From AJMO to AIMO: Turn an Answer into a Written Argument

By Ace Achievers · Published 16 October 2026 · Official information checked 14 October 2026

What changes when a competition asks for written reasoning?

AMT’s AJMO format asks for integer answers and does not require written solutions. AIMO includes questions requiring written solutions. The change is not just a longer worksheet: a student must explain why the argument covers all cases and why its steps follow from the conditions.

For an integer-answer practice question, a correct result may hide an incomplete method. It could come from a promising guess or a small list that happens to include the right case. Written practice exposes that gap. Start with a short explanation the child can understand fully, rather than demanding a formal proof of a very difficult problem immediately.

How do we turn a numerical answer into an argument?

An integer answer is 5. A complete explanation has two parts. First, show that every choice of five works, using the four groups. Second, give a choice of four that fails. The second part proves the threshold cannot be smaller. Showing one successful five-number example would not establish a guarantee.

Now change the available numbers to 1 through 10 and the target to 11. There are five complementary pairs, so the smallest guaranteed number of choices becomes 6. The child should explain the same two parts rather than just add one to the earlier answer.

What should the student actually write?

1. State the objects and restrictions: distinct integers in the stated interval.
2. Explain the grouping and why every available number belongs to one group.
3. Connect the number of choices to the number of groups.
4. Give a counterexample for a smaller claimed threshold.
5. End with the exact conclusion requested.

A useful short version is: “There are four complementary pairs. Five chosen numbers cannot occupy four pairs with at most one in each, so a whole pair is chosen. Four is insufficient because 1,2,3,4 work as a counterexample. Therefore the minimum is five.” Each sentence performs a mathematical job.

How can a parent assess the next level?

Try suitable official AJMO/AIMO comparison material and observe sustained independent work. Can the child recover from a failed approach, account for every case, and explain why a relationship holds? These observations are more useful than assuming a particular certificate automatically selects AIMO.

The official Discovery Series includes an AIMO 2024 collection. The AMT archive also links a historical AIMO paper with solutions. Keep their labels clear: they are AIMO material, not AJMO 2026 papers. Read a worked argument for its structure, then use a fresh, suitable problem to see whether the child can produce a similar explanation.

How do we practise without making writing a barrier?

Let the child explain aloud first, sketch a diagram, and then write a few necessary sentences. Remove repetition and fill gaps in the logic. Do not replace mathematical reasoning with polished but vague prose. If the student is not yet independently handling the underlying problems, continue strengthening the appropriate AMC-level foundations before expecting longer Olympiad arguments.