[/quote]
,
![\[
2^a + 3^b + 1 = 6^c
\]](http://latex.artofproblemsolving.com/1/9/e/19eb5efa2ccdf54cd277339f8d3b99f75168779d.png)
are such that 
,
of prime positive integers 
such that number 
equals to a square of an integer.
satisfy all conditions below:
for all 
for all 
for all 
be real number greater than 
.![\[S=\log_a {bc}+\log_b {ca}+\log_c {ab}\]](http://latex.artofproblemsolving.com/5/9/e/59e1a33017116963165f744b49b8515f126c64ef.png)
for which:
for all 
.
such that 
,
,
are all equal to
,
be real numbers, such that 
.
.
,
(where 
are the digits and 
)
For example the permutation sum of 20253 is 
Let 
and 
be two fivedigit integers with the same permutation sum.
.
be the incircle of triangle 
.
is parallel to side 
and tangent to 
is parallel to side 
and tangent to 
.![\[a\sqrt{1+b^{3}}+b\sqrt{1+c^{3}}+c\sqrt{1+a^{3}}\le 5.\]](http://latex.artofproblemsolving.com/f/f/a/ffaeec844a6bdf10a96e8e9d298ef97ac839d713.png)
Y by
Let 
. Two subsets 
and 
of 
are said to be friends if the following conditions are true:
that are friends such that each one of them contains at least 
elements.[/quote]
. Two subsets 
and 
of 
are said to be friends if the following conditions are true:- They do not share any elements.
- They both have the same number of elements.
- The product of all elements from equals the product of all elements from .
that are friends such that each one of them contains at least 
elements.[/quote]This post has been edited 1 time. Last edited by Titibuuu, Mar 17, 2025, 8:58 AM
