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 whyIts diagonals cross at right angles, cut each other in half and bisect the corner angles; a rectangle has equal diagonals that are not perpendicular, and a square has both properties at once. Opposite angles of a rhombus are equal, so it has two angle sizes, not four — 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.

Its last two digits form a multiple of 4

HintLook only at the ending pair of figures.

The why100 is always a multiple of 4 (4 × 25 = 100), so only the last two digits of a number affect its divisibility by 4. The same idea gives divisibility by 8 from the last three digits, and by 3 from the digit sum instead of the ending.

√(hypotenuse² − known side²)

HintRearrange a² + b² = c² to isolate the missing leg.

The whyPythagoras' theorem states a² + b² = c² for the two legs a, b and hypotenuse c of a right-angled triangle. Rearranged, a = √(c² − b²) — this finds any missing side once the other two are known, which then lets you find the triangle's area.

8

HintTwo negatives side by side act like a positive.

The whySubtracting a negative number is the same as adding its positive: 5 − (−3) = 5 + 3 = 8. Track this sign flip every time you resolve a bracket nested inside other brackets.

k²

HintArea is measured in two dimensions at once.

The whyHalving every length of a shape (k = ½) shrinks its area to (½)² = ¼ of the original — this is why four half-size copies exactly tile the original shape. Cubing the same k gives the volume scale factor for a 3D solid.

10

HintCount the boundary bars needed in each direction separately.

The whyA grid with C columns and R rows of panes needs (C + 1) vertical bars and (R + 1) horizontal bars — always one more than the number of gaps in each direction — giving C + R + 2 bars in total. The same 'one more than the gaps' idea counts fence posts or seats in a row.

Convert the percentage to a decimal by dividing by 100

Hint% literally means 'per hundred' — use that directly.

The why0.024% of 50 million = (0.024 ÷ 100) × 50,000,000 = 0.00024 × 50,000,000 = 12,000. Converting to a decimal first keeps track of where the decimal point lands when the percentage itself is tiny.

60°; 90°; 108°

HintInterior angle formula: (n − 2) × 180° ÷ n.

The whyFrom (n − 2) × 180° ÷ n: n = 3 gives 60°, n = 4 gives 90°, n = 5 gives 108° (and n = 6 gives 120°). When several regular polygons meet edge-to-edge at one point, any leftover gap is 360° minus the sum of the known interior angles there.

See if the pieces recombine into a shape you know

HintLook for symmetry that lets you slot the odd bits together.

The whyFour leftover triangles inside a dissected octagon can often be slid together to exactly fill a small square whose area is easy to state directly — turning several fiddly area calculations into one simple one.

Each piece keeps its original side lengths

HintCongruent means identical in every measurement.

The whyBecause the triangles are congruent, not just similar, every side length from the original rectangle survives unchanged inside the new shape — so its perimeter can be found directly from the rectangle's own dimensions via Pythagoras, without measuring the new figure.

lowest common denominator

HintClear every small fraction at once with a single multiplication.

The whyMultiplying the numerator and denominator of a compound fraction by the LCD of its inner fractions turns every term into a whole number, after which the whole expression simplifies to one straightforward fraction.

(x + y)(x − y)

HintTwo squares subtracted always split into a sum times a difference.

The whyx² − y² = (x + y)(x − y) for any x and y. So if you already know x + y and x − y separately — say from solving two equations — you can find x² − y² by multiplying those two values, with no need to find x and y individually.

Add the equations, then subtract them

HintCombine the pair two different ways, once up and once down.

The whyAdding two equations whose coefficients are mirror images of each other collapses them to a multiple of (x + y); subtracting collapses them to a multiple of (x − y). This avoids solving for x and y separately when only their sum, difference, or something like x² − y² is needed.

4ab

HintExpand both squared brackets and see what survives.

The whyExpanding gives (a² + 2ab + b²) − (a² − 2ab + b²) = 4ab. So knowing a sum and a difference of two numbers gives you their product directly — this is exactly how the shaded area between two touching circles collapses to a clean multiple of π once you know (r₁ + r₂) and (r₁ − r₂).