Difference between revisions of "Principle of Insufficient Reason"
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===Introductory=== | ===Introductory=== | ||
===Intermediate=== | ===Intermediate=== | ||
| + | 1. The fraction, | ||
| + | |||
| + | <math>\frac{ab+bc+ac}{(a+b+c)^2}</math> | ||
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| + | where <math>a,b</math> and <math>c</math> are side lengths of a triangle, lies in the interval <math>(p,q]</math>, where <math>p</math> and <math>q</math> are rational numbers. Then, <math>p+q</math> can be expressed as <math>\frac{r}{s}</math>, where <math>r</math> and <math>s</math> are relatively prime positive integers. Find <math>r+s</math>. | ||
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===Olympaid=== | ===Olympaid=== | ||
Revision as of 16:14, 8 January 2024
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
Problems
Introductory
Intermediate
1. The fraction,
where 
and 
are side lengths of a triangle, lies in the interval ![$(p,q]$](http://latex.artofproblemsolving.com/c/b/b/cbb6084727fb3e1c6027f7742cc15d4802ad8681.png)
, where 
and 
are rational numbers. Then, 
can be expressed as 
, where 
and 
are relatively prime positive integers. Find 
.
Olympaid
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