Foundation to stretch · original practice
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BMO Round 1 has six problems worth 10 marks each in 3½ hours. Give a complete, justified proof. No calculator or protractor. This bank includes classical proof exercises and extended problems; parameter variants practise the same idea. Model solutions support self-review, not automatic BMO marking.
Official guidance & papers ↗Timed rounds select 3 proofs from this bank, with 35 minutes per question. Your score is practice feedback, not a prediction of an official result.
Study mode · browse all 108 questions, with hints and explanations.
For which positive integers is a square number? Prove that there are no others.
Use paper for exploratory work and diagrams; put your final argument in the proof space above. Geometry proofs should label points clearly.