UT Math Club Problem of the Month
September 2010
Problem I (Struggling with Lothar’s Legacy) Let Z+ denote the set of positive integers, let p ∈ Z+ be an odd prime, and define the function
hp : Z+ → Z+
by the following formula:
(
hp (n) =
n+p
n
2
if n is odd
if n is even
Now for each non-negative integer m ≥ 0 let Kp (m) denote the number
of positive integers n ∈ Z+ such that hm (n) = 1, where hm (n) := (h ◦
· · · ◦ h)(n) denotes m-fold function composition (e.g. h2 (n) = (h ◦ h)(n) =
h(h(n))). Determine whether or not the following limit exists:
lim Kp (m)
m→∞
and if it does, calculate the limit.
Problem II (Separation Theorem for Conic Sets) Let C ⊂ Rn be a non-empty set
which satisfies the following property:
If x, y ∈ C and λ, µ ≥ 0 then λx + µy ∈ C.
(We say that C is a conic set if it satisfies this property). Now let z ∈
Rn \C (i.e. z ∈ Rn and z ∈
/ C). Prove that there exists some vector v ∈ Rn
such that hv, zi > 0 and such that hv, ci ≤ 0 for all c ∈ C.
Submission Deadline: Thursday, September 30 by 11:59 PM
1