There are $n$ pigeons, where $n \in \Bbb N$, $n\ge1$. Out of these $n$ pigeons, $k$ are smart and know about the pigeonhole principle, where $k < n/2$. The remaining $n−k$ pigeons are not-so-smart. Each smart pigeon eats at least $6$ worms and at most $10$ worms, and each not-so-smart pigeon eats at most $4$ worms (but might also eat no worm and remain hungry!). At least how many pigeons eat the same number of worms? We are looking for the best lower bound.
2026-03-31 22:43:47.1774997027
Pigeonhole Principle
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Hint: label $11$ holes with the numbers $0$ to $10$. Now put each pigeon in the hole corresponding to the amount of worms said pigeon has eaten. You have $k$ that are distributed amongst the $5$ holes labeled $6$ to $10$ and the other $n-k$ are distributed amongst the remaining $6$ holes (the ones labeled $0$ to $5$). Use the pigeon-hole principle in both cases and take the best lower bound (the highest of both).