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Difference between revisions of "2001 IMO Problems/Problem 6"

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==Solution==
 
==Solution==
 
{{solution}}
 
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First, <math>(KL+MN)-(KM+LN)=(K-N)(L-M)>0</math> as <math>K>N</math> and <math>L>M</math>.  Thus, <math>KL+MN>KM+LN</math>.   
 
First, <math>(KL+MN)-(KM+LN)=(K-N)(L-M)>0</math> as <math>K>N</math> and <math>L>M</math>.  Thus, <math>KL+MN>KM+LN</math>.   
  
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Now, we have:
 
Now, we have:
−
<cmath>(K+L-M+N)(-K+L+M+N)&=&KM+LN</cmath>
+
<cmath>(K+L-M+N)(-K+L+M+N)=KM+LN</cmath>
−
<cmath>-K^2+KM+L^2+LN+KM-M^2+LN+N^2&=&KM+LN</cmath>
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<cmath>-K^2+KM+L^2+LN+KM-M^2+LN+N^2=KM+LN</cmath>
 
<cmath>L^2+LN+N^2&=&K^2-KM+M^2</cmath>
 
<cmath>L^2+LN+N^2&=&K^2-KM+M^2</cmath>
 
So, we have:
 
So, we have:

Revision as of 19:08, 4 December 2015

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

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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: \[(K+L-M+N)(-K+L+M+N)=KM+LN\] \[-K^2+KM+L^2+LN+KM-M^2+LN+N^2=KM+LN\] 

\[L^2+LN+N^2&=&K^2-KM+M^2\] (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 \[(KM+LN) \mid (KL+MN)(KN+LM).\] 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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