Difference between revisions of "2020 INMO Problems/Problem 4"
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Define,<math>S_1=a_1+a_1a_2+\cdots +a_1\cdots a_{n-1}+2</math>. | Define,<math>S_1=a_1+a_1a_2+\cdots +a_1\cdots a_{n-1}+2</math>. | ||
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<math>2^{n-1}< a_1\cdots a_n \le 2^{n}</math>. | <math>2^{n-1}< a_1\cdots a_n \le 2^{n}</math>. | ||
Revision as of 12:33, 5 November 2020
Problem
Let be an integer and let be real numbers such that . Prove that
Solution(1)
For , we want to show that where and . This is equivalent to showing that , which is true.
Suppose, now, that the given inequality is true for , where . Now, consider reals with sum . Then, and , so by induction hypothesis,
This means or as desired. ~biomathematics
Solution(2)
Define,. ,
In general ,.
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Using Tchevbycev inequality we have , .
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.[Applying Induction on successive ].
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.[using GM-AM]
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Since, and , Hence , .
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The RHS inequality is trivial by AM-GM inequality.
For LHS inequality I would like to use induction.
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We have , and .
. Suppose , the statement is true for such that , and .
Now , consider .
Suppose , is median of the sequence, and and .
and .
Our induction step is complete.
This two claim leads and equality for .