IMC 2023

2

Hint1 + 8 = 9.

The why1 + 8 + 27 = 36 = 6² and 1 + 2 + 3 = 6. So a sum of consecutive cubes from 1 is always a perfect square — a triangular number squared.

Equal gradients between pairs

HintWhat does a straight line keep constant?

The whyGradient = rise ÷ run between any two of the points; if P to Q and Q to R give the same value, the three are collinear. It converts a 'which condition guarantees a straight line?' question into one equation.

1

HintWhich positive number divides everything?

The whyThat is the whole game in 'which squares are 9 more than a prime?': n² − 9 = (n − 3)(n + 3), and a prime can only be 1 × itself, so n − 3 = 1, n = 4 and 16 − 9 = 7. One square, one prime.

Count in the sample space

HintDon't reason about 'the other one' — where do probabilities come from?

The whyAt least one head keeps 3 of the 4 equally likely outcomes (HH, HT, TH), and only HH has both — so the answer is ⅓, not ½. The ½ argument silently assumes you know which coin is the head.

even

Hint36 = 2² × 3² and 12 = 2² × 3 — what differs?

The why2² × 5⁴ = (2 × 5²)² = 50². Cubes need every exponent a multiple of 3. To make a product a square, discard the primes with odd exponents.

a² + 2ab + b²

HintDraw a square of side a + b and look at the four regions.

The whyGiven a + b = 7 and a² + b² = 25: 49 = 25 + 2ab, so ab = 12 — the area of a right-angled triangle with legs a and b is then 6, with no need to find the legs themselves.

A rhombus

HintA squashed square.

The whyOpposite angles of a rhombus are equal, so it has two angle sizes, not four; a pentagon with five equal sides can still have five different angles. Equal sides pin down all the angles only in a triangle.

Add the three equations

HintEach letter appears in exactly two of the given facts.

The whyDoubling each mean gives the pair totals; adding them counts every letter twice, so halve for a + b + c and divide by 3. The same 'add all the equations' move works whenever every unknown appears in the same number of equations.

3

HintSort first; the middle has as many on each side.

The whyMedian = the middle of the ordered list; with an even count, average the two middle values. A fifth unknown number cannot move the median of 1, 4, 4, 7 away from 4 — wherever it slots in, position three is a 4.

A right angle

HintThe radius is the shortest route from the centre to the line.

The whyRadius ⟂ tangent, and two circles that touch have their centres and the touching point on one straight line. Together they turn 'touching circles' pictures into right-angled triangles whose sides are sums or differences of radii — then Pythagoras.

multiply

HintNot add — each stage acts on what is already there.

The whyGive away a fifth, then a quarter of what is left: (4/5) × (3/4) = 3/5 of the original remains. Work backwards from what is left at the end to find the starting amount.

243; 343; 256

HintEach is under 400.

The whyAlso 2¹⁰ = 1024, 5⁴ = 625, 3⁴ = 81, 6³ = 216. 'Undo the square root twice' questions are pure recall of these — √√x = 2 means x = 2⁴ = 16.