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.