1960 AHSME Problem 15

Triangle I is equilateral with side A, perimeter P, area K, and circumradius R (radius of the circumscribed circle). Triangle II is equilateral with side a, perimeter p, area k, and circumradius r. If A is different from a, then:

\begin{array}{l} \textbf{(A)}\ P:p = R:r \text{ } \text{only sometimes} \qquad \textbf{(B)}\ P:p = R:r \text{ } \text{always}\qquad \\ \textbf{(C)}\ P:p = K:k \text{ } \text{only sometimes} \qquad \textbf{(D)}\ R:r = K:k \text{ } \text{always}\qquad \textbf{(E)}\ R:r = K:k \text{ } \text{only sometimes} \end{array}

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