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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.

Pick's Quadratics

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

To prove Pick's Theorem does not require any advanced mathematics, just careful reasoning. The Proof of Pick's Theorem is the challenge in the next problem which leads you step by step through the proof. This problem is about a generalistaion of Pick's Theorem.

Pick's Theorem can be generalised as follows:
'For any planar polygon with vertices at lattice points the quadratic formula $i(k)=Ak^2 - Bk +C$ gives the number of $k$-points inside the polygon and the quadratic formula $g(k)= Ak^2 + Bk +C$ gives the number of $k$-points in the closed polygon (including the boundary and the interior points), where $A$ is the area of the polygon.'

This challenge asks you to verify this generalised form of Pick's Theorem for a particular rectangle.

The proof that the given quadratic formulae hold for all polygons is difficult and requires mathematics beyond school level. However it is worth noting that this is the form of Pick's Theorem that generalises to 3 and higher dimensions.