Herzog Competition - 1999

Problem 1

Determine which of the numbers ${1999}^{1999}$ or ${2000}^{1998}$ is the larger.

Problem 2

Prove or disprove: If a sequence of integers in arithmetic progression contains a square, then it must contain infinitely many squares.

Problem 3

A ladybug (beetle of the family Coccinellidae) walks along the real line. She starts at point $a$ and walks toward $b$. Halfway there, she realizes that she forgot to eat an aphid at $a$ and turns back. But halfway back to $a$ she decides to go to $b$ anyway and turns around again, only to change her mind again, halfway to $b$. She continues in this manner. Find all cluster points (i.e. accumulation points) of the resulting sequence of turning points.

Problem 4

Given segment AB in the plane, determine the locus of all points P such that $\angle \mathrm{PAB}+\angle \mathrm{PBA}$ is constant. (Give a detailed description in terms of the constant.)

Problem 5

Let $a,b,c$ be positive real numbers. Prove that

$$(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge 9.$$

Problem 6

Let $p(x)$ be a polynomial with real coefficients, and with all zeros real. Suppose also that $p(x)\ge 0$ for all real numbers $x$. Prove that $p(x)={\{q(x)\}}^2$ for some polynomial $q(x)$ with real coefficients.