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be positive real numbers such that 
. Prove that 
be positive real numbers. Find the minimum value of![\[ T = (x+y+z+t)^2.\left[\dfrac{1}{x(y+z+t)}+\dfrac{1}{y(z+t+x)}+\dfrac{1}{z(t+x+y)}+\dfrac{1}{t(x+y+z)}\right] \]](http://latex.artofproblemsolving.com/c/a/c/cac0967ebdf30c1f79a1ec3e468cc035402333a7.png)
be integer numbers satisfy that: 
.Prove that: 
and its internal angle bisector 
. The line intersects the circumcircle 
of triangle at 
and 
. Circle 
with diameter 
cuts again at 
. Prove that 
is the symmedian line of triangle .
and 
be four points in plane, such that 
and 
.Define the point 
and the line 
such that 
and 
. Line cuts 
at 
and circumcircle of 
at 
. Prove that the circumcircles of triangles 
and 
are tangent at .
circumcenter of 
.
at arc 
containing 
and 
.
.
.
cyclic.
.
.
cyclic.Then,
.In triangle 
.So,
.
is the circumcenter of 
. Let 
be the antipode of 
in 
. Then 
are collinear. Also, as 
and 
, we get that 
is the orthocenter of 
. But 
, giving 
are collinear. Thus 
This easily gives that 
and 
are tangent to each other at 
., then the problem becomes quite well known.
be the diameter of the circumcircle of 
.
,
.
,
and hence 
which implies 
are collinear.
and 
so 
and this again implies that 
.
.
,
is mapped to line 
.
is mapped to 
is mapped to 
is mapped to circle 
.
and 
are right angle, we must have 
which implies that