SMO Senior Preparation: Singapore Olympiad Guide

SMO Senior preparation has to train two genuinely different examinations, because that is what the olympiad is. The habits that produce a good Round 1 are close to the opposite of those that produce a good Round 2.

A long tradition and what it values

The Singapore Mathematical Olympiad has been run by the Singapore Mathematical Society since 1956, and the Senior section is written by Secondary 3 and Secondary 4 students. Eligibility includes a birth-date cutoff that shifts annually, so it is worth confirming before registering.

The tradition has a recognisable character: non-routine problems that do not announce their method, and an emphasis on representing a situation correctly before computing anything. What changes at Senior level is depth rather than philosophy. The problems are longer, the required ideas less immediate, and the second round expects a sustained argument.

Two rounds, two different demands

Round 1 is two and a half hours, mixing multiple-choice and short-answer questions with equal weighting across them. It rewards accurate, reasonably quick work across a spread of problems.

Round 2 is four hours and asks for full written solutions, and only the top ten per cent of Round 1 reach it.

Four hours is long enough that stamina matters. A student prepared only for pace finds the second round deeply uncomfortable: the instinct that serves them well in Round 1 — move on, do not dwell, trust a quick estimate — is precisely wrong for a problem that wants half an hour of sustained development.

Good SMO Senior preparation trains these separately, because practising only one produces a student who is good at half the olympiad.

The move from answers to arguments

For most students this is the hardest part, and it is not primarily about mathematics.

A student can see exactly why a result holds and still write a submission that assumes the thing it was meant to prove. Round 2 marks the argument: steps justified, cases handled, the conclusion actually reached rather than implied.

We work on this directly from early in the course — stating what is given, justifying the step that is not obvious, dealing with the awkward case rather than the convenient one. Students find the first few weeks frustrating and then find it changes how they think about problems generally, including in Round 1.

The four areas at senior level

Number theory moves beyond divisibility rules into modular arithmetic used as a tool, orders, and the standard olympiad results applied rather than recited. Our number theory course follows that progression.

Algebra covers polynomials, functional relationships, inequalities and sequences. The inequality toolkit — particularly AM-GM and Cauchy-Schwarz — recurs often enough that fluency with it is close to compulsory, and our algebra course treats it as a toolkit with an order of attack rather than a list.

Geometry is circles, cyclic quadrilaterals and similar triangles, in configurations that begin to look familiar once enough have been worked. Diagrams drawn properly, because a bad figure hides the answer.

Combinatorics rewards a clear idea over heavy calculation: pigeonhole, invariants and extremal arguments turn apparently impossible problems into short ones. Our combinatorics course covers these systematically.

Represent before you compute

The Singapore tradition places unusual weight on setting a problem up before any calculation, and this is the habit most worth taking from it.

A student in a hurry reads a problem, spots a number, and starts manipulating. A student trained this way first asks what the problem is actually describing: whether a diagram makes the relationship visible, whether naming a quantity turns a wordy condition into one equation, whether a small case reveals the pattern the general case is hiding.

Problems that look intractable frequently become routine once represented properly. Problems that look routine occasionally turn out not to be, which is equally worth discovering before committing twenty minutes.

What the olympiad does and does not lead to

This is worth stating plainly because coaching pages often blur it. National team training and IMO selection run through the Open section, not the Senior section.

SMO Senior is a strong olympiad in its own right, and performing well in it is a genuine achievement. It is not a direct step onto the Singapore IMO team, and a student should not choose it on that basis.

Round 1 awards silver, bronze and honourable mention. Round 2 awards gold with cash prizes for the top thirty, and the top scorer receives the Liew Mai Heng Memorial Prize.

How it fits other olympiads

For a student also preparing for IOQM and RMO, the overlap is nearly complete. The topics are the same four, and Round 2 demands exactly the sustained written argument that RMO does.

Preparation for one is largely preparation for the other, which makes SMO Senior material a sensible addition for students on the Indian pathway who want more problem exposure at the right level. The same is true of the CEMC written-solution contests, which give proof practice at lower stakes.

What to expect from a year of work

Expect the proof writing to take longer to improve than the problem solving. A student typically becomes noticeably better at Round 1 within a few months and takes most of a year to write a Round 2 solution that would earn full marks.

That asymmetry is normal and worth knowing, because a student who judges their progress only by Round 1 scores will conclude the Round 2 work is not paying off, shortly before it does.

Calculators are not permitted in either round. Eligibility, the birth-date cutoff and the exact dates are set by the Singapore Mathematical Society each year, so check sgmathsociety.org for the current cycle rather than relying on figures reproduced elsewhere.