Problem

Source: 2017 Saudi Arabia BMO TST I p4

Tags: fibonacci number, number theory



Fibonacci sequences is defined as $f_1=1$,$f_2=2$, $f_{n+1}=f_{n}+f_{n-1}$ for $n \ge 2$. a) Prove that every positive integer can be represented as sum of several distinct Fibonacci number. b) A positive integer is called Fib-unique if the way to represent it as sum of several distinct Fibonacci number is unique. Example: $13$ is not Fib-unique because $13 = 13 = 8 + 5 = 8 + 3 + 2$. Find all Fib-unique.