sungyub/dapo-math-17k-verl huggingface dataset:
user: 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 $m + n$.

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
user: 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
user: 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
user: Find $n$ so that $4^{4^{4^2}}=2^{8^n}$.

Answer: 11
user: 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
user: 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
user: 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
user: 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
user: $|5x^2-\tfrac25|\le|x-8|$ 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
user: 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
user: Find $n>0$ such that $\sqrt[3]{\sqrt[3]{5\sqrt2+n}+\sqrt[3]{5\sqrt2-n}}=\sqrt2$.

Answer: 7
user: 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
user: 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
user: 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
user: 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
user: 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
user: Find the number of ordered triples of positive integers $(a, b, c)$ such that $abc$ divides $(ab + 1)(bc + 1)(ca + 1)$.

Answer: 16
user: 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
user: 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
user: 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
user: 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
user: 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
user: How many subsets of $\{1,2,3,4,\dots,12\}$ contain exactly one prime number?

Answer: 640
user: 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
user: Find $n$ such that $\dfrac1{2!9!}+\dfrac1{3!8!}+\dfrac1{4!7!}+\dfrac1{5!6!}=\dfrac n{10!}$.

Answer: 92
user: Find $n$ such that\[\frac{1!\cdot2!\cdot3!\cdots10!}{(1!)^2(3!)^2(5!)^2(7!)^2(9!)^2}=15\cdot2^n.\]

Answer: 8
user: Suppose $x$ is a real number satisfying $x^2-990x+1=(x+1)\sqrt x$. Find $\sqrt x+\tfrac1{\sqrt x}$.

Answer: 32
user: 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
user: 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
user: 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
user: Find the sum of all the real values of x satisfying $(x+\frac{1}{x}-17)^2$ $= x + \frac{1}{x} + 17.$

Answer: 35
user: 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
user: 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
user: Find the sum of all positive integers n with the property that the digits of n add up to 2015−n.

Answer: 4004
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: Find the greatest positive integer $n$ so that $3^n$ divides $70! + 71! + 72!.$

Answer: 36
user: 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
user: 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
user: 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
user: How many non-congruent isosceles triangles (including equilateral triangles) have positive integer side
lengths and perimeter less than 20?

Answer: 41
user: 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
user: 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
user: Find the remainder when $8^{2014}$ + $6^{2014}$ is divided by 100.

Answer: 0
user: 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
user: 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
user: 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
user: 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
user: There are digits a and b so that the 15-digit number 7a7ba7ab7ba7b77 is divisible by 99.
Find 10a + b.

Answer: 53
user: 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
user: 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
user: 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
user: 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
user: Find the sum of all perfect squares that divide 2016.

Answer: 210
user: 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
user: 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
user: 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
user: 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
user: 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
user: Find the greatest integer $n < 1000$ for which $4n^3 - 3n$ is the product of two consecutive odd integers.

Answer: 899
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: Find the greatest prime divisor of $29! + 33!$.

Answer: 991
user: 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
user: 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
user: 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
user: 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
user: 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