c\) form an arithmetic progression pass through a common point. The coordinates of that point are \((m, n)\). Please calculate the value of \(m + n\).
Answer: 1
user: What is the greatest number of consecutive integers whose sum is $45$?
Answer: 90
user: What is the greatest number of consecutive integers whose sum is 45?
Answer: 90
user: What is the sum of all real numbers $x$ for which the median of the numbers $4,6,8,17,$ and $x$ is equal to the mean of those five numbers?
Answer: -5
user: The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each triangle has side length $2$ and the third vertices of the triangles meet at the center of the square. The region inside the square but outside the triangles is shaded. Find the area of the shaded region. The original answer is in the form $x - y\sqrt{z}$, please provide the value of $x + y + z$.
Answer: 19
user: For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$?
Answer: 19
user: Let $f(x) = x^{2}(1-x)^{2}$. Find the value of the sum
\[f \left(\frac{1}{2019} \right)-f  \left(\frac{2}{2019} \right)+f \left(\frac{3}{2019} \right)-f \left(\frac{4}{2019} \right)+\cdots + f \left(\frac{2017}{2019} \right) - f \left(\frac{2018}{2019} \right).\]
Answer: 0
user: What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is $\underline{not}$ a divisor of the product of the first $n$ positive integers?
Answer: 996
user: The function $f$ is defined by 
$$f(x) = \lfloor|x|\rfloor - |\lfloor x \rfloor|$$ 
for all real numbers $x$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to the real number $r$. The original answer is a set of distinct integer values that form the range of $f$, in the form \{a, b\}. Please provide the sum of a and b.
Answer: -1
user: A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \frac{3}{7}$, and
\[a_n=\frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}\] for all $n \geq 3$. Then $a_{2019}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find the value of $p+q$.
Answer: 8078
user: A rectangular floor that is $10$ feet wide and $17$ feet long is tiled with $170$ one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit? Provide your answer as an integer.
Answer: 26
user: The figure below shows $13$ circles of radius $1$ within a larger circle. All the intersections occur at points of tangency. Find the area of the region, shaded in the figure, inside the larger circle but outside all the circles of radius $1$. If the answer is in the form $k\sqrt{m}$, what is the value of $k + m$?
Answer: 7
user: For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0.\overline{23}_k = 0.232323..._k$. What is the value of $k$?
Answer: 16
user: What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than $2019$?
Answer: 22
user: Right triangle \(ACD\) with right angle at \(C\) is constructed outwards on the hypotenuse \(\overline{AC}\) of isosceles right triangle \(ABC\) with leg length \(1\), as shown, so that the two triangles have equal perimeters. Find \(\sin(2\angle BAD)\). The original answer is in the form \(\frac{k}{m}\), where k and m are integers, and the fraction is in simplest form. Please provide the value of \(k + m\).
Answer: 16
user: What is the sum of all real numbers $x$ for which the median of the numbers $4, 6, 8, 17,$ and $x$ is equal to the mean of those five numbers?
Answer: -5
user: How many ways are there to paint each of the integers $2, 3, \cdots , 9$ either red, green, or blue so that each number has a different color from each of its proper divisors? Provide your answer as an integer.
Answer: 432