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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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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.

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For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.

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?