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.
with the following property: the 
positive divisors of have a permutation 
such that for 
, the number 
is a perfect square.
, and it is possible to arrange all its members around a round table so that every two neighbors are enemies. Given that there are at most 
unsociable groups, prove that it is possible to partition the company into 
parts so that no two enemies are in the same part.
on a plane with centers 
, respectively. 
is feet of the perpendicular from 
to 
and 
and 
are intersections of line 
with circle 
. Points 
and 
can move around the circle 
and 
, respectively. Find the maximum possible value of 
.
+1 w
and 
, respectively. Randomly picking from bin B, the probability of choosing balls marked X and Y are and , respectively. If the contents of the two bins are merged into one bin, what is the probability of choosing two balls marked X and Y from this bin?
be reals such that 
Prove that
Let be reals such that 
Prove that
to 
) until they have chosen seven such digits, and then consider the resulting seven-digit number by concatenating the digits in the order selected, with the seventh digit appearing last (i.e. 
). Alice wins if and only if the resulting number is the last seven decimal digits of some perfect seventh power. Please determine which player has the winning strategy.
,
and 
Prove that
