.
.![\[\sum ab +\sum ab^2 \le 6\]](http://latex.artofproblemsolving.com/9/f/6/9f632ab1e6df5029bcd4416eb06623123ad2cd8c.png)
be an integer and 
is a sequence of real numbers for which 
and:
Prove that all elements of the sequence are positive.
be the incenter of triangle 
.
,
,
be the orthocenters of triangles 
,
,
,
,
,
,
be an integer. An 
-
of not necessarily different positive integers is expensive if there exists a positive integer 
such that 
a) Find all integers 
there exists an integer 
Y by LuoJi
Let 
be a point inside 
. It is known that there exists a line tangent to the incircles of 
, 
and 
. Prove that if 
is the intersection point of the common external tangents of a random pair of these incircles and Y is the intersection point of common external tangents of some other pair of these three incircles, then 
passes through either 
, 
or 
.
be a point inside 
. It is known that there exists a line tangent to the incircles of 
, 
and 
. Prove that if 
is the intersection point of the common external tangents of a random pair of these incircles and Y is the intersection point of common external tangents of some other pair of these three incircles, then 
passes through either 
, 
or 
.
we get that second tangent l′ from E to incircle of 
.
and 
lie on the same conic.
and 
share the same incircle 
.
and the tangent 
are given. Let 
be the projective involution of 
of involution define point 
by intersection of the second tangent lines from 
to 
there is the involution that swaps 
,
and 
.
that swaps 
,
and 
.
imply that 
.
.
