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 scalene triangle, 
be the median through 
, and 
be the incircle. Let touch 
at point 
and segment 
meet for the second time at point 
. Let 
be the triangle formed by lines and and the tangent to at . Prove that the incircle of triangle is tangent to the circumcircle of triangle .
be real numbers such that 
Prove that
Where 
Where 
are are the lengths of the sides of a 
gon such that
then![$$(n-2)\left[\sum_{i=1}^{n}{\frac{a_i^2}{(1-a_i)^2}}-\frac n{(n-1)^2}\right]\geq(2n-1)\left(\sum_{i=1}^{n}{\frac{a_i}{1-a_i}}-\frac n{n-1}\right)^2.$$](http://latex.artofproblemsolving.com/9/3/e/93e5529518807b92cd9e0382f5fb07a293128b04.png)
be an acute triangle. Tangents on the circumscribed circle of triangle at points 
and 
intersect at point . Let 
and 
be a foot of the altitudes from onto 
and 
and let 
be the midpoint of . Prove: is the orthocenter of the triangle 
.
cuts 
in half.
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
there are elements 
in an arithmetic progression in the order 
such that 
.
(for 
) form an arithmetic progression. If 
is an injective function, then the following properties hold:
, we have 
(i.e. one-to-one).
is an increasing function such that 
.
holds for any set of distinct positive integers that forms an arithmetic progression.
is an injective function, then the following properties hold: is an increasing function such that .
and we focus on only 
, which 
could be decreasing?
. Then, there exists an 
such that 
due to injectivity. WLOG suppose 
. Now, I claim that there exists an 
such that 
for all 
. To show this, assume the contrary, Then, for any 
, there exists an 
such that 
. Then there exists an 
such that 
by the same logic, etc., so we have 
, impossible. So, there exists some 
such that 
for all 
. In particular, 
. Furthermore, since 
, 
and 
. Thus, 
form an arithmetic sequence and 
, as desired.
there are elements in an arithmetic progression in the order such that .
is finite, there are finitely many so that 
. Therefore, for sufficiently large we have that 
, so by our assumption we require 
. But then pick such a , and note the chain![\[ f(k) > f(2k - 1) > f(4k - 3) > f(8k - 7) > \cdots \]](http://latex.artofproblemsolving.com/1/d/2/1d22ac0007dbd2448c76bdc835f0560cc46d5f12.png)
contradicting the well ordering principle.
and 
.
,