IMC 2022

12

HintAn hour has sixty minutes, not a hundred.

The whyMinutes 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

HintCompare ½ and ⅓ in your head first.

The whySame 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

HintTwo conditions: where it crosses, and at what angle.

The whySo 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

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

The whyA 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

HintThink prime.

The why4ⁿ = 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

HintThe string only bends where it touches a coin.

The whyWhatever 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

HintWhich price is the 'original' that the change is measured against?

The whyPercentage 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

HintThe whole of which the arc is a part.

The whyarc ÷ 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

HintWhich one is picked straight out of the list?

The whyThe 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

HintOne shared edge — how many perimeters was it on before?

The whyJoining 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

HintThree of them tile round a point with nothing left over.

The whyIts 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

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

The whyA 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?'.