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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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Roots and Coefficients

If xyz = 1 and x+y+z =1/x + 1/y + 1/z show that at least one of these numbers must be 1. Now for the complexity! When are the other numbers real and when are they complex?

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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.

Quadratic Harmony

Age 16 to 18 Challenge Level:
You might make use of the relationship between the coefficients and the sum and product of the roots of the quadratic equations. Without loss of generality you can take $a \geq b$ and then you'll need to consider separately the case where $a=b$ and where $a> b$.