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 
be given . On the extension of the segment 
beyond 
, a point 
is chosen arbitrarily. Let the line through and 
intersect the line 
at the point 
. Let 
be the intersection of the lines 
and 
. Prove that the line 
is tangent to the circumcircle of the triangle 
.
be any point. Let 
be projection point from to 
. Circumcircle 
cuts circumcircle 
at 
. Let 
be antipode of wrt . Prove that 
are cyclic.
, 
be real numbers, such that 
. Prove that 
.
, angle 
. Point lies on 
such that 
. Line parallel to 
passing trough intersects 
at 
. Prove that 
.
be a parallelogram of center 
. Prove that for any point 
, there exist unique points 
and 
such that is the center of mass of 
.
Compute -![\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}x} \bigg[ \int_{0}^{e^x} \log ( t ) \cos^4 ( t ) \mathrm{d}t \bigg]
\end{align*}](http://latex.artofproblemsolving.com/8/6/f/86fb48db976012106bd30f74cb6ada8c6e6d7800.png)
For 
define 
Determine the open interval(s) (if any) where 
is decreasing and all the open interval(s) (if any) where is increasing.
Determine all the local minima of (if any) and all the local maxima of (if any) 
of positive integers such that
, then it must not contain edge 
.
and since 
, hence 
.
. Then 
. But 
. Hence, 
, giving 
. Hence, to construct 
, we only need to choose 
from the 5 sets 
keeping in mind that 
.
.'s is 
.
and 
Prove that
digit number is called interesting if its digits are distinct and is divisible by 
.