JMC 2017

Alternate angles

HintOne is on each side of the crossing line.

The whyA line crossing two parallel lines makes alternate (Z) angles equal, corresponding (F) angles equal and co-interior (C) angles that sum to 180°. Competition diagrams hide the Z; look for it whenever two lines are marked parallel.

twentieth

Hint5 out of 100 — cancel by 5.

The whySmall percentages are worth knowing as fractions: 5% is 1/20, 10% is 1/10, 20% is 1/5, 25% is 1/4. When 90% of a job is done, a tenth remains.

15

HintAdd all nine numbers, then share the total equally among the rows.

The whyWhatever nine consecutive numbers a magic square uses, the line total is their sum divided by three — for 4 to 12 it is 24. Knowing the line total turns a magic-square question into simple subtraction.

Simplify it fully

HintGet the numbers small before you scale them up again.

The whyCancel 216/480 to 9/20 before finding the same proportion of 800: 9/20 × 800 = 360 is far easier than 216/480 × 800. Common factors are always cheaper to remove early than to carry through a calculation.

45

HintFold a square along its diagonal.

The whyA square cut along a diagonal gives two of them. Whenever two sides from a vertex are equal and the angle between them is a right angle, the other two corners come free; when the angle between equal sides is 60°, the triangle is equilateral instead.

Smaller

HintTry a half: a half of a half.

The why(½)² = ¼, which is less than ½. Squaring makes numbers bigger only above 1; between 0 and 1 it shrinks them, and square-rooting does the opposite (√¼ = ½, bigger). 'Always' statements about squaring are usually false for this reason.

13

HintIt is just above 12, whose square you know is 144.

The whyKnowing squares up to 15² = 225 and cubes up to 6³ = 216 by heart saves time all through the paper. A number divided by its own square root leaves no remainder — the root is a factor.

Sum of their speeds

HintEach rider closes part of the gap every hour.

The whyApproaching each other, the gap closes at the combined speed; travelling the same way, it changes at the difference of the speeds. If one sets off later, first work out how far the other has gone by then, and only then use the combined speed.

The distributive law

HintRead it right-to-left too: it is how you pull a common factor out of two products.

The whyRead it right to left and it becomes a shortcut: 17 × 83 − 17 × 23 = 17 × (83 − 23) = 17 × 60. Whenever two products share a factor, take the factor out before touching the arithmetic.

Their lowest common multiple

HintNot the product — unless the two share no factor at all.

The whyMultiples of both 4 and 9 are the multiples of 36, and there the LCM is the product because 4 and 9 share no factor. Multiples of both 4 and 6 are the multiples of 12, not 24, because 4 and 6 share the factor 2.

6

HintThe cut can pass through every face of the cube at most once.

The whyTriangles, squares, rectangles, pentagons and even regular hexagons can all be cut from a cube — but never a heptagon, since a cube has only six faces to cross. Cross-section questions reward imagining the cut rather than trusting your first instinct.

middle; most common; smallest

HintThree summaries, three different questions about the list.

The whyFor an even-sized list the median is the mean of the two middle values. Check which average a question names before doing anything — the three answer different questions.