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.
. Prove that
in order to make it a perfect square?
is 
and 
. Prove that 
is 
degrees, where 
and 
are the midpoints of 
and 
, respectively.
and 
be positive real numbers such that 
. Find the minimum value of 
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.
![$\sqrt[2]{2}$](http://latex.artofproblemsolving.com/1/8/6/1869ac018fb72a26a12ab14499cbf1b1c13b4a10.png)
, ![$\sqrt[3]{3}$](http://latex.artofproblemsolving.com/1/7/a/17a6bb5dc11696b8650b6d3a11e770f861f4c326.png)
, and ![$\sqrt[5]{5}$](http://latex.artofproblemsolving.com/a/b/9/ab90f2d42dc24065e2a64988fe76f22eb9359799.png)
.
, that is, with nonzero, let 
be the product 
.
and 
. with at least one 
amoung their digits satisfy that is a power of 2?
and 
.
and 
Prove that
,
.
must be either powers of 
or 
work, where each element is either a fixed 
or 
,
