IMC 2026

2

提示Two staircases make a rectangle.

为什么Triangular numbers: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91. The mean of 1 to n is (n + 1) ÷ 2, so those means grow by a half each step.

It factorises as (n − 1)(n + 1)

提示What is 1 the square of?

为什么(n − 1)n(n + 1) is a product of three consecutive numbers, so it is always a multiple of 6; for n not divisible by 2 or 3, (n − 1)(n + 1) is a multiple of 8 and of 3, hence of 24 — which is why 17, 19 and 23 work while 21 and 27 do not.

Clear the fractions first

提示A ratio survives one particular operation on every part.

为什么1 : ½ : ⅓ is 6 : 3 : 2 (multiply by 6), eleven parts in all. A ratio is unchanged when every part is multiplied by the same number — so fractions and decimals never need to stay in it.

Equal to the rectangle's

提示Slide every inward edge outwards.

为什么Each horizontal edge projects onto the top or bottom of the enclosing rectangle, each vertical edge onto a side, with nothing double-counted — so a staircase shape has perimeter 2(width + height). Add any sloping edges separately.

a√2

提示Half a square, cut corner to corner.

为什么So on a figure whose angles are all multiples of 45°, a sloping edge is √2 times its horizontal (= vertical) span, and two sloping edges that together rise 3 have total length 3√2. Split the perimeter into level, upright and sloping parts.

(√3 ÷ 4) s²

提示Half the base times a height found by Pythagoras.

为什么Height = (√3 ÷ 2) s, so area = ½ × s × (√3 ÷ 2) s. A parallelogram's area is base × perpendicular height — never base × slant side. Ratios of areas built from the same √3 cancel it, leaving a whole-number ratio.

Two equal halves

提示What does the line change about the two pieces, and what does it not?

为什么Equal bases plus a shared perpendicular height means equal areas. Two such lines from two vertices split the triangle into four pieces you can pair up.

3xy

提示Expand the cube and collect the mixed terms.

为什么With x + y = 5 and x³ + y³ = 95: 125 = 95 + 15xy, so xy = 2 — no need to find x or y. Its square cousin: (x + y)² = x² + y² + 2xy.

100

提示Write the percentage as a fraction in lowest terms and look at the bottom.

为什么97/100 × N is a whole number only when 100 divides N, because 97 and 100 are coprime. 90% is different: 9/10 × N only needs N to be a multiple of 10. Reduce the percentage to lowest terms and read the denominator.

Division by the number of orders

提示Three items can be lined up in how many ways?

为什么Picking three things in order gives n × (n − 1) × (n − 2) sequences, and each set of three appears in 3 × 2 × 1 = 6 of them, so divide by 6. Then count the favourable sets systematically and divide.

1; even

提示Largest digit plus largest digit plus a carry — how big can that be?

为什么9 + 9 + 1 = 19, so a carry is never more than 1; a column total of 2A or 2A + 1 tells you at once whether a carry came in; a leading digit is never 0. Three facts crack most letter-sum puzzles.

Peel from the outside inwards

提示Which layer was built last?

为什么Move the outer constant, take reciprocals, and the next layer appears; three or four repeats reach x. Simplifying (rather than solving) goes the other way, from the innermost fraction out.