and 
.
.
be a closed map with the property that the inverse image of each point in 
is a compact subset of 
.
is compact whenever 
is compact in 
,
.
such that 
is uniformly continuous on 
.
be a positive integer.Prove that there is a polynomial 
with integer coefficients so that 
,![$$a^{2n+1}+b^{2n+1}+c^{2n+1}=abc[P(a,b)+P(b,c)+P(a,c)]$$](http://latex.artofproblemsolving.com/5/2/0/520813bebb2f297928208f7b7b2eea76d24043ac.png)
.
.
be an equilateral triangle of side length 
.
is chosen on side 
such that the length of 
is 
.
and 
are chosen on sides 
and 
.
.
to their corresponding opposite vertices. The intersections of these line segments form a triangle, labeled 
.
is an equilateral triangle with side length 
.
.
Y by
A mouse initially occupies cage A and is trained to change cages by going through a tunnel whenever an alarm sounds. Each time the alarm sounds, the mouse chooses any of the tunnels adjacent to its cage with equal probability and without being affected by previous choices. What is the probability that after the alarm sounds 23 times the mouse occupies cage B?
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This post has been edited 1 time. Last edited by Qebehsenuef, Apr 30, 2025, 4:46 AM
