[/quote]
,
has circumcircle 
and circumcenter 
.
with center 
intersects the segment 
at points 
and 
,
,
are all different and lie on line 
and 
be the points of intersection of 
be the second point of intersection of the circumcircle of triangle 
and the segment 
.
be the second point of intersection of the circumcircle of triangle 
and the segment 
.
and 
are different and intersect at the point 
.
.
be a point inside a triangle 
such that![\[ \angle APB - \angle ACB = \angle APC - \angle ABC.
\]](http://latex.artofproblemsolving.com/a/6/e/a6e3dc27d0457682fa22af4f918f58d9cd8403bc.png)
,
be the incenters of triangles 
,
,
,
,
meet at a point.
satisfying
Prove
cups labeled 
,
-
Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most 
steps.
Prove that in at most 
steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
for which 
divides 
.
and 
,
cannot be a power of 
.
of natural numbers such that for every pair of natural numbers 
and 
,![\[
\frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2
\]](http://latex.artofproblemsolving.com/1/6/7/167679c1707b957d87311298ea5b72347a9bdc45.png)
such that 
for all 
such that for all real numbers 
and 
,![\[f(xf(y))+f(y)=f(x+y)+f(xy).\]](http://latex.artofproblemsolving.com/d/2/f/d2ff6ed448cf39e7ec9bce6b944965d5e89b9878.png)
table filled with natural numbers, we say it is a divisor table if:
-
,
-
,
for every 
.
is given. Determine the smallest natural number 
,
to tangent 
and 
.
is tangent to two fixed circles.
be its incircle. Denote by 
and 
the points where 
,
and 
the points on sides 
and 
,
the point of intersection of segments 
and 
.
.
.
![\begin{align*}
\frac{a}{c} &= b-\sqrt{b}+c \\
\sqrt{\frac{a}{c}} &= \sqrt{b}+1 \\
\sqrt[4]{\frac{a}{c}} &=\sqrt[3]{b}-1
\end{align*}](http://latex.artofproblemsolving.com/0/c/d/0cd22d36acedbcb10d76d1b884aa35418273ed52.png)
be positive integers. Prove that the expression![\[\frac{\gcd(a+b,ab)}{\gcd(a,b)}\]](http://latex.artofproblemsolving.com/1/b/7/1b78a2027c3b7156fbb566901d2cfd088cc09f00.png)
is always a positive integer, and determine all possible values it can take.
Y by Adventure10, Mango247
Let 
be a sequnce of concentric circles such that the sequence 
, where 
denotes the radius of 
, is increasing and 
as 
. Let 
be a triangle inscribed in 
. extend the rays 
to meet 
in 
and 
respectively and form the triangle 
. Continue this process. Show that the sequence of triangles 
tends to an equilateral triangle as 
be a sequnce of concentric circles such that the sequence 
, where 
denotes the radius of 
, is increasing and 
as 
. Let 
be a triangle inscribed in 
. extend the rays 
to meet 
in 
and 
respectively and form the triangle 
. Continue this process. Show that the sequence of triangles 
tends to an equilateral triangle as 
