2001 IMO Problems/Problem 6

Problem 6

$K > L > M > N$ are positive integers such that $KM + LN = (K + L - M + N)(-K + L + M + N)$. Prove that $KL + MN$ is not prime.

Solution

First, $(KL+MN)-(KM+LN)=(K-N)(L-M)>0$ as $K>N$ and $L>M$. Thus, $KL+MN>KM+LN$.

Similarly, $(KM+LN)-(KN+LM)=(K-L)(M-N)>0$ since $K>L$ and $M>N$. Thus, $KM+LN>KN+LM$.

Putting the two together, we have \[KL+MN>KM+LN>KN+LM\]

Now, we have: \[(K+L-M+N)(-K+L+M+N)=KM+LN\] \[-K^2+KM+L^2+LN+KM-M^2+LN+N^2=KM+LN\] \[L^2+LN+N^2=K^2-KM+M^2\] So, we have: \[(KM+LN)(L^2+LN+N^2)=KM(L^2+LN+N^2)+LN(L^2+LN+N^2)\] \[=KM(L^2+LN+N^2)+LN(K^2-KM+M^2)\] \[=KML^2+KMN^2+K^2LN+LM^2N\] \[=(KL+MN)(KN+LM)\] Thus, it follows that \[(KM+LN) \mid (KL+MN)(KN+LM).\] Now, since $KL+MN>KM+LN$ if $KL+MN$ is prime, then there are no common factors between the two. So, in order to have \[(KM+LN)\mid (KL+MN)(KN+LM),\] we would have to have \[(KM+LN) \mid (KN+LM).\] This is impossible as $KM+LN>KN+LM$. Thus, $KL+MN$ 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
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