CAPS 1995

Find all pairs of functions $f ,g : Z \rightarrow Z$ that satisfy $f (g(x)+y) = g( f (y)+x)$ for all integers $x,y$ and such that $g(x) = g(y)$ only if $x = y$ .

Consider all triangles $ABC$ in the cartesian plane whose vertices are at lattice points (i.e. with integer coordinates) and which contain exactly one lattice point (to be denoted $P$ ) in its interior. Let the line $AP$ meet $BC$ at $E$ . Determine the maximum possible value of the ratio $\frac{AP}{PE}$ .

For each real number $p > 1$ , find the minimum possible value of the sum $x+y$ , where the numbers $x$ and $y$ satisfy the equation $(x+\sqrt{1+x^2})(y+\sqrt{1+y^2}) = p$ .

The diagonals of a convex quadrilateral $ABCD$ are orthogonal and intersect at point $E$ . Prove that the reflections of $E$ in the sides of quadrilateral $ABCD$ lie on a circle.