IMC 2025

6

提示What does 2 × 8 end in?

为什么Powers 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

提示Combine the straight line with the triangle's own total.

为什么If 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

提示Think of paying whole pounds and getting change.

为什么Prices 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

提示0.1 recurring is 1 over what?

为什么A 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

提示Not the numbers themselves.

为什么Expand 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

提示Pythagoras four times, then pair up.

为什么Dropping 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

提示The touching point lies on the line of centres.

为什么Centres 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

提示How many equal shares make the whole?

为什么A 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

提示One shared multiplier scales both parts.

为什么Introduce 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

提示Which multipliers keep the parity and which force it?

为什么So 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

提示Both brackets share a common power.

为什么5⁴(5² − 1) ÷ 2⁶(2² − 1) = 625 × 24 ÷ (64 × 3) = 625 × 8 ÷ 64 = 625/8. Never expand a power you can factorise.

2

提示Each pair got counted in both orders.

为什么Choosing 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.