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 set of integers containing the numbers 
and 
. Suppose further that any integer root of any non-zero polynomial with coefficients in also belongs to . Prove that 
belongs to .
be real numbers such that 
and 
Prove that
people arranged in a circle, and 
coins are distributed among them, where each person has at least 
coins. Each person is then assigned a random index number in 
such that no two people have the same number. Then every minute, if 
is the number of minutes passed, the person with index number congruent to mod will give a coin to the person on his left or right. After some time, everyone has the same number of coins. is this always possible, regardless of the original distribution of coins and index numbers?
be a convex cyclic quadrilateral with circumcircle 
. Let 
produced beyond 
meet 
produced beyond 
, at 
. Let 
be a line through meeting 
and 
at 
and 
respectively, so that 
(respectively 
) are on opposite sides of (resp. 
). Suppose 
and 
are points on the arc 
of not containing 
so that 
are tangent to . Prove that 
are collinear.
.
.
Where 
.![$\sqrt[3]{ab}+ \sqrt[3]{cd} < \sqrt[3]{(a+b+c)(b+c+d)}$](http://latex.artofproblemsolving.com/0/5/0/0502ba65f0545b3af84c41b2549d6256550b473b.png)
.
be a sequence of reals such that 
and 
for all positive integers 
approaches the real number 
