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.
. Prove that
Let 
. Prove that
and 
outside triangle 
such that 
and 
. Connect 
and 
, which intersect at point 
. Let 
intersect 
at point 
. Prove that 
.
be pairwise distinct integers. Show that the equation 
has at most one integral solution.
for real numbers 
not multiple of 
.
of the triangle is extended beyond 
to so that 
. The side 
is extended beyond 
to so that 
. Prove that, if 
, then the triangle is right-angled.
segments 
and 
are parallel. Angle bisectors of 
and 
meet at 
. Angle bisectors of 
and 
meet at 
. Prove that 
is cyclic
, but to prove...
.
Let 
.![$$ \frac{1}{2}\geq \frac{a}{a^2+3 }+ \frac{b}{b^2+3} \geq \frac{1}{4}(\frac{1}{\sqrt[3]{3}}+\sqrt[3]{3}-1)$$](http://latex.artofproblemsolving.com/f/9/1/f9119e071c12778037616c624c7b9ca75356e8e3.png)
![$$ \frac{1}{2}\geq \frac{a}{a^3+3 }+ \frac{b}{b^3+3} \geq \frac{1}{2\sqrt[3]{9}}$$](http://latex.artofproblemsolving.com/2/b/2/2b2d8a011229fe88a81cb75f9b018ad3a6245faa.png)
![$$ \frac{1}{2}\geq \frac{a}{a^2+ab+2}+ \frac{b}{b^2+ ab+2} \geq \frac{4\sqrt[3]{3}+3\sqrt[3]{9}-6}{17}$$](http://latex.artofproblemsolving.com/5/f/6/5f673cd494b1909ca237db63e96f949c24b6f2ad.png)
![$$ \frac{1}{2}\geq \frac{a}{a^3+ab+2}+ \frac{b}{b^3+ ab+2} \geq \frac{\sqrt[3]{3}}{5}$$](http://latex.artofproblemsolving.com/b/c/3/bc343a07c88dbd98ebffdfb78f311b7b82683d65.png)
and 
Prove that
Let 
Prove that
and 
.
and 
be the centers of the incircles of triangles 
and 
respectively. Find the angle between the lines 
and 
.