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 an isosceles trapezoid with 
. Let 
be the midpoint of 
. Denote by 
and 
the circumcircles of the triangles 
and 
, respectively. Let 
be the crossing point of the tangent to at 
with the tangent to at 
. Prove that 
is tangent to .
is a positive integer. Call all positive divisors of which are different from 
and beautiful divisors.We call a special number when it has at least 
beautiful divisors and difference of any beautiful divisors divides as well. Find all special numbers.
we have 
.
.
, where 
, the internal angle bisectors of angles 
, 
, and 
meet the sides 
, 
, and 
at points 
, 
, and 
, respectively. Let 
be the incenter of triangle 
, and let 
be the foot of the perpendicular from to line 
. Prove that if the quadrilateral 
is cyclic, then the center of its circumcircle lies on segment 
.
,
and 
. Interior angle bisector of 
intersects 
at 
, and 
at .Prove that circumcircle of 
passes from circumcenter of 
.
be an infinite sequence of real numbers in the interval 
. Show that there exists a number that occurs exactly once in the sequence![\[ \frac{a_1}{1},\frac{a_2}{2},\frac{a_3}{3},\frac{a_4}{4},\ldots.\]](http://latex.artofproblemsolving.com/3/c/e/3cef10f63e9c2dde90b29c742ece8fd31e0a8e94.png)
with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points 
with integer coordinates such that 
for every pair 
and 
, of equal radius intersect at different points 
and 
. Consider a circle externally tangent to at 
and internally tangent to at point 
. Prove that lines 
and 
intersect at a point lying on .
be inscribed in the circle 
. A line through point 
intersects and 
at points and 
, respectively. Let be the reflection of across the midpoint of , and 
be the reflection of across the midpoint of . Prove that:
of triangle across line 
lies on the circle .
and 
coincide.
on the board. For each procces, the stutent deletes the number x and replaces it with either 
or 
or 
. Is this possible to make the number 
on the board?
and 
.
?
lies inside triangle such that 
and 
. Point is the midpoint of segment . Point lies on segment with 
. Prove that 
.
satisfying
Prove
:![\[
1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n + 1) \mid (4n + 3)(4n + 5) \cdot \ldots \cdot (8n + 3).
\]](http://latex.artofproblemsolving.com/3/1/a/31a0eac3c7ca298c8fa221083dc94e127e705784.png)
is the midline of 
,
.
and parallel to 
so 
.
.
and 
share a base, 
to each, the areas of 
and 
are equal.
(just a shorthand for 
)
and 
,
with the factor 
,![$[APXS]={1\over 4}[ABCD]$](http://latex.artofproblemsolving.com/f/2/a/f2a981b2b6a8440c8d6c5afe5500df95cee153f0.png)
,![$[APOS]=[APXS]$](http://latex.artofproblemsolving.com/b/3/c/b3cab30cf50f676a5357e34544cab29333211f1b.png)
,
)
)