Quadratic functions and graphs

Find a condition which determines whether the hyperbola y^2 - x^2 = k contains any points with integer coordinates.

The illustration shows the graphs of twelve functions. Three of them have equations y=x^2, x=y^2 and x=-y^2+2. Find the equations of all the other graphs.

Here is a pattern composed of the graphs of 14 parabolas. Can you find their equations?

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?

Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

Substitute -1, -2 or -3, into an algebraic expression and you'll get three results. Is it possible to tell in advance which of those three will be the largest ?

An inequality involving integrals of squares of functions.