METHODS · ORIGINAL PRACTICE

AJMO Problem-Solving Strategies: Three Original Worked Examples

By Ace Achievers · Published 16 October 2026 · Official information checked 14 October 2026

The short answer

When a problem looks unfamiliar, make its conditions visible: draw the shape, list cases in a fixed order, or combine number restrictions. These three original examples show a first step, a solution and a way to test whether the method transfers.

These are Ace Achievers teaching examples, not questions from the 2026 AJMO paper. They illustrate techniques and do not predict official difficulty.

1. Geometry: calculate what is easier, then subtract

The problem

An 8 cm by 5 cm rectangle has a 3 cm by 2 cm rectangular corner removed. What is the remaining area?

First step: draw the whole rectangle and the missing corner. Label both sets of dimensions.

The whole area is 8 × 5 = 40 cm². The removed area is 3 × 2 = 6 cm². The remaining area is 34 cm².

The arithmetic is straightforward; the representation helps prevent subtracting lengths from areas. Ask the child to label cm and cm² correctly. They can check by splitting the remaining L-shape into two rectangles and adding the areas.

Transfer task: give the remaining area as 34 cm² and the missing corner’s height as 2 cm. Keeping the outer dimensions, ask for the corner’s width. The missing area is 6 cm², so its width is 3 cm. The child now works backwards instead of only repeating multiplication.

2. Counting: decide whether order changes the outcome

The problem

A child chooses two different badge colours from red, blue, green and white. The order of choosing does not matter. How many pairs are possible?

First step: list pairs by their first colour, keeping one fixed colour order.

Red–blue, red–green, red–white; blue–green, blue–white; green–white. There are 6 pairs.

The organising rule matters more than the list alone. Red–blue is the same pair as blue–red, so listing both would count one choice twice. After the first colour is fixed, only list colours later in the chosen order.

Transfer task: now make one badge for the captain and one for the vice-captain, with different colours. The roles make order matter: 4 choices for captain × 3 for vice-captain = 12. A small change in the wording changes the count.

3. Number restrictions: combine conditions before testing

The problem

How many integers from 200 to 399 inclusive end in 6 and are divisible by 9?

First step: write a candidate as 100h + 10t + 6, with h = 2 or 3 and t from 0 to 9.

For h = 2, the digit sum is t + 8. It is a multiple of 9 only at t = 1, giving 216. For h = 3, the sum is t + 9: t = 0 or 9 gives 306 or 396. There are 3 numbers.

Check that every candidate obeys the range and final-digit condition. A complete case split by hundreds digit is more reliable than guessing multiples from memory. A calculator-free alternate check is dividing the three candidates by 9: 24, 34 and 44.

Transfer task: change the final digit to 5. The same structure gives 225 and 315, so there are 2. Ask the child to derive these from the constraints rather than from the previous answer.

How should we use these examples?

Attempt each problem before opening its explanation in your own practice routine. Then name the method in one sentence, try the transfer task and return to a related question later. If a child succeeds only after you identify the technique, practise method selection in a mixed set.

These techniques connect with the geometry, counting and number-theory teaching in our existing AMC courses. For competition-specific fit, compare with the samples linked from AMT’s AJMO page. See the preparation plan for a routine that combines teaching and independent attempts.

Give practice a clear learning pathway

Explore the complete curriculum, then start the free lessons in your own learning account. These existing AMC and AIMO courses develop related skills; they are not dedicated AJMO courses or official competition entry.

Questions parents ask

Are these the actual 2026 AJMO questions?

No. These are original Ace Achievers teaching examples. They are not reproduced exam questions, official samples or an estimate of the paper’s difficulty.

What should a child try when they do not know how to start?

Restate the conditions, draw or list relevant information, and try a smaller case. Then check which representation reveals a useful relationship.

How do we stop counting twice?

Fix an organising rule and decide whether order matters. Explain why each outcome belongs to exactly one group in the list.

Why change the problem after solving it?

A changed condition tests whether the child understands the method and can adapt it. Repeating the same answer can instead test memory.