AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be a set of Natural numbers from 
to 
.
(
) distinct elements from this set.
. If 
and 
, show that the area of 
is not less than the area of 
, and the pairwise comparison of the sums of elements of all its subsets (with the empty set defined as having a sum of 0), which amounts to 
inequalities, these given comparisons satisfy the following three constraints:
and 
, if the sums of elements of and are greater than those of subsets 
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 
, by 
the midpoint of , and by 
the orthocenter of . Let 
be the point of intersection of the circumcircle 
of the triangle and the half line 
, and 
be the point of intersection (other than 
) of the line 
and the circle . Prove that 
must hold.
the length of the line segment .)
such that 
(where 
), and a functional equation 
that satisfies 
for all , has a finite number of solutions. Find such .
is the function obtained by composing 
times, that is, 
)
are there such that 
and 
?
be an acute triangle with 
, and let 
be the projection from onto 
. Let be a point on the extension of 
past such that 
. Let 
be on the perpendicular bisector of 
such that and 
are on the same side of and![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of across 
and 
be 
and 
, respectively. Let 
and 
such that 
. Let 
and 
intersect the circumcircles of 
and 
at 
and 
, respectively. Let 
and 
intersect 
and 
at 
and 
. Let 
intersect at 
. Prove that 
.
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 , 
. Can the sum of the cosines of two of them be equal to the cosine of the third?
![$cos(a)cos(b)cos(c)[1-tan(a)tan(b)-tan(a)tan(c)-tan(b)tan(c)]= cos(a+b+c) = 0 = cos(a)cos(b)cos(c)-sin(a)sin(b)cos(c)-sin(a)cos(b)sin(c)-cos(a)sin(b)sin(c)$](http://latex.artofproblemsolving.com/5/7/1/571c82b9ec1fa429335b252e3b6b7ea655bd7b26.png)
![$cos(a)cos(b)cos(c)[1-tan(a)tan(b)-tan(a)tan(c)-tan(b)tan(c)]=0$](http://latex.artofproblemsolving.com/1/1/7/117aa0c917ddee598c5a578af0ace34212dafd7a.png)
![$[1-tan(a)tan(b)-tan(a)tan(c)-tan(b)tan(c)]=0$](http://latex.artofproblemsolving.com/8/7/3/8734f2d951e96f90eabab7711161be33e1df3e4b.png)
=0
.
then prove
)
a bit
Because 
,
with equality when 
.
.