Beautiful geometry
by m4thbl3nd3r, Apr 17, 2025, 4:41 PM
Let 
be the circumcircle of triangle 
, 
is the midpoint of 
and 
be the second intersection of 
and . Tangent line of at intersects at 
, let 
be a transversal of and 
be intersections of 
with . Let 
be a tangent line of . Prove that 
is cyclic
be the circumcircle of triangle 
, 
is the midpoint of 
and 
be the second intersection of 
and . Tangent line of at intersects at 
, let 
be a transversal of and 
be intersections of 
with . Let 
be a tangent line of . Prove that 
is cyclic
a
Prove that
,
lie on a line in this order and satisfy 
.
be the midpoints of 
and 
,
is acute, and let 
be its orthocentre. Points 
lie on lines 
and 
,
and 
and 
and 
and 
are the incenter and the 
-
on the circumcircle, and 
be the foot of the
intersects the circumcircle again at 
. For
on the circumcircle of triangle 
intersect 
. Prove
are concyclic.
be a convex pentagon such that 
.
is the circumcenter of triangle 
.
be the circumcenter of triangle 
.
passes through the midpoint of segment 
.
such that for all 
the following holds:![\[f(x)+f(y)=f(xy)+f(f(x)+f(y))\]](http://latex.artofproblemsolving.com/a/c/a/aca098172ea7ce5663e9dd0dd10c2f1495630230.png)
be integers such that there are consecutive integers 
satisfy 
.
is a perfect square.
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 
.
and 
meet at a point belonging to the diameter 
April 2025
November 2024