Prove that
be a triangle with orthocentre 
.
to the circle with diameter 
touch this circle at 
and 
.
and 
be a sequence that consists of an initial block of 
positive distinct integers that then repeat periodically. This means that 
are 
for every positive integer 
.
prime numbers, find for which values of 
terms of the sequence.
Y by mathbuzz, Adventure10
Let 
. Consider a function 
. A subset 
of 
is said to be invariant if for all 
we have 
. The empty set and are also considered as invariant subsets. By 
we define the number of invariant subsets of for the function 
.
i) Show that there exists a function such that 
.
ii) Show that for every
there exists a function such that 
.
. Consider a function 
. A subset 
of 
is said to be invariant if for all 
we have 
. The empty set and are also considered as invariant subsets. By 
we define the number of invariant subsets of for the function 
.i) Show that there exists a function
such that 
.ii) Show that for every
there exists a function such that 
.This post has been edited 3 times. Last edited by Sayan, May 16, 2012, 7:18 AM
to be (vacuously) considered as an invariant subset, for any 
)
,
for all 
.
when 
o
,
has exactly 
subsets..
![$k \in [1,n]$](http://latex.artofproblemsolving.com/a/a/3/aa3a443ab1319cdf02766fc00339f297716ea9ec.png)
consider the function![\[f(x) = \begin{cases}
x + 1 & \text{for} , 1\le x\le n-k-1
\\
1 & \text{for} , x\ge n-k \end{cases} \]](http://latex.artofproblemsolving.com/6/c/6/6c6b534ca8ea910c89abbbbc7e78c4e39ad337d6.png)
and 
(Where 
is subset of 
)
just take a cycle, and to get to a permutation with 
from 
just take 
and a permutation 
of 
such that 
and we should be done.
points and applying the same construction in (i) to the remaining numbers then everything is cool
iff 
.
.
and 
.
,
,
,
,
number of subsets of 
(
use part (i).
