# 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 quantities are the same. This is what enabled us to say WLOG.

## Proof

There is not a proof

## Problems

### Introductory

A hexagon is inscribed in a circle with radius . What is the maximum area of the hexagon? (Solution)

### Intermediate

1. The fraction,

where and are side lengths of a triangle, lies in the interval , where and are rational numbers. Then, can be expressed as , where and are relatively prime positive integers. Find .

(Source: I made it. Solution here. See solution 2 for the solution using this method. See solution 1 for the rigorous solution.)

### Olympiad

Find the maximum value of . (Solution 1 or Solution 2)

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