Difference between revisions of "2001 IMO Problems/Problem 6"
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− | 6 | + | == Problem 6 == |
+ | <math>K > L > M > N</math> are positive integers such that <math>KM + LN = (K + L - M + N)(-K + L + M + N)</math>. Prove that <math>KL + MN</math> is not prime. | ||
+ | |||
+ | ==Solution== | ||
+ | |||
+ | |||
+ | 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>. | ||
+ | |||
+ | Putting the two together, we have | ||
+ | <cmath>KL+MN>KM+LN>KN+LM</cmath> | ||
+ | |||
+ | Now, we have: | ||
+ | <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> | ||
+ | <cmath>L^2+LN+N^2=K^2-KM+M^2</cmath> | ||
+ | So, we have: | ||
+ | <cmath>(KM+LN)(L^2+LN+N^2)=KM(L^2+LN+N^2)+LN(L^2+LN+N^2)</cmath> | ||
+ | <cmath>=KM(L^2+LN+N^2)+LN(K^2-KM+M^2)</cmath> | ||
+ | <cmath>=KML^2+KMN^2+K^2LN+LM^2N</cmath> | ||
+ | <cmath>=(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. | ||
+ | |||
+ | ==See also== | ||
+ | {{IMO box|num-b=5|num-a=6|year=2001}} | ||
+ | |||
+ | [[Category: Olympiad Number Theory Problems]] |
Latest revision as of 15:36, 20 December 2018
Problem 6
are positive integers such that . Prove that is not prime.
Solution
First, as and . Thus, .
Similarly, since and . Thus, .
Putting the two together, we have
Now, we have: So, we have: 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 |