$80$ balls in a row. $50$ of them are yellow and $30$ are blue. Prove that there are at least $2$ blue balls with a distance of exactly $3$ or $6$.
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2026-02-22 21:53:22.1771797202
$80$ balls in a row. $50$ of them are yellow and $30$ are blue. Prove that there are at least $2$ blue balls with a distance of exactly $3$ or $6$.
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Divide the $80$ slots by congruence $\pmod 3$. We remark that there are $27$ slots of the form $3n+ 1$, $27$ of the form $3n+2$ and $26$ of the form $3n$. As we have $30$ blue balls to distribute, at least one of those congruence classes contains at least $10$ blue balls.
Suppose we had a counterexample. Then the gaps between consecutive blue balls in that congruence class would each have to be at least $2$, but we have at least $9$ gaps and $2\times 9+10=28>27$ so this is impossible, and we are done.