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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

Consider a rectangle with coordinates $(0,0), (3,0), (3,2), (0,2)$ and count the number of lattice points (points with whole number coordinates, coloured yellow in the diagram) on the perimeter and inside the rectangle.

Define $k$-points as points with coordinates $({a\over k}, {b\over k})$ where $a, b$ and $k$ are integers. For example, for the rectangle in the diagram the yellow points are the lattice points, $k=1$, and the 2-points are the red and yellow points taken together.

It is known that for any polygon in the plane which has vertices at lattice points the number of $k$-points in the interior of the polygon is $Ak^2 - Bk + C$ and the number of $k$-points in the closed polygon, including the perimeter and the interior, is $Ak^2 + Bk + C$. Verify that these quadratic formulae hold for the given rectangle and find $A$, $B$ and $C$.

Suggest a connection between the coefficients $A$, $B$ and $C$ and the area of the rectangle and the number of $k$-points on the perimeter.

Assume that for any plane polygon there is a quadratic formula for the number of $k$-points inside the polygon given by $Ak^2 -Bk + C$. Explain why, for large $k$, the area of the polygon is given by $$\lim_{k\to \infty} \frac {{\rm number of interior} k-{\rm points}}{k^2}= A.$$