[/quote]
,
to 
.
)
.
and 
,
is not less than the area of 
,
inequalities, these given comparisons satisfy the following three constraints:
and 
,
and 
respectively, then the sum of elements of the union 
is greater than that of 
.
that satisfies all these conditions?
.
be an acute triangle. Denote by 
the foot of the perpendicular line drawn from the point 
to the side 
,
the orthocenter of 
be the point of intersection of the circumcircle 
of the triangle 
,
be the point of intersection (other than 
)
and the circle 
must hold.
such that 
(where 
)
that satisfies 
for all 
is the function obtained by composing 
times, that is, 
)
,
.
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 
be 
and 
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
,
and 
intersect 
and 
at 
intersect 
.
.
sastifying 
such that the relation ![\[|z^{2n}+z^nw^n+w^{2n} | = 2^{2n}+2^n+1 \]](http://latex.artofproblemsolving.com/1/2/2/1229a609403904449c447225094b3d97e063efd4.png)
holds for all positive integers 
,
,
of positive integers 
such that 
.
denotes component-wise function multiplication)
and 
Prove that
Y by SHARKYKESA, Mango247
Let 
be an isosceles triangle with 
and 
. Let 
be the midpoint of 
and 
be the circumcircle of 
. Let 
be a point on . Suppose that the circle centered at passing through 
intersects at distinct points 
and 
. Let 
and 
lie on 
such that 
and 
are tangent to , and denote by 
the intersection of and .
Prove that regardless of the choice of, the triangle 
has constant perimeter.
be an isosceles triangle with 
and 
. Let 
be the midpoint of 
and 
be the circumcircle of 
. Let 
be a point on . Suppose that the circle centered at passing through 
intersects at distinct points 
and 
. Let 
and 
lie on 
such that 
and 
are tangent to , and denote by 
the intersection of and .Prove that regardless of the choice of
, the triangle 
has constant perimeter.
