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 an isosceles triangle with vertex at 
and 
. On side 
select point 
, such that 
. Point 
is the intersection point of the perpendicular bisector of the segment 
and the line passing through the point parallel to 
. The point 
lies on the line such that lies on the segment 
and 
. Prove that a line passing through perpendicular on 
, and a line that pass through perpendicular on , intersect on the line .
be an isosceles triangle with vertex at 
and 
. On side 
select point 
, such that 
. Point 
is the intersection point of the perpendicular bisector of the segment 
and the line passing through the point parallel to 
. The point 
lies on the line such that lies on the segment 
and 
. Prove that a line passing through perpendicular on 
, and a line that pass through perpendicular on , intersect on the line .
,
is infact an isosceles trapezoid.
and 
is intersection of the perpendicular line from 
,
.
be foot from 
to 
the intersection of 
,
foot from 
is cyclic and we can use pop to finish it off by showing 
