Question

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:

$42$

$46$

$122$

$290$

$326$

Answer: A

Concepts:ages sum and difference of cubes Diophantine Equation greatest common divisor

Difficulty rating: 2270

Small Hint:

Let the ages be $D,$ $T,$ and $H,$ and rewrite the linear equation as $2(H-D)=D+T$

Big Hint:

Factor both differences of cubes after rewriting the cubic equation

Solution:

The equations are $\begin{aligned} 3D+T&=2H,\\ 2H^3&=3D^3+T^3. \end{aligned}$ Rewrite them as $2(H-D)=D+T$ and $\begin{gathered} 2(H-D)(H^2+HD+D^2)\\ =(D+T)\\ \qquad\cdot(D^2-DT+T^2). \end{gathered}$ Canceling the equal positive factors gives $\begin{aligned} H^2+HD+DT-T^2&=0,\\ (H+T)(H+D-T)&=0, \end{aligned}$ so $T=H+D.$ The linear equation then gives $H=4D,$ and pairwise relative primality forces $(D,H,T)=(1,4,5).$ The requested sum is $1^2+4^2+5^2=42.$

Therefore, the correct answer is A.

1972 AMC 12 Problem 34

Answer: A

Difficulty rating: 2270

Small Hint:

Big Hint:

Solution:

Problem 34 in Other Years