If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.
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?
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?
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
$$x = 1 + \sqrt 2 + \sqrt 3$$.
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
Tony Cardell, State College Area High School, PA, USA, also sent
in a good solution.