Difference between revisions of "2001 IMO Problems/Problem 6"
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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>. | |
Similarly, <math>(KM+LN)-(KN+LM)=(K-L)(M-N)>0</math> since <math>K>L</math> and <math>M>N</math>. Thus, <math>KM+LN>KN+LM</math>. | Similarly, <math>(KM+LN)-(KN+LM)=(K-L)(M-N)>0</math> since <math>K>L</math> and <math>M>N</math>. Thus, <math>KM+LN>KN+LM</math>. | ||
Revision as of 19:07, 4 December 2015
Problem 6
are positive integers such that 
. Prove that 
is not prime.
Solution
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First, 
as 
and 
. Thus, 
.
Similarly, 
since 
and 
. Thus, 
.
Putting the two together, we have
![\[KL+MN>KM+LN>KN+LM\]](http://latex.artofproblemsolving.com/b/7/3/b73cbb5d01685618dbc9cafdd1d62190452bd70d.png)
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 ![\[(KM+LN) \mid (KL+MN)(KN+LM).\]](http://latex.artofproblemsolving.com/6/7/d/67d2cb1de4dd0212d569c188b7632220ac4bbe9f.png)
Now, since if 
is prime, then there are no common factors between the two. So, in order to have ![\[(KM+LN)\mid (KL+MN)(KN+LM),\]](http://latex.artofproblemsolving.com/0/8/7/0879e3460c5c35fa23b99a9102d3327de7f2cc6e.png)
we would have to have ![\[(KM+LN) \mid (KN+LM).\]](http://latex.artofproblemsolving.com/5/2/0/52003afc5d5c0082dbcfb6f39a1d3007d8fb758c.png)
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 | ||
