Mathematical
Problem-Solving
Strategies
Twelve fundamental strategies used by olympiad mathematicians — from the Extremal Element to Mathematical Induction. Each comes with worked examples at three levels of difficulty.
Extremal Element
Start from the most constrained element
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.
Special & Boundary Cases
Test the extremes first
Before attempting the general case, try the extremes. A special case occurs when a variable takes an extreme value: $n=1$, $x=0$, $k=n$, etc. These cases often reveal whether a claim is true and how to prove it.
Counterexamples
One example says YES — one counterexample says NO
A counterexample is the most direct method to disprove a claim. Instead of explaining why something is wrong, simply find one instance where it fails. One example is enough to refute a universal claim.
Backward Thinking
Start from the goal — work backwards
Like solving a maze from the exit! Instead of starting from the data and trying to reach the conclusion, start from the conclusion and ask: 'What must be true for this to hold?' Often the goal has more structure than the hypotheses.
Symmetry
If it looks the same from everywhere, use it
When a problem has symmetry, some variables 'play the same role'. This often simplifies it dramatically. If $x$, $y$, $z$ play the same role, the extremum must be symmetric. Three types: algebraic, geometric, cyclic.
Invariants
Find what remains constant when everything changes
An invariant is a quantity that doesn't change while something else does. If the start and end states have different invariant values, the transformation is impossible. Invariants can be numerical, modular, or structural.
Pigeonhole Principle
More pigeons than holes → some hole has two pigeons
If $n+1$ objects are placed in $n$ categories, at least one category contains $\geq 2$ objects. Simple to state, but its applications are astonishing. Generalized: if $kn+1$ objects go in $n$ categories, some has $\geq k+1$.
Double Counting
Count the same thing in two different ways
Count the same quantity from two perspectives, then equate the results. This yields a useful relation. Classic: counting handshakes by persons (sum of degrees) or by pairs (twice the number of handshakes).
Reformulation
Say the same thing in a different way
Transform a problem into an equivalent one that is easier to solve. Change variables, translate to a different domain, or reinterpret the question. The key insight: if two problems are equivalent, solving either one solves both.
Auxiliary Construction
Create something new to solve the problem
Add a new element to the problem — a line, point, function, or set — that wasn't in the original statement but helps connect the pieces and reach the solution.
Proof by Contradiction
Assume the opposite — then derive a contradiction
Assume the opposite of what you want to prove, then show this assumption leads to something impossible (a contradiction). If the negation is false, the original statement must be true. One of the most powerful and widely-used proof techniques.
Mathematical Induction
The ladder to infinity: one step at a time
Prove something holds for infinitely many cases with just two steps. Base case: verify for $n_0$. Inductive step: show that if it holds for $n = k$, it holds for $n = k+1$. Then it holds for all $n \geq n_0$.