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Pythagorean Triples

How many right-angled triangles are there with sides that are all integers less than 100 units?

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A tennis ball is served from directly above the baseline (assume the ball travels in a straight line). What is the minimum height that the ball can be hit at to ensure it lands in the service area?

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Square Pegs

Which is a better fit, a square peg in a round hole or a round peg in a square hole?

The Pillar of Chios

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3
rectangles and crescents

Semicircles are drawn on the sides of a rectangle ABCD as shown in the diagram. A circle passing through points ABCD carves out four crescent-shaped regions (coloured yellow and green in the diagram).

Prove that the sum of the areas of the four crescents is equal in area to the rectangle ABCD.