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non-negative real numbers,but 
0.Prove that![\[1\leq \frac{a+b}{a+4b+c}+\frac{b+c}{b+4c+a}+\frac{c+a}{c+4a+b}\leq \frac{4}{3}\]](http://latex.artofproblemsolving.com/8/b/7/8b7d7b8cbcab8f2eed31e17aac717fa3c70c92f1.png)
are positive real numbers, then
with 
. Let point 
be orthocenter 
and 
midpoint 
. It is also known that 
perpendicular to 
. Let 
be a point on the segment 
such that 
and point 
be the intersection of 
and 
. Prove that 
concyclic
be an integer. Consider an 
chessboard consisting of 
unit squares. A configuration of 
rooks on this board is peaceful if every row and every column contains exactly one rook. Find the greatest positive integer 
such that, for each peaceful configuration of rooks, there is a 
square which does not contain a rook on any of its 
unit squares.
be the midpoint of 
, also let 
be the point on 
such that 
and 
be the point such that 
and 
. Hence 
, hence 
, which implies 
, therefore 
. But 
, therefore 
. So and 
are points such that and 
, thus 
. Now 
and 
, so 
. Therefore we conclude that 
. Let 
be the midpoint of . We have 
, thus 
, as desired.
and 
.
.
be an acute triangle with the circumcircle 
. Denote the midpoints 
, 
![$[AC]$](http://latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
, ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
respectively, the projection 
of 
on 
and the projections , of , respectively. Prove that 
.
. Thus,
.
is the circumcenter of 
.
are the feet of perpendiculars from 
,
.
.
are collinear. So 
and circle with diameter 
