Building Tetrahedra

Can you make a tetrahedron whose faces all have the same perimeter?

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?

Polycircles

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

Negatively Triangular

Age 14 to 16Challenge Level

Four straight lines have the following equations:

\begin{align} 3x+8y&=59 \\ x-2y&=1 \\ y-4x&=3 \\ 3y+2x &= -19 \end{align}

How many points of intersections do you expect there to be?

Three of the four lines enclose a triangle which only has negative co-ordinates for all points on that triangle.

Which three lines?