[/quote]
,
be a cyclic pentagon inscribed in a circle of centre 
which has angles 
.
and 
meet at a point belonging to the diameter 
.
and 
are the incenter and the 
,
be the midpoint of arc 
on the circumcircle, and 
be the foot of the
.
intersects the circumcircle again at 
. For
on the circumcircle of triangle 
intersect 
. Prove
are concyclic.
,
lie on a line in this order and satisfy 
.
be the midpoints of 
and 
,
is acute, and let 
be its orthocentre. Points 
and 
lie on lines 
and 
,
and 
and 
and 
.
are the midpoints of 
respectively. 
be the incenter and tangent of angles 
respectively. Prove that 
have a common point
be the circumcircle of triangle 
be the second intersection of 
and 
be a transversal of 
be intersections of 
with 
be a tangent line of 
is cyclic
be integers such that there are consecutive integers 
satisfy 
.
is a perfect square.
a
Prove that
of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly 
coins showing 
t
the process starting with the configuration 
would be 
,
be the number of operations before Harry stops. For example, 
and 
.
possible initial configurations 
,
,
of the
,
,
, 
.
such that
Y by
Let 
and 
be circles with no points in common. Draw their common external tangents 
and 
and their common internal tangents, 
and 
, as shown in the figure. Suppose that couches the circles at 
and 
and that intersects and at X and Y as shown. Prove that 
and 
are equal in length.
and 
be circles with no points in common. Draw their common external tangents 
and 
and their common internal tangents, 
and 
, as shown in the figure. Suppose that couches the circles at 
and 
and that intersects and at X and Y as shown. Prove that 
and 
are equal in length.This post has been edited 1 time. Last edited by parmenides51, Jan 30, 2021, 8:03 AM
