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Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

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Intersecting Circles

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

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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?

A Chordingly

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Jessica Zhang of Woodbridge High School, Arwa Jamil , Year 8, the International School Brunei and Nisha Doshi and Suzanne Abbott from Year 9 the Mount School, York all sent good solutions to this problem.

Let the radius of the big circle be $R$ and the radius of the small circle be $r$. By Pythagoras theorem: $$ R^2 - r^2 = ({1\over 2}AB)^2 $$ The area of the big circle is $\pi R^2$ and the area of the small circle is $ \pi r^2 $. The area of the annulus $A$ is the area of the big circle minus the area of the small circle so: $$ A = \pi R^2 - \pi r^2 = \pi (R^2 - r^2) = \pi({1\over 2}AB)^2 = \pi (AB)^2/4. $$