MathLabs

Problem 1

The function f(n)f(n) is defined on the positive integers and takes non-negative integer values. f(2)=0f(2)=0, f(3)>0f(3)>0, f(9999)=3333f(9999)=3333, and for all m,nm,n: f(m+n)−f(m)−f(n)=0 or 1.f(m+n)-f(m)-f(n)=0 \text{ or } 1. Determine f(1982)f(1982).
Step 7 of 7: Conclusion
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In plain words

Final answer, obtained purely from the superadditivity inequality and two exact data points.

f(1982)=660f(1982)=660
Detailed analysis

The contradiction in Step 6 eliminates 661661, leaving f(1982)=660f(1982)=660 as the only possibility, in agreement with every solution published on the AoPS wiki page for this problem.