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Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

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Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.

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AP Rectangles

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Rectangle Cutting

Stage: 3 Short Challenge Level: Challenge Level:1

Tom and Jerry must each cut their piece of paper in half. Suppose the sides of the original piece of paper have length $2x$ and $2y$, with $x\geq y$.

Then $2x+4y=40$ and $4x+2y=50$ which implies that $6x+6y=90$, therefore the perimeter of the original piece, $4x+4y=60$.

This problem is taken from the UKMT Mathematical Challenges.