Putnam 1985 
Topic: Geometry Calculus

Problem: Let $T$ be an acute triangle. Inscribe a rectangle $R$ in $T$ with one side along a side of $T$. Then inscribe a rectangle $S$ in the triangle formed by the side of $R$ opposite the side on the boundary of $T$, and the other two sides of $T$, with one side along the side of $R$. For any polygon $X$, let $A(X)$ denote the area of $X$. Find the maximum value, or show that no maximum exists, of $\frac{A(R)+A(S)}{A(T)}$, where $T$ ranges over all triangles and $R,S$ over all rectangles as above.

Solution: Label lengths as in the figure. Then \[ \frac{A(R) + A(S)}{A(T)} = \frac{ay + bz}{hx/2}, \] where $h = a + b + c$, the altitude of $T$. By similar triangles, \[ \frac{x}{h} = \frac{y}{b+c} = \frac{z}{c}, \] so \[ \frac{A(R) + A(S)}{A(T)} = \frac{(b+c)x}{h} + \frac{cx}{h} = \frac{2}{h^2}(ab + ac + bc). \] We need to maximize $ab + ac + bc$ subject to $a + b + c = h$. One way to do this is to first fix $a$, so $b + c = h - a$. Then \[ ab + ac + bc = a(h - a) + bc, \] and $bc$ is maximized when $b = c$. We now wish to maximize $2ab + b^2$ subject to $a + 2b = h$. This is a straightforward calculus problem giving $a = b = c = h/3$. Hence the maximum ratio is $\boxed{2/3}$ (independent of $T$).