[/quote]
,
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, 
)
are there such that 
and 
?
be an acute triangle with 
,
be the projection from 
.
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 
be 
and 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
,
and 
intersect 
and 
at 
and 
.
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 
, where 
is a real number. Prove that for all positive 
then prove
Y by
Prove the following:
a) If
is a right prism and 
such that 
and 
are perpendicular each other and concurrent, then the prism is regular.
b) If is a regular prism and 
, then there are so that the lines and are perpendicular each other and concurrent.
a) If
is a right prism and 
such that 
and 
are perpendicular each other and concurrent, then the prism is regular.b) If
is a regular prism and 
, then there are so that the lines and are perpendicular each other and concurrent.This post has been edited 2 times. Last edited by parmenides51, Aug 15, 2024, 7:43 AM
