IMC 2021

The reflex interior angle

HintMore than a half turn, measured on the inside.

The whyThe 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

Hint10, 100, 1000 … each leave remainder 1.

The whyBecause 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

HintLook only at the units digit of a product.

The whyTest claims like this with one small example before trusting them: 3 × 5, 4 × 6, 2 × 2.

Reflection in the x-axis

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

The whyReflect 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

HintIt is a fraction of the full disc.

The whyA 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

HintThe most famous whole-number right-angled triangle.

The whyMultiples 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

HintWhat happens to the lens in the middle if you just add?

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

Substitute the point into both equations

HintWhat is true of any position that lies on a line?

The whyA 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

HintIt is a rate, not a length or a time.

The whyKeep 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

HintWork with the sum of everything, not the average itself.

The whyLet 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

HintTwo minus signs together.

The why2 − (−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

HintHow many quarters fit into 3?

The why3 ÷ ¼ = 3 × 4 = 12: dividing by a small fraction gives a big answer. Turn mixed numbers into improper fractions before you flip anything.