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.

7/11

提示Cross-multiply: set 5 × 11 against 7 × 8.

为什么Cross-multiplying weighs 5 × 11 = 55 against 7 × 8 = 56, and the larger product sits above the larger fraction, so 7/11 wins. The common-denominator route agrees: 55/88 against 56/88. The quick same-top rule (bigger bottom, smaller fraction) only applies when the numerators already match, which is why it cannot settle this pair.

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?'.

It scales by that same factor

提示One multiplier applied to every ingredient carries straight through to the total mix.

为什么A squash recipe's ingredients add up to 240 ml of concentrate; tripling every ingredient triples the total too, giving 720 ml — no need to scale each ingredient separately and re-add. Converted to litres, that's 0.72 l, not 7.2 l or 72 l, which is the easy slip when moving between millilitres and litres.

Cancel common trailing zeros

提示It shrinks both figures to friendlier sizes before you divide.

为什么42,000,000 ÷ 6,000: both numbers end in at least three zeros, so cancel three from each first, leaving the much friendlier 42,000 ÷ 6 = 7,000. Dividing the top and bottom of a division by the same power of ten never changes the answer — it just makes the arithmetic easier to see.

Divide the total amount by the daily rate

提示Find how many single days it takes to cover the whole job.

为什么Saving £15 a week toward a total of £700 needs 700 ÷ 15 ≈ 46.7 weeks — just under a year's 52 weeks, so the honest description is 'just under a year', not 'a year'. The same division — total amount divided by the rate — turns any steady-progress problem into a single number of time units, measured against a sensible benchmark like a week, a month or a year.

difference

提示The gap you get when you subtract one number from another stays fixed.

为什么73 − 48 equals 75 − 50, because both numbers went up by 2; it also equals 70 − 45, because both went down by 3. This 'compensation' trick swaps an awkward subtraction for a friendlier one with the same answer, and it's the quickest way to spot which of several expressions secretly matches a given difference without calculating any of them fully.

Convert it to an improper fraction first

提示Rewrite the mixed number as a single value over one denominator before doing anything else.

为什么To find ⅔ of 3½, convert 3½ to the improper fraction 7/2 first, then multiply: ⅔ × 7/2 = 14/6 = 2⅓. The usual slip is to apply the fraction to the whole-number part only and carry the ½ through untouched, giving 2½. Splitting the mixed number is only safe if you take ⅔ of BOTH parts and add them: 2 + ⅓ = 2⅓. A percentage behaves the same way — 20% of 3¾ is 1/5 × 15/4 = 3/4.

Reverse it with the opposite operation

提示Undo the machine backwards from the answer, one step at a time.

为什么A machine does ×2, then −6, then ×2, then −6, in that order. To work backwards from an output, undo the LAST step first: add 6, halve, add 6, halve. Reversing only the operations but keeping the original order — instead of also reversing the order — sends you to the wrong number.

Multiply them

提示Two chained percentages collapse into one product of decimals.

为什么25% of 60% of a number is the same as working out 0.25 × 0.6 = 0.15, i.e. 15% of the number directly. Chaining doesn't mean adding the percentages (25 + 60 = 85% would be wrong) or finding each 'of' separately and subtracting — it means multiplying the two decimal forms together.

Write x/5 = x + 12 as an equation

提示Turn the sentence into symbols first, then rearrange to solve for the number.

为什么Setting up x/5 = x + 12 and multiplying both sides by 5 gives x = 5x + 60, so −4x = 60 and x = −15. A negative answer isn't a mistake here — some word-to-algebra puzzles are only satisfied by a negative number, and trusting the algebra over an instinct that the answer 'should' be positive is exactly what solves them.

Area ratio equals the length ratio squared

提示If every side scales ×3, the covered surface scales by 3×3, not by 3.

为什么Two similar triangles have corresponding sides in the ratio 2 : 5. Their areas come out in the ratio 4 : 25 (that's 2² : 5²), not 2 : 5. This squared relationship is why doubling a shape's linear size roughly quadruples the material needed to cover it, and why similar 3D solids scale volume by the cube of the length ratio instead.

zero

提示Nothing else can make a multiplication vanish to nothing.

为什么Solving (x − 3)(x + 5)x = 0 gives three answers without expanding a thing: x = 3, x = −5 or x = 0, because a product can only come out as nothing if one of its pieces does. This is the fast way to finish an equation once you've forced one side down to a product of factors — for example, y(y − 4)(y + 4) = 0 hands you all three solutions of y³ = 16y immediately, without multiplying anything out.

Represent the unknown with a letter and reason generally

提示Give the starting value a symbol and follow it through algebraically, instead of picking a single case.

为什么Testing 'the sum of three consecutive integers is always a multiple of 3' with 4, 5, 6 gives 15 — a multiple of 3, which looks convincing but only proves that one case. Writing the three numbers as n − 1, n and n + 1 and adding gives 3n, obviously a multiple of 3 whatever whole number n is — now it holds for every case at once, not just the one you tried.

Pythagoras' theorem

提示The right angle between the two shorter lengths is the clue — square and compare them to the joining one.

为什么Two touching semicircles can hide a right angle between two of their radii; if the larger semicircle has radius 5 and two equal legs of length r meet it at that right angle, then r² + r² = 5², so r² = 12.5. The smaller semicircle's area, ½πr², is then ½π(12.5) = 6.25π — found without ever solving for r itself, since only r² was needed.

Together they equal the whole rectangle's perimeter

提示Pair one corner with its diagonal partner and add their two side-totals — the cuts cancel out.

为什么Splitting a 10 cm by 7 cm rectangle this way, however the two cuts fall, the top-left piece's perimeter plus the bottom-right piece's perimeter always comes to 2 × (10 + 7) = 34 cm. The cut lengths appear once as an addition and once as a subtraction across the diagonal pair, so they always cancel, leaving exactly the original rectangle's perimeter.