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Inequality and continued fraction

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

September 20, 2012
inequalitiescontinued fractionalgebra proposedalgebra

Problem Statement

Suppose N∈NN\in \mathbb N is not a perfect square, hence we know that the continued fraction of N\sqrt{N} is of the form N=[a0,a1,a2,...,an‾]\sqrt{N}=[a_0,\overline{a_1,a_2,...,a_n}]. If a1≠1a_1\neq 1 prove that ai≤2a0a_i\le 2a_0.
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