[/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 
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 
.
.
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)
.
meet at a point belonging to the diameter 
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.
,
.
,
is acute, and let 
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
Let 
be a convex quadrilateral with sides 
, inscribed in a circle. Lines 
and 
are cut in 
(
is between 
and ). Lines 
and 
are cut at 
(
is between and ). The circle of diameter 
cuts the line at 
(
) and 
at 
(
). Show that line 
passes through the midpoint of 
.
be a convex quadrilateral with sides 
, inscribed in a circle. Lines 
and 
are cut in 
(
is between 
and ). Lines 
and 
are cut at 
(
is between and ). The circle of diameter 
cuts the line at 
(
) and 
at 
(
). Show that line 
passes through the midpoint of 
.
