be a cyclic quadrilateral. Let 
be the midpoint of the arc 
of its circumcircle which does not contain 
or 
and 
meet at 
and 
meet at 
and 
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 
.
Y by Adventure10, Mango247
Let 
be the midpoint of 
in 
. Let 
be any point on 
. If the internal angle bisector of 
and 
intersect at 
. Prove that, if 
, then lies on 
be the midpoint of 
in 
. Let 
be any point on 
. If the internal angle bisector of 
and 
intersect at 
. Prove that, if 
, then lies on 
This post has been edited 4 times. Last edited by AlastorMoody, Mar 15, 2019, 7:35 AM
Reason: @Below Thanks! I made a mistake while translating, Sorry! Now try!
Reason: @Below Thanks! I made a mistake while translating, Sorry! Now try!
,
.
;
with 
= 
tangents 
)
or 
= 
= 
= 
or 
is internal bisector of 
= 
+ 
= 
+ 
+ 
+ 
= 
+ 
+ 
= 
+ 
or 
.
,
.
.
will imply 
.
.
.
will become under the inversion the symmedian of the triangle 
,
,
is the intersection of the tangents in 
wrt 
,
be the reflection of 
.
is a parallelogram. Denote 
and 
is cyclic.
as desired.
