Math Olympiad Guide: From First Problem to the OSN

Almost every math olympiad problem, from a school selection round up to the IMO, falls into one of four fields: algebra, combinatorics, number theory and geometry. The split comes from the international olympiad syllabus and is also used in the OSN syllabus, with names and scope adjusted to each school level. At junior high, number theory is often simply called numbers, and some syllabi add data and probability. At primary level the material is organised into five areas that sit closer to the grade 4 and 5 curriculum.

Each field has its own flavour. Number theory and algebra feel the most familiar because the raw material is already known from school: factors, remainders, equations and sequences. Combinatorics feels strange at first since ready-made formulas are rare and every problem asks for a fresh way of arranging things. Geometry rewards a careful drawing and an eye trained to see angle relationships that are not obvious at first glance.

A practical way to pick an entry point is to spend one or two weeks on five to eight introductory problems from each field, noting which one is most enjoyable and which one causes the most dead ends. The enjoyable field becomes the first door, because early enjoyment keeps practice going once problems get harder. When that field feels comfortable, its share of practice shrinks and the next field comes in, so that no field is left far behind when the selection season arrives.

Combinatorics is the field of counting and arranging: how many ways are there, can a certain arrangement exist, and who wins a given game. The basics are the rules of sum and product, permutations, combinations and inclusion-exclusion. Built on top of these are ideas that keep returning in harder problems: the pigeonhole principle, invariants, colouring arguments and extremal arguments.

The pigeonhole principle sounds trivial. Among any 13 people, two share a birth month, because there are only 12 months. The challenge lies in choosing the right pigeonholes for a particular problem. Invariants work differently. Remove two opposite corner squares from an 8 by 8 chessboard and 62 squares remain; can the board be covered by 31 dominoes? Every domino covers one black and one white square, while the two removed squares share a colour, leaving 32 squares of one colour and 30 of the other. The covering is impossible, and the proof takes three sentences.

This field often supplies the deciding problem at national level because each problem calls for a new constructive idea. The most common mistakes are counting something twice and missing a case. The drill: before trusting a formula, list every case in writing, test the result on a small version of the problem that can be counted by hand, and compare. If the formula gives 20 and the hand count gives 18, something has been counted twice, and tracking down where is a valuable exercise in its own right.

Geometry shows up in nearly every round and often arrives as a heavily weighted proof problem. Its material covers triangles and their special lines (altitudes, medians, angle bisectors), similarity and congruence, and circles. At senior high, trigonometry and coordinate geometry join in as extra tools for problems that resist pure angle work.

The most important tool in this field is the diagram. Compass, ruler and protractor are used to draw configurations to scale, since a skewed sketch can make two lines look parallel or hide three points that actually lie on one line. Once the diagram is clean, students practise angle chasing: writing down every angle that can be calculated and hunting for equal pairs. Two circle facts come up constantly. Inscribed angles standing on the same arc are equal. A quadrilateral whose four vertices lie on one circle has opposite angles summing to 180 degrees. Spotting a hidden cyclic quadrilateral frequently unlocks the whole problem.

The signature difficulty in geometry is finding the auxiliary line. The right one usually becomes visible only after a student has seen dozens of similar configurations. The drill: after finishing each geometry problem, record which auxiliary line opened it up and why it was chosen. Within a few months these notes turn into a personal dictionary of configurations. The field is maturing when a student can guess from an accurate diagram which points are collinear or which four points are concyclic, and then prove the guess.

Olympiad problems are designed so that the route to a solution is hidden at first. Feeling stuck is a normal part of training, and the ability to stay with it can be practised like any other technique. The usual list of heuristics: try small cases such as n = 1, 2, 3, 4; draw a picture or build a table; work backwards from what is asked; simplify the problem by dropping one condition; and look for a similar problem solved before.

A small-case example: what is the sum of the first n odd numbers? For n = 1, 2, 3, 4 the sums are 1, 4, 9 and 16. The conjecture is that the sum is always n². That conjecture is then proved, perhaps by induction, or by arranging dots into a square that grows by one layer each time the next odd number is added. Try, conjecture, then prove: this working rhythm carries across all four fields.

The habit that does the most damage is opening the solution too early. A solution read after five minutes of effort gives a brief sense of understanding, and the key idea rarely sticks. A longer-lasting routine is to set a minimum attempt time, say 20 minutes for a district-level problem and longer for later rounds, then stretch it gradually. After reading a solution, close the book and write the solution out unaided the next day. Any problem that cannot be rewritten goes onto the list of missed problems for another try.

At primary level, OSN participants are usually in grade 4 or 5. The material is organised into five areas: numbers, geometry, measurement, data and logic. All of it rests on the primary curriculum, and what sets the problems apart is their non-routine character: the route to the answer has to be found through reasoning. The skill trained at this level is explaining how an answer was found, which prepares students for formal proof when they reach junior high.

Each area has practice that sits close to everyday life. Numbers are trained through number puzzles, sequence patterns, and games with factors and multiples. Geometry through drawing, cutting and rearranging shapes, and counting how many triangles hide inside a figure. Measurement through estimating the length or weight of objects and then measuring them. Data through recording daily things and reading the resulting table. Logic through puzzles, sudoku and strategy board games. The general strategies are shared across all five: structured trial and error, making a table, drawing, and working backwards.

A reasonable plan for primary students starts four to six months before OSN-S, with short practice spells a few times a week. An initial check uses 10 to 15 mixed problems from the five areas with no strict time limit, to see which areas are strong and which need more attention. Timed mock papers come in during the final month, once or twice a week at most. Because the participants are children, parents help most by looking after the atmosphere: keeping the schedule light, praising the courage to try a new strategy, and contacting the class teacher early in the school year, since the OSN pathway begins at school.