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Question

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?

Accepted Answer

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 = 2^459, (b,c) = (0,43) or (43,0). Then a + b + c = 2b + 2c + 16 = 102.