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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.

Intersections

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

In this case, rather than working with the awkward numbers, you may find it easier to substitute $\alpha = 0.99999$ and solve the equations algebraically, and finally substitute $0.99999$ for $\alpha$ to find the numerical solutions.

To explain the solutions think about the geometry associated with these equations.