Find all natural numbers $a$ such that for every positive integer $n$ the number $n(a+n)$ is not a perfect square.

Let $ABCD$ be a rectangle with sides $AB=3$ and $BC=2$. Let $P$ be a point on side $AB$ such that the bisector of $\angle CDP$ passes through the midpoint of $BC$. Find the length of segment $BP$.

We will say that a natural number is acceptable if it has at most $9$ distinct prime divisors. There is a stack of $100!=1\times2\times\cdots\times100$ stones. A legal move consists in removing $k$ stones from the stack, where $k$ is an acceptable number. Two players, Lucas and Nicolas, take turns making legal moves; Lucas starts the game. The one who removes the last stone wins. Determine which of the players has a winning strategy and describe this strategy.

Let $N$ be the number of ordered lists of $9$ positive integers $(a,b,c,d,e,f,g,h,i)$ such that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}+\frac{1}{f}+\frac{1}{g}+\frac{1}{h}+\frac{1}{i}=1.$$Determine whether $N$ is even or odd.

Find all positive integers $n$ that can be represented in the form $$n=\mathrm{lcm}(a,b)+\mathrm{lcm}(b,c)+\mathrm{lcm}(c,a)$$where $a,b,c$ are positive integers.

Given two positive integers $a$ and $b$, an legal move consists in choosing a proper divisor of one of them and adding it to $a$ or adding it to $b$. Two players, Agustin and Ian, take turns making an legal move; Agustin plays first. Whoever gets a number greater than or equal to $2015$ wins the game. Determine which of the players has a winning strategy if $a=3, b=5$. Determine which of the players has a winning strategy if $a=6, b=7$.