Math Competitions: A Beginner's Roadmap

A beginner does not need to learn every contest topic at once. The first roadmap should connect four kinds of work: reliable foundations, one topic at a time, unfamiliar problems, and honest review. Adjust the plan to the official syllabus, age division, format, and rules of the competition you actually intend to enter.

Week 0: Pick the target and take a baseline

Find the organizer's current syllabus and one representative past set. Without studying first, try a small sample under relaxed conditions. Label each question:

• solved with a clear method;
• understood but calculation failed;
• topic not yet learned;
• no useful first move.

This separates a knowledge gap from a problem-selection or arithmetic gap. Keep the baseline; repeat a comparable sample later.

Weeks 1 and 2: Stabilize the foundations

Prioritize operations that appear inside harder questions: factors and multiples, fractions, ratios, percentages, integer signs, simple equations, and careful counting.

Worked example: 49 x 3 + 51 x 3 looks like two products, but the shared factor gives (49 + 51) x 3 = 100 x 3 = 300. The contest skill is noticing structure before calculating.

Use short drills for errors that consume attention, but do not turn every session into speed work. A learner also needs untimed problems where the first move is not announced.

Weeks 3 to 5: Run topic cycles

Choose one topic for several sessions. A useful cycle is:

1. Learn or review one core idea.
2. Solve two direct examples.
3. Solve three problems where the idea is not named.
4. Compare solutions and record the shortest valid route.
5. Revisit one missed problem after a day.

Possible beginner cycles include divisibility, parity, basic combinatorics, angle chasing, sequences, and algebraic factoring. Follow the target competition's scope rather than this list if they differ.

Weeks 6 and 7: Mix topics and practise selection

Use short past-problem sets. Before calculating, write a five-word plan such as "factor the shared term" or "count the complement." This exposes random trial-and-error and makes solution review more useful.

Worked example: To count two-digit numbers with at least one digit 7, count the complement. There are 90 two-digit numbers. There are 8 x 9 = 72 with no 7: eight choices for the tens digit and nine for the ones digit. The answer is 90 - 72 = 18.

After each set, sort misses by topic, first move, execution, or time choice. The largest group sets the next practice target.

Week 8: Rehearse the real format

Use the official time limit and allowed materials. Practise decisions as well as solutions: when to continue, when to mark a question for return, and how much checking the format permits. Do not invent rules from another competition.

Keep a solution-review card

For every important miss, record:

Prompt in one line	First useful observation	Why my route failed	Nearby problem
Shared products around 50	Both terms contain 3	Calculated separately	38 x 7 + 62 x 7

Copying a full official solution can hide the decision that mattered. Reduce the review to the observation that unlocked the problem, then prove it transfers on a nearby example.

At the end of eight weeks, compare a new representative set with the baseline. The useful measures are questions understood, first moves found, recurring error labels, and solutions you can now explain. Those results determine the next cycle.