Herzog Competition - 1988

THE SIXTEENTH HERZOG PRIZE EXAMINATION

November 5, 1988

Problem 1

Determine (with proof) which is larger, ${(\frac{21}{20})}^{100}$ or 121.

Problem 2

Determine (with proof) whether an equilateral triangle can be placed in the plane so that the coordinates of all three vertices are integers.

Problem 3

Call a subset of {1,2,...,n} unfriendly if it contains no two consecutive integers. Prove that the number of unfriendly subsets of {1,2,...,n} is the (n+1)st Fibonacci number Fₙ₊₁. (Here the empty set counts as an unfriendly set. The Fibonacci numbers are defined by the recursion Fₙ₊₁ = Fₙ + Fₙ₋₁ with F₀ = F₁ = 1.)

Problem 4

Determine (with proof) for which integers n

f (n) = n (n + 1) (n + 2) (n + 3) + 1

is a perfect square.

Problem 5

Prove that the series

\sum_{n = 0}^{\infty} (3 x - x^{2}) {(\frac{2 x^{2} - 5 x + 2}{x + 2})}^{n}

converges for x in the interval [0,3]. Let f(x) be the sum of this series. Sketch the graph of f over [0,3].

Problem 6

Let v₁, v₂, ..., vₙ be the (consecutive listed) vertices of a regular polygon inscribed in the unit circle. Prove that

| v_{1} v_{2} | \cdot | v_{1} v_{3} | \cdot | v_{1} v_{4} | \cdot \dots \cdot | v_{1} v_{n} | = n,

where $| v_{1} v_{i} |$ denotes the distance from $v_{1}$ to $v_{i}$ .