meetforever/GRPO_CL huggingface dataset:
user: Spencer is making burritos, each of which consists of one wrap and one filling. He has enough filling for up to four beef burritos and three chicken burritos. However, he only has five wraps for the burritos; in how many orders can he make exactly five burritos?
Answer: 25
user: The sequence $(z_n)$ of complex numbers satisfies the following properties:
[list]
[*]$z_1$ and $z_2$ are not real.
[*]$z_{n+2}=z_{n+1}^2z_n$ for all integers $n\geq 1$.
[*]$\dfrac{z_{n+3}}{z_n^2}$ is real for all integers $n\geq 1$.
[*]$\left|\dfrac{z_3}{z_4}\right|=\left|\dfrac{z_4}{z_5}\right|=2$. [/list]
Find the product of all possible values of $z_1$.
Answer: 65536
user: For how many unordered sets $\{a,b,c,d\}$ of positive integers, none of which exceed $168$, do there exist integers $w,x,y,z$ such that $(-1)^wa+(-1)^xb+(-1)^yc+(-1)^zd=168$?  If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\left\lfloor25e^{-3\frac{|C-A|}C}\right\rfloor$.
Answer: 761474
user: A cafe has 3 tables and 5 individual counter seats. People enter in groups of size between 1 and 4, inclusive, and groups never share a table. A group of more than 1 will always try to sit at a table, but will sit in counter seats if no tables are available. Conversely, a group of 1 will always try to sit at the counter ﬁrst. One morning, $M$ groups consisting of a total of $N$ people enter and sit down. Then, a single person walks in, and realizes that all the tables and counter seats are occupied by some person or group. What is the minimum possible value of $M + N$?
Answer: 16
user: Let $W$ be the hypercube $\{(x_1,x_2,x_3,x_4)\,|\,0\leq x_1,x_2,x_3,x_4\leq 1\}$.  The intersection of $W$ and a hyperplane parallel to $x_1+x_2+x_3+x_4=0$ is a non-degenerate $3$-dimensional polyhedron.  What is the maximum number of faces of this polyhedron?
Answer: 8
user: Let $N$ be a positive integer whose decimal representation contains $11235$ as a contiguous substring, and let $k$ be a positive integer such that $10^k>N$.  Find the minimum possible value of \[\dfrac{10^k-1}{\gcd(N,10^k-1)}.\]
Answer: 89
user: The polynomial $f(x)=x^3-3x^2-4x+4$ has three real roots $r_1$, $r_2$, and $r_3$.  Let $g(x)=x^3+ax^2+bx+c$ be the polynomial which has roots $s_1$, $s_2$, and $s_3$, where $s_1=r_1+r_2z+r_3z^2$, $s_2=r_1z+r_2z^2+r_3$, $s_3=r_1z^2+r_2+r_3z$, and $z=\frac{-1+i\sqrt3}2$.  Find the real part of the sum of the coefficients of $g(x)$.
Answer: -26
user: Find the rightmost non-zero digit of the expansion of $(20)(13!)$.
Answer: 6
user: Let $\{a_n\}_{n\geq 1}$ be an arithmetic sequence and $\{g_n\}_{n\geq 1}$ be a geometric sequence such that the first four terms of $\{a_n+g_n\}$ are $0$, $0$, $1$, and $0$, in that order.  What is the $10$th term of $\{a_n+g_n\}$?
Answer: -54
user: Let $a_1$, $a_2$, $a_3$, $a_4$, $a_5$ be real numbers whose sum is $20$. Determine with proof the smallest possible value of \[ \displaystyle\sum_{1\le i \le j \le 5} \lfloor a_i + a_j \rfloor. \]
Answer: 72
user: Compute the number of ordered quintuples of nonnegative integers $(a_1,a_2,a_3,a_4,a_5)$ such that $0\leq a_1,a_2,a_3,a_4,a_5\leq 7$ and $5$ divides $2^{a_1}+2^{a_2}+2^{a_3}+2^{a_4}+2^{a_5}$.
Answer: 6528
user: Natalie has a copy of the unit interval $[0,1]$ that is colored white. She also has a black marker, and she colors the interval in the following manner: at each step, she selects a value $x\in [0,1]$ uniformly at random, and 

(a) If $x\leq\tfrac12$ she colors the interval $[x,x+\tfrac12]$ with her marker.

(b) If $x>\tfrac12$ she colors the intervals $[x,1]$ and $[0,x-\tfrac12]$ with her marker.

What is the expected value of the number of steps Natalie will need to color the entire interval black?
Answer: 5
user: Find the integer closest to 

\[\frac{1}{\sqrt[4]{5^4+1}-\sqrt[4]{5^4-1}}\]
Answer: 250
user: Find the number of triples of sets $(A, B, C)$ such that:

(a) $A, B, C \subseteq \{1, 2, 3, \dots , 8 \}$.
(b) $|A \cap B| = |B \cap C| = |C \cap A| = 2$.
(c) $|A| = |B| = |C| = 4$.

Here, $|S|$ denotes the number of elements in the set $S$.
Answer: 45360
user: The numbers $2^0, 2^1, \dots , 2{}^1{}^5, 2{}^1{}^6 = 65536$ are written on a blackboard. You repeatedly take two numbers on the blackboard, subtract one form the other, erase them both, and write the result of the subtraction on the blackboard. What is the largest possible number that can remain on the blackboard when there is only one number left?
Answer: 131069
user: Let $f:\mathbb{N}\to\mathbb{N}$ be a function satisfying the following conditions:

(a) $f(1)=1$.
(b) $f(a)\leq f(b)$ whenever $a$ and $b$ are positive integers with $a\leq b$.
(c) $f(2a)=f(a)+1$ for all positive integers $a$.

How many possible values can the $2014$-tuple $(f(1),f(2),\ldots,f(2014))$ take?
Answer: 1007
user: Eli, Joy, Paul, and Sam want to form a company; the company will have 16 shares to split among the $4$ people. The following constraints are imposed:

$\bullet$ Every person must get a positive integer number of shares, and all $16$ shares must be given out. 
$\bullet$ No one person can have more shares than the other three people combined. 

Assuming that shares are indistinguishable, but people are distinguishable, in how many ways can the shares be given out?
Answer: 315
user: We have a calculator with two buttons that displays and integer $x$. Pressing the first button replaces $x$ by $\lfloor \frac{x}{2} \rfloor$, and pressing the second button replaces $x$ by $4x+1$. Initially, the calculator displays $0$. How many integers less than or equal to $2014$ can be achieved through a sequence of arbitrary button presses? (It is permitted for the number displayed to exceed 2014 during the sequence. Here, $\lfloor y \rfloor$ denotes the greatest integer less than or equal to the real number $y$).
Answer: 233
user: For an integer $n$, let $f_9(n)$ denote the number of positive integers $d\leq 9$ dividing $n$.  Suppose that $m$ is a positive integer and $b_1,b_2,\ldots,b_m$ are real numbers such that $f_9(n)=\textstyle\sum_{j=1}^mb_jf_9(n-j)$ for all $n>m$.  Find the smallest possible value of $m$.
Answer: 28
user: Six distinguishable players are participating in a tennis tournament. Each player plays one match of tennis against every other player. The outcome of each tennis match is a win for one player and a loss for the other players; there are no ties. Suppose that whenever $A$ and $B$ are players in the tournament for which $A$ won (strictly) more matches than $B$ over the course of the tournament, it is also the case that $A$ won the match against $B$ during the tournament. In how many ways could the tournament have gone?
Answer: 2048
user: Let $a_1,a_2,\ldots$ be an infinite sequence of integers such that $a_i$ divides $a_{i+1}$ for all $i\geq 1$, and let $b_i$ be the remainder when $a_i$ is divided by $210$.  What is the maximal number of distinct terms in the sequence $b_1,b_2,\ldots$?
Answer: 127
user: Let $A=\{a_1,a_2,\ldots,a_7\}$ be a set of distinct positive integers such that the mean of the elements of any nonempty subset of $A$ is an integer. Find the smallest possible value of the sum of the elements in $A$.
Answer: 1267
user: An [i]up-right path[/i] from $(a, b) \in \mathbb{R}^2$ to $(c, d) \in \mathbb{R}^2$ is a finite sequence $(x_1, y_z), \dots, (x_k, y_k)$ of points in $ \mathbb{R}^2 $ such that $(a, b)= (x_1, y_1), (c, d) = (x_k, y_k)$, and for each $1 \le i < k$ we have that either $(x_{i+1}, y_{y+1}) = (x_i+1, y_i)$ or $(x_{i+1}, y_{i+1}) = (x_i, y_i + 1)$. Two up-right paths are said to intersect if they share any point.

Find the number of pairs $(A, B)$ where $A$ is an up-right path from $(0, 0)$ to $(4, 4)$, $B$ is an up-right path from $(2, 0)$ to $(6, 4)$, and $A$ and $B$ do not intersect.
Answer: 1750
user: There is a heads up coin on every integer of the number line. Lucky is initially standing on the zero point of the number line facing in the positive direction. Lucky performs the following procedure:

$\bullet$ Lucky looks at the coin (or lack thereof) underneath him. 

$\bullet \, - \, $ If the coin is heads, Lucky flips it to tails up, turns around, and steps forward a distance of one unit.

$ \qquad a -$ If the coin is tails, Lucky picks up the coin and steps forward a distance of one unit facing the same direction.

$ \qquad a -$ If there is no coin, Lucky places a coin heads up underneath him and steps forward a distance of one unit facing the same direction.

He repeats this procedure until there are 20 coins anywhere that are tails up. How many times has Lucky performed the procedure when the process stops?
Answer: 6098
user: Suppose that $(a_1,\ldots,a_{20})$ and $(b_1,\ldots,b_{20})$ are two sequences of integers such that the sequence $(a_1,\ldots,a_{20},b_1,\ldots,b_{20})$ contains each of the numbers $1,\ldots,40$ exactly once.  What is the maximum possible value of the sum \[\sum_{i=1}^{20}\sum_{j=1}^{20}\min(a_i,b_j)?\]
Answer: 5530
user: Let \[ A = \frac{1}{6}((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3) \].

Compute $2^A$.
Answer: 72
user: Find the number of ordered quadruples of positive integers $(a,b,c,d)$ such that $a,b,c,$ and $d$ are all (not necessarily distinct) factors of $30$ and $abcd>900$.
Answer: 1940
user: Let $a,b,c,d,e$ be nonnegative integers such that $625a+250b+100c+40d+16e=15^3$.  What is the maximum possible value of $a+b+c+d+e$?
Answer: 153
user: Let $Q$ be a polynomial \[Q(x)=a_0+a_1x+\cdots+a_nx^n,\] where $a_0,\ldots,a_n$ are nonnegative integers.  Given that $Q(1)=4$ and $Q(5)=152$, find $Q(6)$.
Answer: 254
user: Find the number of ordered pairs of integers $(a,b)\in\{1,2,\ldots,35\}^2$ (not necessarily distinct) such that $ax+b$ is a "quadratic residue modulo $x^2+1$ and $35$", i.e. there exists a polynomial $f(x)$ with integer coefficients such that either of the following $\textit{equivalent}$ conditions holds:

[list]
[*] there exist polynomials $P$, $Q$ with integer coefficients such that $f(x)^2-(ax+b)=(x^2+1)P(x)+35Q(x)$;
[*] or more conceptually, the remainder when (the polynomial) $f(x)^2-(ax+b)$ is divided by (the polynomial) $x^2+1$ is a polynomial with integer coefficients all divisible by $35$.
[/list]
Answer: 225
user: In triangle $ABC$, $AB=2$, $AC=1+\sqrt{5}$, and $\angle CAB=54^{\circ}$. Suppose $D$ lies on the extension of $AC$ through $C$ such that $CD=\sqrt{5}-1$. If $M$ is the midpoint of $BD$, determine the measure of $\angle ACM$, in degrees.
Answer: 63
user: Let $N=30^{2015}$.  Find the number of ordered 4-tuples of integers $(A,B,C,D)\in\{1,2,\ldots,N\}^4$ (not necessarily distinct) such that for every integer $n$, $An^3+Bn^2+2Cn+D$ is divisible by $N$.
Answer: 24
user: Compute the number of sequences of integers $(a_1,\ldots,a_{200})$ such that the following conditions hold.

[list]
[*] $0\leq a_1<a_2<\cdots<a_{200}\leq 202.$
[*] There exists a positive integer $N$ with the following property: for every index $i\in\{1,\ldots,200\}$ there exists an index $j\in\{1,\ldots,200\}$ such that $a_i+a_j-N$ is divisible by $203$.
[/list]
Answer: 20503
user: The fraction $\tfrac1{2015}$ has a unique "(restricted) partial fraction decomposition'' of the form \[\dfrac1{2015}=\dfrac a5+\dfrac b{13}+\dfrac c{31},\] where $a$, $b$, and $c$ are integers with $0\leq a<5$ and $0\leq b<13$.  Find $a+b$.
Answer: 14
user: For each positive integer $n$ and non-negative integer $k$, define $W(n,k)$ recursively by 
\[
W(n,k) = 
\begin{cases}
	n^n & k = 0 \\
	W(W(n,k-1), k-1) & k > 0.
\end{cases}
\]
Find the last three digits in the decimal representation of $W(555,2)$.
Answer: 875
user: Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0), (2,0), (2,1),$ and $(0,1)$. $R$ can be divided into two unit squares, as shown; the resulting figure has seven edges.
[asy]
size(3cm);
draw((0,0)--(2,0)--(2,1)--(0,1)--cycle); draw((1,0)--(1,1)); 
 [/asy]
Compute the number of ways to choose one or more of the seven edges such that the resulting figure is traceable without lifting a pencil. (Rotations and reflections are considered distinct.)
Answer: 81
user: Compute the sum of all integers $1 \le a \le 10$ with the following property: there exist integers $p$ and $q$ such that $p$, $q$, $p^2+a$ and $q^2+a$ are all distinct prime numbers.
Answer: 20
user: Determine the number of triples $0 \le k,m,n \le 100$ of integers such that \[ 2^mn - 2^nm = 2^k. \]
Answer: 22
user: An infinite sequence of real numbers $a_1, a_2, \dots$ satisfies the recurrence
\[ a_{n+3} = a_{n+2} - 2a_{n+1} + a_n \]
for every positive integer $n$. Given that $a_1 = a_3 = 1$ and $a_{98} = a_{99}$, compute $a_1 + a_2 + \dots + a_{100}$.
Answer: 3
user: Define $\phi^!(n)$ as the product of all positive integers less than or equal to $n$ and relatively prime to $n$. Compute the remainder when \[ \sum_{\substack{2 \le n \le 50 \\ \gcd(n,50)=1}} \phi^!(n) \] is divided by $50$.
Answer: 30
user: For a positive integer $n$, denote by $\tau(n)$ the number of positive integer divisors of $n$, and denote by $\phi(n)$ the number of positive integers that are less than or equal to $n$ and relatively prime to $n$. Call a positive integer $n$ $\emph{good}$ if $\varphi (n) + 4\tau (n) = n$. For example, the number $44$ is good because $\varphi (44) + 4\tau (44)= 44$.

Find the sum of all good positive integers $n$.
Answer: 172
user: Nine pairwise noncongruent circles are drawn in the plane such that any two circles intersect twice. For each pair of circles, we draw the line through these two points, for a total of $\binom 92 = 36$ lines. Assume that all $36$ lines drawn are distinct. What is the maximum possible number of points which lie on at least two of the drawn lines?
Answer: 462
user: Let $a,b$ and $c$ be real numbers such that
\begin{align*}
a^2+ab+b^2&=9 \\
b^2+bc+c^2&=52 \\
c^2+ca+a^2&=49.
\end{align*}
Compute the value of $\dfrac{49b^2+39bc+9c^2}{a^2}$.
Answer: 52
user: A contest has six problems worth seven points each. On any given problem, a contestant can score either $0$, $1$, or $7$ points. How many possible total scores can a contestant achieve over all six problems?
Answer: 28
user: Starting with an empty string, we create a string by repeatedly appending one of the letters $H$, $M$, $T$ with probabilities $\frac 14$, $\frac 12$, $\frac 14$, respectively, until the letter $M$ appears twice consecutively. What is the expected value of the length of the resulting string?
Answer: 6
user: Let $V = \left\{ 1, \dots, 8 \right\}$. How many permutations $\sigma : V \to V$ are automorphisms of some tree?

(A $\emph{graph}$ consists of some set of vertices and some edges between pairs of distinct vertices.
It is $\emph{connected}$ if every two vertices in it are connected by some path of one or more edges.
A $\emph{tree}$ $G$ on $V$ is a connected graph with vertex set $V$ and exactly $|V|-1$ edges,
and an $\emph{automorphism}$ of $G$ is a permutation $\sigma : V \to V$ such that
vertices $i,j \in V$ are connected by an edge if and only if $\sigma(i)$ and $\sigma(j)$ are.)
Answer: 30212
user: Determine the smallest positive integer $n \ge 3$ for which
\[ A \equiv 2^{10n} \pmod{2^{170}} \]
where $A$ denotes the result when the numbers $2^{10}$, $2^{20}$, $\dots$, $2^{10n}$ are written in decimal notation and concatenated (for example, if $n=2$ we have $A = 10241048576$).
Answer: 14
user: For positive integers $n$, let $S_n$ be the set of integers $x$ such that $n$ distinct lines, no three concurrent, can divide a plane into $x$ regions (for example, $S_2=\{3,4\}$, because the plane is divided into 3 regions if the two lines are parallel, and 4 regions otherwise). What is the minimum $i$ such that $S_i$ contains at least 4 elements?
Answer: 4
user: Find the number of ordered pairs of integers $(a, b)$ such that $a, b$ are divisors of 720 but $ab$ is not.
Answer: 2520
user: Determine the number of integers $2 \le n \le 2016$ such that $n^n-1$ is divisible by $2$, $3$, $5$, $7$.
Answer: 9
user: How many equilateral hexagons of side length $\sqrt{13}$ have one vertex at $(0,0)$ and the other five vertices at lattice points? 

(A lattice point is a point whose Cartesian coordinates are both integers. A hexagon may be concave but not self-intersecting.)
Answer: 216
user: Let
\[
A = \lim_{n \rightarrow \infty} \sum_{i=0}^{2016} (-1)^i \cdot \frac{\binom{n}{i}\binom{n}{i+2}}{\binom{n}{i+1}^2}
\]
Find the largest integer less than or equal to $\frac{1}{A}$.

The following decimal approximation might be useful: $ 0.6931 < \ln(2) < 0.6932$, where $\ln$ denotes the natural logarithm function.
Answer: 1
user: Let $\Delta A_1B_1C$ be a triangle with $\angle A_1B_1C = 90^{\circ}$ and $\frac{CA_1}{CB_1} = \sqrt{5}+2$. For any $i \ge 2$, define $A_i$ to be the point on the line $A_1C$ such that $A_iB_{i-1} \perp A_1C$ and define $B_i$ to be the point on the line $B_1C$ such that $A_iB_i \perp B_1C$. Let $\Gamma_1$ be the incircle of $\Delta A_1B_1C$ and for $i \ge 2$, $\Gamma_i$ be the circle tangent to $\Gamma_{i-1}, A_1C, B_1C$ which is smaller than $\Gamma_{i-1}$.

How many integers $k$ are there such that the line $A_1B_{2016}$ intersects $\Gamma_{k}$?
Answer: 4030
user: Kristoff is planning to transport a number of indivisible ice blocks with positive integer weights from the north mountain to Arendelle. He knows that when he reaches Arendelle, Princess Anna and Queen Elsa will name an ordered pair $(p,q)$ of nonnegative integers satisfying $p + q \le 2016$. Kristoff must then give Princess Anna \emph{exactly} $p$ kilograms of ice. Afterward, he must give Queen Elsa $\emph{exactly}$ $q$ kilograms of ice.

What is the minimum number of blocks of ice Kristoff must carry to guarantee that he can always meet Anna and Elsa's demands, regardless of which $p$ and $q$ are chosen?
Answer: 18
user: Let $A$ denote the set of all integers $n$ such that $1 \le n \le 10000$, and moreover the sum of the decimal digits of $n$ is $2$. Find the sum of the squares of the elements of $A$.
Answer: 7294927
user: Define $\phi^{!}(n)$ as the product of all positive integers less than or equal to $n$ and relatively prime to $n$. Compute the number of integers $2 \le n \le 50$ such that $n$ divides $\phi^{!}(n)+1$.
Answer: 12
user: For positive integers $n$, let $c_n$ be the smallest positive integer for which $n^{c_n}-1$ is divisible by $210$, if such a positive integer exists, and $c_n = 0$ otherwise. What is $c_1 + c_2 + \dots + c_{210}$?
Answer: 329
user: Determine the remainder when
$$\sum_{i=0}^{2015} \left\lfloor \frac{2^i}{25} \right\rfloor$$
is divided by 100, where $\lfloor x \rfloor$ denotes the largest integer not greater than $x$.
Answer: 14
user: Call a positive integer $N \ge 2$ ``special'' if for every $k$ such that $2 \leq k \leq N$, $N$ can be expressed as a sum of $k$ positive integers that are relatively prime to $N$ (although not necessarily relatively prime to each other). How many special integers are there less than $100$?
Answer: 50
user: On the Cartesian plane $\mathbb{R}^2$, a circle is said to be \textit{nice}
if its center is at the origin $(0,0)$ and it passes through at least one lattice point
(i.e. a point with integer coordinates).
Define the points $A = (20,15)$ and $B = (20,16)$. How many nice circles intersect the open segment $AB$?

For reference, the numbers $601$, $607$, $613$, $617$, $619$, $631$, $641$, $643$, $647$, $653$, $659$, $661$, $673$, $677$, $683$, $691$ are the only prime numbers between $600$ and $700$.
Answer: 10
user: $\textbf{(Lucas Numbers)}$ The Lucas numbers are defined by $L_0 = 2$, $L_1 = 1$, and $L_{n+2} = L_{n+1} + L_n$ for every $n \ge 0$. There are $N$ integers $1 \le n \le 2016$ such that $L_n$ contains the digit $1$. Estimate $N$.

An estimate of $E$ earns $\left\lfloor 20 - 2|N-E| \right\rfloor$ or $0$ points, whichever is greater.
Answer: 1984
user: For positive integers $a,b$, $a\uparrow\uparrow b$ is defined as follows: $a\uparrow\uparrow 1=a$, and $a\uparrow\uparrow b=a^{a\uparrow\uparrow (b-1)}$ if $b>1$.

Find the smallest positive integer $n$ for which there exists a positive integer $a$ such that $a\uparrow\uparrow 6\not \equiv a\uparrow\uparrow 7$ mod $n$.
Answer: 329
user: Define the sequence $a_1, a_2 \dots$ as follows: $a_1=1$ and for every $n\ge 2$,
\[
a_n = \begin{cases}
n-2 & \text{if } a_{n-1} =0 \\
a_{n-1} -1 & \text{if } a_{n-1} \neq 0
\end{cases}
\]
A non-negative integer $d$ is said to be {\em jet-lagged} if there are non-negative integers $r,s$ and a positive integer $n$ such that $d=r+s$ and that $a_{n+r} = a_n +s$. How many  integers in $\{1,2,\dots, 2016\}$ are jet-lagged?
Answer: 51
user: Kelvin the Frog was bored in math class one day, so he wrote all ordered triples $(a, b, c)$ of positive integers such that $abc=2310$ on a sheet of paper. Find the sum of all the integers he wrote down. In other words, compute $$\sum_{\stackrel{abc=2310}{a,b,c\in \mathbb{N}}} (a+b+c),$$ where $\mathbb{N}$ denotes the positive integers.
Answer: 49140
user: Find the number of ordered triples of positive integers $(a, b, c)$ such that
\[6a + 10b + 15c = 3000.\]
Answer: 4851
user: Let $P$ and $A$ denote the perimeter and area respectively of a right triangle with relatively prime integer side-lengths. Find the largest possible integral value of $\frac{P^2}{A}$

[color = red]The official statement does not have the final period.[/color]
Answer: 45
user: Sam spends his days walking around the following $2\times 2$ grid of squares.
\begin{tabular}[t]{|c|c|}\hline
1&2\\ \hline
4&3 \\ \hline
\end{tabular}
Say that two squares are adjacent if they share a side. He starts at the square labeled $1$ and every second walks to an adjacent square. How many paths can Sam take so that the sum of the numbers on every square he visits in his path is equal to $20$ (not counting the square he started on)?
Answer: 167
user: Consider the graph in $3$-space of
\[0 = xyz(x + y)(y + z)(z + x)(x - y)(y - z)(z - x).\]
This graph divides $3$-space into $N$ connected regions. What is $N$?
Answer: 48
user: Kelvin and $15$ other frogs are in a meeting, for a total of $16$ frogs. During the meeting, each pair of distinct frogs becomes friends with probability $\frac{1}{2}$. Kelvin thinks the situation after the meeting is [I]cool[/I] if for each of the $16$ frogs, the number of friends they made during the meeting is a multiple of $4$. Say that the probability of the situation being cool can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime. Find $a$.
Answer: 1167
user: Sean is a biologist, and is looking at a strng of length $66$ composed of the letters $A$, $T$, $C$, $G$. A [i]substring[/i] of a string is a contiguous sequence of letters in the string. For example, the string $AGTC$ has $10$ substrings: $A$, $G$, $T$, $C$, $AG$, $GT$, $TC$, $AGT$, $GTC$, $AGTC$. What is the maximum number of distinct substrings of the string Sean is looking at?
Answer: 2100
user: Kelvin the Frog and $10$ of his relatives are at a party. Every pair of frogs is either [i]friendly[/i] or [i]unfriendly[/i]. When $3$ pairwise friendly frogs meet up, they will gossip about one another and end up in a [i]fight[/i] (but stay [i]friendly[/i] anyway). When $3$ pairwise unfriendly frogs meet up, they will also end up in a [i]fight[/i]. In all other cases, common ground is found and there is no fight. If all $\binom{11}{3}$ triples of frogs meet up exactly once, what is the minimum possible number of fights?
Answer: 28
user: Kelvin the Frog likes numbers whose digits strictly decrease, but numbers that violate this condition in at most one place are good enough. In other words, if $d_i$ denotes the $i$th digit, then $d_i\le d_{i+1}$ for at most one value of $i$. For example, Kelvin likes the numbers $43210$, $132$, and $3$, but not the numbers $1337$ and $123$. How many $5$-digit numbers does Kelvin like?
Answer: 14034
user: Let $ABCD$ be a convex quadrilateral with $AB=5$, $BC=6$, $CD=7$, and $DA=8$. Let $M$, $P$, $N$, $Q$ be the midpoints of sides $AB$, $BC$, $CD$, $DA$ respectively. Compute $MN^2-PQ^2$.
Answer: 13