Numbers $1$ through $2014$ are written on a board. A valid operation is to erase two numbers $a$ and $b$ on the board and replace them with the greatest common divisor and the least common multiple of $a$ and $b$. Prove that, no matter how many operations are made, the sum of all the numbers that remain on the board is always larger than $2014$ $\times$ $\sqrt[2014]{2014!}$

A pair of positive integers $(a,b)$ is called charrua if there is a positive integer $c$ such that $a+b+c$ and $a\times b\times c$ are both square numbers; if there is no such number $c$, then the pair is called non-charrua. a) Prove that there are infinite non-charrua pairs. b) Prove that there are infinite positive integers $n$ such that $(2,n)$ is charrua.

Let $ABCD$ be a rectangle and $P$ a point outside of it such that $\angle{BPC} = 90^{\circ}$ and the area of the pentagon $ABPCD$ is equal to $AB^{2}$. Show that $ABPCD$ can be divided in 3 pieces with straight cuts in such a way that a square can be built using those 3 pieces, without leaving any holes or placing pieces on top of each other. Note: the pieces can be rotated and flipped over.

Show that the number $n^{2} - 2^{2014}\times 2014n + 4^{2013} (2014^{2}-1)$ is not prime, where $n$ is a positive integer.

Let $ABCD$ be an inscribed quadrilateral in a circumference with center $O$ such that it lies inside $ABCD$ and $\angle{BAC} = \angle{ODA}$. Let $E$ be the intersection of $AC$ with $BD$. Lines $r$ and $s$ are drawn through $E$ such that $r$ is perpendicular to $BC$, and $s$ is perpendicular to $AD$. Let $P$ be the intersection of $r$ with $AD$, and $M$ the intersection of $s$ with $BC$. Let $N$ be the midpoint of $EO$. Prove that $M$, $N$, and $P$ lie on a line.

Let $F$ be a family of subsets of $S = \left \{ 1,2,...,n \right \}$ ($n \geq 2$). A valid play is to choose two disjoint sets $A$ and $B$ from $F$ and add $A \cup B$ to $F$ (without removing $A$ and $B$). Initially, $F$ has all the subsets that contain only one element of $S$. The goal is to have all subsets of $n - 1$ elements of $S$ in $F$ using valid plays. Determine the lowest number of plays required in order to achieve the goal.