Try a short problem first. Then use the topic map to plan what to learn next.
Jump to the full topic map · Continue the four-subject practice plan
Readiness check: explain, then calculate
You need fractions, basic algebra and divisibility before the harder topics. Try these without a calculator. The reasoning matters more than speed.
- Which is larger, 5/8 or 7/12?
- Solve 3(x − 2) = 2x + 5.
- Can a sum of three odd integers be even?
Worked solutions
1. Use the positive common denominator 24: 5/8 = 15/24 and 7/12 = 14/24. Therefore 5/8 is larger. Cross multiplication also works because both denominators are positive.
2. Expand first: 3x − 6 = 2x + 5. Subtract 2x and add 6 to get x = 11. Substitution gives 27 on both sides, which checks the answer.
3. Write the integers as 2a + 1, 2b + 1 and 2c + 1. Their sum is 2(a + b + c + 1) + 1, which is odd. This covers negative odd integers as well.
Choose your next step
If a solution felt unfamiliar, practise that school-mathematics skill first. If all three were comfortable, begin with the short proofs in proof-writing and the remainder argument in number theory. A country’s grade label does not by itself measure readiness.
These are teaching examples written for this guide, not past-paper questions or an official marking scheme.
Topic map and learning goals
| Unit | Topics | What mastery looks like |
|---|---|---|
| Number sense | Integers, fractions, rational and irrational numbers, absolute value, powers, roots, estimation | Manipulate expressions without losing sign or domain restrictions |
| Elementary algebra | Identities, factorisation, linear and quadratic equations, simultaneous equations | Rewrite a problem into a useful equivalent form |
| Sets and functions | Sets, inclusion, mappings, domain, range, injectivity, surjectivity, composition | State precisely which inputs and outputs are allowed |
| Geometry basics | Angles, parallel lines, congruence, similarity, Pythagoras, area | Justify relationships from given information rather than a drawing’s appearance |
| Mathematical statements | “For all”, “there exists”, implication, converse, necessary and sufficient conditions | Distinguish an example from a proof and a converse from the original claim |
| Notation | Summation, products, divisibility, congruence, interval and set notation | Read and write solutions consistently |
Teach these alongside short problems. Finishing a school textbook is useful preparation, but does not by itself demonstrate Olympiad readiness.