[/quote]
,
such that![\[
f(xy) = f(x)f(y) \;-\; f(x + y) \;+\; 1,
\quad \forall x, y \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/7/a/5/7a51a7be2306d18f4e80db1f5aac6e9bbf25c25c.png)
such that for any positive integer 
,
.
and 
lie respectively on sides 
and 
of a triangle 
.
and 
cross at 
.
and 
cross again at point 
satisfying 
and their sum is an even number, consider the quadratic polynomial:![\[
P(x) = x^2 - (m^2 - m + 1)x + (m^2 - n^2 - m)(n^2 + 1).
\]](http://latex.artofproblemsolving.com/d/5/1/d51aaddfddfe06cc91abfdd3476a155770a8a5fd.png)
are positive integers but are not perfect squares.
such that![\[
ad = bc, \quad a + d = 2^k, \quad b + c = 2^m
\]](http://latex.artofproblemsolving.com/1/1/d/11d16732e3a77af46ad17adecf104a442a5d7f67.png)
for some positive integers 
and 
.
onto 
.
,
respectively. Finally, let the rays 
,
meet the circumcircle of 
in 
respectively. Prove that the tangent from 
meet on the circumcircle of 
be a triangle with circum-circle 
.
be a point in the interior of triangle 
.
meet 
respectively. Suppose 
is the point of intersection of 
with 
;
is the point of intersection of 
with 
.
is parallel to 
.
such that 
is a perfect square but none of the integers 
are perfect squares.
.
and 
intersect at 
.
be the midpoint of 
is chosen on the extension of 
beyond 
is chosen on the segment 
such that 
.
and 
.
bisects 
.
.
be the points where the incircle touches the sides 
respectively. Let 
meet the line 
respectively. Further assume that both 
are outside the triangle 
are concyclic.
.
and 
,
Y by MS_asdfgzxcvb
Let triangle 
have incenter 
and intouch triangle 
. Let the circumcircle of intersect 
at 
and have center 
. Let 
be the midpoint of arc 
on the circumcircle. Suppose quadrilateral 
is cyclic such that 
lies on 
. Show that 
.
have incenter 
and intouch triangle 
. Let the circumcircle of intersect 
at 
and have center 
. Let 
be the midpoint of arc 
on the circumcircle. Suppose quadrilateral 
is cyclic such that 
lies on 
. Show that 
.
are collinear. 
so 
.
hence 
is also on the polar of 
as desired
