IMC 2017

2, 3, 7 or 8

HintFour of the ten digits never appear at the end of a square.

The whyThe last digit of n² depends only on the last digit of n, so ten small squares settle it: a square always ends in 0, 1, 4, 5, 6 or 9. Questions asking how many squares end in a certain digit, or whether a total could be a square, are decided by this list alone.

2

HintOne number bucks the "primes are odd" rule.

The whyBecause every other prime is odd, two primes can only add to an odd number if one of them is the even one. So to test whether an odd number is a sum of two primes, take away that even prime and ask whether what remains is prime.

Multiplied by four

HintThe area formula uses the radius twice.

The whyArea = πr², so doubling r multiplies the area by 2² = 4 — and the region added by that doubling is three quarters of the bigger circle, a fact many shaded-fraction questions rest on.

The range

HintHow spread out the data is, in one number.

The whyThe range is a measure of spread, not an average. For a set of different whole numbers the smallest possible range is one less than the count, and only when the numbers are consecutive.

9

HintEach digit is worth ten times as much in one number as in the other.

The whyWith tens digit a and units digit b the number is 10a + b and its reversal is 10b + a; subtracting gives 9(a − b), so only the gap between the two digits matters. Writing digits as an expression is what unlocks every reversal puzzle.

3

HintBoth are small powers of the same number; find it before you touch the exponents.

The whyOnce both sides are powers of one base, the bases can be dropped and the exponents equated, using (aᵐ)ⁿ = aᵐⁿ on each side. The answer can be a fraction — a non-integer exponent is not a sign of a mistake.

540

HintSplit it from one corner and count the triangles.

The why180(n − 2) with n = 5. Any five-sided shape — irregular, even with a reflex corner — has this total. A regular pentagon therefore has interior angles of 108° and exterior angles of 72°.

Subtract the fractions

HintBoth levels are measured against the same container.

The whyThe amount used equals the difference of the two fractions of the full battery, so a common denominator gives it as one fraction — and the full capacity is the amount used divided by that fraction. Reverse-fraction questions are the unitary method in disguise.

5 and 6

HintWhich two square numbers sit either side of it?

The whyTrap a root between neighbouring squares: 25 < 30 < 36. To decide which side it is closer to, try the halfway value — 5.5² = 30.25, so √30 is just under 5.5. "Which is closest to" questions with a root in the answer are settled this way.

360

HintCut it into two triangles.

The whyWhen a quadrilateral's angles are given as a ratio, share 360° in that ratio: each part is 360 divided by the total of the ratio's parts. The same works for any polygon once you know its angle sum.

60; 120; 135

HintHow the tiles meet: triangles six to a point, hexagons three, octagons with squares.

The whyHalving an equilateral triangle of side 2 gives a right-angled triangle with sides 1, √3 and 2 — so the height of an equilateral triangle of side s is (√3/2)s and its area is (√3/4)s².

Its angle's share of 360°

HintThe bigger the turn at the centre, the bigger the slice — in exact proportion.

The whyCorner sectors of one radius at every vertex of a polygon add up to (angle sum ÷ 360) circles — half a circle for a triangle, one whole circle for a quadrilateral.