1977 AHSME Problems/Problem 12

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Problem 12

Al's age is $16$ more than the sum of Bob's age and Carl's age, and the square of Al's age is $1632$ more than the square of the sum of Bob's age and Carl's age. What is the sum of the ages of Al, Bob, and Carl?

$\text{(A)}\ 64 \qquad  \text{(B)}\ 94 \qquad  \text{(C)}\ 96 \qquad  \text{(D)}\ 102 \qquad  \text{(E)}\ 140$


Solution

Solution by e_power_pi_times_i

Denote Al's age, Bob's age, and Carl's age by $a$, $b$, and $c$, respectively. Then, $a = 16 + b + c$ and $a^2 = 1632 + b^2 + c^2$. Substituting the first equation into the second, $(16 + b + c)^2 = b^2 + c^2 + 2bc + 32b + 32c + 256 = b^2 + c^2 + 1632$. Thus, $bc + 16b + 16c = 688$, and $(b+16)(c+16) = 944$. Since $944 = 2^4\cdot59$, $(b,c) = (0,43)$ or $(43,0)$. Then $a + b + c = 2b + 2c + 16 = \boxed{\textbf{(D)}\ 102}$.

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