US-TST 2000

Let $a, b, c$ be nonnegative real numbers. Prove that $$\frac{a+b+c}{3} - \sqrt[3]{abc} \leq \max\{(\sqrt{a} - \sqrt{b})^2, (\sqrt{b} - \sqrt{c})^2, (\sqrt{c} - \sqrt{a})^2\}.$$

Let $ABCD$ be a cyclic quadrilateral and let $E$ and $F$ be the feet of perpendiculars from the intersection of diagonals $AC$ and $BD$ to $AB$ and $CD$ , respectively. Prove that $EF$ is perpendicular to the line through the midpoints of $AD$ and $BC$ .

Let $p$ be a prime number. For integers $r, s$ such that $rs(r^2 - s^2)$ is not divisible by $p$ , let $f(r, s)$ denote the number of integers $n \in \{1, 2, \ldots, p - 1\}$ such that $\{rn/p\}$ and $\{sn/p\}$ are either both less than $1/2$ or both greater than $1/2$ . Prove that there exists $N > 0$ such that for $p \geq N$ and all $r, s$ , $$\left\lceil \frac{p-1}{3} \right\rceil \le f(r, s) \le \left\lfloor \frac{2(p-1)}{3} \right\rfloor.$$

Let $n$ be a positive integer. Prove that $$\binom{n}{0}^{-1} + \binom{n}{1}^{-1} + \cdots + \binom{n}{n}^{-1} = \frac{n+1}{2^{n+1}} \left( \frac{2}{1} + \frac{2^2}{2} + \cdots + \frac{2^{n+1}}{n+1} \right).$$

Let $n$ be a positive integer. A $corner$ is a finite set $S$ of ordered $n$ -tuples of positive integers such that if $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ are positive integers with $a_k \geq b_k$ for $k = 1, 2, \ldots, n$ and $(a_1, a_2, \ldots, a_n) \in S$ , then $(b_1, b_2, \ldots, b_n) \in S$ . Prove that among any infinite collection of corners, there exist two corners, one of which is a subset of the other one.

Let $ABC$ be a triangle inscribed in a circle of radius $R$ , and let $P$ be a point in the interior of triangle $ABC$ . Prove that $$\frac {PA}{BC^{2}} + \frac {PB}{CA^{2}} + \frac {PC}{AB^{2}}\ge \frac {1}{R}.$$ Alternative formulation: If $ABC$ is a triangle with sidelengths $BC=a$ , $CA=b$ , $AB=c$ and circumradius $R$ , and $P$ is a point inside the triangle $ABC$ , then prove that

$\frac {PA}{a^{2}} + \frac {PB}{b^{2}} + \frac {PC}{c^{2}}\ge \frac {1}{R}$ .