2014 Canadian MO Problems/Problem 5
Problem
Fix positive integers 
and 
. 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 
to all of them or subtract 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 
of the numbers on the blackboard are all simultaneously divisible by 
.
Solution
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