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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?

Polynomial Relations

Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.

More Polynomial Equations

Find relationships between the polynomials a, b and c which are polynomials in n giving the sums of the first n natural numbers, squares and cubes respectively.

Janusz Asked

Age 16 to 18
Challenge Level
For $y = ax + b$, if you can use the given conditions and write $b$ in terms of $a$, then for any value of $a$ there will be a linear function with this property.

For the second part use the formula relating the sum of the roots and the coefficients and then work systematically to consider all cases.