1976 AHSME Problem 10

If m,~n,~p, and q are real numbers and f(x)=mx+n and g(x)=px+q, then the equation f(g(x))=g(f(x)) has a solution

\begin{array}{l} \textbf{(A) }\text{for all choices of }m,~n,~p, \text{ and } q\qquad\\ \textbf{(B) }\text{if and only if }m=p\text{ and }n=q\qquad\\ \textbf{(C) }\text{if and only if }mq-np=0\qquad\\ \textbf{(D) }\text{if and only if }n(1-p)-q(1-m)=0\qquad\\ \textbf{(E) }\text{if and only if }(1-n)(1-p)-(1-q)(1-m)=0 \end{array}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.