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.
of 
, define[list]
;
;![\[
f(\pi) = (-1)^{\tau(\pi)} \cdot 2^{\sigma(\pi)} \cdot \frac{\sigma(\pi)}{\sigma(\pi) + 1}.
\]](http://latex.artofproblemsolving.com/f/f/0/ff0e3a0f6b372e096ba92121227f26e22dcafecd.png)
[/list]
permutations of .
be an odd prime number prove that there exists a positive integer 
such that and 
are both primitive roots modulo
be a set of size 
integers. Show that exist some 
such that
for some constant 
.
be distinct positive integers. Prove that
Further, determine when equality holds.
be a positive integer. Find the smallest positive integer 
such that for any set 
of points in the interior of the unit square, there exists a set of rectangles such that the following hold: is not in the interior of any rectangle. is in the interior of at least one of the rectangles
such that for all 
and 
then 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
be the 
altitude of the triangle 
and 
be foots of perpendicular lines through 
on 
, respectively. Construct the parallelogram 
and altitudes 
of the triangle . Prove that 
lies on 
permutations and for 
only the relative orders 
are allowed. Hence we divide by 3 and we find 
.
![$ a,b,c \in[0,1],ab+bc+ca=1. $](http://latex.artofproblemsolving.com/a/a/7/aa7c35c03319810bad861b81a14a6819c83b8892.png)
Prove that
prove by Cauchy Inequality?
Prove that
,
players named 
is to be sent for batting. If 
and 
Then find the total number of batting orders if other players could go in any order.
