### Bang's Theorem

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

### 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?

### 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?

# Root to Poly

##### Stage: 4 Challenge Level:

Congratulations to Fok Chi Kwong from Yuen Long Merchants Association Secondary School, Hong Kong on this solution.

We may find the required polynomial by starting from the expression :

$$x = 1 + \sqrt 2 + \sqrt 3$$.

Squaring both sides and simplifying, we get

$x - 1 = \sqrt 2+ \sqrt 3$ $x^2 - 2x + 1 = 5 + 2\sqrt 6$ $x^2 - 2x - 4 = 2\sqrt 6$ $(x^2 - 2x - 4)^2 = 24$ $x^4 - 4x^3 + 4x^2 - 8x^2 + 16x + 16 = 24$ $x^4 - 4x^3 - 4x^2 + 16x - 8 = 0$

Thus $p(x) = x^4 - 4x^3 - 4x^2 + 16x - 8$ is the required polynomial.

Tony Cardell, State College Area High School, PA, USA, also sent in a good solution.