2001 IMO Problems/Problem 6

Problem 6

are positive integers such that . Prove that is not prime.

Solution

This problem needs a solution. If you have a solution for it, please help us out by adding it. First, as and . Thus, .

Similarly, since and . Thus, .

Putting the two together, we have

Now, we have:

\[(K+L-M+N)(-K+L+M+N)&=&KM+LN\] (Error compiling LaTeX. Unknown error_msg)

\[-K^2+KM+L^2+LN+KM-M^2+LN+N^2&=&KM+LN\] (Error compiling LaTeX. Unknown error_msg)

\[L^2+LN+N^2&=&K^2-KM+M^2\] (Error compiling LaTeX. Unknown error_msg)

So, we have:

\[(KM+LN)(L^2+LN+N^2)&=&KM(L^2+LN+N^2)+LN(L^2+LN+N^2)\] (Error compiling LaTeX. Unknown error_msg)

\[&=&KM(L^2+LN+N^2)+LN(K^2-KM+M^2)\] (Error compiling LaTeX. Unknown error_msg)

\[&=&KML^2+KMN^2+K^2LN+LM^2N\] (Error compiling LaTeX. Unknown error_msg)

\[&=&(KL+MN)(KN+LM)\] (Error compiling LaTeX. Unknown error_msg)

Thus, it follows that Now, since if is prime, then there are no common factors between the two. So, in order to have we would have to have This is impossible as . Thus, must be composite.

See also

2001 IMO (Problems) • Resources
Preceded by Problem 5	1 • 2 • 3 • 4 • 5 • 6	Followed by Problem 6
All IMO Problems and Solutions