and 
,
cookies from one of the piles, of which he eats 
and puts 
for which Marica has a winning strategy.
be non-negative real numbers such that 
Prove that
Y by Adventure10
Given is an acute triangle 
. 
is an arbitrary point at the side 
and 
is midpoint of 
. The foots of the perpendiculars from 
to 
and 
are points 
and 
. Prove that center of the circumcircle of triangle 
lies on the fixed line for all points from the side .
. 
is an arbitrary point at the side 
and 
is midpoint of 
. The foots of the perpendiculars from 
to 
and 
are points 
and 
. Prove that center of the circumcircle of triangle 
lies on the fixed line for all points from the side .
meets 
at 
.
is the diameter of circle 
so we only need to prove that 
is the projection of 
onto 
meets circle 
at 
.
thus 
is a concyclic quadrilateral. Hence 
which implies that 
is the tangent of 
.
as 
.
be an arbitary point on 
as 
varies over the segment. Similarly, define 
as circumcenter for point 
Now prove that 
is ![\(\angle [90^{\circ} + B] \)](http://latex.artofproblemsolving.com/6/f/3/6f3d60a5dd3c6ca4ff5e97f0cd942aef934af464.png)
i.e., it does not depend upon the location of 
which has fixed angle measures the required circumcenters lie on a fixed line. [\Hide]
such that 
,
the reflection of 
about 
.
reflection of 
so 
is cyclic. Clearly 
is cyclic.
means 
cyclic. Hence the circumcentre lies on the perpendicular bisector of 
,
