It is well known that 
.
be positive numbers such that 
.![\[T = \dfrac{bc}{\sqrt{a}+3}+\dfrac{ca}{\sqrt{b}+3}+\dfrac{ab}{\sqrt{c}+3} \]](http://latex.artofproblemsolving.com/6/d/5/6d570d06679f17452790b90171428d645e97fe9f.png)
be the incenter of a triangle 
with 
,
be the midpoint of the arc 
of the circumcircle of the triangle. The perpendicular line to 
passing through 
at point 
.
intersects the circumcircle of triangle 
at point 
.
is tangent to the circumcircle of triangle 
such that there exist infinitely many rational numbers 
satisfying![\[
\left| \frac{q}{p} - \frac{\sqrt{5} - 1}{2} \right| < \frac{\alpha}{p^2}.
\]](http://latex.artofproblemsolving.com/1/1/6/1162c6a7215bff2f9fdaa042a6952f86524e9e92.png)
be a triangle with incenter 
,
and circumcircle 
.
be the midpoints of sides 
,
,
and let 
be the tangency points of 
be the intersections of line 
with line 
and line 
,
be the midpoint of arc 
of 
.
bisects 
.
be the set of integers, and let 
be a function. Prove that there are infinitely many integers 
such that the function 
defined by 
is not bijective.
,
such that 
.
be the orthocenter of acute triangle 
be the foot of the altitude from 
to 
,
be the reflection of 
.
intersects line 
.
.
lie on a circle 
and point 
and 
are tangent to 
are collinear, and (iii) 
.
bisects 
.
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 
.
disjoint directed cycles. The number of invariant subsets are equivalent to the subsets of the set whose elements are the 
,
,
,
,
number of subsets of 
(
use part (i).
