JMC 2017：全部卡片

Alternate angles

提示One is on each side of the crossing line.

为什么A 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.

60

提示The sale price is four fifths of what it was, so undo that multiplication.

为什么Cutting by 20% multiplies by 0.8, so undoing it means dividing: 48 ÷ 0.8 = 60. Adding 20% back on to £48 gives £57.60, which is not the same thing. Multiplier thinking also chains: a 10% rise followed by a 10% fall is × 1.1 × 0.9 = × 0.99, a 1% loss overall. Worth knowing as fractions too: 5% is 1/20, 10% is 1/10, 20% is 1/5, 25% is 1/4.

15

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

为什么Whatever 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

提示Get the numbers small before you scale them up again.

为什么Cancel 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

提示Fold a square along its diagonal.

为什么A 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

提示Try a half: a half of a half.

为什么(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

提示It is just above 12, whose square you know is 144.

为什么Knowing 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

提示Each rider closes part of the gap every hour.

为什么Approaching 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

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

为什么Read 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

提示Not the product — unless the two share no factor at all.

为什么Multiples 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

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

为什么Triangles, 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

提示Three summaries, three different questions about the list.

为什么For 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.

Adding them

提示Multiplying by 1 never changes a number at all.

为什么Multiplying by 1 leaves a number exactly as it is, so once one of the numbers involved is 1, addition (or even subtraction) can beat multiplication comfortably. This flips as soon as both numbers grow past 2, e.g. 4 × 5 = 20 easily beats 4 + 5 = 9. Never assume × or ÷ automatically gives the biggest answer — check with the actual numbers involved.

First work out how many minutes reach the next o'clock time

提示Split the duration into two chunks around that point.

为什么For instance, adding 1 hour 25 minutes to 9:50 am is easier as 10 minutes to reach 10:00 am, then the remaining 1 hour 15 minutes on top, giving 11:15 am. Splitting a time calculation at the next o'clock reading avoids minute totals over 60, which are easy to miscount.

Convert each one to a decimal or a common form

提示You can't compare fractions by eye if their denominators differ.

为什么Turning expressions like 1/2 + 1/3 and 3/4 × 2/5 into decimals (0.83 and 0.3) puts them on the same footing, so ordering is simple. Estimating first — is the answer bigger or smaller than 1? — narrows things down before exact working.

Half of it

提示Slide the apex along the side above the base: base and height never change, so neither does the space inside.

为什么A rectangle split along a diagonal always gives two congruent right-angled triangles, each covering exactly half the rectangle's area — so if you know a triangle's area, doubling it gives you the area of the rectangle sharing its base and height, and vice versa.

18

提示A minute is 60 seconds; find three tenths of that.

为什么1 minute = 60 seconds, so any fraction of a minute converts to seconds by multiplying that fraction by 60 — one tenth is 6 seconds, three tenths is 18. The same multiply-the-whole-amount idea works for any 'fraction of a unit' conversion, like finding a fraction of an hour in minutes.

Add its digits — if that sum is a multiple of 9, so is the number

提示Try it on 9, 18, 27, 36 — notice a pattern in what they add up to.

为什么This digit-sum test also works for 3 (digit sum divisible by 3). It's especially handy for spotting multiples of a compound number like 45 = 5 × 9: since 5 and 9 share no common factor, a number is a multiple of 45 exactly when it's a multiple of both 5 and 9 — check the units digit for 5, and the digit sum for 9.

2

提示Splitting the polygon into triangles from one vertex always gives two fewer triangles than sides.

为什么The interior angles of an n-sided polygon sum to (n − 2) × 180° — a pentagon (5 sides) splits into 3 triangles for 540°, a heptagon (7 sides) splits into 5 triangles for 900°. This formula, together with angles at a point summing to 360°, unlocks most polygon angle-chasing questions.

Also coloured

提示Picture folding the paper along that line — everything marked lands on something else marked.

为什么With two lines of symmetry (say horizontal and vertical), a coloured square off both lines belongs to a matched set of four — its reflections in each line, and the reflection of those again. This is why symmetric colouring almost always comes in pairs or fours rather than single squares.

Always even

提示Two times anything is automatically a multiple of two.

为什么Doubling any whole number, however large or however odd it started, produces a multiple of 2 — an even number, with no exceptions. This is a handy check in problems built from a small set of coin, weight or step values: if two equal 'unknown' amounts are combined, their total must be even, which can rule out several possibilities at once.

Pair each even number with the odd number just below it

提示Every matched couple, like (8 − 7) or (20 − 19), comes to exactly 1.

为什么Pairing 8 − 7, 20 − 19, and so on, each pair contributes exactly 1, so the whole calculation collapses to (number of pairs) × 1 — turning a long addition-and-subtraction into simple counting. This trick works whenever two lists run in step with a constant gap between matching terms.

7

提示Check the last digits of 1² through 9² and see which ones actually turn up.

为什么Squares can only end in 0, 1, 4, 5, 6 or 9 — 2, 3, 7 and 8 never appear as a square's last digit. This quick check rules out wrong answers fast: a number ending in 7 can never be a perfect square, whatever else about it looks promising.

Write the perimeter as a multiple of the unknown length, then solve

提示Count how many of that one edge go around the outside before setting up your equation.

为什么If a large shape's outline is made up of, say, 12 copies of a single edge length x, its perimeter is 12x — so a known total perimeter turns straight into a one-step equation for x. Once x is known, any other perimeter built from the same repeating edge follows immediately.

eliminate; choose

提示Subtracting one equation from the other often removes a variable cleanly.

为什么Subtracting a 'total items' equation from a 'total parts' equation (like counting vehicles versus counting their wheels) often leaves a simpler equation in two unknowns, which usually has more than one whole-number solution. The extra worded clue — an inequality between two of the unknowns — is what narrows several candidates down to exactly one.