Understand a concept.
Try it. Take the next step.
Start with a lesson you can almost do. Read a worked example, try a short practice session, then write a proof on paper. You can use every lesson without an account.
Start with odd and even numbers →Choose a concept
New to Olympiad thinking? Try parity → divisibility → remainders → greatest common divisors. You can also explore algebra, geometry or counting independently when the prerequisites feel familiar.
Odd and even: parity
Use odd and even numbers to rule out impossible outcomes and build a short proof.
Learn and practiseDivisibility and prime factors
Explain divisibility tests and use prime factors to decide what divides a number.
Learn and practiseRemainders and modular arithmetic
Replace large numbers by small remainders and recognise repeating power cycles.
Learn and practiseGreatest common divisors and the Euclidean algorithm
Find the greatest common divisor efficiently and explain why the remainder method works.
Learn and practiseAlgebraic identities and clever factorisation
Use identities to simplify calculations and turn expressions into useful products.
Learn and practiseAngle chasing in triangles
Use triangle angle sums, isosceles triangles and exterior angles with clear reasons.
Learn and practiseCounting without missing or repeating cases
Choose between addition and multiplication, and distinguish ordered choices from unordered pairs.
Learn and practiseThe pigeonhole principle
Choose useful boxes and prove that some group must contain enough objects.
Learn and practiseNo matching lesson yet. Try another subject or clear the search.
From short answers to Olympiad proofs
These eight lessons introduce useful ways of thinking. They are a starting sequence, not a complete IMO training course. A correct choice is useful feedback, but a competition proof also needs reasons written in a logical order.
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