[/quote]
,
be non-negative real variables with 
Prove that
Equality holds iff 
or 
where 
,
such that 
.
be the circle through 
tangent to 
at 
the circle through 
at 
.
be the second intersection of 
lies on 
Prove that
be reals such that 
and 
Prove that
and 
with integer coefficients is called 
if from the divisibility of both differences 
and 
by 
,
and 
are divisible by 100. Does there exist such an important pair of polynomials 
,
,
and 
is also important?
satisfy:
![$[(x-2)^2+(y-4)^2][ (x-1)^2 + (y-2)^2] = 36$](http://latex.artofproblemsolving.com/b/1/0/b108dc2cac5b7491fb83418eb36c7184f6d5d30c.png)
be real numbers such that 
.
such that: 
.
and 
.
grid, taking turns. Petya goes first. On his turn, a player writes an uppercase Russian letter in an empty cell (each cell can contain only one letter). When all cells are filled, Petya is declared the winner if there are four consecutive cells horizontally spelling the word ``ПЕТЯ'' (PETYA) from left to right, or four consecutive cells vertically spelling ``ПЕТЯ'' from top to bottom. Can Petya guarantee a win regardless of Vasya's moves?
.
Also, find the values of 
and 
.
Y by Adventure10, Mango247
Let 
be a acute-angled triangle with centroid 
, the angle bisector of 
intersects 
in 
. Let 
and 
be points in 
where 
and 
. Let 
be the midpoint of 
, let 
be a point in the line 
such that 
(where is the segment 
), prove that:
be a acute-angled triangle with centroid 
, the angle bisector of 
intersects 
in 
. Let 
and 
be points in 
where 
and 
. Let 
be the midpoint of 
, let 
be a point in the line 
such that 
(where is the segment 
), prove that:
be the midpoint of 
intersect at point 
then in 
note that 
and 
is parallelogram.Also by centroid
.
we have that 
also 
and 
.
and let 
also let 
)
which implies that 
we need to prove that 
or in this case 
is cyclic and 
