Maths Olympiad Prep

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Combinatorics Difficulty 4.6 AIME Prove it United States

Problem:
A pair of successive numbers in the same row is called a switch pair if one number in the pair is even and the other is odd.
Prove that the number of switch pairs in row nn is at most twice the number of odd numbers in row nn.

Solution

Solution:
Each switch pair contains an odd number, and each odd number can belong to at most two switch pairs (since it has only two neighbors).

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