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On the blackboard are written the $400$ integers $1, 2, 3, \cdots , 399, 400$

Source: May Olympiad 2016 L1 P5

March 14, 2021
combinatorics

Problem Statement

On the blackboard are written the 400400 integers 1,2,3,⋯ ,399,4001, 2, 3, \cdots , 399, 400. Luis erases 100100 of these numbers, then Martin erases another 100100. Martin wins if the sum of the 200200 erased numbers equals the sum of those not deleted; otherwise, he wins Luis. Which of the two has a winning strategy? What if Luis deletes 101101 numbers and Martín deletes 9999? In each case, explain how the player with the winning strategy can ensure victory.
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