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.

125; 216; 343

提示Five squared times five, six squared times six, seven squared times seven.

为什么The cubes up to ten — 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 — are worth knowing by sight, and the tenth is exactly a thousand. Questions about sums of consecutive cubes, or about which numbers below a limit are cubes, are then lists of a handful of values rather than calculations.

Its digit sum is divisible by 3

提示Sum the individual figures, not the whole value itself.

为什么141 = 3 × 47 fails to be prime because 1 + 4 + 1 = 6 divides by 3 — the digit-sum test spots non-primes fast without needing a division. The same idea, checked against 9 instead of 3, is the test for divisibility by 9.

60

提示How many seconds make up one minute?

为什么300 downloads counted in a minute is 300 ÷ 60 = 5 downloads per second — the same divide-by-60 step turns any per-minute rate (heartbeats, footsteps, words typed) into a per-second one, and multiplying by 60 reverses it.

0; 1; 4; 9

提示0², 1², 2² and 3² all land back inside a single digit.

为什么These four values matter whenever a puzzle asks whether some digit, or a product of two digits, is itself square — checking against just 0, 1, 4, 9 is faster than testing a square root each time.

They are all congruent

提示Each one is a copy of its neighbour turned through a quarter of a full turn.

为什么Because they're congruent right-angled triangles, the shaded band between the two squares splits into four equal pieces — so one triangle's area is a quarter of (big square's area − small square's area), without needing individual side lengths.

Nothing — the product is unchanged

提示Seven threes and three sevens land on the same total.

为什么This is the commutative property: a × b = b × a for any numbers a and b. A calculator whose keys have been scrambled still gives a correct answer for a calculation only when reordering its two factors truly leaves the result unchanged.

1 : k²

提示Area is a two-dimensional measurement, so it scales with length squared.

为什么Doubling every side (k = 2) always multiplies area by 4, and tripling sides multiplies area by 9 — true for any pair of similar shapes, from triangles to hexagons, not just squares or circles.

6

提示Each triangle can only contribute at most three of its own edges to the boundary.

为什么The overlap can be a triangle, quadrilateral, pentagon or hexagon depending on how the two triangles cross — but it can never have seven or more sides, since between them the two triangles only have six edges to contribute.

One half

提示The three interior angles of any triangle add to 180°.

为什么180° is exactly half of the 360° in a full turn, so however the triangle is shaped, three same-radius sectors placed at its corners always combine into half a circle — a shortcut for area problems that dot circles around a triangle's vertices.

360°/n

提示All the exterior angles of any polygon add up to one full turn.

为什么A regular pentagon's exterior angle is 360°/5 = 72°, so its interior angle is 180° − 72° = 108°. Composite figures that glue a regular polygon onto a square or triangle almost always start from this calculation.

It appears on both sides and cancels

提示Expand fully — the same xy term is hiding in every expansion, so it drops out.

为什么Once the xy term cancels, what's left is a linear equation in x and y — the standard shortcut whenever two rectangles built from linear expressions are given equal area, turning what looks like a quadratic problem into ordinary simultaneous equations.

(√3/4)l²

提示Split it into two right-angled halves; the height works out to (√3/2) times the side.

为什么This comes from area = ½ × base × height, with height = (√3/2)l. A hexagon or star built from equilateral triangles of different sizes becomes a few applications of this one formula rather than fresh area work each time.

x + r

提示Range is defined as the gap between the biggest and smallest values in a set.

为什么Writing the largest value as x + r turns a 'how many lists satisfy…' problem into an algebraic one: fix the smallest value as x, and every other constraint (mean, median, mode) becomes an equation or inequality in x and the other unknowns.

1/√3

提示Bisect an equilateral triangle to get a right-angled triangle with sides 1, √3 and 2.

为什么That bisected equilateral triangle gives every standard trig ratio at 30° and 60° without a calculator — rationalised, 1/√3 = √3/3. It comes up whenever a problem asks for a perpendicular distance created by a 60° angle.