AT-OEMO-BEG 2018

Let $a, b$ and $c$ denote positive real numbers. Prove that $\frac{a}{c}+\frac{c}{b}\ge \frac{4a}{a + b}$ . When does equality hold?

(Walther Janous)

Let $ABC$ be an acute-angled triangle, $M$ the midpoint of the side $AC$ and $F$ the foot on $AB$ of the altitude through the vertex $C$ . Prove that $AM = AF$ holds if and only if $\angle BAC = 60^o$ .

(Karl Czakler)

For a given integer $n \ge 4$ we examine whether there exists a table with three rows and $n$ columns which can be filled by the numbers $1, 2,...,, 3n$ such that $\bullet$ each row totals to the same sum $z$ and $\bullet$ each column totals to the same sum $s$ . Prove: (a) If $n$ is even, such a table does not exist. (b) If $n = 5$ , such a table does exist.

(Gerhard J. Woeginger)

For a positive integer $n$ we denote by $d(n)$ the number of positive divisors of $n$ and by $s(n)$ the sum of these divisors. For example, $d(2018)$ is equal to $4$ since $2018$ has four divisors $(1, 2, 1009, 2018)$ and $s(2018) = 1 + 2 + 1009 + 2018 = 3030$ . Determine all positive integers $x$ such that $s(x) \cdot d(x) = 96$ .

(Richard Henner)