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There were two solutions from Madras College, one from Thomas, James, Mike and Euan and the other from Sue Liu which is reproduced below.
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This triangle is a right angled isosceles triangle, the
hypotenuse being \sqrt{2}b. We draw a line through A and point
M, the midpoint of the line BC. We draw the line B`C` giving
another right angles isosceles triangle AB`C`, similar to
triangle ABC but with sides a and hypotenuse \sqrt{2}a. Now N
is the point where the line B`C` meets the line AM, and P is
the point where BC` meets B`C (also on AM).
It is clear that triangle PBC is similar to triangle P
B`C` and the enlargement factor from B`C` to BC is b/a. So
the line PM is b/a times as long as the line PN. Also the
line AN is half the length of B`C`, so it is \sqrt{2}a/2. The
line AM is half the length of BC so it is \sqrt{2}b/2.
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Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.
The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?
A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?