Difference between revisions of "2001 IMO Problems/Problem 6"
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Thus, it follows that <cmath>(KM+LN) \mid (KL+MN)(KN+LM).</cmath> | Thus, it follows that <cmath>(KM+LN) \mid (KL+MN)(KN+LM).</cmath> | ||
Now, since <math>KL+MN>KM+LN</math> if <math>KL+MN</math> is prime, then there are no common factors between the two. So, in order to have <cmath>(KM+LN)\mid (KL+MN)(KN+LM),</cmath> we would have to have <cmath>(KM+LN) \mid (KN+LM).</cmath> This is impossible as <math>KM+LN>KN+LM</math>. Thus, <math>KL+MN</math> must be composite. | Now, since <math>KL+MN>KM+LN</math> if <math>KL+MN</math> is prime, then there are no common factors between the two. So, in order to have <cmath>(KM+LN)\mid (KL+MN)(KN+LM),</cmath> we would have to have <cmath>(KM+LN) \mid (KN+LM).</cmath> This is impossible as <math>KM+LN>KN+LM</math>. Thus, <math>KL+MN</math> must be composite. | ||
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==See also== | ==See also== | ||
{{IMO box|num-b=5|num-a=6|year=2001}} | {{IMO box|num-b=5|num-a=6|year=2001}} | ||
[[Category: Olympiad Number Theory Problems]] | [[Category: Olympiad Number Theory Problems]] | ||
Revision as of 10:44, 10 February 2015
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,
.
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 | ||
