Real(ly) Numbers

If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?

Overturning Fracsum

Solve the system of equations to find the values of x, y and z: xy/(x+y)=1/2, yz/(y+z)=1/3, zx/(z+x)=1/7

Bang's Theorem

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

System Speak

Stage: 4 and 5 Challenge Level:

The answers to this problem are:

either $a = 3/2$, $b = 2/3$, $c = 3$, $d = 1$ and $e = 4$

or $a = -3/2$, $b = -2/3$, $c = -3$, $d = -1$ and $e = -4$

Roderick (Simon Langton Grammar School) provided a correct solution based on an interesting trial and improvement technique. Students at Smithdon High School - Robert , Jack and Matthew and John of Hethersett High School relied on a more traditional method of proving their solution to the problem, the essence of which is reproduced below.

Let $a = N$

hence $b = 1/N$

and $c = 2/b$ i.e. $c = 2N$

and $d = 3/c$ i.e. $d = 3/2N$

and $e = 4/d$ i.e. $e = 4/(3/2n)$ i.e. $e = 8N/3$.

But $e = 6/N$.

Hence $6/N = 8N/3$

i.e $8N^2 = 18$

i.e $a = N = +$ or $- 3/2$ and so on....