IMC 2024

97

提示Count down from 100 and test each odd number.

为什么Primes 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

提示Rotate a parallelogram by a half turn.

为什么So 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)

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

为什么So 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

提示Order of operations still applies up in the exponent.

为什么A 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

提示10 = 2 × 5 — when does the product first contain the awkward one?

为什么n! = 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)

提示Which coordinate stays put on the mirror line?

为什么A 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

提示How many faces can one flat cut create?

为什么One 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

提示Decrease means less than 1 — how much less?

为什么Up 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

提示Try 36: list the pairs.

为什么A 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

提示Which lengths does the area formula actually use?

为什么A 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

提示It is not just the curve.

为什么Perimeter 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

提示The usual puzzle, with a different operation.

为什么Compare 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

提示Same signs divide to a positive result; different signs divide to this instead.

为什么So (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)

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

为什么This 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)

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

为什么So 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

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

为什么For 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)

提示Factor out the a that both terms share.

为什么This 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

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

为什么So 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)

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

为什么So 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

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

为什么So 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

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

为什么Adding 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

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

为什么So 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

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

为什么Congruent 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

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

为什么Because 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

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

为什么This 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.