Difference between revisions of "2001 IMO Shortlist Problems/A2"
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− | We proceed with a proof by contradiction. Suppose the statement were false. Then, there exists a sequence <math>a_0, a_1, \ldots</math> of positive integers for which there are only finitely many <math>a_n</math> with <math>1+a_n> a_{n-1}\sqrt[n]{2}</math>. Let the largest such <math>n</math> be <math>N-1</math>, so that <math>1+a_n\le a_{n-1}\sqrt[n]{2}</math> whenever <math>n\le N</math>. Then, it is clear that <math>1+a_{n+N}\le a_{n+N-1}\sqrt[n]{2}</math> for all nonnegative <math>n</math>. Therefore, define <math>b_n=a_{n+N}</math>. If there does not exist a sequence <math>b_0, b_1, \ldots</math> of positive integers for which <math>1+b_n\le b_{n-1}\sqrt[n]{2}</math>, it is clear that there will not exist any sequence <math>a_0, a_1, \ldots</math> for which that property is eventually true. | + | We proceed with a proof by contradiction. Suppose the statement were false. Then, there exists a sequence <math>a_0, a_1, \ldots</math> of positive integers for which there are only finitely many <math>a_n</math> with <math>1+a_n> a_{n-1}\sqrt[n]{2}</math>. Let the largest such <math>n</math> be <math>N-1</math>, so that <math>1+a_n\le a_{n-1}\sqrt[n]{2}</math> whenever <math>n\le N-1</math>. Then, it is clear that <math>1+a_{n+N}\le a_{n+N-1}\sqrt[n]{2}</math> for all nonnegative <math>n</math>. Therefore, define <math>b_n=a_{n+N}</math>. If there does not exist a sequence <math>b_0, b_1, \ldots</math> of positive integers for which <math>1+b_n\le b_{n-1}\sqrt[n]{2}</math>, it is clear that there will not exist any sequence <math>a_0, a_1, \ldots</math> for which that property is eventually true. |
Thus, I claim there does not exist a sequence of positive integers <math>b_0, b_1, \ldots</math> for which <math>1+b_n\le b_{n-1}\sqrt[n]{2}</math>. Again, suppose there does exist such a sequence. Then, define <math>x_0=b_0</math> and <math>x_n=x_{n-1}\sqrt[n]{2} -1</math>. It is clear that <math>x_n\ge b_n</math> for all <math>n</math>. I claim that this sequence will always become eventually negative. Note that | Thus, I claim there does not exist a sequence of positive integers <math>b_0, b_1, \ldots</math> for which <math>1+b_n\le b_{n-1}\sqrt[n]{2}</math>. Again, suppose there does exist such a sequence. Then, define <math>x_0=b_0</math> and <math>x_n=x_{n-1}\sqrt[n]{2} -1</math>. It is clear that <math>x_n\ge b_n</math> for all <math>n</math>. I claim that this sequence will always become eventually negative. Note that |
Latest revision as of 03:38, 17 June 2019
Problem
Let be an arbitrary infinite sequence of positive numbers. Show that the inequality holds for infinitely many positive integers .
Solution
We proceed with a proof by contradiction. Suppose the statement were false. Then, there exists a sequence of positive integers for which there are only finitely many with . Let the largest such be , so that whenever . Then, it is clear that for all nonnegative . Therefore, define . If there does not exist a sequence of positive integers for which , it is clear that there will not exist any sequence for which that property is eventually true.
Thus, I claim there does not exist a sequence of positive integers for which . Again, suppose there does exist such a sequence. Then, define and . It is clear that for all . I claim that this sequence will always become eventually negative. Note that , which becomes negative if and only if does. In other words, becomes zero if is unbounded. However, is eventually less than , so this sum is indeed unbounded and the proof is complete.