IMC 2024

97

HintCount down from 100 and test each odd number.

The whyPrimes above 5 end in 1, 3, 7 or 9, but that only rules numbers out: 91 = 7 × 13 and 87 = 3 × 29 pass the ending test and still fail. The two-digit primes run from 11 to 97, twenty-one of them.

two

HintRotate a parallelogram by a half turn.

The whySo an angle list with no repeats cannot be a kite or a parallelogram; a rectangle needs four right angles; a trapezium needs one pair of adjacent angles summing to 180°. Read the angle list before naming the shape.

√(x² + y²)

HintDrop a perpendicular to the horizontal axis and use the right-angled triangle.

The whySo every point with x² + y² = 169 sits on the circle of radius 13 centred at the origin: (13, 0), (5, 12), (12, 5), (0, 13) — but (7, 7) is only √98 away.

first

HintOrder of operations still applies up in the exponent.

The whyA power of a power multiplies the exponents: (aᵐ)ⁿ = aᵐⁿ, so (2³)⁴ = 2¹²; but 2^(3²) = 2⁹. Dividing powers of one base subtracts exponents: 5⁷ ÷ 5⁴ = 5³.

5

Hint10 = 2 × 5 — when does the product first contain the awkward one?

The whyn! = 1 × 2 × ⋯ × n contains the factor 5 (and 2) once n ≥ 5, so 5!, 6!, 7! … all end in 0, and the units digit of 1! + 2! + … + 20! is the units digit of 1 + 2 + 6 + 24. The remainder on dividing by 5 or by 10 is read from that digit.

y; (b, a)

HintWhich coordinate stays put on the mirror line?

The whyA point reflected in the y-axis keeps its y and flips x; in y = x the coordinates swap. Chaining reflections just needs the rules applied in order, then the shape they trace.

7

HintHow many faces can one flat cut create?

The whyOne cut, one new face per piece — a piece has at most one more face than the solid you started with. Slice a corner off a cuboid and count both pieces.

× 0.8

HintDecrease means less than 1 — how much less?

The whyUp 20% is × 1.2, down 20% is × 0.8; 'twice as much as' becomes = 2 × (…). Turning every percentage phrase into a multiplier makes the word problem a one-line equation.

√N, paired with itself

HintTry 36: list the pairs.

The whyA square has an odd number of factors precisely because its root has no partner. So 3600 = 2⁴ × 3² × 5² (45 factors) has 22 factors above 60 and 22 below. Counting 'factors bigger than the root' is half of (count − 1).

They are equal

HintWhich lengths does the area formula actually use?

The whyA line from a vertex to the midpoint of the opposite side splits any triangle into two equal-area halves. Shaded-fraction questions are mostly this fact in disguise.

Two radii plus the arc

HintIt is not just the curve.

The whyPerimeter of a sector = 2r + arc. Cut a sector of radius one quarter of the side from each corner of a square: each side keeps two radius-lengths, so the boundary is four arcs plus eight radii — four sector perimeters exactly.

A multiplicative magic square

HintThe usual puzzle, with a different operation.

The whyCompare a row with a diagonal that shares a cell: the shared factor cancels and a single unknown drops out. Non-zero entries mean you may divide freely.

Negative

HintSame signs divide to a positive result; different signs divide to this instead.

The whySo (positive) ÷ (negative) and (negative) ÷ (positive) are both negative, while (negative) ÷ (negative) is positive — the same rule as multiplying directed numbers. Getting the sign right first avoids picking the wrong size of answer afterwards.

n (the number underneath)

HintSquaring is the exact opposite of taking a square root — one undoes the other.

The whyThis is why Pythagoras' theorem works cleanly even when a triangle's shorter sides are given as surds like √7 and √13: squaring each one just returns 7 and 13, so the hypotenuse-squared is their sum, with no decimal estimate needed along the way.

The middle term (the median)

HintConsecutive integers are evenly spaced, exactly like a simple arithmetic sequence.

The whySo the mean of the three youngest of seven consecutive ages fixes the middle one of those three; shifting to the three oldest shifts every age up by the same fixed amount, so the new mean is the old mean plus that shift — no need to list every age.

Divide the total quantity by the total number of items

HintThis is the unitary method — work out what one single unit is worth first.

The whyFor example, splitting kilometres travelled ÷ litres of fuel used gives kilometres per litre directly — cancel any shared trailing zeros in the two totals before dividing, to keep the arithmetic simple.

a × (a + 1)

HintFactor out the a that both terms share.

The whyThis turns a calculation like 847 × 847 + 847 into 847 × 848 — one multiplication by a nearby round-ish number instead of squaring a large one; the same shared-factor trick works whenever a value is added to a multiple of itself.

Keeps the common denominator as small as possible

HintLook inside each individual fraction for a shared factor before combining them.

The whySo instead of multiplying two large, unsimplified denominators together, you multiply two much smaller ones — e.g. 18/24 simplifies to 3/4 first, so it combines with any other fraction over a far smaller shared denominator.

Twice the frame width (2w)

HintThe border covers both opposite sides of the square, not just one edge.

The whySo the frame's area is (outer side)² − (outer side − 2w)²; using w instead of 2w undercounts the border by exactly one width all round — an easy slip under time pressure.

1

HintApply Pythagoras' theorem once for each new unit-length step.

The whySo after n such unit steps from the start, the squared distance is exactly n and the true distance is √n — the classic 'square-root spiral' construction, which builds a segment of any length √n using nothing but unit lengths and right angles.

Adding all three equations together

HintLook at what happens to every unknown once every given relationship is combined.

The whyAdding all three makes every unknown appear exactly twice, giving twice the sum of the three unknowns; dividing by two gives that total directly, which then substitutes into any one of the original equations to isolate a single unknown.

sum; difference

HintOne case has the circles sitting side by side; the other has one circle nested inside the other.

The whySo two externally-touching circles of radii 3 and 5 have centres 8 apart, while internally-touching circles of the same two radii have centres just 2 apart — the smaller gap shows one circle sitting inside the space of the other.

RHS — right angle, hypotenuse, side

HintIt is one of the four standard triangle congruence tests, and it only applies when there's a 90° corner.

The whyCongruent triangles then guarantee every remaining side and angle matches too — here that forces the two right angles' feet to land at the same point, which is exactly the extra fact Pythagoras' theorem needs inside the smaller triangle.

equilateral

HintEach of the six triangles has a 60° angle at the centre, and its two other angles match.

The whyBecause a regular hexagon's radius equals its side length, all three sides of each central triangle are equal — that is why the hexagon splits into six equilateral triangles specifically, not merely six isosceles ones as most regular polygons do.

(100 + x)(100 − y) = 10000

HintWrite each percentage change as a multiplier out of 100, then multiply the two together.

The whyThis turns a percentage-reversal problem into a factor-pair search: every valid whole-number x comes from a factor of 10000 that is less than 100, so counting those factors counts the possible values of x directly — no trial and error needed.