be six positive real numbers. Prove that
for which the following statement holds: there exists at least one triple 
of integers such that
and all triples 
are integers.
Prove that
Where 
Prove that
Let 
Prove that
Let 
Prove that
and 
. Prove that
Where 
,
where 
is a parallelogram.
.![$[CT]$](http://latex.artofproblemsolving.com/f/3/f/f3f8e9913dc696e543e6a34d169ec3deb276c84d.png)
,
to the midpoint of ![$[DT]$](http://latex.artofproblemsolving.com/6/d/d/6dd49550ab42ec2f047eaf4b357ac4e7d1387b1e.png)
,
to the midpoint of ![$[AT]$](http://latex.artofproblemsolving.com/6/e/3/6e397da3c2a38b457f7b3504c5cb2a488a64f3a6.png)
and point 
to the midpoint of ![$[BT]$](http://latex.artofproblemsolving.com/6/a/4/6a4b1bcea7921e762a359321ab861b6937f04782.png)
.
.
with edges 
.
and radius 
.
.
with edge 
cm.
and tangent to the plane 
.
,
and 
.
.
and 
,
is a regular tetrahedron.
.
A sphere of radius 
is circumscribed about tetrahedron 
Given that 
,
,
,![$$r^2+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge \frac{9\sqrt[3]{4}}{4}$$](http://latex.artofproblemsolving.com/6/c/6/6c665b4611f5fc8ebeba03ac649bfee4e2634950.png)
with equality at ![$a=b=c=\sqrt[3]{2}.$](http://latex.artofproblemsolving.com/b/b/9/bb9c9671b23cb45e5f85ae01bf25d566c2e43e91.png)
Y by Mango247, Mango247, Mango247
Let 
be a triangle and 
its circumscribed circle. The internal bisector of angle 
meets 
at point 
. Let 
be the circle tangent to 
and . Similarly, let 
be the circle tangent to 
and . Prove that the tangency points of and with 
coincide.
be a triangle and 
its circumscribed circle. The internal bisector of angle 
meets 
at point 
. Let 
be the circle tangent to 
and . Similarly, let 
be the circle tangent to 
and . Prove that the tangency points of and with 
coincide.
be the midpoint of arc 
fixes 
and so swaps 
to itself, it fixes 
at the point at distance 
.
and 
respectively. Now deffine 
,
and 
,
.
be the midpont of minor arc 
.
and 
,
, but as 
is tangent to 
in 
and 
.
are concyclic. Thus 
, and we're done.
