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.

The number with the smallest absolute value

HintIgnore the sign and compare sizes.

The whyTwo negative numbers can look far apart yet one sits nearer zero — always strip the sign and compare what remains. The same idea also settles "which is bigger" once every option is negative: the smaller absolute value is the larger number.

Split it into place-value chunks that are already multiples of that divisor

HintThink about grouping digits, not checking one at a time.

The whySpot that many multiples of the divisor already sit inside the number's place-value blocks — millions, thousands, hundreds — so only a small remainder needs checking directly, saving a full long division.

Divide the total surplus by the surplus per item

HintSet up one equation using the count as the unknown.

The whyWrite (surplus per item) × n = total surplus, then solve for n. This turns a wordy description into a one-line equation before any numbers are plugged in — the same move works for any "each one always contributes X extra" scenario.

multiplying

HintWhat's the opposite operation to dividing?

The whySo a ÷ (b ÷ c) becomes a × c ÷ b — useful when a division chain is nested in brackets, because the innermost fraction flips to a multiplication as you unwrap it outward.

1

HintOne of the two claims must be false, and the other true.

The whyThis settles the first contradicting pair straight away; anyone else's statement is then checked against what that pair's truth-values force — a chain of only-one-can-be-right deductions, not full case-by-case guessing.

Equals 2 — no fraction arithmetic needed

HintCheck whether one mixed number is simply twice the other first.

The whySpotting that relationship first (comparing whole-number parts and denominators) is faster than converting both to improper fractions and cancelling — the full method still works, but the shortcut skips it entirely when it applies.

Factor out a common term to leave a perfect square inside the root

HintLook for what multiplies both parts before you try to simplify directly.

The whyRewriting the expression so a shared factor comes out front often turns a messy subtraction under a root into (factor) × (perfect square), which then square-roots cleanly instead of needing an estimate.

Cancel the matching factors from both sides first

HintSpot which numbers repeat on each side of the equation.

The whySpotting a shared factor on both sides and dividing it away first turns a big multiplication into a much smaller one — the two remaining products must still be equal, so solve that smaller equation instead.

Multiply each blend's share of the mix by its own percentage, then add the results

HintThink of it as a weighted average of the two mixtures.

The whyEach blend contributes to the final mix in proportion to how much of it is used, and each blend itself is a percentage split — multiplying those two percentages together and summing across blends gives the true overall percentage, avoiding a false halfway average.

Add all three equations, then eliminate to find the one unknown whose coefficient differs

HintCombine every relationship at once rather than solving one at a time.

The whySumming every equation is still the right first move - it collapses three relationships into one - but the coefficients rarely land equal, so what comes out is a multiple of the total plus one leftover unknown. Pin that unknown down first by eliminating between two of the original equations, then substitute it into the summed equation and the total drops out. The saving is real: you find one unknown, never all three.

Combine the area equations so the hidden lengths cancel, leaving an equation in the total area alone

HintTry adding or multiplying the region relationships together and see what disappears.

The whyThis often produces a quadratic in the total once you substitute back in — solve it, discard any root too small to contain the known regions, and the genuine total remains. No individual side length needs to be found at all.

Use a rough bound from the total cost to rule out the too-large options

HintCheck which candidate count actually stays under the given maximum.

The whyBecause the shortfall repeats every fixed number of items, several counts satisfy the pence condition alone; only one keeps the overall spend under the actual amount stated, so a quick size check (not more algebra) picks it out.

hypotenuse; leg

HintPicture the two centres, the surface, and a right angle between them.

The whyThe line joining the two centres has length r + R (sum of radii, the hypotenuse); the vertical offset between where each touches the surface equals the difference of the radii (one leg); the horizontal offset is the third leg, found with Pythagoras.