Sit a Practice Test
The USAMO practice sets draw six problems at a time from the hand-verified USAMO pool (number theory, combinatorics, geometry, algebra, inequalities, games). Answers are numeric so they can be checked — but write the PROOF on paper as you would on the real exam.
Treat each set as a session: hours for six problems, no calculator. After finishing, read every solution in full and compare the write-up to yours line by line: did you justify the step the solution justifies? Did you handle the equality/edge case? The gap between "I saw the idea" and "I wrote a 7" is the entire game.
State what you will prove. Define your notation. Label lemmas and prove each before using it. Justify every non-obvious step with a named tool. Handle equality cases and degenerate configurations. End with a clear conclusion sentence.
Graders read for GAPS: an unproved claim, a case not covered, a "clearly" that is not clear. Proof by contradiction must state the negated assumption precisely; induction must state the hypothesis and prove the base. Diagrams help but are not proofs — every geometric fact you read off a picture must be justified (angle chase, similar triangles, power of a point) in words. Numbered steps are your friend.
USAMO problems are points: for a complete proof, – for a complete proof with a minor gap, – for substantial progress (a correct key lemma), for the answer alone. A partial solution that clearly states what was proved and what was not can earn more than a confident but flawed "complete" one — honesty about gaps is rewarded.