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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?

Janusz Asked

In y = ax +b when are a, -b/a, b in arithmetic progression. The polynomial y = ax^2 + bx + c has roots r1 and r2. Can a, r1, b, r2 and c be in arithmetic progression?

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.

Interpolating Polynomials

Age 16 to 18 Challenge Level:
This problem required making logical arguments that come together to form a proof. It can be hard to put all the steps in the right place but we received great solutions from Michael from the UK, Kaan from the German Highschool, Istanbul, Turkey and Alex from England. They all used very similar ideas and Michael produced a very well written and formatted word document explaining the ideas. The full solution is in PDF format at the following link: Interpolating Polynomials Soln1.pdf 

Thanks for the fantastic ideas you sent in!