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
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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. ! Misplaced alignment tab character &.)
\[-K^2+KM+L^2+LN+KM-M^2+LN+N^2&=&KM+LN\] (Error compiling LaTeX. ! Misplaced alignment tab character &.)
\[L^2+LN+N^2&=&K^2-KM+M^2\] (Error compiling LaTeX. ! Misplaced alignment tab character &.)
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. ! Misplaced alignment tab character &.)
\[&=&KM(L^2+LN+N^2)+LN(K^2-KM+M^2)\] (Error compiling LaTeX. ! Misplaced alignment tab character &.)
\[&=&KML^2+KMN^2+K^2LN+LM^2N\] (Error compiling LaTeX. ! Misplaced alignment tab character &.)
\[&=&(KL+MN)(KN+LM)\] (Error compiling LaTeX. ! Misplaced alignment tab character &.)
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 |