ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: The map below shows an east/west road connecting the towns of Acorn, Centerville, and Midland, and a
north/south road from Centerville to Drake. The distances from Acorn to Centerville, from Centerville to
Midland, and from Centerville to Drake are each 60 kilometers. At noon Aaron starts at Acorn and
bicycles east at 17 kilometers per hour, Michael starts at Midland and bicycles west at 7 kilometers per
hour, and David starts at Drake and bicycles at a constant rate in a straight line across an open ﬁeld. All
three bicyclists arrive at exactly the same time at a point along the road from Centerville to Midland. Find
the number of kilometers that David bicycles. For the map go to http://www.purplecomet.org/welcome/practice
Answer: 65
Problem: Julius has a set of ﬁve positive integers whose mean is 100. If Julius removes the median of the set of ﬁve
numbers, the mean of the set increases by 5, and the median of the set decreases by 5. Find the maximum
possible value of the largest of the ﬁve numbers Julius has.
Answer: 269
Problem: Find the sum of all perfect squares that divide 2016.
Answer: 210
Problem: Jeremy wrote all the three-digit integers from $100$ to $999$ on a blackboard. Then Allison erased each of the
$2700$ digits Jeremy wrote and replaced each digit with the square of that digit. Thus, Allison replaced every
1 with a 1, every 2 with a 4, every 3 with a 9, every 4 with a 16, and so forth. The proportion of all the digits Allison wrote that were ones is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find $m + n.$
Answer: 87
Problem: Let $T_k = \frac{k(k+1)}{2}$ be the $k$-th triangular number. The infinite series
$$\sum_{k=4}^{\infty}\frac{1}{(T_{k-1} - 1)(Tk - 1)(T_{k+1} - 1)}$$
has the value $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 451
Problem: Let $a_0$, $a_1$, ..., $a_6$ be real numbers such that $a_0 + a_1 + ... + a_6 = 1$ and
$$a_0 + a_2 + a_3 + a_4 + a_5 + a_6 =\frac{1}{2}$$
$$a_0 + a_1 + a_3 + a_4 + a_5 + a_6 = \frac{2}{3}$$
$$a_0 + a_1 + a_2 + a_4 + a_5 + a_6 =\frac{7}{8}$$
$$a_0 + a_1 + a_2 + a_3 + a_5 + a_6 =\frac{29}{30}$$
$$a_0 + a_1 + a_2 + a_3 + a_4 + a_6 =\frac{143}{144}$$
$$a_0 + a_1 + a_2 + a_3 + a_4 + a_5 =\frac{839}{840}$$

The value of $a_0$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 5041
Problem: The positive integer $m$ is a multiple of 111, and the positive integer $n$ is a multiple of 31. Their sum is 2017. Find $n - m$.
Answer: 463
Problem: Find the number of rearrangements of the letters in the word MATHMEET that begin and end with the same letter such as TAMEMHET.
Answer: 540
Problem: Find the greatest integer $n < 1000$ for which $4n^3 - 3n$ is the product of two consecutive odd integers.
Answer: 899
Problem: The Stromquist Comet is visible every 61 years. If the comet is visible in 2017, what is the next leap year when the comet will be visible?
Answer: 2444
Problem: Omar has four fair standard six-sided dice. Omar invented a game where he rolls his four dice over and over again until the number 1 does not appear on the top face of any of the dice. Omar wins the game if on that roll the top faces of his dice show at least one 2 and at least one 5. The probability that Omar wins the game is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 819
Problem: Find the number of functions $f$ that map the set $\{1,2, 3,4\}$ into itself such that the range of the function $f(x)$ is the same as the range of the function $f(f(x))$.
Answer: 148
Problem: Find the sum of all values of $a + b$, where $(a, b)$ is an ordered pair of positive integers and $a^2+\sqrt{2017-b^2}$ is a perfect square.
Answer: 67
Problem: Let $a_1 = 1 +\sqrt2$ and for each $n \ge 1$ dene $a_{n+1} = 2 -\frac{1}{a_n}$. Find the greatest integer less than or equal to the product $a_1a_2a_3 ... a_{200}$.
Answer: 283
Problem: On a typical morning Aiden gets out of bed, goes through his morning preparation, rides the bus, and walks from the bus stop to work arriving at work 120 minutes after getting out of bed. One morning Aiden got out of bed late, so he rushed through his morning preparation getting onto the bus in half the usual time, the bus ride took 25 percent longer than usual, and he ran from the bus stop to work in half the usual time it takes him to walk arriving at work 96 minutes after he got out of bed. The next morning Aiden got out of bed extra early, leisurely went through his morning preparation taking 25 percent longer than usual to get onto the bus, his bus ride took 25 percent less time than usual, and he walked slowly from the bus stop to work taking 25 percent longer than usual. How many minutes after Aiden got out of bed did he arrive at work that day?
Answer: 126
Problem: Eight red boxes and eight blue boxes are randomly placed in four stacks of four boxes each. The probability that exactly one of the stacks consists of two red boxes and two blue boxes is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find $m + n$.
Answer: 843
Problem: Paul starts at $1$ and counts by threes: $1, 4, 7, 10, ... $. At the same time and at the same speed, Penny counts backwards from $2017$ by fives: $2017, 2012, 2007, 2002,...$ . Find the one number that both Paul and Penny count at the same time.
Answer: 757
Problem: The set of positive real numbers $x$ that satisfy $2 | x^2 - 9 | \le 9 | x | $ is an interval $[m, M]$. Find $10m + M$.
Answer: 21
Problem: Let $x$, $y$, and $z$ be real numbers such that
$$12x - 9y^2 = 7$$
$$6y - 9z^2 = -2$$
$$12z - 9x^2 = 4$$
Find $6x^2 + 9y^2 + 12z^2$.
Answer: 9
Problem: Find the number of three-element subsets of $\{1, 2, 3,...,13\}$ that contain at least one element that is a multiple of $2$, at least one element that is a multiple of $3$, and at least one element that is a multiple of $5$ such as $\{2,3, 5\}$ or $\{6, 10,13\}$.
Answer: 63
Problem: Find the least positive integer $m$ such that $lcm(15,m) = lcm(42,m)$. Here $lcm(a, b)$ is the least common multiple of $a$ and $b$.
Answer: 70
Problem: Find the greatest prime divisor of $29! + 33!$.
Answer: 991
Problem: Consider an alphabetized list of all the arrangements of the letters in the word BETWEEN. Then BEEENTW would be in position $1$ in the list, BEEENWT would be in position $2$ in the list, and so forth. Find the position that BETWEEN would be in the list.
Answer: 46
Problem: In the $3$-dimensional coordinate space nd the distance from the point $(36, 36, 36)$ to the plane that passes
through the points $(336, 36, 36)$, $(36, 636, 36)$, and $(36, 36, 336)$.
Answer: 200
Problem: The expression $\left(1 + \sqrt[6]{26 + 15\sqrt3} -\sqrt[6]{26 - 15\sqrt3}\right)^6= m + n\sqrt{p}$ , where $m, n$, and $p$ are positive integers, and $p$ is not divisible by the square of any prime. Find $m + n + p$.
Answer: 171
Problem: Leaving his house at noon, Jim walks at a constant rate of $4$ miles per hour along a $4$ mile square route returning to his house at $1$ PM. At a randomly chosen time between noon and $1$ PM, Sally chooses a random location along Jim's route and begins running at a constant rate of $7$ miles per hour along Jim's route in the same direction that Jim is walking until she completes one $4$ mile circuit of the square route.
The probability that Sally runs past Jim while he is walking is given by $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 64
Problem: A right circular cone has a height equal to three times its base radius and has volume 1. The cone is inscribed inside a sphere as shown. The volume of the sphere is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

[center][img]https://snag.gy/92ikv3.jpg[/img][/center]
Answer: 581
Problem: For real numbers $a, b$, and $c$ the polynomial $p(x) = 3x^7 - 291x^6 + ax^5 + bx^4 + cx^2 + 134x - 2$ has $7$ real roots whose sum is $97$. Find the sum of the reciprocals of those $7$ roots.
Answer: 67
Problem: Caden, Zoe, Noah, and Sophia shared a pizza. Caden ate 20 percent of the pizza. Zoe ate 50 percent more of the pizza than Caden ate. Noah ate 50 percent more of the pizza than Zoe ate, and Sophia ate the rest of the pizza. Find the percentage of the pizza that Sophia ate.
Answer: 5
Problem: Find the minimum value of $4(x^2 + y^2 + z^2 + w^2) + (xy - 7)^2 + (yz - 7)^2 + (zw - 7)^2 + (wx - 7)^2$ as $x, y, z$, and $w$ range over all real numbers.
Answer: 96
Problem: Let $a$ be a solution to the equation $\sqrt{x^2 + 2} = \sqrt[3]{x^3 + 45}$. Evaluate the ratio of $\frac{2017}{a^2}$ to $a^2 - 15a + 2$.
Answer: 6
Problem: Let $ABCDE$ be a pentagon with area $2017$ such that four of its sides $AB, BC, CD$, and $EA$ have integer length. Suppose that $\angle A = \angle B = \angle C = 90^o$, $AB = BC$, and $CD = EA$. The maximum possible perimeter of $ABCDE$ is $a + b \sqrt{c}$, where $a$, $b$, and $c$ are integers and $c$ is not divisible by the square of any prime. Find $a + b + c$.
Answer: 178
Problem: There are integers $a_1, a_2, a_3,...,a_{240}$ such that $x(x + 1)(x + 2)(x + 3) ... (x + 239) =\sum_{n=1}^{240}a_nx^n$. Find the number of integers $k$ with $1\le k \le  240$ such that ak is a multiple of $3$.
Answer: 159
Problem: Let $a, b$, and $c$ be real numbers. Let $u = a^2 + b^2 + c^2$ and $v = 2ab + 2bc + 2ca$. Suppose $2018u = 1001v + 1024$. Find the maximum possible value of $35a - 28b - 3c$.
Answer: 32
Problem: Five girls and five boys randomly sit in ten seats that are equally spaced around a circle. The probability that there is at least one diameter of the circle with two girls sitting on opposite ends of the diameter is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 118
Problem: If a and b are in the interval $\left(0, \frac{\pi}{2}\right)$ such that $13(\sin a + \sin b) + 43(\cos a + \cos b) = 2\sqrt{2018}$, then $\tan a + \tan b = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 69
Problem: Find the number of positive integers $k \le 2018$ for which there exist integers $m$ and $n$ so that $k = 2^m + 2^n$.
For example, $64 = 2^5 + 2^5$, $65 = 2^0 + 2^6$, and $66 = 2^1 + 2^6$.
Answer: 66
Problem: Two identical blue blocks, two identical red blocks, two identical green blocks, and two identical purple blocks are placed next to each other in a row. Find the number of distinct arrangements of these blocks where no blue block is placed next to a red block, and no green block is placed next to a purple block.
Answer: 248
Problem: Bradley is driving at a constant speed. When he passes his school, he notices that in $20$ minutes he will be exactly $\frac14$ of the way to his destination, and in $45$ minutes he will be exactly $\frac13$ of the way to his destination. Find the number of minutes it takes Bradley to reach his destination from the point where he passes his school.
Answer: 245
Problem: Find the positive integer $k$ such that the roots of $x^3 - 15x^2 + kx -1105$ are three distinct collinear points in the complex plane.
Answer: 271
Problem: If you roll four standard, fair six-sided dice, the top faces of the dice can show just one value (for example, $3333$), two values (for example, $2666$), three values (for example, $5215$), or four values (for example, $4236$). The mean number of values that show is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 887
Problem: The following diagram shows a grid of $36$ cells. Find the number of rectangles pictured in the diagram that contain at least three cells of the grid.
[img]https://cdn.artofproblemsolving.com/attachments/a/4/e9ba3a35204ec68c17a364ebf92cc107eb4d7a.png[/img]
Answer: 345
Problem: One right pyramid has a base that is a regular hexagon with side length $1$, and the height of the pyramid is $8$. Two other right pyramids have bases that are regular hexagons with side length $4$, and the heights of those pyramids are both $7$. The three pyramids sit on a plane so that their bases are adjacent to each other and meet at a single common vertex. A sphere with radius $4$ rests above the plane supported by these three pyramids. The distance that the center of the sphere is from the plane can be written as $\frac{p\sqrt{q}}{r}$ , where $p, q$, and $r$ are relatively prime positive integers, and $q$ is not divisible by the square of any prime. Find $p+q+r$.
Answer: 88
Problem: Aileen plays badminton where she and her opponent stand on opposite sides of a net and attempt to bat a birdie back and forth over the net. A player wins a point if their opponent fails to bat the birdie over the net. When Aileen is the server (the first player to try to hit the birdie over the net), she wins a point with probability $\frac{9}{10}$ . Each time Aileen successfully bats the birdie over the net, her opponent, independent of all previous hits, returns the birdie with probability $\frac{3}{4}$ . Each time Aileen bats the birdie, independent of all previous hits, she returns the birdie with probability $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 73
Problem: Line segment $\overline{AB}$ has perpendicular bisector $\overline{CD}$, where $C$ is the midpoint of $\overline{AB}$. The segments have lengths $AB = 72$ and $CD = 60$. Let $R$ be the set of points $P$ that are midpoints of line segments $\overline{XY}$ , where $X$ lies on $\overline{AB}$ and $Y$ lies on $\overline{CD}$. Find the area of the region $R$.
Answer: 1080
Problem: In $\vartriangle ABC$ points $D, E$, and $F$ lie on side $\overline{BC}$ such that $\overline{AD}$ is an angle bisector of $\angle BAC$, $\overline{AE}$ is a median, and $\overline{AF}$ is an altitude. Given that $AB = 154$ and $AC = 128$, and $9 \times DE = EF,$ find the side length $BC$.
Answer: 94
Problem: A triangle with side lengths $16$, $18$, and $21$ has a circle with radius $6$ centered at each vertex. Find $n$ so that the total area inside the three circles but outside of the triangle is $n\pi$.
[img]https://4.bp.blogspot.com/-dpCi7Gai3ZE/XoEaKo3C5wI/AAAAAAAALl8/KAuCVDT9R5MiIA_uTfRyoQmohEVw9cuVACK4BGAYYCw/s200/2018%2Bpc%2Bhs2.png[/img]
Answer: 90
Problem: Let $ABCD$ be a square with side length $6$. Circles $X, Y$ , and $Z$ are congruent circles with centers inside the square such that $X$ is tangent to both sides $\overline{AB}$ and $\overline{AD}$, $Y$ is tangent to both sides $\overline{AB}$ and $\overline{BC}$, and $Z$ is tangent to side $\overline{CD}$ and both circles $X$ and $Y$ . The radius of the circle $X$ can be written $m -\sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.
Answer: 195
Problem: One afternoon a bakery finds that it has $300$ cups of  flour and $300$ cups of sugar on hand. Annie and Sam decide to use this to make and sell some batches of cookies and some cakes. Each batch of cookies will require $1$ cup of flour and $3$ cups of sugar. Each cake will require $2$ cups of flour and $1$ cup of sugar. Annie thinks that each batch of cookies should sell for $2$ dollars and each cake for $1$ dollar, but Sam thinks that each batch of cookies should sell for $1$ dollar and each cake should sell for $3$ dollars. Find the difference between the maximum dollars of income they can receive if they use Sam's selling plan and the maximum dollars of income they can receive if they use Annie's selling plan.
Answer: 210
Problem: Find the number of ordered quadruples of positive integers $(a,b,c, d)$ such that $ab + cd = 10$.
Answer: 58
Problem: Let $a, b$, and $c$ be integers simultaneously satisfying the equations $4abc + a + b + c = 2018$ and $ab + bc + ca = -507$. Find $|a| + |b|+ |c|$.
Answer: 46
Problem: A jeweler can get an alloy that is $40\%$ gold for $200$ dollars per ounce, an alloy that is $60\%$ gold for $300$ dollar per ounce, and an alloy that is $90\%$ gold for $400$ dollars per ounce. The jeweler will purchase some of these gold alloy products, melt them down, and combine them to get an alloy that is $50\%$ gold. Find the minimum number of dollars the jeweler will need to spend for each ounce of the alloy she makes.
Answer: 240
Problem: On $\vartriangle ABC$ let $D$ be a point on side $\overline{AB}$, $F$ be a point on side $\overline{AC}$, and $E$ be a point inside the triangle so that $\overline{DE}\parallel  \overline{AC}$ and $\overline{EF} \parallel \overline{AB}$. Given that $AF = 6, AC = 33, AD = 7, AB = 26$, and the area of quadrilateral $ADEF$ is $14$, nd the area of $\vartriangle ABC$.
Answer: 143
Problem: Rectangle $ABCD$ has side lengths $AB = 6\sqrt3$ and $BC = 8\sqrt3$. The probability that a randomly chosen point inside the rectangle is closer to the diagonal $\overline{AC}$ than to the outside of the rectangle is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 17
Problem: Five lighthouses are located, in order, at points $A, B, C, D$, and $E$ along the shore of a circular lake with a diameter of $10$ miles. Segments $AD$ and $BE$ are diameters of the circle. At night, when sitting at $A$, the lights from $B, C, D$, and $E$ appear to be equally spaced along the horizon. The perimeter in miles of pentagon $ABCDE$ can be written $m +\sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.
Answer: 95
Problem: Suppose $x$ and $y$ are nonzero real numbers simultaneously satisfying the equations
$x + \frac{2018}{y}= 1000$ and $ \frac{9}{x}+ y = 1$.
Find the maximum possible value of $x + 1000y$.
Answer: 1991
Problem: Suppose that $a$ and $b$ are positive real numbers such that $3\log_{101}\left(\frac{1,030,301-a-b}{3ab}\right) = 3 - 2 \log_{101}(ab)$. Find $101 - \sqrt[3]{a}-  \sqrt[3]{b}$.
Answer: 0
Problem: Find the three-digit positive integer $n$ for which $\binom n3  \binom n4 \binom n5 \binom n6 $ is a perfect square.
Answer: 489
Problem: Let $a$ and $b$ be real numbers such that $\frac{1}{a^2} +\frac{3}{b^2} = 2018a$ and $\frac{3}{a^2} +\frac{1}{b^2} = 290b$. Then $\frac{ab}{b-a }= \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 13
Problem: A complex number $z$ whose real and imaginary parts are integers satisfies $\left(Re(z) \right)^4 +\left(Re(z^2)\right)^2 + |z|^4 =(2018)(81)$, where $Re(w)$ and $Im(w)$ are the real and imaginary parts of $w$, respectively. Find $\left(Im(z) \right)^2$
.
Answer: 225
Problem: Triangle $ABC$ has $AB = AC$. Point $D$ is on side $\overline{BC}$ so that $AD = CD$ and $\angle BAD = 36^o$. Find the degree measure of $\angle BAC$.
Answer: 84
Problem: Let $a, b, c$, and $d$ be real numbers such that $a^2 + b^2 + c^2 + d^2 = 3a + 8b + 24c + 37d = 2018$. Evaluate $3b + 8c + 24d + 37a$.
Answer: 1215
Problem: Find the greatest integer $n$ such that $10^n$ divides $$\frac{2^{10^5} 5^{2^{10}}}{10^{5^2}}$$
Answer: 999
Problem: Evaluate $1 + 2 - 3 - 4 + 5 + 6 - 7 - 8 + ... + 2018 - 2019$.
Answer: 0
Problem: The diagram shows some squares whose sides intersect other squares at the midpoints of their sides. The shaded region has total area $7$. Find the area of the largest square.
[img]https://cdn.artofproblemsolving.com/attachments/3/a/c3317eefe9b0193ca15f36599be3f6c22bb099.png[/img]
Answer: 56
Problem: Evaluate
$$\frac{(2 + 2)^2}{2^2} \cdot \frac{(3 + 3 + 3 + 3)^3}{(3 + 3 + 3)^3} \cdot \frac{(6 + 6 + 6 + 6 + 6 + 6)^6}{(6 + 6 + 6 + 6)^6}$$
Answer: 108
Problem: The following diagram shows four adjacent $2\times 2$ squares labeled $1, 2, 3$, and $4$. A line passing through the lower left vertex of square $1$ divides the combined areas of squares $1, 3$, and $4$ in half so that the shaded region has area $6$. The difference between the areas of the shaded region within square $4$ and the shaded region within square $1$ is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
[img]https://cdn.artofproblemsolving.com/attachments/7/4/b9554ccd782af15c680824a1fbef278a4f736b.png[/img]
Answer: 49
Problem: Let N be the greatest positive integer that can be expressed using all seven Roman numerals $I, V, X, L, C,D$, and $M$ exactly once each, and let n be the least positive integer that can be expressed using these numerals exactly once each. Find $N - n$. Note that the arrangement $CM$ is never used in a number along with the numeral $D$.
Answer: 222
Problem: Find the remainder when $\prod_{n=3}^{33}2n^4 - 25n^3 + 33n^2$ is divided by $2019$.
Answer: 0
Problem: The letters $AAABBCC$ are arranged in random order. The probability no two adjacent letters will be the same is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 124
Problem: Find the positive integer $n$ such that $32$ is the product of the real number solutions of $x^{\log_2(x^3)-n} = 13$
Answer: 15
Problem: Suppose that $a, b, c$, and $d$ are real numbers simultaneously satisfying
$a + b - c - d = 3$
$ab - 3bc + cd - 3da = 4$
$3ab - bc + 3cd - da = 5$
Find $11(a - c)^2 + 17(b -d)^2$.
Answer: 63
Problem: Find the number of ordered pairs of integers $(x, y)$ such that $$\frac{x^2}{y}- \frac{y^2}{x}= 3 \left( 2+ \frac{1}{xy}\right)$$
Answer: 2
Problem: Harold has $3$ red checkers and $3$ black checkers. Find the number of distinct ways that Harold can place these checkers in stacks. Two ways of stacking checkers are the same if each stack of the rst way matches a corresponding stack in the second way in both size and color arrangement. So, for example, the $3$ stack arrangement $RBR, BR, B$ is distinct from $RBR, RB, B$, but the $4$ stack arrangement $RB, BR, B, R$ is the same as $B, BR, R, RB$.
Answer: 131
Problem: There are positive integers $m$ and $n$ such that $m^2 -n = 32$ and  $\sqrt[5]{m +\sqrt{n}}+  \sqrt[5]{m -\sqrt{n}}$ is a real root of the polynomial $x^5 - 10x^3 + 20x - 40$. Find $m + n$.
Answer: 388
Problem: Let $a, b, c$, and $d$ be prime numbers with $a \le b \le c \le d > 0$. 
Suppose $a^2 + 2b^2 + c^2 + 2d^2 = 2(ab + bc - cd + da)$. Find $4a + 3b + 2c + d$.
Answer: 51
Problem: Of the students attending a school athletic event, $80\%$ of the boys were dressed in the school colors, $60\%$ of the girls were dressed in the school colors, and $45\% $ of the students were girls. Find the percentage of students attending the event who were wearing the school colors.
Answer: 71
Problem: Let $a$ and $b$ positive real numbers such that $(65a^2 + 2ab + b^2)(a^2 + 8ab + 65b^2) = (8a^2 + 39ab + 7b^2)^2$. Then one possible value of $\frac{a}{b}$ satises $2 \left(\frac{a}{b}\right) = m +\sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.
Answer: 2636
Problem: In the subtraction PURPLE $-$ COMET $=$ MEET each distinct letter represents a distinct decimal digit, and no leading digit is $0$. Find the greatest possible number represented by PURPLE.
Answer: 103184
Problem: Rectangle $ABCD$ has sides $AB = 10$ and $AD = 7$. Point $G$ lies in the interior of $ABCD$ a distance $2$ from side $\overline{CD}$ and a distance $2$ from side $\overline{BC}$. Points $H, I, J$, and $K$ are located on sides $\overline{BC}, \overline{AB}, \overline{AD}$, and $\overline{CD}$, respectively, so that the path $GHIJKG$ is as short as possible. Then $AJ = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 2
Problem: A container contains five red balls. On each turn, one of the balls is selected at random, painted blue, and returned to the container. The expected number of turns it will take before all five balls are colored blue is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 149
Problem: Suppose $a$ is a real number such that $\sin(\pi \cdot  \cos a) = \cos(\pi \cdot \sin a)$. Evaluate $35 \sin^2(2a) + 84 \cos^2(4a)$.
Answer: 21
Problem: Find the number of ordered triples of sets $(T_1, T_2, T_3)$ such that
1. each of $T_1, T_2$, and $T_3$ is a subset of $\{1, 2, 3, 4\}$,
2. $T_1 \subseteq T_2 \cup T_3$,
3. $T_2 \subseteq T_1 \cup T_3$, and
4. $T_3\subseteq T_1 \cup T_2$.
Answer: 625
Problem: Find the number of ordered triples of positive integers $(a, b, c)$, where $a, b,c$ is a strictly increasing arithmetic progression, $a + b + c = 2019$, and there is a triangle with side lengths $a, b$, and $c$.
Answer: 336