BdMO TST 2021 NT Exam P2, IMO SL N2 - Some islands are not connected
- Anindya Biswas
- Posts:264
- Joined:Fri Oct 02, 2020 8:51 pm
- Location:Magura, Bangladesh
- Contact:
For each prime $p$, there is a kingdom of $p$-Landia consisting of $p$ islands numbered $1, 2,\dots, p$. Two distinct islands numbered $n$ and $m$ are connected by a bridge if and only if $p$ divides $(n^2-m + 1)(m^2-n + 1)$. The bridges may pass over each other, but cannot cross. Prove that for infinitely many $p$ there are two islands in $p$-Landia not connected by a chain of bridges.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
— John von Neumann
— John von Neumann