be an acute triangle with incentre 
and 
.
and 
intersect the circumcircle of 
and 
,
such that 
and 
are parallelograms (with 
,
)
be the point of intersection of lines 
and 
.
,
,
denote the foot of the altitude from 
to 
.
with diameter 
intersect altitude 
at distinct points 
and 
.
passing through point 
at 
.
meets 
.
intersects 
at 
intersects 
.
:o
so 
and since 
is just 
therefore 
so 
feet and area 
such that![\[
\sqrt{x - 2011} + \sqrt{2011 - x} + 10
\]](http://latex.artofproblemsolving.com/4/8/b/48b82d13b733947420a0d763c6af74636770835a.png)
is an integer.
to 
have the digit 
and are divisible by 
?
Y by
Let 
be a rectangle and 
any point. Let 
be the foots of the perpendiculars from to the lines 
, respectively. Let the perpendicular bisectors of 
and 
intersect at 
, and those of 
and 
intersect at 
. Prove that the lines 
and 
are parallel.
be a rectangle and 
any point. Let 
be the foots of the perpendiculars from to the lines 
, respectively. Let the perpendicular bisectors of 
and 
intersect at 
, and those of 
and 
intersect at 
. Prove that the lines 
and 
are parallel.Attachments:
This post has been edited 4 times. Last edited by buratinogigle, Apr 17, 2025, 12:10 AM
Reason: Typo
Reason: Typo
