ILL 1990 Problem 94

Given integer $n > 1$ and real number $t \geq 1$ . $P$ is a parallelogram with four vertices $(0, 0), (0, t), (tF_{2n+1}, tF_{2n}), (tF_{2n+1}, tF_{2n} + t)$ . Here, ${F_n}$ is the $n$ -th term of Fibonacci sequence defined by $F_0 = 0, F_1 = 1$ and $F_{m+1} = F_m + F_{m-1}$ . Let $L$ be the number of integral points (whose coordinates are integers) interior to $P$ , and $M$ be the area of $P$ , which is $t^2F_{2n+1}.$ i) Prove that for any integral point $(a, b)$ , there exists a unique pair of integers $(j, k)$ such that $j(F_{n+1}, F_n) + k(F_n, F_{n-1}) = (a, b)$ , that is, $jF_{n+1} + kF_n = a$ and $jF_n + kF_{n-1} = b.$ ii) Using i) or not, prove that $|\sqrt L-\sqrt M| \leq \sqrt 2.$