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.
is acute triangle. 
is orthocenter, 
is the midpoint of 
, 
is the midpoint of smaller arc . Point 
is on 
such that, 
is perpendicular to 
. Prove that: 
is paralelogram
is the midpoint of , 
bisects 
and intersects 
at , 
is the incenter of 
, 
bisects 
and intersects the circumcircle of 
at 
, 
is parallel to and intersects at 
. Prove that 
written on the board.At every step,Mikail chooses the two numbers 
and substitutes them with 
and 
where 
is the unchosen number on the board. Prove that at least 
negative number must be remained on the board at any step.
positive real numbers,prove the inequality:![$\sqrt[7]{\frac{a}{b+c}+\frac{b}{a+c}}+\sqrt[7]{\frac{b}{a+c}+\frac{c}{a+b}}+\sqrt[7]{\frac{c}{a+b}+\frac{a}{b+c}} \geq 3$](http://latex.artofproblemsolving.com/a/d/4/ad44b49424a2a1891161c4d6bf099e1f6dc63bee.png)
.
have the same incenter .Prove that 
is concyclic iff 
is concurrent
and satisfy 
, then 
is an even number.
in order to make it a perfect square?
are 
and 
Find the product of all possible values of 
such that 
is a perfect square and 
is a prime number.
be real numbers less than 2.Then prove that 
with 
is integer
can be prime only if 
so has at most two elements has integer root and non integer root. Then 
where 
is second non-integer root is prime then or 
is prime, then 
can be 
or 
then both 
and can not be integer at same time, because their difference is less than so has at most three elements
has integer root 
and 
are prime values, so can have 
values
and 
Prove that
