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2014 Canadian MO Problems/Problem 5

Revision as of 21:29, 26 November 2023 by Tomasdiaz (talk | contribs) (Created page with "== Problem == Fix positive integers <math>n</math> and <math>k\ge 2</math>. A list of n integers is written in a row on a blackboard. You can choose a contiguous block of inte...")
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Problem

Fix positive integers $n$ and $k\ge 2$. A list of n integers is written in a row on a blackboard. You can choose a contiguous block of integers, and I will either add $1$ to all of them or subtract $1$ from all of them. You can repeat this step as often as you like, possibly adapting your selections based on what I do. Prove that after a finite number of steps, you can reach a state where at least $n-k+2$ of the numbers on the blackboard are all simultaneously divisible by $k$.

Solution

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