IMC 2019

By two faces

HintNothing is added; something is hidden — on both sides.

The whyEvery join hides one face from each cube, so a shape built from k cubes with j face-to-face joins has surface area 6k − 2j. The shape with the most joins has the smallest surface area — count joins, not exposed faces.

The units digit of n

HintCube 12 and 22 and compare the endings.

The whyThe same is true of every power, so a cube ends in 8 only when the number cubed ends in 2, and a square ends in 9 only when its root ends in 3 or 7. Unlike squares, cubes can end in any digit — each of 0 to 9 appears exactly once among the cubes of 0 to 9.

The triangular numbers

Hint1, then 1 + 2, then 1 + 2 + 3 …

The whyThe nth triangular number is n(n + 1)/2 — the sum of the whole numbers from 1 to n. They run odd, odd, even, even, so a triangular number just after an even number is odd.

positive; negative

HintSame signs, different signs, and the one number that swallows everything.

The whyThe sign rules apply to × and ÷ only — adding two negatives gives a negative, and subtracting a negative is adding. Order-of-operations questions packed with brackets and minus signs are testing whether you apply each rule to the right operation.

9

HintA thousand millions: add the zeros.

The whyA billion has nine zeros. Dividing it by a smaller round number is best done by cancelling powers of ten first — 10⁹ ÷ (4 × 10⁴) = 10⁵ ÷ 4 = 25 000 — so the zeros are never miscounted.

multiplies

HintWrite (a²)³ out in full and look at how many a's you get.

The whyCompare 8⁵ with 4⁷ by writing both as powers of 2: 2¹⁵ against 2¹⁴. Keeping everything as a power of the smallest base is the whole technique for index questions with no calculator.

Watch the square's colour

HintEvery single move flips something that only has two states.

The whyColour the board like a chessboard: every move — a step to a neighbour, or a reflection in a line running between rows or columns — swaps black for white. After an odd number of moves the token is on the opposite colour to where it started; after an even number, the same colour. Parity halves the search before you list anything.

Factorise x out

HintBoth terms contain x — what does that let you write?

The whyFrom ab = 2a + b: a(b − 2) = b, so a = b ÷ (b − 2); check that b = 2 is impossible in the original before dividing by the bracket.

7/9

HintNot tenths — the denominator that makes one digit repeat forever.

The whyA single recurring digit d is d/9; a recurring pair ab is ab/99. Add recurring decimals as fractions — the digit-by-digit method has no right-hand end to start from.

Take the square root

HintAreas came from lengths by one operation; go back the other way.

The whyAreas of similar shapes are in the ratio of the squares of their sides, so going from areas back to sides means taking a square root: areas 1 : 7 means sides 1 : √7 — not 1 : 3.5.

13

HintThree consecutive odd primes.

The whySo 123 123, 987 987 — any block-repeated six-digit number — is divisible by 7, 11 and 13, because it equals the block × 1001.

Longest side < sum of the other two

HintTry 2, 3 and 10 with real sticks.

The whyThe triangle inequality — and every length must be positive. When side lengths are given as expressions in n, solving "two sides equal" is only half the job: each solution must then be checked against both conditions, and any that fails is discarded.