[/quote]
,
be a triangle with 
let 
be the circumcircle of 
and let 
be its radius. Point 
is chosen on 
such taht 
and point 
is the foot of the perpendicular from 
.
mets 
.
is chosen on line 
such that 
and 
lie on a line in that order. Finally, let 
be a point satisfying 
and 
.
,
has two real roots, so that the one is the square of the other.
,
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
.
.
such that: ![\[ x = \sqrt{ x - \frac{1}{x} } + \sqrt{ 1 - \frac{1}{x} } \]](http://latex.artofproblemsolving.com/e/c/f/ecf7fb3c2805290f2cb07b7fdd7c73e1c7a55b99.png)
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)
.
and 
meet at a point belonging to the diameter 
and 
are the incenter and the 
-
be the midpoint of arc 
on the circumcircle, and 
.
intersects the circumcircle again at 
. For
on the circumcircle of triangle 
intersect 
. Prove
are concyclic.
,
lie on a line in this order and satisfy 
.
and 
,
is acute, and let 
be its orthocentre. Points 
and 
,
and 
and 
and 
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 
Y by Adventure10, Mango247, Mango247, Mango247
Let 
be one of the two points of intersection of two circles with centers 
respectively.The tangents at to the two circles meet the circles again at 
. Let a point 
be located so that 
is a parallelogram. Show that is also the circumcenter of triangle 
.
be one of the two points of intersection of two circles with centers 
respectively.The tangents at to the two circles meet the circles again at 
. Let a point 
be located so that 
is a parallelogram. Show that is also the circumcenter of triangle 
.
.
the second intersection point of 
.
is perpendicular to 
and 
respectively.(because AB,AC,AQ are the radical axes of 
,
.
is perpendicular at 
,
.
(because of the parallelogram).So it suffices to prove that 
which is true because these angles have their lines perpendicular.
.
then 
lies on perpendicular bisector of ![$ [AC]$](http://latex.artofproblemsolving.com/5/b/0/5b08b2b92472414250205866b93405af4772e86c.png)
,
.
,
,
is the perpendicular bisector of 
,
is also perpendicular bisector of 
.
,
, let 
and 
we get 
is parallelogram we have 
, hence we get 
