Find all pairs of positive integers satisfying is a perfect square

Find all integer pairs satisfy that:

Let be a polynomial with integer coefficients. Suppose there exist infinitely many integer pairs such that . Prove that the graph of is symmetric about a point (i.e., it has a center of symmetry).

Let be the set of the first perfect squares of nonzero integers. Suppose that for some . Here stands for the set . Prove that holds for every .

For positive integers , find all possible positive integer values of

Let be a convex hexagon, such that the triangles and are equilateral and the diagonals and are concurrent. Prove that or

A frog is located on a unit square of an infinite grid oriented according to the cardinal directions. The frog makes moves consisting of jumping either one or two squares in the direction it is facing, and then turning according to the following rules:
i) If the frog jumps one square, it then turns to the right;
ii) If the frog jumps two squares, it then turns to the left.

Is it possible for the frog to reach the square exactly squares north of the initial square after some finite number of moves if it is initially facing:
a) North;
b) East?

Find all functions such that
(Albania)

(i) ,
(ii)
(iii)