MathLabs

Problem 5

Given positive integers mm and nn, find the smallest positive integer kk such that among any kk people, either there are 2m2m people who can be divided into mm pairs of mutually acquainted people, or there are 2n2n people who can be divided into nn pairs of mutually unacquainted people.
Step 5 of 5: Restore symmetry and state the answer
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r(m,n)=2(m+n)−min⁡{m,n}−1r(m,n)=2(m+n)-\min\{m,n\}-1
Detailed analysis

If m≥nm\ge n, the value is 2m+n−1=2(m+n)−n−12m+n-1=2(m+n)-n-1. If n≥mn\ge m, interchange the roles. Therefore the general answer is r(m,n)=2(m+n)−min⁡{m,n}−1r(m,n)=2(m+n)-\min\{m,n\}-1.