Theorems for maximum sums of reciprocals and inscribed polygon areas

[recon]

I need to know the names of theorems related to the following two problems:

1. What is the maximum sum less than 1 but more than 0 that can be formed from $$\frac{1}{p} + \frac{1}{q} + \frac{1}{r}$$, where p, q and r are positive integers?

2. What is the maximum perimeter and area of an inscribed quadrilateral and triangle in a circle with a fixed radius r?

 

[recon]

I realize that there may be no established theorems for the above 2 problems, so can anyone please suggest how I can go about solving them?

 

[recon]

OK, I'm going to shamelessly bump this. *BUMP*

Can someone tell me if it is even possible to solve the second problem? The first one has been solved already.

 

[VietDao29]

Hi,
I am trying to find the maximum perimeter an inscribed quadrilateral, but I haven't succeeded yet.
About the maximum area an inscribed quadrilateral, I suggest you using:
$$S_{ABC} = \frac{1}{2} \times AB \times AC \times \sin{BAC}$$
Call A, B, C, D the points on the circle.
Try to figure out the $$S_{AOB}, S_{BOC}, S_{COD}, S_{DOA}$$ using the above function.
OA = OB = OC = OD = R
And $$\sin{90} = 1 \mbox{is max}$$
So an inscribed quadrilateral has the max erea is the inscribed square.
That's it.
Hope it help,
PS: Can you give me the answer for number 1?
Is it
$$\frac{41}{42}$$?
Thanks,
Viet Dao,

 

[recon]

PS: Can you give me the answer for number 1?
Is it
$$\frac{41}{42}$$?
Thanks,
Viet Dao,

Thanks for the help. And yes, that is the answer to question 2, not 1.