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

Square Pegs

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

The Old Goats

A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't fight each other but can reach every corner of the field?

A Chordingly

Age 11 to 14 Challenge Level:
Prove that the area of the annulus is ${1\over 4}\pi AB^2$ where $AB$ is a tangent to the inner circle.

annulus with AB drawn