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.
it is fixed positiv number.
, where 
are integer numbers .
such that:
be a triangle with circumcenter 
. Let 
be the midpoints of 
and 
respectively and let 
be the projection of on 
. Let 
be the projection of 
on 
. Let 
intersect 
at points 
and 
. Let 
intersct at points 
. Let 
intersect 
at 
and 
intersect at 
. Then show that 
are concurrent.
and 
respectively be the incentre, circumcentre and circumcircle of triangle 
. Points 
are chosen on , such that 
, and point 
is the foot of the altitude from 
to 
. If 
are similarly defined, prove that lines 
and 
concurr on 
.
be a sequence that consists of an initial block of 
positive distinct integers that then repeat periodically. This means that 
are distinct positive integers and 
for every positive integer 
. The terms of the sequence are not known and the goal is to find the period . To do this, at each move it possible to reveal the value of a term of the sequence at your choice. is one of the first 
prime numbers, find for which values of there exist a strategy that allows to find revealing at most 
terms of the sequence.
such that, for all real numbers 
with 
, 
of positive integers for which there exist positive integers 
and 
such that
holds for all integers 
(Note that 
denotes the greatest common divisor of integers 
and 
)
other than a polynomial satisfying
Prove that
and let 
and 
be tangency points of the incircle of 
,
and 
respectively. Let the circle with centre 
and 
.
,
and 
,
.
and 
be feet of perpendiculars from 
and 
respectively. Prove that 
,
and 
are concurrent.
s
(the constant 
) .Prove that
are interior angles of a triangle, the condition 
from which it follows that
where 
,
,
and 
with 
.
,
,
thereby
(the constant 
from which it follows that
Since 
,
,
when 
;
when 
,
.
is 
if 
,
and 
,
,
.
.
) .Then
(the constant 
(the constant 
from which it follows that
Since that 
thereby 
,
.
has no maximum. 
from which it follows that
where 
and 
with 
.
.
.
.
(the constant 
(the constant 
Where 
Where 
Where 
Prove that
Where 
and 
Prove that 
Let 
Prove that 
Let 
Prove that 
Let 
Prove that 
Let 
Prove that 
Let 
Prove that 
Let 
Prove that 
Let 
Prove that 
Prove that 
Let 
Prove that 
.![$abc\geq 3\sqrt[3]{abc}+2$](http://latex.artofproblemsolving.com/7/a/b/7ab2f6a8d1099cb193ad35dfce08897d7eb25f33.png)
.
,
which is equivalent to 
which asserts 
or 
.![$(ab+bc+ca)^{2}\geq 3abc(a+b+c)\geq 9abc\sqrt[3]{abc}=144$](http://latex.artofproblemsolving.com/a/2/c/a2c95a4971acdf865fcaf77902f9242f512ba420.png)
.
![$$ab+bc+ca\geq 3\sqrt[3]{(abc)^2}\geq12$$](http://latex.artofproblemsolving.com/2/c/9/2c988b04ff2bc784b81538dd2cbe02628a7cfd41.png)
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
Let 
and 
Prove that
such that 
Prove that
be non-negative numbers such that 
In 
and 
Prove that
and 
prove that:
Ukraine2008？
Prove that
Let 
P
Prove that
we are done!
Let 
,
,
be positive real numbers such that 
.
Let 
be positive real numbers such that 
![\[\frac{1}{a+2}+\frac{1}{b+2}+\frac{1}{c+2}\leq 1.\]](http://latex.artofproblemsolving.com/6/9/b/69b62a5866ba96e43b0c59f89daae9710a896c45.png)
.Prove that
,
,
by Muirhead 
.
be positive real numbers . Prove that
Where 
be positive real numbers such that 
Prove that
