IMC 2021

The reflex interior angle

提示More than a half turn, measured on the inside.

为什么The formula counts the angle on the inside of the shape at every vertex; where the boundary turns inward, that inside angle is bigger than 180°. At a square's corner seen from outside, the inside angle of the surrounding polygon is 360° − 90°.

same

提示10, 100, 1000 … each leave remainder 1.

为什么Because 10, 100, 1000 … are all 1 more than a multiple of 9, a number and its digit sum differ by a multiple of 9 — so they share a remainder, and the same is true for division by 3. It also drives 'casting out nines': the remainder of a product by 9 comes from the remainders of its factors.

Always odd

提示Look only at the units digit of a product.

为什么Test claims like this with one small example before trusting them: 3 × 5, 4 × 6, 2 × 2.

Reflection in the x-axis

提示The first coordinate is untouched, so the mirror runs along that direction.

为什么Reflect in the x-axis: y changes sign. Reflect in the y-axis: x changes sign. Reflect in y = x: swap the coordinates. Applied to a whole line, y = 3x − 5 becomes −y = 3x − 5, that is y = −3x + 5: the gradient and the intercept both flip sign.

2

提示It is a fraction of the full disc.

为什么A semicircle of diameter 10 has area 12.5π; a quarter circle of radius 4 has area 4π. Leave π in the answer — the options are written with it.

5

提示The most famous whole-number right-angled triangle.

为什么Multiples of 3-4-5 (6-8-10, 9-12-15, 15-20-25) and of 5-12-13 (10-24-26) appear constantly, often hidden inside a diagram. Spot the triple and the square root is never needed.

Sum minus the overlap

提示What happens to the lens in the middle if you just add?

为什么Inclusion–exclusion for two regions: total = first + second − overlap. Forgetting the subtraction counts the shared region twice.

Substitute the point into both equations

提示What is true of any position that lies on a line?

为什么A point on a line makes the line's equation true. So a shared point gives one true equation per line — two lines, two equations — which is usually exactly enough to pin down two unknown constants.

Average speed

提示It is a rate, not a length or a time.

为什么Keep the units consistent before anything else: a 'three minutes less' clue needs the hours converted to minutes (or 3 minutes written as 1/20 of an hour).

total = mean × count

提示Work with the sum of everything, not the average itself.

为什么Let the unknown count appear on both sides: the added numbers raise the total and the count at once, and the fixed mean ties them together in one equation.

adding; positive

提示Two minus signs together.

为什么2 − (−3) = 5 and (−2) × (−3) = 6, but −2 − 3 = −5 and (−2) + (−3) = −5: only two signs that act on each other cancel. Work nested brackets from the inside out.

reciprocal

提示How many quarters fit into 3?

为什么3 ÷ ¼ = 3 × 4 = 12: dividing by a small fraction gives a big answer. Turn mixed numbers into improper fractions before you flip anything.