Problem

Source: Iran 3rd round 2012-Algebra exam-P2

Tags: inequalities, continued fraction, algebra proposed, algebra



Suppose $N\in \mathbb N$ is not a perfect square, hence we know that the continued fraction of $\sqrt{N}$ is of the form $\sqrt{N}=[a_0,\overline{a_1,a_2,...,a_n}]$. If $a_1\neq 1$ prove that $a_i\le 2a_0$.