ISL 1992

Prove that for any positive integer $m$ there exist an infinite number of pairs of integers $(x, y)$ such that

(i) $x$ and $y$ are relatively prime; (ii) $y$ divides $x^2 + m$ ; (iii) $x$ divides $y^2 + m.$ (iv) $x + y \leq m + 1-$ (optional condition)

Let $\mathbb{R}^+$ be the set of all non-negative real numbers. Given two positive real numbers $a$ and $b,$ suppose that a mapping $f: \mathbb{R}^+ \mapsto \mathbb{R}^+$ satisfies the functional equation:

$$f(f(x)) + af(x) = b(a + b)x.$$

Prove that there exists a unique solution of this equation.

The diagonals of a quadrilateral $ABCD$ are perpendicular: $AC \perp BD.$ Four squares, $ABEF,BCGH,CDIJ,DAKL,$ are erected externally on its sides. The intersection points of the pairs of straight lines $CL, DF, AH, BJ$ are denoted by $P_1,Q_1,R_1, S_1,$ respectively (left figure), and the intersection points of the pairs of straight lines $AI, BK, CE DG$ are denoted by $P_2,Q_2,R_2, S_2,$ respectively (right figure). Prove that $P_1Q_1R_1S_1 \cong P_2Q_2R_2S_2$ where $P_1,Q_1,R_1, S_1$ and $P_2,Q_2,R_2, S_2$ are the two quadrilaterals.

Alternative formulation: Outside a convex quadrilateral $ABCD$ with perpendicular diagonals, four squares $AEFB, BGHC, CIJD, DKLA,$ are constructed (vertices are given in counterclockwise order). Prove that the quadrilaterals $Q_1$ and $Q_2$ formed by the lines $AG, BI, CK, DE$ and $AJ, BL, CF, DH,$ respectively, are congruent.

Attachments:

A convex quadrilateral has equal diagonals. An equilateral triangle is constructed on the outside of each side of the quadrilateral. The centers of the triangles on opposite sides are joined. Show that the two joining lines are perpendicular.

Alternative formulation. Given a convex quadrilateral $ABCD$ with congruent diagonals $AC = BD.$ Four regular triangles are errected externally on its sides. Prove that the segments joining the centroids of the triangles on the opposite sides are perpendicular to each other.

Original formulation: Let $ABCD$ be a convex quadrilateral such that $AC = BD.$ Equilateral triangles are constructed on the sides of the quadrilateral. Let $O_1,O_2,O_3,O_4$ be the centers of the triangles constructed on $AB,BC,CD,DA$ respectively. Show that $O_1O_3$ is perpendicular to $O_2O_4.$