Quadratic functions and graphs

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

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?

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.

An inequality involving integrals of squares of functions.

Take a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed?