[/quote]
,
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 
.
sastifying 
be a set of Natural numbers from 
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 midpoint of 
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.
the length of the line segment 
such that 
(where 
)
that satisfies 
for all 
is the function obtained by composing 
times, that is, 
)
be an acute triangle with 
,
be the projection from 
.
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 
.
.
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 
Y by
Let the quadrilateral 
be inscribed in a circle with center 
, with obtuse angles 
and 
. Let 
be the intersection point of the lines 
and 
. 
and 
are the feet of the perpendiculars drawn from the point on 
and 
, respectively. Let 
be the intersection point of the lines 
and . Let 
be the intersection point of the lines 
and . Denote by 
the midpoint of the segment 
. Prove that the points 
lie on the same line.
be inscribed in a circle with center 
, with obtuse angles 
and 
. Let 
be the intersection point of the lines 
and 
. 
and 
are the feet of the perpendiculars drawn from the point on 
and 
, respectively. Let 
be the intersection point of the lines 
and . Let 
be the intersection point of the lines 
and . Denote by 
the midpoint of the segment 
. Prove that the points 
lie on the same line.This post has been edited 3 times. Last edited by parmenides51, Jun 21, 2022, 3:06 AM
