Herzog Competition - 1978

Problem 1

(MAA E 1359) Given 8 positive integers $a_1<a_2<\cdots <a_8\le 15$ . Prove that there exists a k such that aᵢ − aⱼ = k has at least 3 solutions with i > j.

Problem 2

(L.M. Kelly) Let P(x) denote a polynomial with integral coefficients. Prove that it is impossible for P(1000) = 1000, P(2000) = 2000, and P(3000) = 1000.

Problem 3

(Vera T. Sós) Let 0 < α < 1 be irrational. Put the n fractional parts ⟨α⟩, ⟨2α⟩, ⟨3α⟩, …, ⟨nα⟩ in increasing order. Show there is at most 3 distinct distances between successive terms.

Problem 4

If P and Q lie in the interior of a regular tetrahedron, show that the angle PAQ is less than 60° where A is any vertex of the tetrahedron.

Problem 5

(E. Nordhaus) Let S be a set of distinct positive integers such that

1. no integer of S exceeds 100
2. every three integers selected from S can be the sides of a (non-degenerate) triangle.

What is the maximum number of integers S can contain?

Problem 6

(MAA E 1381) Show that

$$1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots +\frac{x^{2n}}{\left(2n\right)!}=0$$

has no real roots.