be a triangle such that 
.
lie on segments 
,
and 
are the angle bisectors of triangle 
.
has circumcircle 
and circumcenter 
.
with center 
intersects the segment 
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 
.
.
.
Y by cubres
Let 
be an acute triangle with 
, midpoint 
of side 
, altitude 
(
), and orthocenter 
. A circle passes through points 
and 
, is tangent to line 
, and intersects the circumcircle of triangle at a second point 
. The circumcircle of triangle 
intersects line 
at a second point 
. Prove that the lines 
and 
are perpendicular.
be an acute triangle with 
, midpoint 
of side 
, altitude 
(
), and orthocenter 
. A circle passes through points 
and 
, is tangent to line 
, and intersects the circumcircle of triangle at a second point 
. The circumcircle of triangle 
intersects line 
at a second point 
. Prove that the lines 
and 
are perpendicular.
,
,
.
is cyclic.
is cyclic, so 
also 
,
colinear. 
,
inscribed in the unit circle, so that
Now we find the coordinate of 
,
,
Now we find the coordinate of 
.
are concyclic, we have
Now 
,
,
Now
So we write this as
And since 
is pure imaginary, we have
Then we have the vectors
and so
which is pure imaginary. 
has to be the isosceles trapezoid point. Since 
,
is cyclic. Indeed,![\[\angle QXD=90^{\circ}-\angle QAD=\angle QPA'=\angle QPD,\]](http://latex.artofproblemsolving.com/b/1/7/b17aec3b296dc5c518f6210dc4543d8caf5052af.png)
as desired. 
intersects the circumcircle of 
.
,
is cyclic. Hence 
is cyclic, implying 
,
.
be the intersection, different from 
and 
.
,
is an isosceles trapezoid.
be the point such that 
.
,
are concyclic.
,
is the orthocenter of triangle 
.
,
.
,
,
,
be such that 
and redefine 
.
is cyclic with diameter 
,
touches 
.
,
is cyclic. By Theorem 3.1 in 
(where 
)
by Lemma 2.2, we need 
.
and 
by Lemma 2.2, which implies that 
