IMC 2022

12

提示An hour has sixty minutes, not a hundred.

为什么Minutes are sixtieths, not hundredths: 0.1 h = 6 min, 0.25 h = 15 min, 0.75 h = 45 min. The trap is reading 0.15 hours as 15 minutes.

The one with the larger denominator

提示Compare ½ and ⅓ in your head first.

为什么Same top, bigger bottom, smaller fraction; same bottom, bigger top, bigger fraction. And a fraction is above 1 exactly when its numerator beats its denominator — the first sort for a 'which is smallest?' list.

Perpendicular bisector of it

提示Two conditions: where it crosses, and at what angle.

为什么So to find the mirror line from a point and its image: the midpoint gives a point on the line, and the perpendicular gradient is −1 ÷ the segment's gradient (a gradient of 2 needs −½). Then y = mx + c through the midpoint.

−1

提示Steep up against gentle down — the product is a fixed negative number.

为什么A gradient of 3 pairs with −⅓; a gradient of 1 pairs with −1. Parallel lines share a gradient instead. Both facts turn 'reflect this point in the line' into arithmetic.

2

提示Think prime.

为什么4ⁿ = 2²ⁿ and 8ᵐ = 2³ᵐ; likewise 9ⁿ = 3²ⁿ. Once everything is in one base, halving means subtracting 1 from the exponent (2¹⁴ ÷ 2 = 2¹³), never halving the exponent.

One full circle, 2πr

提示The string only bends where it touches a coin.

为什么Whatever the layout, the string turns through 360° in total, so its arcs together make exactly one circle of radius r; the straight parts equal the gaps between the centres.

Percentage profit

提示Which price is the 'original' that the change is measured against?

为什么Percentage change is always measured against the starting value — cost price for profit, the original for an increase. Selling for £72 at a 20% profit means cost × 1.2 = 72, so the cost was £60.

Arc length ÷ circumference

提示The whole of which the arc is a part.

为什么arc ÷ 2πr = angle ÷ 360 = sector area ÷ πr². Any one of the three fractions gives the other two, so a radius-4 sector with arc 6 has area (6 ÷ 8π) × 16π = 12 with no angle needed.

mode; third; spread

提示Which one is picked straight out of the list?

为什么The mean need not be in the list, and the median of an even count need not be either; the mode forces a repeat. Setting mode, median and mean against each other is solved by writing the sorted list as a, b, m, ?, ? and translating each rule.

It falls by 2k

提示One shared edge — how many perimeters was it on before?

为什么Joining hides the shared edge twice over — once from each shape: 20 squares of side 3 in a row have separate perimeters totalling 240, and 19 joins of length 3 take off 19 × 6, leaving 126. Count the joins, not the shapes.

120

提示Three of them tile round a point with nothing left over.

为什么Its side equals the distance from centre to vertex, and dropping a perpendicular from one vertex onto a neighbouring side's line gives a 30-60-90 triangle with sides in the ratio 1 : √3 : 2 — the source of every √3 in hexagon answers.

List the factor pairs

提示A product of whole numbers has only a few ways to happen.

为什么A product equal to a small whole number has finitely many whole-number factor pairs (including negative ones, which the 'positive' condition then discards). Rearranging xy − 3x − 3y = 3 into (x − 3)(y − 3) = 12 is the standard move for 'how many solutions?'.