1973 AMC 12 Problem 35

Use symmetry across the vertical diameter to see that the upper semicircle is divided into five equal chords

Write $s=2\sin18^\circ$ and $d=2\sin54^\circ,$ then relate each to the square of the other

Let $K$ be the left endpoint of the horizontal diameter. Reflection across the vertical diameter shows that $KM=NR=s$ and $QN=MP=s.$ Together with the given equal chords, the upper semicircle is split into five equal arcs. Each subtends $36^\circ$ at $O.$ Thus $$\begin{aligned} s&=2\sin18^\circ,\\ d&=2\sin54^\circ\\ &=2\cos36^\circ. \end{aligned}$$

Using the double-angle identities, $$\begin{aligned} d&=2(1-2\sin^218^\circ)\\ &=2-s^2,\\ s&=2\cos72^\circ\\ &=4\cos^236^\circ-2\\ &=d^2-2. \end{aligned}$$ Adding these equations gives $$d+s=d^2-s^2=(d-s)(d+s).$$ Since $d+s\gt0,$ it follows that $d-s=1.$ Substituting $d=s+1$ into $d=2-s^2$ yields $$s^2+s=1, \qquad s=\frac{\sqrt5-1}{2}.$$ Therefore $$ds=s(s+1)=1$$ and $$\begin{aligned} d^2-s^2 &=(d-s)(d+s)\\ &=2s+1\\ &=\sqrt5. \end{aligned}$$ All three equations are necessarily true.

Therefore, the correct answer is E.