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.
such that 
is a perfect square.
be real positive numbers such that: 
. Prove that: 
and 
and 
are real numbers for which the following inequality is satisfied :![\[\left|\sum_{i=1}^{n} b_i\cos(ka_i)\right|<\frac{1}{k}\]](http://latex.artofproblemsolving.com/5/e/4/5e4b3bdc2d5b71e42ba3e6339845794625080eb5.png)
. Prove that 
.
, 
be the orthocenter of it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
. Prove that the points 
and lie on the same line.
and 
be positive real numbers such that 
. Prove that
When equality holds?
and 
lie on the sides 
and 
of the square 
respectively, and 
. The circle through 
intersects 
and 
again at 
and 
, respectively. Prove that 
, 
, 
and 
are the points of tangency of incircle with the center of 
to , 
and 
respectively. Let be the foot of the perpendicular from to 
. is on 
such that 
. If 
is the orthocenter of 
, prove that 
bisects 
.
, let 
be a variable point on the line 
such that 
lies between 
and and the incircles of the triangles 
and 
intersect at two distinct points 
and 
Prove that the line 
passes through a point independent of .
with real coefficients such that for all reals such that 
we have the following relations![\[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]](http://latex.artofproblemsolving.com/4/4/9/4491a53f9c6f96c18e42eb984b8e83566140bfdb.png)
be a triangle with 
and 
, and let be the midpoint of . Let be a variable point of the circumcircle of , and let be the point on the segment 
such that 
. It is given that the line through and perpendicular to intersects the line 
at a unique point . lies on this circle for all possible positions 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) 
be complex numbers satisfying 
. Show that there exist some among the complex numbers such that the modulus of the sum of these complex numbers is not less than 
.
hence your claim is done.
as vectors, and put them all in the plane starting at the origin. By pidgeonhole, one of our three sectors has at least 1/3 of the total length of the vectors. Wlog this sector is the one with rays in the 0 radian and 2pi/3 radian directions as its boundaries.
