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 circumscribed about circle 
, and let 
be the orthocenter of 
. The circle touches line 
at 
. The tangent to the circle 
at meets at 
. Let 
be the midpoint of 
, and let the line 
meet again at 
. The tangent to parallel to meets the line 
at 
. Prove that 
is tangent to .
is positive integer and 
are primes solve the system: 
and 
, and square 
. Point 
and 
are on the circle, and 
is tangent to the circle at point 
. Let 
represent the midpoint of 
and 
represent the intersection between and circle. Prove that 
.
be an integer. The sequence 
is defined via 
, 
and![\[ a_{n+2} = ka_{n+1} + a_n \]](http://latex.artofproblemsolving.com/b/5/e/b5e3e6f9f0b0709113498266e41b372299a2986f.png)
for any positive integer . Prove that there are finitely many pairs 
of positive integers such that 
.
such that 
and there exists an integer 
such that 
is a multiple of 
.
as 
fir 
. Prove that 
for all 
.
![$\color[rgb]{0.35,0.35,0.35}\displaystyle\sum_{n=0}^\infty\frac{\sin (nx)}{3^n}$](http://latex.artofproblemsolving.com/a/f/3/af33799c6924b61378db323645ea3e1b9c4e8c9d.png)
if ![$\color[rgb]{0.35,0.35,0.35}\sin x=1/3$](http://latex.artofproblemsolving.com/c/e/a/cea5c46445caa512d94d655350170fb19518a2c8.png)
and ![$\color[rgb]{0.35,0.35,0.35} 0\le x\le \pi/2$](http://latex.artofproblemsolving.com/9/8/b/98b7d75c944002b8b9c11dbe8cca0d402e6fbb67.png)
. be triangle with circumcenter 
and orthocenter . 
intersect again at 
respectively. Lines through parallel to 
intersects 
at 
respectively. Point such that 
is a parallelogram. Prove that lines 
and are concurrent at a point on 
.
is built on the side 
of an equilateral triangle to the outside. Let be the intersection point of and 
. Prove that 
.
be real numbers so that 
and 
Show that
![$a,b,c\in[0,1]$](http://latex.artofproblemsolving.com/8/b/d/8bd8f441b63556fdf318ad140bbd3e8e358e9e1d.png)
,
,![\[f(a,b,c)=(1-a^2)^2+(1-b^2)^2+(1-c^2)^2.\]](http://latex.artofproblemsolving.com/7/c/1/7c1611d9b80f5f694afed1894acd57a93aa5b31c.png)
Prove that
,s.t :
is isosceles in 
: 
: 