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A problem was asked at Putnam Competition in 2003 (Problem 3), about finding the minimum Value of $|\sin x+ \cos x + \tan x + \cot x +\sec x +\csc x|$ where $x$ is Real.

the question paper and solutions.

I was thinking if there was any other simpler way to solve this problem. What strategy one should follow to determine the average value of above function?

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    Proceeding from @J.M.'s comments. As $x \to 0^+$, the function is lower bounded by $\csc x$, which in turn is at least $\frac{1}{x}$. Since the integral $\int_{0}^{a} \frac{1}{x}$ diverges for any a > 0, the average value of this function, over say $[0,2\pi]$, is also infinite.2011-09-12

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Please have a look at this: http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2325342&sid=03b52017fa0ef5480a4573048ae117c1#p2325342

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    That is definitely a good read (pretty much related to 'solutions given on AMC.MAA.ORG site'), but what about the **average value** of $|\sin x+ \cos x + \tan x + \cot x +\sec x +\csc x|$ ? I have not even an Idea to how to start working on this!2011-09-12