I found three difficult problems for me, involving binomial coefficients. They are extremely interesting I think, but I don't know if I have enough knowledge to manage. Seem really hard, can you help me with them?
Prove that every $z\in\mathbb{N}$ we can represent (in one and only one way) by $ z={x+y+1\choose 2}+x $ for some $x,y\in\mathbb{N}$.
Show that the number: $ \sum_{k\ge 0}{n\choose k}F_{m+k} $ is some Fibonacci number for $m,n\in\mathbb{N}$
How many sequences consisting of $0$ and $1$ are there, where we have $2n$ zeros and $2n$ ones, but before $n$-th $1$ in sequence we have at most $n$ zeros?