AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be a triangle and 
its circumcentre. A line tangent to the circumcircle of the triangle 
intersects sides 
at 
and 
at 
. Let 
be the image of 
under 
. Prove that the circumcircle of the triangle 
is tangent to the circumcircle of triangle .
be a semicircle with diameter and midpoint 
. Let 
be a point on different from and 
.
touches in a point 
, the segment 
in a point , and additionally the segment 
. The circle 
touches in a point and additionally the segments 
and .
and 
are perpendicular.
with no parallel sides is inscribed in a circle 
. Circles 
are inscribed in triangles 
, respectively. Common external tangents are drawn between 
and 
, and 
, and 
, and and , not containing any sides of quadrilateral . A quadrilateral whose consecutive sides lie on these four lines is inscribed in a circle 
. Prove that the lines joining the centers of and , and , and the centers of and all intersect at one point.
, 
, 
, 
are integers such that 
.
and 
of 
such that for each integer 
with 
, we have![\[ n\mid a_i - b_i - c_i
\]](http://latex.artofproblemsolving.com/1/a/e/1ae8d54f068b646ac7a1e43d75e7f26510cfdc95.png)
be an odd prime and 
be integers so that the integers 
are divisible by . divides each of .
be real numbers such that 
Prove that 
Where 
be a triangle with circumcircle 
. Let 
and 
respectively denote the midpoints of the arcs and that do not contain the third vertex. Let 
denote the midpoint of arc 
(the arc 
including ). Let 
be the incenter of . Let 
be the circle that is tangent to and internally tangent to at , and let 
be the circle that is tangent to and internally tangent to at . Show that the line 
, and the lines through the intersections of and , meet on .
friends, 
people are seated around a circular table in a fixed order. Ali places apples for serving directly in front of himself and wants to distribute them among everyone. Since Ali and his friends dislike eating alone and won't start unless everyone receives an apple at the same time, in each step, each person who has at least one apple passes one apple to the first person to their right who doesn't have an apple (in the clockwise direction). such that after some number of steps, the situation reaches a point where each person has exactly one apple.
be an arbitrary triangle and is the circumcenter of 
.Points 
lie on 
,respectively such that the reflection of WRT 
is tangent to circumcircle of 
.Prove that the circumcircle of triangle 
is tangent to circumcircle of triangle .
and 
are marked on the bisector of 
of the triangle so that 
. Let be the midpoint of 
. Prove that 
.
, let be intersection of the bisector of and 
. Notice that 
, because 
. is the midpoint of . We have that 
.
Hence, we are done.
.
, lies on the circumcircle 
of 
and 
is a diameter.
, so 
is the midpoint of arc 
of . I will show 
. to 
meet at 
. Since is a diameter, 
is also a diameter. onto ,
.
is a harmonic quadrilateral (since it is a kite inscribed in a circle), so 
.
, so 
and thus is the midpoint of .
as is the midpoint of arc of .
and 
Prove that
Prove that 
