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Problem 964

AMC 12 late, AIME early

Combinatorics Difficulty 4.8 Prove it Berkeley Math Circle: Monthly Contest 5 · United States

Aerith and Bob play rounds of pool. At some point Bob had won more rounds than Aerith, but now Aerith has won $85\%$ of their rounds. Show that at some point, Aerith had won exactly $75\%$ of their rounds.

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:

At any time, let $a$ and $b$ represent the respective number of rounds won by Aerith and Bob. Now, after $n$ games, $a-3b = n(85\% - 3 \cdot 15\%) > 0$ . However, at some point in the past $b$ was greater than $a$ , so $a-3b$ was negative. Consider the first subsequent moment when $a-3b$ was non-negative. At that moment,
$(a-1)-3b < 0 \leq a-3b,$
so as $a$ and $b$ were integers, $a-3b=0$ and $a=75\% \cdot (a+b)$ , as desired.

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