What a complete olympiad solution actually looks like

You solve the problem. You write it up. It comes back with three marks out of seven, and the comment is not that the idea was wrong. The idea was right. Something else was missing, and nobody ever told you exactly what.

So here is what a complete solution does, and what an incomplete one leaves out. A complete solution is not a longer solution, and it is not a more formal one. Formality is a style; completeness is a property. It is the property that a reader who does not want to believe you cannot find a single place to stand.

The test that defines the word

Imagine handing your write-up to someone whose job is to disbelieve it. They read one sentence at a time and ask the same question of each: does this follow from what came before, or from something every reader already knows? Your solution is complete when that reader reaches the end without once having to supply a step themselves.

That test is why I know why this is true and I have written a proof are different states, and why the distance between them is not effort. It is a small number of specific omissions, and they repeat.

The four beats

Almost every finished solution, in any topic, moves through four beats in this order.

1. The model. Say what the objects are, in the language you intend to argue in. Nothing is proved here; the problem is translated. Skipping this is what makes an otherwise correct argument unreadable.
2. The engine. The one real idea, applied to a specific named object rather than gestured at. Most engines are one or two sentences. If yours is a paragraph, it is probably two engines and one of them is unnecessary.
3. The exhaustion. Every branch the engine opens gets closed. This is the beat people skip, and it is where marks actually live.
4. The return. The last sentence says the thing the problem asked, with its quantifier back in place.

Notice that only beat two is what most people mean by solving. The other three are the write-up, and they are more than half the marks.

The beats on one problem

Take among any six people, three all know each other or three are mutual strangers.

Beat one: people become points, and each pair gets one of two colours according to whether they are acquainted. Beat two: fix one person, look at the five pairs involving them, and note that two colours cannot split five things evenly, so three of those pairs match. That is pigeonhole, and it is one line. Beat three: among those three people, either some pair matches the colour you already have, or no pair does, and each of those two situations produces the trio you were asked for. Beat four: say it back for every six-person gathering, not for the picture you drew.

The claim itself is not in doubt, and it is worth knowing why: there are 32,768 ways to two-colour the pairs among six people, and a machine can check that every last one of them contains a matching trio. What is in doubt is whether your write-up proves it. Beats one and four are the ones a strong student drops as obvious. Beat three is the one that is genuinely hard to notice you dropped.

Gap one: the branch you did not notice you skipped

In the six-person problem, the tempting write-up handles the case where a matching pair turns up among your three people, feels finished, and stops. The other branch is not a formality. There are colourings in which the trio you need exists only in that second branch, so a proof that omits it is not a proof with a hole in the presentation. It is a proof of nothing.

The cheapest way to catch this class of error is to ask what the problem's numbers are protecting. Five people are not enough: seat five people in a ring, let each be acquainted with their two neighbours and a stranger to the other two, and no trio matches. Since the statement is false for five, any argument that never uses the sixth person is wrong, whatever it concludes.

Gap two: a list without the "and no others"

Find every positive integer $n$ for which $n+1$ divides $n^2+1$ is a complete-solution problem disguised as a search. The answer is $n = 1$, and finding it takes ten seconds of trial. The solution is the argument that nothing else can work: $n^2+1$ exceeds $(n+1)(n-1)$ by exactly two, so $n+1$ has to divide two, and for a positive $n$ that pins it down.

The pairs with $x^y = y^x$ and $x \ne y$ are the same shape. Anyone can produce $(2,4)$ and $(4,2)$ by inspection. Nobody gets marks for producing them. A search of every pair up to three hundred also finds only those two, and a search proves nothing at all, which is precisely the point: the missing half of the write-up is the reason the search would still come up empty at a million.

Whenever a problem says find all, your solution has two halves and you have probably written one.

Gap three: the equality case and the degenerate configuration

Prove $x^2 + 1 \ge 2x$, with equality exactly when $x = 1$ is deliberately built so that the easy half is genuinely easy. The difference is $(x-1)^2$, a square, so the inequality is immediate. The words if and only if then ask for a second, separate claim: that the equality happens at $x=1$ and nowhere else. Two directions, two sentences.

The same gap in geometry is the degenerate configuration: the point that might land outside the segment, the triangle that might be obtuse, the circle that might be tangent instead of secant. If your argument used the picture, the picture is a case, and the other pictures are the cases you did not do.

Gap four: words doing work that arguments should do

Clearly, obviously, it is easy to see, and similarly are the four words that mark the exact spot where a solution stopped being complete. They are not banned; used honestly they save a reader time. Used dishonestly they mean the author could not do the step and hoped nobody would look.

The one worth its own paragraph is without loss of generality. It is legitimate exactly when you can name the symmetry that makes the cases interchangeable, and say it in the sentence: relabelling, reflecting, swapping two variables the statement treats identically. If you cannot name it, you are not reducing the cases, you are choosing one.

Gap five: proving the neighbouring statement

The last gap is the hardest to self-diagnose, because the work is real and the proof is valid. It is just a proof of something else: the converse, or one instance of the claim, or the version with an extra assumption you quietly needed.

Two opposite corners removed from a chessboard, and a claim that the rest cannot be tiled by dominoes is the cleanest illustration. You can spend an hour failing to tile it and have proved nothing whatsoever, because I could not do it and it cannot be done are different statements. Colour the board and one line settles it: every domino covers one square of each colour, opposite corners share a colour, so removing them leaves thirty of one and thirty-two of the other, and no tiling can balance that. The colouring is the whole proof, and it arrives in beat one.

The five-minute pass over your own write-up

Do this before you hand anything in. It is mechanical, which is why it works under pressure.

1. Circle every case split. Count the branches, then count the branches you closed. These are usually different numbers.
2. Find every clearly, obviously and similarly. Either delete the word, or replace it with the sentence it was standing in for.
3. Read your last sentence alone. Does it contain the quantifier from the problem statement, or does it talk about the objects you happened to name halfway through?
4. List the hypotheses the problem gave you and find where each was used. An unused hypothesis almost always means you proved a different, easier statement.
5. Read the first sentence and the last sentence together. If they are not about the same things, the model and the conclusion have drifted apart.

Point four catches more real errors than the other four combined, and almost nobody does it.

Where to practise this

Recognition and write-up are different skills and want different practice. For write-up, the fastest way to improve is to solve fewer problems and finish more of them: take a problem you already solved, apply the five-minute pass, and rewrite. Do that ten times and the beats stop being a checklist.

The problem archive has the statements above in full, with the topic, difficulty and expected time on each, and every problem in the Olympiad Set carries progressive hints, a worked solution and a marking scheme so you can compare your write-up against what a grader is actually looking for. The daily problem posts a fresh one every day in three tiers with no account needed, and the technique pages name the engines so you can tell beat two from the rest of the argument. If you would rather find out which of the five gaps is yours than guess, the diagnostic is the shortest route to an answer.