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.
and 
play a series of of 
games. Each game either results in a win or loss for . Total number of ways in which can win the series is:
+1 w
and 
with integer coefficients is called 
if from the divisibility of both differences 
and 
by 
, it follows that both 
and 
are divisible by 100. Does there exist such an important pair of polynomials 
, 
, such that the pair 
and 
is also important?
+1 w
with 
such that
.
be real numbers such that 
. Show that
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?
satisfy 
. Prove that
Also, find the values of for which the equality happens.
such that 
are the lengths of a right triangle.
, the midpoint of the 
side is 
and 
is a point on the 
side such that 
is bisector of the angle 
. Show that is perpendicular to 
be reals such that 
and 
Prove that
satisfy:
![$[(x-2)^2+(y-4)^2][ (x-1)^2 + (y-2)^2] = 36$](http://latex.artofproblemsolving.com/b/1/0/b108dc2cac5b7491fb83418eb36c7184f6d5d30c.png)
such that: 
.
and 
.
and 
.
![\[
\frac{(x+y+z)^5 - x^5 - y^5 - z^5}{(x+y)(y+z)(z+x)} = 0
\]](http://latex.artofproblemsolving.com/7/8/f/78f8e6c2570684631dc288144e13568d71e68ec8.png)
.
is parallel to 
.
which implies that triangle 
is isosceles with 
.
,
with respect to 
be a point on 
such 
.
and 
.
.
and 
.
such that, 
.
intersect at 
.
is the angle bisector of 
of isosceles 
,
.
.
.
,
,
is cyclic. So, 
and 
.
.
.
is also cyclic. As, 
,
,
.
.
.
.
,
,
- cyclic with diameter 