Herzog Competition - 1977

Herzog Prize Examination — 1977

Exam date: November 12, 1977

Problem 1

(D. Moran) Let M be an n × n matrix of integers whose inverse is also a matrix of integers. Prove that the number of odd entries in M is at least n and at most n² − n + 1.

Problem 2

(A.M.M.E 1297) Having chosen 0 < a₁, b₁ < 1 define recursively

$a_{n + 1} = a_{1} (1 - a_{n} - b_{n}) + a_{n}$

and

$b_{n + 1} = b_{1} (1 - a_{n} - b_{n}) + b_{n}$

Prove that $lim_{n} a_{n}$ and $lim_{n} b_{n}$ both exist and evaluate these limits.

Problem 3

(L. Kelly) Consider the binary homogeneous quadratic form x² + bxy + y². Suppose it is known that this form produces perfect integral squares for all positive integral choices of x and y. Prove that b must be ±2.

Problem 4

(L. Kelly) A solid sphere rolls on a plane π always touching a fixed line L. Find the locus of its center.

Problem 5

(L. Kelly) Show that if all the distances between pairs of points of a seven point subset of the unit disc are at least 1, then the points must be the vertices of a regular inscribed hexagon and the center of the circle.

Problem 6

(A.M.M.E 1342) If x,y > 0, prove that

$x^{y} + y^{x} > 1$