[/quote]
,
such that 
and 
are coprime if and only if 
?
,
be the projection from 
onto 
be a point on the extension of 
past 
.
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 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
and 
intersect 
and 
at 
.
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 ![\[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]](http://latex.artofproblemsolving.com/0/6/2/0623b7c328af6d088cfc5dd288afe3e17a94ff29.png)
for all reals 
and 
.
to 
.
)
Y by HWenslawski
Circle 
is an exscribed circle of triangle 
tangent to side 
at point 
and extensions of sides 
and 
at points 
and 
respectively. A circle with diameter intersects the segment 
at the points 
(the point 
belongs to the segment 
). Prove that 
and 
intersect at the center of the circle .
is an exscribed circle of triangle 
tangent to side 
at point 
and extensions of sides 
and 
at points 
and 
respectively. A circle with diameter intersects the segment 
at the points 
(the point 
belongs to the segment 
). Prove that 
and 
intersect at the center of the circle .This post has been edited 1 time. Last edited by parmenides51, Jul 22, 2021, 3:31 PM
Reason: typo: exscribed instead of inscribed, see #3
Reason: typo: exscribed instead of inscribed, see #3
and 
- intersection of 
and 
.![$$\left[\angle LBR=\angle DBC,\ \angle BLR=90^\circ-\frac12\angle BAC=\angle BDC\right]\implies \triangle BLR\sim\triangle BDC$$](http://latex.artofproblemsolving.com/a/2/c/a2c7f277416bbc7b7926ce70dc53b438634cc8e5.png)
quadrilateral 
is cyclic. Hence 
.
.
.
,
are midpoints of sides 
respectively, then 
.
be the intersection of 
with 
Hence
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