AT-OEMO-FED1 2015

Let $a$ , $b$ , $c$ , $d$ be positive numbers. Prove that $$(a^2 + b^2 + c^2 + d^2)^2 \ge (a+b)(b+c)(c+d)(d+a)$$ When does equality hold?

(Georg Anegg)

Let $ABC$ be an acute-angled triangle with $AC < AB$ and circumradius $R$ . Furthermore, let $D$ be the foot ofthe altitude from $A$ on $BC$ and let $T$ denote the point on the line $AD$ such that $AT = 2R$ holds with $D$ lying between $A$ and $T$ . Finally, let $S$ denote the mid-point of the arc $BC$ on the circumcircle that does not include $A$ .

Prove: $\angle AST = 90^\circ$ .

(Karl Czakler)

Alice and Bob play a game with a string of $2015$ pearls.

In each move, one player cuts the string between two pearls and the other player chooses one of the resulting parts of the string while the other part is discarded.

In the first move, Alice cuts the string, thereafter, the players take turns.

A player loses if he or she obtains a string with a single pearl such that no more cut is possible.

Who of the two players does have a winning strategy?

(Theresia Eisenkölbl)

A police emergency number is a positive integer that ends with the digits $133$ in decimal representation. Prove that every police emergency number has a prime factor larger than $7$ .

(In Austria, $133$ is the emergency number of the police.)

(Robert Geretschläger)