Competition mathematics demands a fundamentally different approach to problem-solving than standard curriculum mathematics. In an exam, you are typically applying a method you have been taught to a problem that is designed to be solvable. In a competition, you are frequently facing a problem for which no standard method exists—where the challenge is not executing a technique, but discovering one.
The students who succeed at the highest levels of competition mathematics are not necessarily those with the most knowledge. They are those who have developed a toolkit of general strategies that can be applied when no specific technique comes to mind. Here are five of the most powerful.
1. Work with a Simpler Case
When a problem seems intractable, strip it down. If it involves a large number, try the same problem with a small number. If it involves a general case, try a specific case. If it has multiple variables, fix all but one.
This is not about solving the simpler problem instead—it is about finding the pattern. By working through several small cases, you often discover a structure or invariant that generalises. A problem that asks about all positive integers might reveal its secret when you check the cases n = 1, 2, 3, and 4.
The key insight is that mathematical structures tend to repeat. What is true for small cases is often true in general, and understanding why it works in the small case frequently illuminates the general proof.
2. Draw a Picture
This sounds obvious, but it is remarkable how many students attempt to solve geometry, combinatorics, and even number theory problems purely symbolically. A diagram does not just record information—it reveals relationships that are invisible in algebraic notation.
In geometry, of course, a diagram is essential. But in combinatorics, drawing out small cases—literally sketching the arrangements—can reveal symmetries and counting strategies. In number theory, plotting values or using a number line can make modular arithmetic relationships click.
The rule is simple: if you are stuck, draw something. Even a rough, inaccurate sketch can unlock an insight that pages of algebra will not.
3. Look for Invariants
An invariant is a quantity or property that does not change under the operations described in the problem. Finding an invariant is often the key to proving that a certain configuration is impossible, or that a process must terminate.
For example, if a problem involves a game where players take turns modifying a position, ask yourself: what stays the same no matter what move is made? The parity of a sum, the colouring of a chessboard, the value of an expression modulo some number—these are all classic invariants.
The strategy is to identify what cannot change, and then use that to constrain what is possible. If the target state has a different invariant than the starting state, you have immediately proven that the target is unreachable. This is one of the most elegant and powerful tools in competition mathematics.
4. Consider Extremal Cases
When a problem asks you to prove that something is always true, or to find the best possible result, consider the extreme cases. What happens when a value is as large as possible? As small as possible? When variables are equal? When one variable dominates?
Extremal reasoning is the backbone of optimisation problems. If you are trying to find the maximum of a quantity, ask: what configuration would make it large? What prevents it from being larger? The answer to the second question often reveals the structure that bounds the quantity—and proving that bound is usually the heart of the problem.
This strategy also applies to proof by contradiction. Assume the opposite of what you want to prove, and then push that assumption to its extreme. If the extreme case leads to a contradiction, the assumption must be false.
5. Exploit Symmetry
Symmetry is everywhere in competition mathematics, and learning to exploit it is one of the highest-leverage skills you can develop. If a problem has a symmetric structure—if swapping certain elements leaves the problem unchanged—then the solution must respect that symmetry.
This manifests in many ways. In counting problems, symmetric configurations are often overcounted, and dividing by the symmetry group gives the correct count. In optimisation, symmetric configurations are frequently extremal. In algebraic problems, symmetric expressions can often be simplified using symmetric substitutions.
The deeper insight is that symmetry is not just about visual patterns—it is a structural property of the problem itself. Learning to recognise it, whether in a geometric figure, an algebraic expression, or a combinatorial arrangement, will transform your problem-solving.
Putting It All Together
These five strategies are not independent. A single problem might require working with simpler cases to discover an invariant, exploiting symmetry to simplify the counting, and considering extremal cases to complete the proof. The art of competition mathematics is knowing which strategy to reach for, and when.
The only way to develop this judgement is through practice. Work through problems that are just beyond your current ability. When you are stuck—and you will be—resist the temptation to look up the solution immediately. Sit with the problem. Try each strategy. Let your mind wander. The struggle itself is where the learning happens.
Competition mathematics is hard. That is the point. But it is also one of the most intellectually rewarding activities available to a young mathematician. Every problem you solve expands not just your knowledge, but your capacity to think.