These paragraphs are from Steve Olson's book Count Down: Six Kids Vie for Glory at the World's Toughest Math Competition.
On page 170, the author said:
The sixth and last problem on the Forty-second Olympiad —
by tradition the hardest of all — looked deceptively straightforward to the competitors.Let $a > b > c> d$ be positive integers and suppose that $ac +bd =(b + d + a -c)(b + d -a + c)$. Prove that $ab + cd$ is not
prime.
On page 174:
Gabriel's answer to problem six demonstrated his power as a
mathematician. "Gabe's solution was overkill," says Stankova,
"but he solved the problem the way a mathematician would solve it." In his solution he used a mathematical idea called a ring — a set of mathematical objects, any two of which can be added or multiplied to yield another member of the set.
Edit
On page 210:
Gabriel's use of imaginary numbers in problem six was directly linked to the famous equation $e^{\pi i} = -1$. The number omega ($\omega$)
is defined as $\omega = e^{2\pi i/3}$. So $\omega^2 = e^{2\pi i/3}\times e^{2\pi i/3}=e^{2\pi i/3+2\pi i/3}=e^{4\pi i/3}$ (because the exponents of $e$ can be added together when the two numbers are multiplied). By the same token, $\omega^3 =e^{6\pi i/3}=e^{2\pi i}=e^{\pi i}\times e^{\pi i}= -1 \times -1 = 1$.
Thus the set of numbers $1, -1, \omega, -\omega, \omega^2$, and $-w^2$ are related in a particular way. If you multiply any two of them together, you get another member of the set. Gabriel used the powerful properties of this group to crack problem six.
The original problem: https://www.imo-official.org/problems.aspx (2001)
My Question: How did he solve the problem in a ring theoretical method?
Best Answer
Gabriel's original solution. Posted with the permission of the author.