Filter Problems

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The function f $(x)$ on the interval [a, b] is equal to the maximum of several functions of the form $y = C \times 10^{- | x-d |}$ $($where d and C are different, and all C are positive$)$. It is given that f $(a)$ = f $(b)$. Prove that the sum of the lengths of the sections on which the function increases is equal to the sum of the lengths of the sections on which the function decreases.

Prove that for every natural number n $>$ 1 the equality: [$n^{1 / 2}] + [n^{1/ 3}] + … + [n^{1 / n}] = [log_{2}n] + [log_{3}n] + … + [log_{n}n]$ is satisfied.

Are there any irrational numbers a and b such that the degree of $a^b$ is a rational number?

The function f is such that for any positive x and y the equality f $($xy$)$ = f $($x$)$ + f $($y$)$ holds. Find f $($2007$)$ if f $($1/2007$)$ = 1.