[/quote]
,
w![$ \boxed{\ \left(\frac{n}{n+1}\right)^{n+1}<\sqrt [n+1]{(n+1) !}-\sqrt [n]{n!}\ }$](http://latex.artofproblemsolving.com/d/f/2/df2545bd7a1f1759e55190e60648d5cf65a2488a.png)
be the incircle of 
.
amd 
inside 
,
at 
,
.
,
,
.
are concyclic.
such that ![\[p^q+q^p+p+q-5pq\]](http://latex.artofproblemsolving.com/b/a/b/bab8a0ee3555cf1136b6de56179012b920d0a2ad.png)
is a perfect square.
be positive numbers such that 
.![\[T = \dfrac{bc}{\sqrt{a}+3}+\dfrac{ca}{\sqrt{b}+3}+\dfrac{ab}{\sqrt{c}+3} \]](http://latex.artofproblemsolving.com/6/d/5/6d570d06679f17452790b90171428d645e97fe9f.png)
with circumcircle 
,
on 
be points on lines 
,
,
,
concur at a point 
be a point on line 
,
,
be a point on line 
be a point on line 
,
,
concur at a single point.
be non-negative real numbers such that 
Prove that
Prove that 
.
.
.
.
is a point on segment 
intersects 
and 
at 
and 
,
and 
intersect at 
.
and 
,
is cyclic.
be any interior point on 
,
at 
respectively. Prove that 
.
with 
,
and 
.
at 
.
and 
is perpendicular to 
Y by
Let 
be a positive integer. Egor has 
cards with the number “
” written on them, and cards with the number “
” written on them. Egor wants to paint each card red or blue so that no subset of cards of the same color has the sum of the numbers equal to . Find the greatest such that Egor will not be able to paint the cards in such a way.
be a positive integer. Egor has 
cards with the number “
” written on them, and cards with the number “
” written on them. Egor wants to paint each card red or blue so that no subset of cards of the same color has the sum of the numbers equal to . Find the greatest such that Egor will not be able to paint the cards in such a way.
