Problems

Consider the Pascal’s triangle: it starts with \(1\), then each entry in the triangle is the sum of the two numbers above it. Prove that the diagonals of the Pascal’s triangle sum up to Fibonacci numbers.

Prove for any \(m,n\) that \(F_{m+n} = F_{m-1}F_n + F_mF_{n+1}\).

Denote by \(GCD(m,n)\) the greatest common divisor of numbers \(m,n\), namely the largest possible \(d\) which divides both \(n\) and \(m\). Prove for any \(m,n\) that \[GCD(F_n,F_m) = F_{GCD(m,n)}.\]