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Fitting In

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ

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Look Before You Leap

Can you spot a cunning way to work out the missing length?

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Triangle Midpoints

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

Doubly Isosceles

Age 14 to 16 Short Challenge Level:

Triangle PQR is isosceles, with PQ = PR = 2.

Triangle QRS is isosceles, with QR = RS = 1.

Find the length QS.








This problem is taken from the World Mathematics Championships
You can find more short problems, arranged by curriculum topic, in our short problems collection.