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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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I keep three circular medallions in a rectangular box in which they just fit with each one touching the other two. The smallest one has radius 4 cm and touches one side of the box, the middle sized one has radius 9 cm and touches two sides of the box and the largest one touches three sides of the box. What is the radius of the largest one?

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Square Mean

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

Rudolff's Problem

Stage: 4 Challenge Level: Challenge Level:1

Rudolff's book, called Coss, written in 1525, was the first German algebra book. The reason for the title is that cosa is a 'thing', a term used for the unknown. Algebraists were called cossists, and algebra the cossic art, for many years. Here is the 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?