You may also like

problem icon

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?

problem icon

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?

problem icon

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

Age 16 to 18 Challenge Level:

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 $$\lim_{k\to \infty} \frac {{\rm number of interior} k-{\rm points}}{k^2}= A.$$