Let $p(k)$ be the probability that if we choose a uniformly random subset $S$ of $\{1, 2, \ldots, 18\},$ then $|S| \equiv k \pmod{5}.$ Evaluate $$\sum_{k=0}^4 \left|p(k)-\frac{1}{5}\right|.$$

Divide a regular $8960$-gon into non-overlapping parallelograms. Suppose that $R$ of these parallelograms are rectangles. What is the minimum possible value of $R$?

Consider a $4 \times 4$ grid of squares. We place coins in some of the grid squares so that no two coins are orthogonally adjacent, and each $2 \times 2$ square in the grid has at least one coin. How many ways are there to place the coins?

Given the list \[A=[9,12,1,20,17,4,10,7,15,8,13,14],\] we would like to sort it in increasing order. To accomplish this, we will perform the following operation repeatedly: remove an element, then insert it at any position in the list, shifting elements if necessary. What is the minimum number of applications of this operation necessary to sort $A$?

The intramural squash league has 5 players, namely Albert, Bassim, Clara, Daniel, and Eugene. Albert has played one game, Bassim has played two games, Clara has played 3 games, and Daniel has played 4 games. Assuming no two players in the league play each other more than one time, how many games has Eugene played?

Given $x_0\in\mathbb R$, $f,g:\mathbb R\to\mathbb R$, we define the $\emph{non-redundant binary tree}$ $T(x_0,f,g)$ in the following way: [list=1] [*]The tree $T$ initially consists of just $x_0$ at height $0$. [*]Let $v_0,\dots,v_k$ be the vertices at height $h$. Then the vertices of height $h+1$ are added to $T$ by: for $i=0,1,\dots,k$, $f(v_i)$ is added as a child of $v_i$ if $f(v_i)\not\in T$, and $g(v_i)$ is added as a child of $v_i$ if $g(v_i)\not\in T$. [/list] For example, if $f(x)=x+1$ and $g(x)=x-1$, then the first three layers of $T(0,f,g)$ look like: [asy] size(100); draw((-0.1,-0.2)--(-0.4,-0.8),EndArrow(size=3)); draw((0.1,-0.2)--(0.4,-0.8),EndArrow(size=3)); draw((-0.6,-1.2)--(-0.9,-1.8),EndArrow(size=3)); draw((0.6,-1.2)--(0.9,-1.8),EndArrow(size=3)); label("$0$",(0,0)); label("$1$",(-.5,-1)); label("$-1$",(.5,-1)); label("$2$",(-1,-2)); label("$-2$",(1,-2));[/asy] If $f(x)=1024x-2047\lfloor x/2\rfloor$ and $g(x)=2x-3\lfloor x/2\rfloor+2\lfloor x/4\rfloor$, then how many vertices are in $T(2016,f,g)$?

In concurrent computing, two processes may have their steps interwoven in an unknown order, as long as the steps of each process occur in order. Consider the following two processes: \begin{tabular}{c|cc} Process & $A$ & $B$\\ \hline Step 1 & $x\leftarrow x-4$ & $x\leftarrow x-5$\\ Step 2 & $x\leftarrow x\cdot3$ & $x\leftarrow x\cdot4$\\ Step 3 & $x\leftarrow x-4$ & $x\leftarrow x-5$\\ Step 4 & $x\leftarrow x\cdot3$ & $x\leftarrow x\cdot4$ \end{tabular} One such interweaving is $A1$, $B1$, $A2$, $B2$, $A3$, $B3$, $B4$, $A4$, but $A1$, $A3$, $A2$, $A4$, $B1$, $B2$, $B3$, $B4$ is not since the steps of $A$ do not occur in order. We run $A$ and $B$ concurrently with $x$ initially valued at $6$. Find the minimal possible value of $x$ among all interweavings.

David is taking a true/false exam with $9$ questions. Unfortunately, he doesn’t know the answer to any of the questions, but he does know that exactly $5$ of the answers are True. In accordance with this, David guesses the answers to all $9$ questions, making sure that exactly $5$ of his answers are True. What is the probability he answers at least $5$ questions correctly?

Catherine has a plate containing $300$ circular crumbling mooncakes, arranged as follows: [asy] unitsize(10); for (int i = 0; i < 16; ++i){ for (int j = 0; j < 3; ++j){ draw(circle((sqrt(3)*i,j),0.5)); draw(circle((sqrt(3)*(i+0.5),j-0.5),0.5)); } } dot((16*sqrt(3)+.5,.75)); dot((16*sqrt(3)+1,.75)); dot((16*sqrt(3)+1.5,.75)); [/asy] (This continues for $100$ total columns). She wants to pick some of the mooncakes to eat, however whenever she takes a mooncake all adjacent mooncakes will be destroyed and cannot be eaten. Let $M$ be the maximal number of mooncakes she can eat, and let $n$ be the number of ways she can pick $M$ mooncakes to eat (Note: the order in which she picks mooncakes does not matter). Compute the ordered pair ($M$, $n$).

Define a [i]rooted tree[/i] to be a tree $T$ with a singular node designated as the [i]root[/i] of $T$. (Note that every node in the tree can have an arbitrary number of children.) Each vertex adjacent to the root node of $T$ is itself the root of some tree called a [i]maximal subtree[/i] of $T$. Say two rooted trees $T_1$ and $T_2$ are [i]similar[/i] if there exists some way to cycle the maximal subtrees of $T_1$ to get $T_2$. For example, the first pair of trees below are similar but the second pair are not. How many rooted trees with $2019$ nodes are there up to similarity? [center] [img=500x100]https://i.imgur.com/8axcDvz.png[/img] [/center]

Assume that $ R$, a recursive, binary relation on $ \mathbb{N}$ (the set of natural numbers), orders $ \mathbb{N}$ into type $ \omega$. Show that if $ f(n)$ is the $ n$th element of this order, then $ f$ is not necessarily recursive. [i]L. Posa[/i]

In the following list of numbers (given in their binary representations), each number appears an even number of times, except for one number that appears exactly three times. Find the number that appears exactly three times. Leave the answer in its binary representation. \begin{tabular}{cccccc} 010111 & 000001 & 100000 & 011000 & 110101 & 100001 \\ 010100 & 011111 & 111001 & 010001 & 010100 & 101100 \\ 010001 & 011011 & 011111 & 011011 & 100000 & 000001 \\ 110011 & 001000 & 111101 & 100001 & 101100 & 110011 \\ 111111 & 011000 & 001000 & 101000 & 111111 & 101000 \\ 010111 & 100011 & 111001 & 100011 & 110101 & 011111 \\ 100000 & 010100 & 010001 & 101100 & 010111 & 011011 \\ 011000 & 111101 & 111111 & 100001 & 101000 & 100011 \\ 011011 & 010111 & 110011 & 111111 & 000001 & 010001 \\ 101000 & 111001 & 010100 & 110101 & 011000 & 110101 \\ 001000 & 000001 & 100000 & 111101 & 100011 & 001000 \\ 111001 & 110011 & 100001 & 011111 & 101100 \end{tabular}

Consider an undirected, connected graph $G$ with vertex set $\{v_1,v_2,\ldots, v_6\}$. Starting at the vertex $v_1$, an ant uses a DFS algorithm to traverse through $G$ under the condition that if there are multiple unvisited neighbors of some vertex, the ant chooses the $v_i$ with smallest $i$. How many possible graphs $G$ are there satisfying the following property: for each $1\leq i\leq 6$, the vertex $v_i$ is the $i^{\text{th}}$ new vertex the ant traverses?

Answer the following questions : [b](a)[/b] Evaluate $~~\lim_{x\to 0^{+}} \Big(x^{x^x}-x^x\Big)$ [b](b)[/b] Let $A=\frac{2\pi}{9}$, i.e. $40$ degrees. Calculate the following $$1+\cos A+\cos 2A+\cos 4A+\cos 5A+\cos 7A+\cos 8A$$ [b](c)[/b] Find the number of solutions to $$e^x=\frac{x}{2017}+1$$

Define \[f(h,t) = \begin{cases} 8h & h = t \\ (h-t)^2 & h \neq t. \end{cases}\] Cody plays a game with a fair coin, where he begins by flipping it once. At each turn in the game, if he has flipped $h$ heads and $t$ tails and $h + t < 6$, he can choose either to stop and receive $f(h,t)$ dollars or he can flip the coin again; if $h + t = 6$ then the game ends and he receives $f(h,t)$ dollars. If Cody plays to maximize expectancy, how much money, in dollars, can he expect to win from this game?

Consider the grid of numbers shown below. 20 01 96 56 16 37 48 38 64 60 96 97 42 20 98 35 64 96 40 71 50 58 90 16 89 Among all paths that start on the top row, move only left, right, and down, and end on the bottom row, what is the minimum sum of their entries?

You are presented with a mystery function $f:\mathbb N^2\to\mathbb N$ which is known to satisfy \[f(x+1,y)>f(x,y)\quad\text{and}\quad f(x,y+1)>f(x,y)\] for all $(x,y)\in\mathbb N^2$. I will tell you the value of $f(x,y)$ for \$1. What's the minimum cost, in dollars, that it takes to compute the $19$th smallest element of $\{f(x,y)\mid(x,y)\in\mathbb N^2\}$? Here, $\mathbb N=\{1,2,3,\dots\}$ denotes the set of positive integers.

Consider the natural implementation of computing Fibonacci numbers: \begin{tabular}{l} 1: \textbf{FUNCTION} $\text{FIB}(n)$: \\ 2:$\qquad$ \textbf{IF} $n = 0$ \textbf{OR} $n = 1$ \textbf{RETURN} 1 \\ 3:$\qquad$ \textbf{RETURN} $\text{FIB}(n-1) + \text{FIB}(n-2)$ \end{tabular} When $\text{FIB}(10)$ is evaluated, how many recursive calls to $\text{FIB}$ occur?

Sophia writes an algorithm to solve the graph isomorphism problem. Given a graph $G=(V,E)$, her algorithm iterates through all permutations of the set $\{v_1, \dots, v_{|V|}\}$, each time examining all ordered pairs $(v_i,v_j)\in V\times V$ to see if an edge exists. When $|V|=8$, her algorithm makes $N$ such examinations. What is the largest power of two that divides $N$?

We have a collection of $1720$ balls, half of which are black and half of which are white, aligned in a straight line. Our task is to make the balls alternating in color along the line. The following greedy algorithm accomplishes that task for $2n$ balls: \begin{tabular}{l} 1: \textbf{FOR} $i$ \textbf{IN} $[2,3,\dots,2n]$ \\ 2: $\quad$ \textbf{IF} balls $i-1$ and $i$ have the same color: \\ 3: $\quad\quad$ $j\gets$ smallest index greater than $i$ for which balls $i-1$ and $j$ have different colors \\ 4: $\quad\quad$ swap balls $i$ and $j$ \end{tabular} Given a configuration $C$ of our $1720$ balls, let $\hat{\sigma}(C)$ denote the number of swaps the greedy algorithm takes, and let $\sigma(C)$ denote the minimum number of swaps actually necessary to perform the task. Find the maximum value over all configurations $C$ of $\hat{\sigma}(C)-\sigma(C)$.

Max flips $2020$ fair coins. Let the probability that there are at most $505$ heads be $p$. Estimate $-\log_2(p)$ to 5 decimal places, in the form $x.abcde$ where $x$ is a positive integer and $a, b, c, d, e$ are decimal digits.

The continent of Trianglandia is an equilateral triangle of side length $9$, divided into $81$ triangular countries of side length $1$. Each country has the resources to choose at most $1$ of its $3$ sides and build a “wall” covering that entire side. However, since all the countries are at war, no two countries are willing to have their walls touch, even at a corner. What is the maximum number of walls that can be built in Trianglandia?

For how many distinct ordered triples $(a,b,c)$ of boolean variables does the expression $a \lor (b \land c)$ evaluate to true?

Ryan has three distinct eggs, one of which is made of rubber and thus cannot break; unfortunately, he doesn't know which egg is the rubber one. Further, in some 100-story building there exists a floor such that all normal eggs dropped from below that floor will not break, while those dropped from at or above that floor will break and cannot be dropped again. What is the minimum number of times Ryan must drop an egg to determine the floor satisfying this property?

How many distinct spanning trees does the graph below have? Recall that a $\emph{spanning tree}$ of a graph $G$ is a subgraph of $G$ that is a tree and containing all the vertices of $G$. [center][img]http://i.imgur.com/NMF12pE.png[/img][/center]