[/quote]
,
such that 
and 
are coprime if and only if 
?
be an acute triangle with 
,
be the projection from 
be a point on the extension of 
past 
.
be on the perpendicular bisector of 
such that ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](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 cyclic quadrilateral with the circumcircle 
. The points 
and 
are symmetric to 
with respect to the midpoints of 
and 
. The circumcircle of the triangle 
intersects at 
and 
. Prove that 
is the diameter of .
be a cyclic quadrilateral with the circumcircle 
. The points 
and 
are symmetric to 
with respect to the midpoints of 
and 
. The circumcircle of the triangle 
intersects at 
and 
. Prove that 
is the diameter of .This post has been edited 1 time. Last edited by parmenides51, Dec 25, 2020, 5:39 PM
as translation 
by vector 
where 
are circumcenters of 
and so 
equal 
between 
which is 
