NMO Selection Tests for the Balkan and International Mathematical Olympiads

Each side of one rectangle meets the contour of the other rectangle at exactly two points situated on consecutive sides. Let $A_0{A}_1{A}_2{A}_3$ and $B_0{B}_1{B}_2{B}_3$ be circular labellings of the two rectangles such that the segments $A_i{A}_{i+1}$ and $B_i{B}_{i+1}$ have equal lengths and meet at a point labelled $C_i$. The $C_i$ are four alternative vertices of the octagon, so the area of the latter is greater than or equal to the area of the quadrangle $C_0{C}_1{C}_2{C}_3$. Notice that $C_i$ is equally distanced from the lines $A_{i+2}{A}_{i+3}$ and $B_{i+2}{B}_{i+3}$ to deduce that the lines $C_0{C}_2$ and $C_1{C}_3$ are perpendicular to one another. Consequently, $$\text{area }{C}_0{C}_1{C}_2{C}_3=\frac{1}{2}\cdot C_0{C}_2\cdot C_1{C}_3\ge \frac{1}{2}\cdot A_1{A}_2\cdot A_0{A}_1=\frac{1}{2}$$