1972 AHSME Problems/Problem 34

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Problem 34

Three times Dick's age plus Tom's age equals twice Harry's age. Double the cube of Harry's age is equal to three times the cube of Dick's age added to the cube of Tom's age. Their respective ages are relatively prime to each other. The sum of the squares of their ages is

$\textbf{(A) }42\qquad \textbf{(B) }46\qquad \textbf{(C) }122\qquad \textbf{(D) }290\qquad  \textbf{(E) }326$


\[t=2h-3d\] \[3d^3+t^3=2h^3\]

First, substitute in t into the second equation and get $3d^3+8h^3-36h^2d+54hd^2-27d^3=2h^3$. That turns into $h^3-6h^2d+9hd^2-4d^3=0$ which is factored into $(h-4d)(h-d)^2 =0.$ WLOG, $d=1$ and consequently $h=4$. Then $t=8-3=5$. Everything appears to be relatively prime already. The answer is thus $1+16+25=\boxed{\textbf{(A) }42}.$ ~lopkiloinm