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 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
Many thanks to Robert Simons for this question:
"I am exactly $n$ times my daughter's age. In $m$ years I shall
be exactly $(n-1)$ times her age. In $m^2$ 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 $m^3$ years. How old does
this make me?"