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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.

Always Two

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

As some solutions are easily spotted, you may be able to find solutions by trial and error. You still need to prove that the solutions you have found are the only ones.

Write down three simultaneous equations from the statement of the problem.

Take care to keep the symmetry of the expressions.

In pairs subtract one equation from another.

What can you deduce if the product of two linear factors is zero?