2008 iTest Problems/Problem 90
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Problem
For positive reals, let . Find the minimum value of .
Solution 1
By the Trivial Inequality (with equality happening if ), Add to both sides and use the reciprocal property to get Since , multiplying both sides by this value would not change the inequality sign, and doing so results in By using similar steps, we find that
Let , , and , making . Note that . By the Cauchy-Schwarz Inequality, , so . Equality happens if , which is possible if . If , then .
Therefore, the minimum value of (which happens if ) is , so the minimum value of is .
Solution 2 (Titu Spam)
Note that By Titu, Thus, reciprocating and Titu gives Reciprocating once again gives with clearly obtained at therefore the minimum is
See Also
2008 iTest (Problems) | ||
Preceded by: Problem 89 |
Followed by: Problem 91 | |
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