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.
th power.
let 
denote the intersection of diagonals 
and 
, and let 
denote the intersection of diagonals 
and 
. Prove that if 
and 
, then 
bisects angle 
.
are 
and 
, respectively. The excenter and exradius relative to vertex 
is 
and 
, respectively. Let 
denote the circumradius. Prove that if 
, then 
.
-gon, and at each vertex there is a black or a white cap. (Caps of the same colour are indistinguishable from each other.) Under one of the caps 
dollars are hidden, and there is nothing under the other caps. The host rotates the table, and then Ádám chooses a cap, and take what is underneath. Ádám's accomplice, Béla is working at the company behind this game. Béla is responsible for the placement of the dollars under the caps, however, the colors of the caps are chosen by a different collegaue. After placing the money under a cap, Béla
such that for all positive integers 
,![\[
f(mn) = \mathrm{lcm}(m, n) \cdot \gcd(f(m), f(n)),
\]](http://latex.artofproblemsolving.com/6/0/7/6071692e2716901e88cc1cfc19ed60388c10701d.png)
where 
and 
denote the least common multiple and the greatest common divisor of 
and 
, respectively.
has a closed walk of length at most 
passing through each edge of 
at least once.
satisfying the following condition: for any real numbers 
and 
, the number 
is equal to 
or 
contains no term of the form 
and 
.
?
, similarly to 
, by SOS
be a triangle with 
. Let 
, 
, and be the midpoints of the sides 
, 
, and 
respectively. A circle 
passing through 
and tangent to at meets the segments 
and 
at 
and , respectively. The points 
and 
are symmetric to and about and , respectively. The line 
meets 
and 
at and 
, respectively. The line 
meets again at . Prove that 
.
with incenter . It is known that 
is center of the ex-circle tangent to . Likewise, 
and 
are the centers of the ex-circles tangent to and , respectively. Prove that is the orthocenter of the triangle 
.
with incenter . It is known that is center of the ex-circle tangent to . Likewise, and are the centers of the ex-circles tangent to and , respectively. Prove that is the orthocenter of the triangle .
is the orthic triangle of by angle chase so we're done.
perpendicular with 
via 
and 
,
.