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.

Each factor is odd

HintOne even number in the mix is enough to change everything.

The whyA single even factor drags a 2 into the product and makes it even, so an odd product rules every even factor out. That is the contrapositive of the familiar rule, and it is the direction competition questions actually use — 3 × 5 × 7 stays odd, but 3 × 4 × 5 cannot.

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.

12π

HintWork out what each circle would cover on its own, then compare.

The whyInclusion–exclusion runs backwards just as well: covered = first + second − overlap. Each circle has area π × 6² = 36π, so on their own they account for 72π; the covered region is only 60π, and the missing 12π is exactly the piece that was counted twice. The same rearrangement finds the overlap of any two regions once the total covered area is known.

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.

Any order — each number just keeps the sign glued to it

HintRegroup the terms that combine nicely; the + or − in front of a term travels with it.

The whyFor example, 8 − 3 + 5 can be rearranged to 8 + 5 − 3, always giving 10 — grouping the positives and negatives separately before combining once often saves work, especially with larger or messier numbers.

Convert each clue into a day-offset, then count from a known day

HintGive every phrase a plus or minus shift in days, and add them up starting at something certain.

The whyTreat 'today' as day 0. 'Yesterday' is −1, 'the day before X' is (X − 1), and 'two days after Y' is (Y + 2). Chain the offsets together in the order the sentence gives them, then convert the final number back into a day of the week.

Let one quantity be x, express the rest in terms of x, then solve

HintPick one unknown as the anchor, and rewrite the other amounts using that same letter.

The whyIf B is three-fifths of C, and A is half of B, write everything in terms of C: B = (3/5)C and A = (1/2)(3/5)C = (3/10)C. Adding A + B + C to the known total gives one equation in one unknown, which can then be solved directly.

First find all the factors of that number, in order

HintWork out the complete set of divisors before you go hunting for what's missing.

The whyNumbers not in that list are simply everything else, counted off in order — e.g. if the factors of 20 up to 20 are 1, 2, 4, 5, 10, 20, the non-factors below 20 are 3, 6, 7, 8, 9, and so on, continuing past whichever factors come next as you climb higher.

ab + k(a + b) + k²

HintMultiply out two brackets that share the same added constant.

The whyThis turns an unfamiliar product into a known one. If 24 × 28 = 672 is already known, adding 6 to each factor gives 30 × 34 = 672 + 6 × (24 + 28) + 36 = 1020 — much quicker than multiplying two large numbers from scratch.

15

HintMultiply the two percentages' decimal forms together, then convert back to a percentage.

The whyPercentages of percentages combine by multiplication, not addition: 0.25 × 0.60 = 0.15, so 25% of 60% of a number is 15% of it — not 85% (25+60) and not 35% (60−25). The same idea extends to any chain: three consecutive percentage operations multiply three decimals together.

Assign the smallest square a convenient length, such as 1

HintGive the tiniest repeating unit an easy value so every area works out as a whole number.

The whyIf the smallest squares in the diagram have side-length 1, every other length and area in the figure becomes a whole number instead of an unknown ratio — turn the shaded area into a fraction of the total, then convert that fraction to a percentage.

Bisects the base and meets it at right angles

HintSymmetry splits the triangle into two identical mirror-image halves.

The whyThis works because reflecting the triangle in that altitude maps the shape onto itself, swapping the two base vertices — so the altitude must land exactly at the base's midpoint, and Pythagoras' Theorem then applies directly in either half using half the base and the equal side.

Expand several repeats of each recurring block first

HintWrite out enough copies of the repeating pattern to see clearly which side of the target number you land on, and by how much.

The whyRecurring decimals hide their real size in the first few digits, so extend the pattern before judging which side of a target number they fall on — a value like 0.161616… can look bigger than 0.162 at a glance, but writing out several repeats (0.16161616…) shows the true value is smaller.

a² : b²

HintArea scales with the square of the length scale factor, not the factor itself.

The whySimilar shapes multiply every length by the same scale factor k, but area — being length × length — scales by k². A shape enlarged by scale factor 3 has 9 times the area, not 3 times, and a similar solid’s volume would scale by k³.

Rewrite each term as one single fraction

HintFind a common denominator inside every set of brackets before multiplying across.

The why1 + 2/x becomes (x + 2)/x once written over a common denominator; do this to every factor first, and terms often cancel across the brackets, leaving a far simpler result than multiplying everything out piece by piece.

coprime

Hint7 and 4 have no common factor other than 1.

The whyThis is a general rule: if a × n is a multiple of b, and a and b share no common factor, then n itself must be a multiple of b — because none of b's prime factors can come from a, so they all have to come from n. The same idea shows a fraction is already in lowest terms when top and bottom share no factor.

The walkway's speed (in the direction of travel)

HintThe moving surface itself contributes extra distance covered per second.

The whySet up one equation per scenario using distance = speed × time: for the walkway moving, running speed + walkway speed = distance/time; for the stationary walkway, running speed alone = distance/time. Subtracting the equations isolates the walkway’s own speed, and the same idea gives walking speed too.