ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Each time you click a toggle switch, the switch either turns from [i]off[/i] to [i]on[/i] or from [i]on[/i] to [i]off[/i]. Suppose that you start with three toggle switches with one of them [i]on[/i] and two of them [i]off[/i]. On each move you randomly select one of the three switches and click it. Let $m$ and $n$ be relatively prime positive integers so that $\frac{m}{n}$ is the probability that after four such clicks, one switch will be [i]on[/i] and two of them will be [i]off[/i]. Find $m+n$.
Answer: 142
Problem: Find the least positive integer $n$ so that both $n$ and $n+1$ have prime factorizations with exactly four (not necessarily distinct) prime factors.
Answer: 135
Problem: For some integers $a$ and $b$ the function $f(x)=ax+b$ has the properties that $f(f(0))=0$ and $f(f(f(4)))=9$. Find $f(f(f(f(10))))$.
Answer: 10
Problem: A circle in the first quadrant with center on the curve $y=2x^2-27$ is tangent to the $y$-axis and the line $4x=3y$. The radius of the circle is $\frac{m}{n}$ where $M$ and $n$ are relatively prime positive integers. Find $m+n$.
Answer: 11
Problem: At the 4 PM show, all the seats in the theater were taken, and 65 percent of the audience was children. At the 6 PM show, again, all the seats were taken, but this time only 50 percent of the audience was children. Of all the people who attended either of the shows, 57 percent were children although there were 12 adults and 28 children who attended both shows. How many people does the theater seat?
Answer: 520
Problem: A rectangle has side lengths $6$ and $8$. There are relatively prime positive integers $m$ and $n$ so that $\tfrac{m}{n}$ is the probability that a point randomly selected from the inside of the rectangle is closer to a side of the rectangle than to either diagonal of the rectangle. Find $m + n$.
Answer: 203
Problem: Floyd looked at a standard $12$ hour analogue clock at $2\!:\!36$. When Floyd next looked at the clock, the angles through which the hour hand and minute hand of the clock had moved added to $247$ degrees. How many minutes after $3\!:\!00$ was that?
Answer: 14
Problem: Suppose $x,y$ and $z$ are integers that satisfy the system of equations \[x^2y+y^2z+z^2x=2186\] \[xy^2+yz^2+zx^2=2188.\] Evaluate $x^2+y^2+z^2.$
Answer: 245
Problem: Let $z$ be a complex number satisfying $(z+\tfrac{1}{z})(z+\tfrac{1}{z}+1)=1$. Evaluate $(3z^{100}+\tfrac{2}{z^{100}}+1)(z^{100}+\tfrac{2}{z^{100}}+3)$.
Answer: 36
Problem: A picture with an area of $160$ square inches is surrounded by a $2$ inch border. The picture with its border is a rectangle twice as long as it is wide. How many inches long is that rectangle?
Answer: 24
Problem: Six children stand in a line outside their classroom. When they enter the classroom, they sit in a circle in random order. There are relatively prime positive integers $m$ and $n$ so that $\tfrac{m}{n}$ is the probability that no two children who stood next to each other in the line end up sitting next to each other in the circle. Find $m + n$.
Answer: 13
Problem: There is a pile of eggs. Joan counted the eggs, but her count was way off by $1$ in the $1$'s place. Tom counted in the eggs, but his count was off by $1$ in the $10$'s place. Raoul counted the eggs, but his count was off by $1$ in the $100$'s place. Sasha, Jose, Peter, and Morris all counted the eggs and got the correct count. When these seven people added their counts together, the sum was $3162$. How many eggs were in the pile?
Answer: 439
Problem: Find the least six-digit palindrome that is a multiple of $45$. Note that a palindrome is a number that reads the same forward and backwards such as $1441$ or $35253$.
Answer: 504405
Problem: How many ordered triples $(a, b, c)$ of positive integers satisfy $a \le b  \le c$ and $a  \cdot b\cdot  c = 1000$?
Answer: 19
Problem: In how many rearrangements of the numbers $1, \ 2, \ 3, \ 4, \ 5,\  6, \ 7, \ 8,\  9$ do the numbers form a $\textit{hill}$, that is, the numbers form an increasing sequence at the beginning up to a peak, and then form a decreasing sequence to the end such as in $129876543$ or $258976431$?
Answer: 254
Problem: In the following diagram two sides of a square are tangent to a circle with diameter $8$. One corner of the square lies on the circle. There are positive integers $m$ and $n$ so that the area of the square is $m +\sqrt{n}$. Find $m + n$.
[asy]
import graph;
size(4.4cm); 
real labelscalefactor = 0.5;
pen dotstyle = black;
filldraw((-0.707106781,0.707106781)--(-0.707106781,-1)--(1,-1)--(1,0.707106781)--cycle,gray, linewidth(1.4));
draw(circle((0,0),1), linewidth(1.4));
[/asy]
Answer: 134
Problem: For positive integers $m$ and $n$, the decimal representation for the fraction $\tfrac{m}{n}$ begins $0.711$ followed by other digits. Find the least possible value for $n$.
Answer: 45
Problem: Find the least three digit number that is equal to the sum of its digits plus twice the product of its digits.
Answer: 397
Problem: How many seven-digit positive integers do not either start or end with $7$?
Answer: 7200000
Problem: The diagram below shows the first three figures of a sequence of figures. The first figure shows an equilateral triangle $ABC$ with side length $1$. The leading edge of the triangle going in a clockwise direction around $A$ is labeled $\overline{AB}$ and is darkened in on the figure. The second figure shows the same equilateral triangle with a square with side length $1$ attached to the leading clockwise edge of the triangle. The third figure shows the same triangle and square with a regular pentagon with side length $1$ attached to the leading clockwise edge of the square. The fourth figure in the sequence will be formed by attaching a regular hexagon with side length $1$ to the leading clockwise edge of the pentagon. The hexagon will overlap the triangle. Continue this sequence through the eighth figure. After attaching the last regular figure (a regular decagon), its leading clockwise edge will form an angle of less than $180^\circ$ with the side $\overline{AC}$ of the equilateral triangle. The degree measure of that angle can be written in the form $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

[asy]
size(250);
defaultpen(linewidth(0.7)+fontsize(10));
pair x[],y[],z[];
x[0]=origin;
x[1]=(5,0);
x[2]=rotate(60,x[0])*x[1];
draw(x[0]--x[1]--x[2]--cycle);
for(int i=0;i<=2;i=i+1)
{
y[i]=x[i]+(15,0);
}
y[3]=rotate(90,y[0])*y[2];
y[4]=rotate(-90,y[2])*y[0];
draw(y[0]--y[1]--y[2]--y[0]--y[3]--y[4]--y[2]);
for(int i=0;i<=4;i=i+1)
{
z[i]=y[i]+(15,0);
}
z[5]=rotate(108,z[4])*z[2];
z[6]=rotate(108,z[5])*z[4];
z[7]=rotate(108,z[6])*z[5];
draw(z[0]--z[1]--z[2]--z[0]--z[3]--z[4]--z[2]--z[7]--z[6]--z[5]--z[4]);
dot(x[2]^^y[2]^^z[2],linewidth(3));
draw(x[2]--x[0]^^y[2]--y[4]^^z[2]--z[7],linewidth(1));
label("A",(x[2].x,x[2].y-.3),S);
label("B",origin,S);
label("C",x[1],S);[/asy]
Answer: 1088
Problem: For what value of $x$ does the following determinant have the value $2013$?
\[\left|\begin{array}{ccc}5+x & 8 & 2+x \\ 1 & 1+x & 3 \\ 2 & 1 & 2\end{array}\right|\]
Answer: 497
Problem: After Jennifer walked $r$ percent of the way from her home to the store, she turned around and walked home, got on her bicycle, and bicycled to the store and back home.  Jennifer bicycles two and a half times faster than she walks.  Find the largest value of $r$ so that returning home for her bicycle was not slower than her walking all the way to and from the store without her bicycle.
Answer: 60
Problem: How many four-digit positive integers have no adjacent equal even digits? For example, count numbers such as $1164$ and $2035$ but not $6447$ or $5866$.
Answer: 7801
Problem: Find $n$ so that $4^{4^{4^2}}=2^{8^n}$.
Answer: 11
Problem: One of the two Purple Comet! question writers is an adult whose age is the same as the last two digits of the year he was born. His birthday is in August. What is his age today?
Answer: 56
Problem: For positive integer $n$ let $a_n$ be the integer consisting of $n$ digits of $9$ followed by the digits $488$. For example, $a_3 = 999,488$ and $a_7 = 9,999,999,488$. Find the value of $n$ so that an is divisible by the highest power of $2$.
Answer: 6
Problem: In how many ways can you write $12$ as an ordered sum of integers where the smallest of those integers is equal to $2$? For example, $2+10$, $10+2$, and $3+2+2+5$ are three such ways.
Answer: 70
Problem: A quarry wants to sell a large pile of gravel. At full price, the gravel would sell for $3200$ dollars. But during the first week the quarry only sells $60\%$ of the gravel at full price. The following week the quarry drops the price by $10\%$, and, again, it sells $60\%$ of the remaining gravel. Each week, thereafter, the quarry reduces the price by another $10\%$ and sells $60\%$ of the remaining gravel. This continues until there is only a handful of gravel left. How many dollars does the quarry collect for the sale of all its gravel?
Answer: 3000
Problem: $|5x^2-\tfrac25|\le|x-8|$ if and only if $x$ is in the interval $[a, b]$. There are relatively prime positive integers $m$ and $n$ so that $b -a =\tfrac{m}{n}$ . Find $m + n$.
Answer: 18
Problem: Right triangle $ABC$ has a right angle at $C$. Point $D$ on side $\overline{AB}$ is the base of the altitude of $\triangle ABC$ from $C$. Point $E$ on side $\overline{BC}$ is the base of the altitude of $\triangle CBD$ from $D$. Given that $\triangle ACD$ has area $48$ and $\triangle CDE$ has area $40$, find the area of $\triangle DBE$.
Answer: 200
Problem: Find $n>0$ such that $\sqrt[3]{\sqrt[3]{5\sqrt2+n}+\sqrt[3]{5\sqrt2-n}}=\sqrt2$.
Answer: 7
Problem: The vertices of hexagon $ABCDEF$ lie on a circle. Sides $AB = CD = EF = 6$, and sides $BC = DE = F A = 10$. The area of the hexagon is $m\sqrt3$. Find $m$.
Answer: 94
Problem: The first number in the following sequence is $1$. It is followed by two $1$'s and two $2$'s. This is followed by three $1$'s, three $2$'s, and three $3$'s. The sequence continues in this fashion.
\[1,1,1,2,2,1,1,1,2,2,2,3,3,3,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,\dots.\]
Find the $2014$th number in this sequence.
Answer: 13
Problem: Start with a three-digit positive integer $A$. Obtain $B$ by interchanging the two leftmost digits of $A$. Obtain $C$ by doubling $B$. Obtain $D$ by subtracting $500$ from $C$. Given that $A + B + C + D = 2014$, find $A$.
Answer: 344
Problem: For positive integers $m$ and $n$, let $r(m, n)$ be the remainder when $m$ is divided by $n$. Find the smallest positive integer $m$ such that
\[r(m, 1) + r(m, 2) + r(m, 3) +\cdots+ r(m, 10) = 4.\]
Answer: 120
Problem: Five men and five women stand in a circle in random order. The probability that every man stands next to at least one woman is $\tfrac m n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 59
Problem: Find the number of ordered triples of positive integers $(a, b, c)$ such that $abc$ divides $(ab + 1)(bc + 1)(ca + 1)$.
Answer: 16
Problem: Find the number of subsets of $\{1,3,5,7,9,11,13,15,17,19\}$ where the elements in the subset add to $49$.
Answer: 22
Problem: The twenty-first century began on January $1$, $2001$ and runs through December $31$, $2100$. Note that March $1$, $2014$ fell on Saturday, so there were five Mondays in March $2014$. In how many years of the twenty-first century does March have five Mondays?
Answer: 44
Problem: Consider the sequences of six positive integers $a_1,a_2,a_3,a_4,a_5,a_6$ with the properties that $a_1=1$, and if for some $j > 1$, $a_j = m > 1$, then $m-1$ appears in the sequence $a_1,a_2,\dots,a_{j-1}$. Such sequences include $1,1,2,1,3,2$ and $1,2,3,1,4,1$ but not $1,2,2,4,3,2$. How many such sequences of six positive integers are there?
Answer: 203
Problem: Let $n$ be a positive integer such that $\lfloor\sqrt n\rfloor-2$ divides $n-4$ and $\lfloor\sqrt n\rfloor+2$ divides $n+4$. Find the greatest such $n$ less than $1000$. (Note: $\lfloor x\rfloor$ refers to the greatest integer less than or equal to $x$.)
Answer: 956
Problem: A jar was filled with jelly beans so that $54\%$ of the beans were red, $30\%$ of the beans were green, and $16\%$ of the beans were blue. Alan then removed the same number of red jelly beans and green jelly beans from the jar so that now $20\%$ of the jelly beans in the jar are blue. What percent of the jelly beans in the jar are now red?
Answer: 55
Problem: How many subsets of $\{1,2,3,4,\dots,12\}$ contain exactly one prime number?
Answer: 640
Problem: Let $x$, $y$, $z$ be positive real numbers satisfying the simultaneous equations
\begin{align*}x(y^2+yz+z^2)&=3y+10z\\y(z^2+zx+x^2)&=21z+24x\\z(x^2+xy+y^2)&=7x+28y.\end{align*}
Find $xy+yz+zx$.
Answer: 31
Problem: Find $n$ such that $\dfrac1{2!9!}+\dfrac1{3!8!}+\dfrac1{4!7!}+\dfrac1{5!6!}=\dfrac n{10!}$.
Answer: 92
Problem: Find $n$ such that\[\frac{1!\cdot2!\cdot3!\cdots10!}{(1!)^2(3!)^2(5!)^2(7!)^2(9!)^2}=15\cdot2^n.\]
Answer: 8
Problem: Suppose $x$ is a real number satisfying $x^2-990x+1=(x+1)\sqrt x$. Find $\sqrt x+\tfrac1{\sqrt x}$.
Answer: 32
Problem: Let $S=2^3+3^4+5^4+7^4+\cdots+17497^4$ be the sum of the fourth powers of the first $2014$ prime numbers. Find the remainder when $S$ is divided by $240$.
Answer: 93
Problem: The Bell Zoo has the same number of rhinoceroses as the Carlton Zoo has lions. The Bell Zoo has three more elephants than the Carlton Zoo has lions. The Bell Zoo has the same number of elephants as the Carlton Zoo has rhinoceroses. The Carlton Zoo has two more elephants than rhinoceroses. The Carlton Zoo has twice as many monkeys as it has rhinoceroses, elephants, and lions combined, and it has two more penguins than monkeys. The Bell Zoo has two-thirds as many monkeys as the Carlton Zoo has penguins. The Bell Zoo has two more penguins than monkeys but only half as many lions as penguins. The total of the numbers of rhinoceroses, elephants, lions, monkeys, and penguins in the Bell Zoo is $48$. Find the total of the numbers of rhinoceroses, elephants, lions, monkeys, and penguins in the Carlton Zoo.
Answer: 57
Problem: Let $f$ be a real-valued function such that $4f(x)+xf\left(\tfrac1x\right)=x+1$ for every positive real number $x$. Find $f(2014)$.
Answer: 403
Problem: Find the sum of all the real values of x satisfying $(x+\frac{1}{x}-17)^2$ $= x + \frac{1}{x} + 17.$
Answer: 35
Problem: Evaluate
$\frac{\log_{10}20^2 \cdot \log_{20}30^2 \cdot \log_{30}40^2 \cdot \cdot \cdot \log_{990}1000^2}{\log_{10}11^2 \cdot \log_{11}12^2 \cdot \log_{12}13^2 \cdot \cdot \cdot \log_{99}100^2}$
.
Answer: 768
Problem: Let a,b,c, and d be real numbers such that \[a^2 + 3b^2 + \frac{c^2+3d^2}{2} = a + b + c+d-1.\] Find $1000a + 100b + 10c + d$.
Answer: 527
Problem: Find the sum of all positive integers n with the property that the digits of n add up to 2015−n.
Answer: 4004
Problem: Gwendoline rolls a pair of six-sided dice and records the product of the two values rolled. Gwendoline
repeatedly rolls the two dice and records the product of the two values until one of the values she records
appears for a third time. What is the maximum number of times Gwendoline will need to roll the two dice?
Answer: 37
Problem: Larry and Diane start $100$ miles apart along a straight road. Starting at the same time, Larry and Diane
drive their cars toward each other. Diane drives at a constant rate of 30 miles per hour. To make it
interesting, at the beginning of each 10 mile stretch, if the two drivers have not met, Larry ﬂips a fair coin.
If the coin comes up heads, Larry drives the next 10 miles at 20 miles per hour. If the coin comes up tails,
Larry drives the next 10 miles at 60 miles per hour. Larry and Diane stop driving when they meet. The expected number of times that Larry ﬂips the coin is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find $m + n.$
Answer: 413
Problem: Janet played a song on her clarinet in 3 minutes and 20 seconds. Janet’s goal is to play the song at a 25%
faster rate. Find the number of seconds it will take Janet to play the song when she reaches her goal.
Answer: 160
Problem: Let $A = {1,2,3,4,5}$ and $B = {0,1,2}$. Find the number of pairs of functions ${{f,g}}$ where both f and g map the set A into the set B and there are exactly two elements $x \in A$ where $f(x) = g(x)$. For example, the function f that maps $1 \rightarrow 0,2 \rightarrow 1,3 \rightarrow 0,4 \rightarrow 2,5 \rightarrow 1$ and the constant function g which maps each element of A to 0 form such a pair of functions.
Answer: 9720
Problem: In an abandoned chemistry lab Gerome found a two-pan balance scale and three 1-gram weights, three
5-gram weights, and three 50-gram weights. By placing one pile of chemicals and as many weights as
necessary on the pans of the scale, Gerome can measure out various amounts of the chemicals in the pile.
Find the number of diﬀerent positive weights of chemicals that Gerome could measure.
Answer: 129
Problem: In the ﬁgure below $\angle$LAM = $\angle$LBM = $\angle$LCM = $\angle$LDM, and $\angle$AEB = $\angle$BFC = $\angle$CGD = 34 degrees. Given that $\angle$KLM = $\angle$KML, ﬁnd the degree measure of $\angle$AEF. This is #8 on the 2015 Purple comet High School. For diagram go to http://www.purplecomet.org/welcome/practice
Answer: 107
Problem: Cindy and Neil wanted to paint the side of a staircase in the six-square pattern shown below so that each
of the six squares is painted a solid color, and no two squares that share an edge are the same color. Cindy
draws all n patterns that can be colored using the four colors red, white, blue, and green. Neil looked at
these patterns and claimed that k of the patterns Cindy drew were incorrect because two adjacent squares
were colored with the same color. This is because Neil is color-blind and cannot distinguish red from
green. Find $n + k$. For picture go to http://www.purplecomet.org/welcome/practice
Answer: 1166
Problem: A courtyard has the shape of a parallelogram ABCD. At the corners of the courtyard there stand poles AA', BB', CC', and DD', each of which is perpendicular to the ground. The heights of these poles are AA' = 68 centimeters, BB' = 75 centimeters, CC' = 112 centimeters, and DD' = 133 centimeters. Find the distance in centimeters between the midpoints of A'C' and B'D'.
Answer: 14
Problem: The product $20! \cdot 21! \cdot 22! \cdot \cdot \cdot 28!$ can be expressed in the form $m$ $\cdot$ $n^3$, where m and n are positive integers, and m is not divisible by the cube of any prime. Find m.
Answer: 825
Problem: Seven people of seven diﬀerent ages are attending a meeting. The seven people leave the meeting one at a
time in random order. Given that the youngest person leaves the meeting sometime before the oldest
person leaves the meeting, the probability that the third, fourth, and ﬁfth people to leave the meeting do so in order of their ages (youngest to oldest) is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find $m + n$.
Answer: 25
Problem: The Fahrenheit temperature ($F$) is related to the Celsius temperature ($C$) by $F = \tfrac{9}{5} \cdot C + 32$. What is the temperature in Fahrenheit degrees that is one-ﬁfth as large if measured in Celsius degrees?
Answer: 50
Problem: Find the greatest positive integer $n$ so that $3^n$ divides $70! + 71! + 72!.$
Answer: 36
Problem: For integers a, b, c, and d the polynomial $p(x) =$ $ax^3 + bx^2 + cx + d$ satisﬁes $p(5) + p(25) = 1906$. Find the minimum possible value for $|p(15)|$.
Answer: 47
Problem: The repeating decimal $2.0151515\ldots$ can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 199
Problem: Deﬁne the determinant $D_1$ = $|1|$, the determinant $D_2$ =     
$|1 1|$ 
$|1 3|$      
, and the determinant $D_3=$        
|1 1 1|
|1 3 3|
|1 3 5|         
.
In general, for positive integer n, let the determinant $D_n$ have 1s in every position of its ﬁrst row and ﬁrst
column, 3s in the remaining positions of the second row and second column, 5s in the remaining positions of the third row and third column, and so forth. Find the least n so that $D_n$ $\geq$ 2015.
Answer: 12
Problem: How many non-congruent isosceles triangles (including equilateral triangles) have positive integer side
lengths and perimeter less than 20?
Answer: 41
Problem: The complex number w has positive imaginary part and satisﬁes $|w| = 5$. The triangle in the complex plane with vertices at $w, w^2,$ and $w^3$ has a right angle at $w$. Find the real part of $w^3$.
Answer: 71
Problem: The Purple Plant Garden Store sells grass seed in ten-pound bags and ﬁfteen-pound bags. Yesterday half
of the grass seed they had was in ten-pound bags. This morning the store received a shipment of 27 more
ten-pound bags, and now they have twice as many ten-pound bags as ﬁfteen-pound bags. Find the total
weight in pounds of grass seed the store now has.
Answer: 1890
Problem: The ﬁgure below has only two sizes for its internal angles. The larger angles are three times the size of the
smaller angles. Find the degree measure of one of the larger angles. For figure here: http://www.purplecomet.org/welcome/practice

P.S The figure has 9 sides.
Answer: 252
Problem: Find the remainder when $8^{2014}$ + $6^{2014}$ is divided by 100.
Answer: 0
Problem: How many subsets of {1,2,3,4,5,6,7,8,9,10,11,12} have the property that no two of its elements diﬀer by more than 5? For example, count the sets {3}, {2,5,7}, and {5,6,7,8,9} but not the set {1,3,5,7}.
Answer: 256
Problem: On the inside of a square with side length 60, construct four congruent isosceles triangles each with base
60 and height 50, and each having one side coinciding with a diﬀerent side of the square. Find the area of
the octagonal region common to the interiors of all four triangles.
Answer: 360
Problem: Given that x, y, and z are real numbers satisfying $x+y +z = 10$ and $x^2 +y^2 +z^2 = 50$, ﬁnd the maximum possible value of $(x + 2y + 3z)^2 + (y + 2z + 3x)^2 + (z + 2x + 3y)^2$.
Answer: 1250
Problem: Suppose that the vertices of a polygon all lie on a rectangular lattice of points where adjacent points on
the lattice are a distance 1 apart. Then the area of the polygon can be found using Pick’s Formula: $I + \frac{B}{2}$ −1, where I is the number of lattice points inside the polygon, and B is the number of lattice points on the boundary of the polygon. Pat applied Pick’s Formula to ﬁnd the area of a polygon but mistakenly interchanged the values of I and B. As a result, Pat’s calculation of the area was too small by 35. Using the correct values for I and B, the ratio n = $\frac{I}{B}$ is an integer. Find the greatest possible value of n.
Answer: 15
Problem: There are digits a and b so that the 15-digit number 7a7ba7ab7ba7b77 is divisible by 99.
Find 10a + b.
Answer: 53
Problem: Ten spherical balls are stacked in a pyramid. The bottom level of the stack has six balls each with radius 6 arranged
in a triangular formation with adjacent balls tangent to each other. The middle level of the stack has three
balls each with radius 5 arranged in a triangular formation each tangent to three balls in the bottom level.
The top level of the stack has one ball with radius 6 tangent to the three balls in the middle level. The
diagram shows the stack of ten balls with the balls in the middle shaded. The height of this stack of balls is m +$\sqrt{n}$, where m and n are positive integers. Find $m + n.$
Answer: 304
Problem: You have a collection of small wooden blocks that are rectangular solids measuring $3$×$4$×$6$. Each of the six faces of each block is to be painted a solid color, and you have three colors of paint to use. Find the
number of distinguishable ways you could paint the blocks. (Two blocks are distinguishable if you cannot
rotate one block so that it looks identical to the other block.)
Answer: 243
Problem: Let $x$ be a real number between 0 and $\tfrac{\pi}{2}$ for which the function  $3\sin^2 x + 8\sin x \cos x + 9\cos^2 x$ obtains its maximum value, $M$. Find the value of $M + 100\cos^2x$.
Answer: 91