AT-OEMO-BEG 2013

Find all natural numbers $n> 1$ for which the following applies: The sum of the number $n$ and its second largest divisor is $2013$ .

(R. Henner, Vienna)

The following figure is given: Determine the number of paths from the starting square $A$ to the target square $Z$ , where a path consists of steps from a square to its top or right neighbor square .

(W. Janous, WRG Ursulinen, Innsbruck)

Let $a$ and $b$ be real numbers with $0\le a, b\le 1$ . Prove that $$\frac{a}{b + 1}+\frac{b}{a + 1}\le 1$$ When does equality holds?

(K. Czakler, GRG 21, Vienna)

Let $ABC$ be an acute-angled triangle and $D$ a point on the altitude through $C$ . Let $E$ , $F$ , $G$ and $H$ be the midpoints of the segments $AD$ , $BD$ , $BC$ and $AC$ . Show that $E$ , $F$ , $G$ , and $H$ form a rectangle.

(G. Anegg, Innsbruck)