Use this glossary while reading the lessons and country guides.
Proof
A complete argument showing why a statement follows from its assumptions, including every necessary case.
Counterexample
One example that disproves a statement claiming something is true in every case.
Lemma
A smaller result proved and then used inside a larger argument.
If and only if
Two implications: each condition is both necessary and sufficient for the other.
Positive integer
One of 1, 2, 3, … . Zero and negative numbers are excluded.
Gcd / coprime
The greatest positive common divisor; two integers are coprime when their gcd is 1.
Congruence modulo m
Two integers leave the same remainder on division by m; equivalently, their difference is divisible by m.
Invariant
A quantity or property that stays unchanged under each allowed move.
Pigeonhole principle
If more objects than boxes are placed in the boxes, at least one box contains two or more objects.
Preliminary / qualifier
An early selection round. Whether it is open entry or by nomination depends on the organiser.
Answer key / solutions
A key may give final answers only. Worked solutions explain how those answers are obtained.
National team / delegation
The group selected to represent a country at the international event. A domestic medal does not automatically give a team place.