IMC 2018

10⁶

提示A thousand thousands.

为什么10³ × 10³. To get a feel for the size: a million seconds is about eleven and a half days. Money questions with hundreds of millions of items are a small number times a power of ten — multiply the small numbers, then attach the zeros and count them.

The two base angles

提示Fold along the line of symmetry — which two corners land on each other?

为什么Mark both base angles the moment you see two equal sides; in a chain of isosceles triangles each apex angle is an exterior angle of the next.

Round both numbers first

提示The word "approximately" means an exact division is a waste of time.

为什么Round to numbers that divide nicely — 30 out of 240 is one eighth — and only look harder if two options are close. Non-calculator papers reward a well-chosen rounding far more than long division.

It stays the same

提示Test it: 2 < 3, multiply both by 5, then by −5.

为什么Multiplying or dividing by a positive number keeps an inequality true in the same direction, so a double inequality with fractions is best cleared by multiplying every part by a common denominator. Only a negative multiplier reverses the signs.

(a + b)(a − b)

提示Try a = 5, b = 3: 25 − 9 = 16 = 8 × 2 — what are 8 and 2 made of?

为什么The difference of two squares. It tells you which whole numbers are a difference of squares: every odd number is (since 2k + 1 = (k + 1)² − k²), every multiple of 4 is, but a number that is twice an odd number never is — the two brackets would have to be one odd and one even.

6

提示Every side is also a radius.

为什么The three long diagonals of a regular hexagon meet at the centre and split it into six equilateral triangles, each with the hexagon's side as all three of its edges.

1

提示Divide one by nine and watch the same remainder come back every time.

为什么From 1/9 = 0.111… you get every ninth: 2/9 = 0.222…, and so on up to 9/9 = 0.999…, which equals 1. Dividing by 90 shifts the recurring block one place: 1/90 = 0.0111….

100a + 10b + c

提示Each column is worth ten times the next one along.

为什么Reversing a three-digit number changes it by a multiple of 99, and the middle digit never matters.

3

提示Find the smallest power of ten that 8 divides — that fixes the tail length.

为什么Because 1000 = 8 × 125, a number is a multiple of 8 exactly when the number formed by its last three digits is. Combine it with the digit-sum test for 9 and you have a test for 72, since 8 and 9 share no factor.

The solution x = 0

提示Dividing by something that might be nothing is never safe.

为什么Move everything to one side and factorise instead: x² − 5x = x(x − 5) = 0 gives both x = 0 and x = 5. Counting "how many values of x" questions hinge on not throwing a root away.

√2

提示Pythagoras on either half of the square.

为什么Half a square is a right-angled isosceles triangle, so the diagonal is √(s² + s²). A square of side 3 placed corner-first spans 3√2; a product of two such spans loses the surd because (√2)² = 2.

8; 27; 64

提示1, 8, then a jump past 25.

为什么The first few cubes — 1, 8, 27, 64, 125 — are worth knowing by sight. Questions about sums of consecutive cubes, or about which numbers below a limit are cubes, are lists of five or six values rather than calculations.