IMC 2020：全部卡片

Same as adding

提示Two minus signs together — think of removing a debt.

为什么a − (−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

提示Not 6 × 4 — choose a pair that shares nothing.

为什么Because 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

提示What do the two amounts have in common?

为什么A 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

提示The tenth doubling starting from 1.

为什么The 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

提示Look for what top and bottom share before doing any arithmetic.

为什么Write 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

提示Build the numbers up one cell at a time from the start.

为什么Label 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

提示Two facts, two letters.

为什么Give 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

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

为什么Halving 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

提示Three tests, three divisors: digit sum, alternating sum, tail.

为什么Divisibility 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

提示The radius is the shortest route from the centre to the line.

为什么Whenever 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

提示Its square must be just under 2; test 1.5 first.

为什么1.4² = 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

提示Start from the one thing you know for certain — the end.

为什么Undo 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.