IMC 2025

6

HintWhat does 2 × 8 end in?

The whyPowers of 2, 3, 7 and 8 all have units-digit cycles of length 4; 4 and 9 have length 2; 0, 1, 5, 6 never change. The remainder on dividing by 5 (or 10) is read straight from that units digit — 8¹⁰ ends in 4, so it leaves remainder 4 on division by 5.

The sum of the two opposite interior angles

HintCombine the straight line with the triangle's own total.

The whyIf the base angles are b, the exterior angle at the base is 180 − b and also equals apex + b. Isosceles triangles hand you two equal base angles for free — say so before you write anything else.

The number of items

HintThink of paying whole pounds and getting change.

The whyPrices like 99p and £2.99 are whole pounds minus a penny, so 25 such items cost 25p less than a whole number of pounds. Read the count off the pence, then a second equation on the pounds finds how many of each.

Ninths

Hint0.1 recurring is 1 over what?

The whyA single repeating digit d is d/9: 0.1 recurring = 1/9, 0.6 recurring = 2/3, 0.9 recurring = 1. So a sum of them is a fraction over 9, which may itself be a recurring decimal.

sum

HintNot the numbers themselves.

The whyExpand the brackets, then substitute the given product rather than solving for a and b — most 'find the sum' questions never need the numbers themselves.

Opposite corners: equal sums of squares

HintPythagoras four times, then pair up.

The whyDropping perpendiculars from P to the sides makes four right-angled triangles sharing legs; adding the two 'opposite' Pythagoras equations gives PA² + PC² = PB² + PD². Distances 6, 8 and 10 to three corners in order force the fourth to be √(36 + 100 − 64) = √72.

r + R

HintThe touching point lies on the line of centres.

The whyCentres and touching point are collinear, so the centre distance is r + R when the circles touch from outside and R − r when one sits inside the other.

A multiple of 4

HintHow many equal shares make the whole?

The whyA ratio a : b in whole parts makes the total a multiple of a + b: 3 : 2 → multiple of 5, 5 : 1 → multiple of 6, 7 : 3 → multiple of 10. Matching totals to ratios is then a divisibility check, not an equation.

3n and 5n

HintOne shared multiplier scales both parts.

The whyIntroduce one letter for the common multiplier: 3n red and 5n blue counters; add 4 red and the ratio is 1 : 1, so 3n + 4 = 5n and n = 2.

even; odd; 2

HintWhich multipliers keep the parity and which force it?

The whySo 5n + 3 even forces n odd; then 7n + 2 is odd and 4n + 1 is odd for every n. Parity questions are settled by checking n = 1 and n = 2 — nothing else changes the pattern.

Factorise each bracket

HintBoth brackets share a common power.

The why5⁴(5² − 1) ÷ 2⁶(2² − 1) = 625 × 24 ÷ (64 × 3) = 625 × 8 ÷ 64 = 625/8. Never expand a power you can factorise.

2

HintEach pair got counted in both orders.

The whyChoosing 2 from 6 gives 6 × 5 ÷ 2 = 15, from 7 it is 21. Constraints such as 'not next to each other' are handled by listing by the first chosen item and counting what remains legal.