IMC 2017

2, 3, 7 or 8

提示Four of the ten digits never appear at the end of a square.

为什么The 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

提示Odd plus odd is even, so one of the two primes must be the single even one.

为什么Every prime except 2 is odd, and odd + odd = even, so an odd sum of two primes has to use the even prime. That turns the whole question into one primality test: 13 works because 13 − 2 = 11 is prime, while 51 fails because 51 − 2 = 49 = 7 × 7. The same parity argument shows 2 and 3 are the only two primes that differ by 1.

Multiplied by four

提示The area formula uses the radius twice.

为什么Area = 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

提示How spread out the data is, in one number.

为什么The 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

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

为什么With 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

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

为什么Once 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

提示Split it from one corner and count the triangles.

为什么180(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

提示Both levels are measured against the same container.

为什么The 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

提示Which two square numbers sit either side of it?

为什么Trap 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.

144

提示Cut the shape into two triangles for the total, then share it among ten equal parts.

为什么When 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

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

为什么Halving 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°

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

为什么Corner 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.

2 and 5

提示Think about which primes are hiding inside every power of ten.

为什么This is why fractions with denominators like 4, 20 or 25 always terminate — each is built only from 2s and 5s — while a denominator with any other prime factor (3, 7, 11…) forces a recurring decimal. Recognising terminating denominators lets you convert several such fractions to decimals and add them by lining up place value, rather than hunting for a common denominator.

Compare the first digit where they differ

提示Look past the matching part — the answer sits at the next new value along.

为什么Recurring-decimal options often share their first several digits, so scan along until you find the first place they disagree — the larger value there decides which decimal is bigger, no matter how the pattern continues afterwards. Writing out a few extra repeats of the recurring block on each option makes this place easy to spot.

360

提示A full turn all the way around a point.

为什么This is different from angles on a straight line (180°) or in a triangle (180°) — it's the 'full turn' fact, and it often combines with parallel-line angle rules (alternate or corresponding angles) when a diagram is built from several shapes meeting at one vertex.

k²:1

提示Squaring the side ratio gives the area ratio.

为什么For similar figures, every length scales by k, so area — built from two lengths — scales by k². Volume, built from three lengths, would scale by k³. It's the same idea as a circle's radius doubling and its area quadrupling, just for any similar shape, not only circles.

Add the equations covering every other unknown once, then subtract from the total.

提示Find the combination of clues that leaves only your target hidden.

为什么This works because summing the right subset counts every unknown except the target exactly once; subtracting that subset's total from the grand total cancels everything else and leaves the target alone. It turns what looks like a large system into one line of arithmetic — spotting the right subset is the whole skill.

a² + b² = c², where c is the hypotenuse and a, b are the other two sides

提示It relates the triangle's longest edge to the two shorter ones, squared and added.

为什么It only holds for a right-angled triangle, with c always the side opposite the right angle (the hypotenuse) — the longest side. Once you have any two sides you can find the third; for the hypotenuse you add the squares, for a shorter side you subtract before rooting.

Write each relationship as its own equation

提示Turn every sentence of the problem into symbols before combining them.

为什么Two independent facts about the same two unknowns give two equations — write both first, in terms of the same two letters, before trying to solve either. Once you have the pair, substitute one equation into the other to reduce it to a single-unknown equation, which may turn out to be a quadratic rather than linear.

zero

提示This is exactly why factorising an equation into a product helps you solve it.

为什么This is the zero-product rule, the reason factorising solves equations: rewrite an equation as (something) × (something else) = 0, and each factor gives its own mini-equation to solve. When the equation comes from a real situation — ages, counts, lengths — one of the resulting values is often impossible, so the context, not the algebra, decides which root is the real answer.

SAS (side, angle, side)

提示The one equal turn must sit trapped between the two equal edges, not anywhere else.

为什么SAS needs the angle to be the included angle — sitting exactly where the two known sides meet — otherwise two sides and a non-included angle can, in some cases, describe two different triangles (the ambiguous case). Once triangles are proved congruent, every corresponding side and angle can be assumed equal without further checking.

Their sum

提示You don't need to isolate each letter separately to answer some questions.

为什么Sometimes a question only wants a combined total, not the individual values. If subtracting two equations happens to leave equal coefficients on both remaining letters, dividing the whole equation by that coefficient gives their sum straight away — a shortcut worth spotting before grinding through full elimination and substitution.

Every pair of positions they could occupy

提示Work through the numbered spots and cross off any two that sit side by side.

为什么For a small number of positions, list every pair of slots the two restricted people could take, cross out any pair that's adjacent, and count what's left — that count becomes one factor in a bigger calculation, since the two people can usually swap between their chosen positions and everyone else can be arranged independently in the remaining spots.

Multiply the number of choices at each step

提示Think of the decisions as independent — the total ways is a product, not a sum, of how many options exist at every stage.

为什么This is the fundamental counting principle: if the first placement can be made in a ways, the next in b ways (given the first is fixed), and so on, the total number of arrangements is a × b × c × … . It's the same reasoning that turns "three choices, then two, then one" into a product rather than a running total, and it applies to arranging the remaining items after any fixed or restricted items are already placed.

numerator; denominator

提示Picture the fractions in a chain — top of one term meets bottom of the next, and they wipe each other out.

为什么Because each fraction is built as (something + 1) over that something, the value on top of one fraction reappears on the bottom of the next, so multiplying them together leaves it out entirely. Only the very first denominator and the very last numerator survive the chain — spotting this turns a long product into a single fraction, which is then easy to test against whole-number targets.