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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?

Overturning Fracsum

Can you solve the system of equations to find the values of x, y and z?

Building Tetrahedra

Can you make a tetrahedron whose faces all have the same perimeter?


Age 14 to 18
Challenge Level

Solve the two pairs of simultaneous equations:

\begin{eqnarray} x + 0.99999y & = & 2.99999 \\ 0.99999x + y & = & 2.99998 \end{eqnarray} and \begin{eqnarray} x + 1.00001y & = & 2.99999 \\ 0.99999 x + y & = & 2.99998. \end{eqnarray}

Explain why the solutions are so different and yet the pairs of equations are nearly identical.

In this question a small perturbation in one of a pair of equations makes a big change in the solutions. Considering the geometrical properties of the lines represented by the equations helps to de-mystify the results.