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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?

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Our Ages

I am exactly n times my daughter's age. In m years I shall be exactly (n-1) times her age. In m2 years I shall be exactly (n-2) times her age. After that I shall never again be an exact multiple of her age. Ages, n and m are all whole numbers. How old am I? Now suppose there is some wishful thinking in the above assertion and I have to admit to being older, and indeed that I will be an exact multiple of her age in m3 years. How old does this make me?

Deep Roots

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Find integer solutions to: $${{\sqrt{a + b\sqrt{x}}} + {\sqrt{c + d\sqrt{x}}}} ={ 1}$$

Hint: As everything on the left is positive and the total is 1, there are only a few cases to consider.