[/quote]
,
be non-negative real variables with 
Prove that
Equality holds iff 
or 
where 
,
is the point on 
such that 
.
be the circle through 
tangent to 
at 
the circle through 
.
be the second intersection of 
.
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
In a triangle 
, points 
and 
are considered on the sides 
and 
respectively, such that the areas of the triangles 
and 
are equal. Prove that 
Note:
is the area of triangle 
.
, points 
and 
are considered on the sides 
and 
respectively, such that the areas of the triangles 
and 
are equal. Prove that 
Note:
is the area of triangle 
.This post has been edited 1 time. Last edited by parmenides51, Sep 4, 2018, 9:39 PM
Reason: typo
Reason: typo
