[/quote]
,
be nonnegative real numbers such that 
Prove that
Let 
Prove that
and 
with conditions,
is a prime number.
is a perfect square.
is also perfect square.
such that the numbers![\[n, f(n),f(f(n)),\dots,f^{m-1}(n)\]](http://latex.artofproblemsolving.com/b/3/0/b303b6dd25b8e172f41737ffa6c136dcc4db2266.png)
are distinct modulo 
for all integers 
with 
.
is defined by 
and 
for 
.
such that for every prime number 
and natural number 
,
is a complete residue system modulo 
for every natural number 
and 
.
be nonnegative real numbers such that 
.
,![\[abc\sqrt{1-d}+bcd\sqrt{1-a}+cda\sqrt{1-b}+dab\sqrt{1-c}\leq 1.\]](http://latex.artofproblemsolving.com/6/9/8/698270a4b6b055b71f9191de370eb674f1ff72f5.png)
be the circumcenter of triangle 
and let 
be the height from 
)
and 
be the midpoints of 
and 
respectively. Let 
and 
be the circumcircles of triangles 
and 
.
.
for 
and 
Prove that
,
bisects the angle between 
and the bisector of 
.
.
.
be a point on 
,
be a line passing through 
.
at 
and the circumcircle of triangle 
at 
)
and 
are tangent at 
prove that![\[a^r(a-b)(a-c) + b^r(b-a)(b-c) + c^r (c-a)(c-b) \geq 0 \]](http://latex.artofproblemsolving.com/0/6/c/06cdb779fc271d73a80bdec409c777e833dd2980.png)
(b) Find an inequality for non negative 
on the greater side.
for non negative 
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 
to 
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).
