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
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
If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.
We had good solutions from Andrei in
Bucharest, and from Matt and Andrew at the Perse School
The solution in the first case is $x = 2, y = 1$ and in the
second case is $x = - 199998, y = 199999$
For the first two lines one gradient is a little under minus one
and the other gradient a little over.
For the second two lines both gradients are a little above minus
All four lines cut the $y$-axis very near to $3$.
Because the pairs of lines in each case are nearly parallel the
slight and unique change in each line's gradient away from minus
one causes the intersection to occur in very different places.