APMO · 8. Counting and extremal combinatorics
Counting by contributions and double counting, the extremal principle (maximal sets, first critical index) and the pigeonhole principle for sums, with five fully proved APMO problems.
Evaluate sums over configurations by counting the contribution of each element.
Prove bounds by double counting incidences and reach them with symmetric constructions.
Use a maximal set or the first index where a running sum passes a threshold.
Apply the pigeonhole principle to families of subset sums.
Write complete answers to maximum problems: a bound plus a fully checked example.