IMC 2020

Same as adding

HintTwo minus signs together — think of removing a debt.

The whya − (−b) = a + b. A bracket like (3 − 8 − 4) is worked out first and is negative, and taking away that negative adds its size. Order-of-operations questions with nested minus signs are won by evaluating each bracket fully before touching the sign in front of it.

Split it into coprime factors

HintNot 6 × 4 — choose a pair that shares nothing.

The whyBecause 8 and 3 share no factor, a number divisible by both is divisible by 24; 6 and 4 overlap (both even), so passing both does not guarantee 24. The tests for 8 (last three digits) and 3 (digit sum) then finish the job.

30% of the difference

HintWhat do the two amounts have in common?

The whyA percentage distributes over a subtraction: p% of A − p% of B = p% of (A − B). Finding the difference first usually turns two awkward calculations into one easy one.

1024

HintThe tenth doubling starting from 1.

The whyThe powers of 2 up to 2¹⁰ — 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 — are worth knowing by heart. "Which of these powers is largest?" questions often cannot be settled by index tricks alone, and knowing the values lets you evaluate each option in seconds.

Cancel the repeated number

HintLook for what top and bottom share before doing any arithmetic.

The whyWrite each part as a product first: a repeated number can be cancelled from top and bottom the moment both are products of it. A large number repeated in a fraction is an invitation to cancel, never to multiply out.

Add the counts of the squares that lead into it

HintBuild the numbers up one cell at a time from the start.

The whyLabel the start with 1, then give every square the sum of the labels on the squares you could have come from. When each square can be entered from the two before it, the labels are the Fibonacci numbers: 1, 1, 2, 3, 5, 8 …

Two equations in two unknowns

HintTwo facts, two letters.

The whyGive each kind a letter and turn each total into a line of algebra. Often the question wants only a combination of the unknowns, and subtracting the equations delivers it without solving fully.

At least 1 and less than 10

HintFrom the smallest one-digit number up to, but not reaching, the smallest two-digit one.

The whyHalving 1.2 × 10⁴ gives 0.6 × 10⁴, which is not standard form: move the point to make 6 × 10³. Whenever a calculation leaves a number part outside the allowed range, adjust the power of ten to compensate.

9; 11; 125

HintThree tests, three divisors: digit sum, alternating sum, tail.

The whyDivisibility tests are the fast way to find a missing digit or rule out an option without doing the multiplication: pick the test for a factor you know the number has.

90

HintThe radius is the shortest route from the centre to the line.

The whyWhenever a circle or semicircle touches a straight edge, draw the radius to the touching point — it makes a right angle, which usually creates a right-angled triangle or a square you can measure.

1.4

HintIts square must be just under 2; test 1.5 first.

The why1.4² = 1.96 and 1.5² = 2.25, so 1.4 < √2 < 1.5. Comparing lengths like 3 + 2√2 and 5 + √2 without a calculator uses exactly this trap, or the cruder 1 < √2 < 2.

Work backwards

HintStart from the one thing you know for certain — the end.

The whyUndo each step in reverse order — subtract what was added, double what was halved. Working forwards means guessing a starting value; working backwards from the end is a single calculation.

Pick convenient side lengths for the whole shape

HintYou get to choose numbers that make the arithmetic easy — the answer is a ratio, not a measurement.

The whyBecause the question only wants a fraction, the actual size of the figure is free to choose — pick a side length that divides nicely (a multiple of 4, say) and every other length and area falls out as a whole number. Subtract whatever unshaded piece is easy to calculate (often a triangle or two) from the total area rather than trying to add up all the shaded pieces directly.

Apply Pythagoras to the horizontal and vertical gaps between them

HintTurn the coordinate difference into the two shorter sides of a right triangle.

The whyThe difference in x-coordinates and the difference in y-coordinates are the two legs of a right-angled triangle whose hypotenuse is the distance you want; square each difference, add, then square-root. Comparing SQUARED distances (skipping the final root) is enough to tell whether several points are the same distance apart, which is how you spot a square or rhombus among plotted points without ever touching a surd.

n − 1

HintCount the gaps between consecutive objects, not the objects themselves.

The whyLine up n objects; there are exactly n − 1 gaps between consecutive ones, and each gap is one neighbouring pair. This is quicker than listing every possible pair (which would be far more, since most pairs are NOT neighbours), and it works for any row length without drawing it out.

Check just the units digit of each choice

HintEvery option ends in a distinct single figure — work that out, not the whole sum.

The whyIf the possible answers all differ only in their final digit, you don't need the whole sum — just add the units digits of each term (using known units-digit patterns for squares, e.g. 3² ends in 9, 9² ends in 1) and read off which final digit matches. This turns a long calculation into a few seconds of mental arithmetic.

Find the part covered by exactly one range

HintMost numbers sit inside two or more of the given intervals — you want the section that isn't shared.

The whySketch or list the ranges on a line and mark where they overlap; wherever two or more ranges cover the same numbers, those numbers can't be the answer, because the question said it fits in exactly one. Only the section left over — belonging to a single range — pins the value down uniquely.

3 : 4 : 5

HintRight-angled triangles with whole-number sides often use this exact trio.

The whyAny right-angled triangle whose sides are in the ratio 3 : 4 : 5 (or a multiple of it, like 6 : 8 : 10) satisfies Pythagoras' theorem exactly, because 3² + 4² = 5². Spotting this ratio inside a diagram — even when the actual lengths are scaled up — lets you find a missing side by simple proportion instead of squaring and rooting.

2; 5

HintAny other prime factor — like 3, 7 or 13 — forces the decimal to recur forever.

The whyEvery terminating decimal is really a fraction over some power of 10, and 10 = 2 × 5 — so only 2s and 5s are allowed in the denominator once it is fully cancelled. A denominator like 12 = 2² × 3 fails the test because of that stray 3, and its decimal repeats forever instead of stopping.

x² equals half of y²

HintApply Pythagoras where the two legs meeting at the right angle are the same length.

The whyPythagoras gives x² + x² = y², so 2x² = y², meaning x² = y² ÷ 2. This lets you jump straight from a hypotenuse to the area of the triangle (½ × x × x) without ever finding x itself — useful whenever a right-angled isosceles triangle is built from smaller identical shapes.

x² − a²

HintMultiply out fully — the middle terms cancel each other out.

The whyThis is the difference-of-two-squares identity: multiplying a sum by a difference of the same two terms always removes the 'middle' terms, leaving just the squares subtracted. It's the reason (x+1)(x−1) is LARGER than (x+2)(x−2) for the same x — a bigger a means a bigger square subtracted, so the expression shrinks as a grows.

The sum of their two radii

HintEach centre sits exactly one radius-length back from the shared touching point.

The whyIf circle A has radius r₁ and circle B has radius r₂ and they touch externally (each outside the other), the straight line joining their centres passes through the touching point and has length r₁ + r₂ — the touching point splits that line into two radii, one from each centre. This is different from two circles touching internally, where the distance between centres is the DIFFERENCE of the radii.

360° ÷ n

HintThe exterior angles of any regular polygon share the total all the way around a point.

The whyThe exterior angles of ANY convex polygon add up to 360°, and in a REGULAR polygon all n of them are equal, so each one is 360° ÷ n. The interior angle is then 180° minus that value, since an interior angle and its exterior angle sit on a straight line together.

Factor out the smaller power of 2

HintPull the lower exponent's term outside a bracket, leaving a simple number behind.

The whyBoth terms are multiples of the smaller power: 2ⁿ + 2ⁿ⁺¹ = 2ⁿ × 1 + 2ⁿ × 2 = 2ⁿ(1 + 2) = 3 × 2ⁿ. Spotting the common factor turns an awkward sum of two powers into a single power times a small whole number, which is essential for solving equations where powers of the same base appear on both sides.

(θ/360) × πr²

HintTreat the whole circle as one area and take only the matching slice of it.

The whyA full circle is 360°, so a sector spanning θ° is that fraction of the whole circle's area: (θ/360) × πr². The same fraction idea applies to arc length — a sector's arc is (θ/360) of the full circumference, 2πr.