AoPS Academy
for which 
is a perfect square.
, define 
as the smallest positive integer satisfying the following property: for any integer 
coprime with , we have 
.
, and integers 
all coprime with , prove that there exists a non-empty subset 
of 
such that![\[\prod_{i \in I} a_i \equiv 1 \pmod{n}.\]](http://latex.artofproblemsolving.com/8/f/a/8fadde1e14e28dda0cb2371b69e783fd7e572352.png)
Proposed by Zhenqian Peng from High School Affiliated to Renmin University of China
and 
functions are given with following properties:
is strict increasing and for each 
there holds 
or 
.
if holds and 
if holds.
and 
such that for all 
we have 
![$f:[-1,1]\to\mathbb{R}$](http://latex.artofproblemsolving.com/5/a/e/5aee4d7ba286e59465514a88e4d50c68e96d2a4b.png)
be a continuous function such that
for every 
in ![$[-1,1],$](http://latex.artofproblemsolving.com/3/a/9/3a9a9276194df4e3ca7e915b63a202e7920fd4a4.png)
and
exists and is finite. is unique, and express 
in closed form.
is a random variable that takes on only nonnegative integer values, with ![$E[X]=1,$](http://latex.artofproblemsolving.com/e/8/2/e826f5b79a2adeacdaccd3884658e94a3b1f8e22.png)
![$E[X^2]=2,$](http://latex.artofproblemsolving.com/5/9/1/5915ee4a36f8615ddc8075a197f3b44a902936c6.png)
and ![$E[X^3]=5.$](http://latex.artofproblemsolving.com/d/1/1/d11c925af9f188f0140a89a0fbd6c83e1dd4fc12.png)
(Here ![$E[Y]$](http://latex.artofproblemsolving.com/a/2/9/a29ea940a5d54ad2a5d668a596f9db0c88fa1123.png)
denotes the expectation of the random variable 
) Determine the smallest possible value of the probability of the event 
and 
be real numbers in the interval 
, and let 
be a continuous real-valued function such that 
for all real . Prove that 
for some constant 
.
such that![\[f'(x)=\frac{f(x+n)-f(x)}n\]](http://latex.artofproblemsolving.com/8/5/7/857920b9c714ca96bb50dfdebcad18d37287db99.png)
and all positive integers 
![$\max_{a,\,b,\,c>0\atop a+b+c=1}(a+\sqrt{b}+\sqrt[3]{c})$](http://latex.artofproblemsolving.com/e/d/8/ed81d27343068e602406c53746b21abdb401f44a.png)
The sourse: 
