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.
and 
be positive real numbers such that 
. Prove that![\[
\frac{x^{3}}{(1 + y)(1 + z)}+\frac{y^{3}}{(1 + z)(1 + x)}+\frac{z^{3}}{(1 + x)(1 + y)}
\geq \frac{3}{4}.
\]](http://latex.artofproblemsolving.com/5/6/5/56568d27a296f1c900fabbb1042549cd2e552e29.png)
+1 w
, ![\[ \frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1. \]](http://latex.artofproblemsolving.com/b/2/f/b2ff456632d4dd3b9ed93e4cba437bdd34d54cd8.png)
, 
, 
cut three common chords concurrent at 
. Consider two distinct circles 
, 
which are internally tangent to all 
. Determine, with proof, which of the following two statements is true. is the insimilicenter of and is the exsimilicenter of and .
+2 w
be a parallelogram of center 
. Prove that for any point 
, there exist unique points 
and 
such that is the center of mass of 
.
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 
.
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) 
board in such a way that a chess king from any cell attacks no more than two marked ones? (The cell on which a king stands is also considered to be attacked by this king.)
board in such a way that a chess king from any cell attacks no more than two marked ones? (The cell on which a king stands is also considered to be attacked by this king.)
grid by 
grid and note that the above problem is same as. grid. There are atmost 
marked cells in every 
subsquare.Find the largest number of marked cells.
tiles of form 
. There are 
tiles of 
shape and there are 
squares of form 
. Note that 
shaped tile and 
tile each can have atmost marked cells while tile of form 
can have atmost 1 marked cell. So we have atmost 
marked cells.
marked cells.
gives 
as ans for grid so we are done. 
board. Some of its cells are colored black in such a way that every 
board of 