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Problem 1088

AIME late

Algebra Difficulty 5.0 Prove it Berkeley Math Circle: Monthly Contest 5 · United States

An iguana writes the number $1$ on the blackboard. Every minute afterwards, if the number $x$ is written, the iguana erases it and either writes $\frac{1}{x}$ or $x+1$ . Can the iguana eventually write the number $\frac{20}{17}$ ?

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Solution:

Yes. First, the iguana writes
$1 \rightarrow 2 \rightarrow \frac{1}{2} \rightarrow \frac{3}{2} \rightarrow \frac{2}{3}$
Then, the iguana adds $1$ to arrive at $\frac{17}{3}$ . Finally, finish with
$\frac{17}{3} \rightarrow \frac{3}{17} \rightarrow \frac{20}{17} .$

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.