be a positive integer. Claire writes 
distinct positive real numbers 
in a row on a blackboard. In a 
William can erase a number 
and replace it with either 
or 
at the same location. His goal is to perform a sequence of moves such that after he is done, the number are strictly increasing from left to right.
independent of 
such that William can always reach his goal in at most 
moves.
independent of 
moves.
a friendly number if there exists a positive rational number 
such that 
.
,
such that 
is on the same side of 
as 
and 
intersects 
at 
.
on 
such that 
is equilateral.
Y by
Let 
be a triangle with side 
and the circles 
which pass through the point 
, having the centers in 
respectively. A variable line through intersects the circles in 
and 
for the second time.
a) Prove that the perpendicular bisectors of the segments![$[MN]$](//latex.artofproblemsolving.com/1/6/2/162e03b9cf481dcfb3d5bdf078be84feab5d2f6e.png)
have one thing in common.
b) Determine the maximum length of the segment.
be a triangle with side 
and the circles 
which pass through the point 
, having the centers in 
respectively. A variable line through intersects the circles in 
and 
for the second time.a) Prove that the perpendicular bisectors of the segments
![$[MN]$](http://latex.artofproblemsolving.com/1/6/2/162e03b9cf481dcfb3d5bdf078be84feab5d2f6e.png)
have one thing in common.b) Determine the maximum length of the segment
.This post has been edited 2 times. Last edited by parmenides51, Dec 8, 2022, 9:13 PM
.
cuts the circle 
in the point 
.
cuts the circle 
in the point 
.
has one thing in common, goes through the fixed point 
.
,
.
