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.
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.
be a quadrilateral inscribed in a circle 
such that 
. Let 
and 
be the midpoints of 
and 
respectively. The line 
meets again at 
. Prove that the tangent at to , the line and the line 
are concurrent.
be a continuous function. A chord is defined as a segment of integer length, parallel to the x-axis, whose endpoints lie on the graph of 
. It is known that the graph of contains exactly 
chords, one of which has length 2025. Find the minimum possible value of .
has been partitioned into disjoint pairs 
(
) so that for all 
, 
equals 
or 
. Prove that the sum ![\[ |a_1-b_1|+|a_2-b_2|+\cdots +|a_{999}-b_{999}| \]](http://latex.artofproblemsolving.com/0/4/4/04422a7d2266b92a6b3f215b698f078cda85091f.png)
ends in the digit 
.
be an acute-angled triangle with circumcircle 
. A circle is internally tangent to at 
and also tangent to 
at 
. Let and 
intersect at 
and 
respectively. Let and be points on line such that 
is the midpoint of 
and 
is the midpoint of 
. Lines 
and 
meet at 
and intersect again at 
and 
respectively. The ray 
meets the circumcircle of triangle 
again at 
.
. 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.
, 
, 
, 
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 .
are concurrent. Then we should be done.
intersect 
at 
. Note that 
are symmedians of triangles 
respectively. So![\[\frac{FZ}{ZE}=\frac{FX}{XD}\frac{DY}{YE}=\frac{FP^2}{PD^2}\frac{DP^2}{PE^2}=\frac{FP^2}{PE^2}
\]](http://latex.artofproblemsolving.com/9/8/2/98262bb7cbab3f15be194e0f0f4f2685c2441eca.png)
Thus 
passes through and we are done.
hence 
collinear.
and 
Prove that
Prove that 
at 
and 
respectively. Let point 
on the incircle and inside 
.
.
and 
simultaneously or located opposite sides from 
