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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?

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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

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Bang's Theorem

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


Stage: 4 and 5 Challenge Level: Challenge Level:1

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 one.

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.