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 positive integer and let 
be an irrational number. Show that for every integer 
, there exists a positive integer 
such that 
.
and 
be positive integers with 
. In a group of people, each one or always
, what is the parity of people who speak to is a subset of size 
of the set of people. The answer can only
” or “
”. and is it possible to determine which people speak the truth and
be pairwise different positive real numbers such that![\[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\]](http://latex.artofproblemsolving.com/8/a/e/8ae0bd1f55eb2fb0b7209b62739a9b98f462ee56.png)
is an integer for every 
Prove that 
of positive integers that satisfy
be a monic polynomial with complex coefficients such that there exist a polynomial 
with complex coefficients for which ![\[P(x^2-2)=P(x)Q(x).\]](http://latex.artofproblemsolving.com/a/6/2/a62b06890b6bbcc5c9634f6de8c7ae5fe9f37828.png)
Determine all complex numbers that could be the root of .
, there exist integers 
such that, if we let 
, then the following conditions hold:
we have 
; the number 
is divisible by ; and the numbers 
each have a prime factor not dividing .
is irreducible for every natural number 
.
be positive real numbers satisfying![\[
a^2 + c^2 = b(a + c).
\]](http://latex.artofproblemsolving.com/9/c/0/9c0730d15b1177ee23d3f65bc2fe8b2cc85d0c62.png)
Let![\[
m = \min \left( \frac{a^2 + ab + b^2}{ab + bc + ca} \right).
\]](http://latex.artofproblemsolving.com/e/b/6/eb6f340defa4b38e35292062b202d138b9de843b.png)
Find the value of 
.
be a function such that
for all 
. Determine all such functions 
.
be real numbers such that 
Prove that 
Where 
. Find all 
such that for all 
is arbitrarily taken on side 
of a triangle 
. Let 
be the intersection of the perpendiculars from the midpoints of segments 
and 
to lines 
and 
, respectively. For which position of is the distance 
minimal?
