Invariant

An invariant refers to a property of a situation that remains the same after multiple given operations.

Problems

$\bullet$ The positive integers $1$ through $10$ are written on a blackboard. At any given point, Evan can erase any three numbers $a$, $b$, and $c$ and replace them with $\sqrt{a^{2}+b^{2}+c^{2}}$. What is the greatest number that can appear on the board at any given point?

$\bullet$ 2011 IMO Problem 2 (it is highly recommended that students watch the video solution, given the difficulty of the IMO)


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