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 
for all 
with parallel sides 
and 
, assume that there exist points 
on line 
outside segment , and 
inside segment 
such that 
. Denote by 
the point of intersection of and 
, and by 
the point of intersection of and . Let 
be the midpoint of segment , assume it does not lie on line . Prove that belongs to the circumcircle of 
if and only if belongs to the circumcircle of 
.
with an area of 
, with 
. If 
, Hence find the value of 
.
be real numbers such that ![\[ \frac{1}{x^2-1} + \frac{1}{y^2-1} + \frac{1}{z^2-1} = 1. \]](http://latex.artofproblemsolving.com/8/f/5/8f5d49c9b44e3db3cd85fea7b6521e1dc02d4269.png)
Prove that ![\[ \frac{1}{x+1} + \frac{1}{y+1} + \frac{1}{z+1} \le 1. \]](http://latex.artofproblemsolving.com/c/0/4/c0438d6eba686d0a11fd4decb4afa9a14aa50bac.png)
such that, for every positive integer 
the integer
is a multiple of 
(Note that 
denotes the greatest integer less than or equal to 
For example, 
and 
)
.
, there exist indices 
such that 
is divisible by .
, find the minimum possible value of 
.
, 
, 
, and 
be the orthocenters of triangles 
, 
, 
, and 
, respectively. Let 
be the centroid of triangle . is the foot of onto the plane .
be the foot of onto the plane . Notice that 
is perpendicular to plane 
. Thus 
, which implies 
. Similarly, 
. Thus 
, as required. and . Notice the centroids of triangles , , and all coincide at some point 
. Since is the foot of onto the plane , it follows that 
. By the claim, we also get that 
. Thus all five points lie on the sphere with diameter 
, concluding the proof.
.
.
.
such that 
.![$$2 \sqrt[4]{\frac{a}{b+c-a}}\ge 2 +\frac{2a^2-b^2-c^2}{(a+b)(a+c)}.$$](http://latex.artofproblemsolving.com/f/0/7/f0731fbe5c83aedc8474975f930f7194c08cc070.png)
and 
Prove that![$$3\left(\sqrt[3] 2+\frac{1}{\sqrt[3] 2} -1\right) \geq a+b+c\geq \frac{3+\sqrt{11}}{2}$$](http://latex.artofproblemsolving.com/f/6/6/f66042b4d07212741245953a8940280c94a52988.png)
![$$\frac{3}{2}\left(4+\sqrt[3] 4-\sqrt[3] 2\right) \geq a+b+c+ab+bc+ca\geq \frac{3(1+\sqrt{11})}{2}$$](http://latex.artofproblemsolving.com/0/5/6/056db8bf4d80242d705c265b6a64e063c8fca15a.png)
Prove that
is given. It is known that triangles 
,
,
,
are all acute. Prove that the orthocenters of these triangles, together with the centroid of triangle 