Quote:
Originally Posted by Chronos 
I assumed his question was more complicated than that.
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That's because you assumed the functions involved are continuous! It's pretty trivial in that case
If however they are assumed only integrable (as they should - ps. I love Hacker's problems, coz they are almost never "well posed", and that's exactly what makes a problem interesting
) than we can just apply Cauchy-Schwartz:
Q(uite)E(asily)D(oneth).
For 2, this came to my mind. Consider a tangent to the circle. Rotating this by the touching point, for an angle s ε (0,π) we obtain a chord of the circle, of length say l(s)>0. This function is defined on (0,π), is continuous, and can be extended to a function continuous on [0,π] by the (obvious) l(0)=l(π)=0. So it must attain a maximum in (0,π). We now see it is increasing for s<π/2 and decreasing for s>π/2, so the maximum is attained for s=π/2 - this chord is exactly the diameter.
