Principle of Insufficient Reason

Statement

In the absence of any relevant evidence, agents should distribute their credence equally among all the possible outcomes under consideration. I.e., If there is not enough reason for two quantities to be different (i.e. they're indistinguishable), the extrema occurs when the quantities are the same. So basically, a symmetric expression is maximized or minimized when the $2$ quantities are the same. This is what enabled us to say WLOG.

Proof

Problems

Introductory

A hexagon is inscribed in a circle with radius $1$ . What is the maximum area of the hexagon? (Solution to Principle of Insufficient Reason Introductory Problem 1

Intermediate

1. The fraction,

$\frac{ab+bc+ac}{(a+b+c)^2}$

where $a,b$ and $c$ are side lengths of a triangle, lies in the interval $(p,q]$ , where $p$ and $q$ are rational numbers. Then, $p+q$ can be expressed as $\frac{r}{s}$ , where $r$ and $s$ are relatively prime positive integers. Find $r+s$ .