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P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

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Just Opposite

A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square?

Platonic Planet

Age 14 to 16
Challenge Level
How much of the planet's surface can the alien see if she is in the middle of one of the faces?
What about on one of the edges?
Or at a vertex?

Try to picture a route around the planet either in three dimensions or on a net. Can you be sure every point on the surface is visible from your route? Can you make a more efficient route? Some diagrams might help!