and 
.![\[\sum_{i=1}^n (a_i-a_{i+1})^3\leq \frac{1}{2\sqrt{3}} (\sum_{i=1}^n a_i )^3,\]](http://latex.artofproblemsolving.com/8/1/0/810f07e3394370ce06aec1e7025bcf5cc5433d60.png)
where 
.
and walks randomly. The probability of any point reaching all adjacent points is equal. Find the probability of the ant reaching 
without passing through 
with 
prime and ![\[a^p+b^p=p!.\]](http://latex.artofproblemsolving.com/f/e/b/feb27894b1f6497559eeb094cfe92256bc26688e.png)
be the incenter of triangle 
.
,
,
be the orthocenters of triangles 
,
,
,
,
,
,
Y by
Let 
and 
be monic quadratic trinomials, and let 
and 
be the vertices of the parabolas 
and 
, respectively. Let 
denote the minimum value of the function 
. It is known that the differences 
and 
are equal positive numbers. Find the angle between the line 
and the 
-axis.
and 
be monic quadratic trinomials, and let 
and 
be the vertices of the parabolas 
and 
, respectively. Let 
denote the minimum value of the function 
. It is known that the differences 
and 
are equal positive numbers. Find the angle between the line 
and the 
-axis.This post has been edited 1 time. Last edited by egxa, Apr 18, 2025, 5:25 PM
