68. Let $AB, CD$ be two perpendicular chords of a circle with center $O$ and radius $r$. The circle is divided into four parts by these chords, which are labeled in clockwise order as $X, Y, Z, W$. Find the maximum and minimum values of $\frac{A(X)+A(Z)}{A(Y)+A(W)}$, where $A(U)$ denotes the area of $U$. (1988 IMO Shortlist)
68. Let's assume the center of the circle falls in the $Z$ as shown in figure (1). Then, as the chord $AB$ moves parallel, the shaded area in figure (2) is greater than the area without shading on its left, so $A(X)+A(Z)$ increases, while $A(Y)+A(W)$ decreases (note that the sum of the areas of $X, Y, Z, W$ is a constant $\pi r^{2}$), thus the ratio $\frac{A(X)+A(Z)}{A(Y)+A(W)}$ increases. Therefore, when point $A$ coincides with point $C$, it is possible to reach the maximum value. In figure (3), the hypotenuse $BD$ of the right triangle $ABD$ is the diameter. Suppose $\triangle ABD$ has the maximum area when $OA$ is the height, at this time $A(Z)$ is the largest, and $A(X)+A(Z)$ is also the largest, with a value of $\frac{1}{2} \pi r^{2}+r^{2}$. While $A(Y)+A(W)$ is the smallest, with a value of $\frac{1}{2} \pi r^{2}-r^{2}$. Therefore, the maximum value of $\frac{A(X)+A(Z)}{A(Y)+A(W)}$ is $\frac{\frac{1}{2} \pi r^{2}+r^{2}}{\frac{1}{2} \pi r^{2}-r^{2}}=\frac{\pi+2}{\pi-2}$.