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 
by 
and 
for all 
, where 
Show that 
is an integer for all 
.
, is defined in plane polar coordinates 
by the inequality
, where 
and 
.
from 
to 
and 
from 
to 
but in vain.
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.
containers placed along the track. Prove that there exists a position starting at which the car, beginning with an empty tank of gas, can complete a loop around the track without running out of gas. The tank of gas is assumed to be large enough.
there exist nonnegative integers 
and integers 
such that 
such that there exists a unique integer 
such that 
with the following property:![\[ n!\mid a^n + 1
\]](http://latex.artofproblemsolving.com/e/2/3/e23d39aab685888993d355d8c93d4e6475f53487.png)
which are larger than 
and less than 
Prove that
Is there any reason why we shouldn't write 
instead?
Is there any reason why we shouldn't write instead?
, 
or a 
notation.
and 
Prove that
Prove that