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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?

Pair Squares

The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.

Quadratic Harmony

Age 16 to 18
Challenge Level
For what values of $a$ and $b$ (where $a$ and $b$ are positive integers) do the two equations: $$x^2-a x+b=0$$ $$x^2-b x+a=0$$ both have positive integer solutions? You may be able to find some values of $a$ and $b$ by trial and error. Can you prove that these are the only possible values?