AT-OEMO-BEG 2011

Let $x$ be the smallest positive integer for which $2x$ is the square of an integer, $3x$ is the third power of an integer, and $5x$ is the fifth power of an integer. Find the prime factorization of $x$ .

(St. Wagner, Stellenbosch University)

Let $p$ and $q$ be real numbers. The quadratic equation $$x^2 + px + q = 0$$ has the real solutions $x_1$ and $x_2$ . In addition, the following two conditions apply: (i) The numbers $x_1$ and $x_2$ differ from each other by exactly $1$ . (ii) The numbers $p$ and $q$ differ from each other by exactly $1$ . Show that then $p$ , $q$ , $x_1$ and $x_2$ are integers.

(G. Kirchner, University of Innsbruck)

Let $ABC$ be an isosceles triangle with $AC = BC$ . On the arc $CA$ of its circumcircle, which does not contain $B$ , there is a point $P$ . The projection of $C$ on the line $AP$ is denoted by $E$ , the projection of $C$ on the line $BP$ is denoted by $F$ . Prove that the lines $AE$ and $BF$ have equal lengths.

(W. Janous, WRG Ursulincn, Innsbruck)