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.