AT-OEMO-BEG 2009

A positive integer number is written in red on each side of a square. The product of the two red numbers on the adjacent sides is written in green for each corner point. The sum of the green numbers is $40$ . Which values are possible for the sum of the red numbers?

(G. Kirchner, University of Innsbruck)

Let $x$ and $y$ be nonnegative real numbers. Prove that $(x +y^3) (x^3 +y) \ge 4x^2y^2$ . When does equality holds?

(Task committee)

There are any number of stamps with the values $134$ , $135$ , $...$ , $142$ and $143$ cents available. Find the largest integer value (in cents) that cannot be represented by these stamps.

(G. Woeginger, TU Eindhoven, The Netherlands)

The center $M$ of the square $ABCD$ is reflected wrt $C$ . This gives point $E$ . The intersection of the circumcircle of the triangle $BDE$ with the line $AM$ is denoted by $S$ . Show that $S$ bisects the distance $AM$ .

(W. Janous, WRG Ursulinen, Innsbruck)