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 
as 
.
intersects 
at 
on 
such that 
is equilateral.
Y by Adventure10, Mango247
Given a circle 
and a point 
on its exterior, two tangents to the circle are drawn through , with 
and 
being the points of tangency. We take a point 
on the minor arc 
of . Let 
be the intersection of 
with the line perpendicular to that goes through , and let 
be the intersection of 
with the line perpendicular to that goes through .
Show that, by varying on the minor arc , all of the lines 
pass through the same point.
and a point 
on its exterior, two tangents to the circle are drawn through , with 
and 
being the points of tangency. We take a point 
on the minor arc 
of . Let 
be the intersection of 
with the line perpendicular to that goes through , and let 
be the intersection of 
with the line perpendicular to that goes through .Show that, by varying
on the minor arc , all of the lines 
pass through the same point.This post has been edited 2 times. Last edited by fprosk, Nov 18, 2015, 3:33 PM
Reason: changed the year
Reason: changed the year
the midpoint of 
.
is always a parallelogramm, hence 
,
is cyclic. Hence, we get
which implies that 
.
,
.
,
