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Overturning Fracsum

Solve the system of equations to find the values of x, y and z: xy/(x+y)=1/2, yz/(y+z)=1/3, zx/(z+x)=1/7

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Bang's Theorem

If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.

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Rudolff's Problem

A group of 20 people pay a total of £20 to see an exhibition. The admission price is £3 for men, £2 for women and 50p for children. How many men, women and children are there in the group?

Gift of Gems

Stage: 4 Challenge Level: Challenge Level:1

Four jewellers possessing respectively eight rubies, ten saphires, a hundred pearls and five diamonds, presented, each from his own stock, one piece to each of the rest in token of regard; and they thus became owners of stock of precisely equal value. Tell me the prices of their gems and the values of their stocks.

[This problem comes from Lilavati , a standard work on Hindu mathematics written by Bhasakaracharyya, who lived in the twelfth century of the Christian era. The book is written as instruction for a young and beautiful woman called Lilavati and it is thought that she was Basakaracharyya's daughter.]