[/quote]
,
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, 
)
are there such that 
and 
?
be an acute triangle with 
,
be the projection from 
onto 
.
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 
and 
and 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
,
and 
intersect 
and 
at 
and 
.
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 
,
,
is a positive integer.
of positive integers 
such that 
.
denotes component-wise function multiplication)
and 
Prove that
distinct points 
connected if the line 
points from 
such that there exists a set 
is connected with exactly 
other points from 
, where 
is a real number. Prove that for all positive 
then prove
Y by CrazyMathMan
Rectangle 
is split by two straight lines, intersecting at point 
, into four rectangles.
a) Prove that if the point lies on the diagonal 
, then the areas of the upper left and lower right rectangles are equal (they are shaded in the figure).
b) Let it be known that the areas of the upper left and lower right rectangles are equal. Does the point then necessarily lie on diagonal ?
is split by two straight lines, intersecting at point 
, into four rectangles.a) Prove that if the point
lies on the diagonal 
, then the areas of the upper left and lower right rectangles are equal (they are shaded in the figure).b) Let it be known that the areas of the upper left and lower right rectangles are equal. Does the point
then necessarily lie on diagonal ?This post has been edited 1 time. Last edited by parmenides51, Jun 16, 2021, 6:30 PM
