← All Strategies
◈
Strategy 01 of 12Extremal Element
Start from the most constrained element
Finite SetsOptimizationContradiction
Instead of looking at all elements simultaneously, pick the most extreme one — the largest, smallest, first, or last. The extremal element has fewer possibilities than the rest, and that constraint gives you information.
When to use it
- →When you see words like 'minimum', 'maximum', 'first', 'last'
- →When you have a finite set of elements
- →When a proof by contradiction is needed
- →When elements have a natural ordering (size, time)
How to think (step by step)
- 1Identify the extreme: Which element is largest/smallest/first/last?
- 2Ask: What can this element NOT do?
- 3Use the constraint: That limitation gives you information
- 4Repeat: If needed, look at the next extreme element
Practice Problems
Three problems at increasing difficulty — try each before revealing the hint or solution.
📘Basic
Given a finite set of positive integers such that the sum of any two distinct elements differs from the sum of any other two distinct elements. Prove that the set contains at most one even number.
📙Intermediate
In a tournament, \(n\) players each play against every other player exactly once, with no draws. Prove that the players can be arranged in a sequence \(A_1, A_2, \ldots, A_n\) such that \(A_i\) beat \(A_{i+1}\) for every \(i\).
📕Advanced
Let \(S\) be a set of \(2n\) points in general position in the plane (no three collinear). Prove that there exists a line through two points of \(S\) such that \(n-1\) points of \(S\) lie on each side.