IMC 2023

2

提示1 + 8 = 9.

为什么1 + 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

提示What does a straight line keep constant?

为什么Gradient = 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

提示Which positive number divides everything?

为什么That 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

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

为什么At 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

提示36 = 2² × 3² and 12 = 2² × 3 — what differs?

为什么2² × 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

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

为什么Given 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

提示A squashed square.

为什么Its 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

提示Each letter appears in exactly two of the given facts.

为什么Doubling 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

提示Sort first; the middle has as many on each side.

为什么Median = 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

提示The radius is the shortest route from the centre to the line.

为什么Radius ⟂ 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

提示Not add — each stage acts on what is already there.

为什么Give 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

提示Each is under 400.

为什么Also 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

提示Look only at the ending pair of figures.

为什么100 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)

提示Rearrange a² + b² = c² to isolate the missing leg.

为什么Pythagoras' 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

提示Two negatives side by side act like a positive.

为什么Subtracting 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

提示Area is measured in two dimensions at once.

为什么Halving 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

提示Count the boundary bars needed in each direction separately.

为什么A 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

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

为什么0.024% 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°

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

为什么From (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

提示Look for symmetry that lets you slot the odd bits together.

为什么Four 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

提示Congruent means identical in every measurement.

为什么Because 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

提示Clear every small fraction at once with a single multiplication.

为什么Multiplying 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)

提示Two squares subtracted always split into a sum times a difference.

为什么x² − 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

提示Combine the pair two different ways, once up and once down.

为什么Adding 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

提示Expand both squared brackets and see what survives.

为什么Expanding 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).