[/quote]
,
such that 
and 
are coprime if and only if 
?
be an acute triangle with 
,
be the projection from 
onto 
.
be a point on the extension of 
past 
.
be on the perpendicular bisector of 
such that 
are on the same side of ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of 
and 
be 
and 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
,
and 
intersect 
and 
at 
and 
.
intersect 
.
.
chessboard into 
non-overlapping rectangles subject to the following conditions:
is the number of white squares in the 
-
.
.
denote the determinant of a square matrix 
.![$g:\left[0, \frac{\pi}{2}\right] \rightarrow R$](http://latex.artofproblemsolving.com/5/3/4/534fcbbc20e62de2dfa2579cd0642663c9fc3380.png)
be the function defined by
where 
.
be a quadratic polynomial whose roots are the maximum and minimum values of the function 
and 
.
and orthocenter 
,
and that line 
is not parallel to line 
is perpendicular to 
.
be the points on the circumcircle of triangle 
is a diameter, and 
and 
are parallel. Let 
.
and 
are perpendicular.
and 
are perpendicular.
such that![\[(a^2-b^2)(b^2-c^2) = abc.\]](http://latex.artofproblemsolving.com/7/1/8/7188261c40a45f5e15533100432d36abbf515efd.png)
.
that 
is a prime.
sastifying 
such that 
be a monic polynomial with integer coefficient. And suppose there exist 4 distinct integer 
such that 
.
such that 
such that ![\[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]](http://latex.artofproblemsolving.com/0/6/2/0623b7c328af6d088cfc5dd288afe3e17a94ff29.png)
for all reals 
and 
.
to 
.
)
Y by
Let 
be a triangle with 
. Let 
be the midpoint of 
, and let 
be a point on segment such that 
. Let 
be the point of intersection, different from 
, of the circumcircle of triangle 
and line 
. Define 
and 
as the points of intersection of line 
with 
and 
, respectively. Prove that is the midpoint of 
.
be a triangle with 
. Let 
be the midpoint of 
, and let 
be a point on segment such that 
. Let 
be the point of intersection, different from 
, of the circumcircle of triangle 
and line 
. Define 
and 
as the points of intersection of line 
with 
and 
, respectively. Prove that is the midpoint of 
.
,
.
which yield the midpoint.
and also let 
hit 
and consider 
is an isosceles trapezoid.
from the angles we get 
isosceles and thus from arcs 
,![\[ -1=(A, C: M, \infty_{AC}) \overset{G}{=} (A, B; M, F) \overset{E}{=} (Q, B; P, \infty_{BC}) \implies BP=PQ \]](http://latex.artofproblemsolving.com/b/8/f/b8f56dea24231c232c025c42e0f2ec5815356085.png)
Thus we are done 
.
be the reflection of 
be the midpoint of 
.
are collinear.
are collinear
Hence 
are collinear because they all lie on line 
sends 
,
is a parallelogram.
,
is an isoceles trapezoid.
.
,
