Herzog Competition - 1987

Michigan State University — Herzog Mathematical Competition

November 7, 1987

Problem 1

Prove that $\frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}$ is not an integer for any n.

Problem 2

Hills — If {aₙ} is a sequence of positive real numbers such that

\sum_{n = 1}^{\infty} e^{- a_{n}}

diverges, show that

\sum_{n = 1}^{\infty} {a_{n}}^{- k}

also diverges for any fixed positive integer k.

Problem 3

Asic — Consider all differentiable functions, f(x), defined for all real x such that f(0) = f′(0) = 1 and |f′(x) − 1| ≤ |x|.

By continuity each such function is nonnegative in a small interval around x = 0. Prove even more: there is an interval in which all the functions are nonnegative and find the length of the largest such interval.

Problem 4

Herzog — Let α = {0,1,3,4,9,···} be the set of all those nonnegative integers which when written in base 3 have no digits equal to 2. Prove

(a) no three numbers in α are in arithmetic progression;

(b) if b is any positive integer not in α then there exists two numbers in α, a₁ and a₂ with a₁ < a₂ < b and a₁, a₂, b are in arithmetic progression.

Problem 5

S = {P₁,P₂,···,Pₙ} is a set of n points in the plane with no two distances d(PᵢPⱼ) equal. Each point is joined by a segment to its nearest neighbor.

Prove: (a) no subset of these segments forms a closed polygon; (b) the number of these segments with a common endpoint is at most 5. Can 5 be attained?

Problem 6

Sonneborn — Five professors attending a seminar find it hard to stay awake. On one occasion

(a) each professor fell asleep exactly twice during the lecture and

(b) each two professors were simultaneously asleep at least once.

Prove that at least three must have been simultaneously asleep at some time during the lecture.