Competition problem — Number theory

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Number theory Difficulty 5.9 AIME, harder Prove it

8. Show that if $p_{k} / q_{k}$ is a convergent of the simple continued fraction expansion of $\sqrt{d}$ then $\left|p_{k}^{2}-d q_{k}^{2}\right|<1+2 \sqrt{d}$ .

Solution

None

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.