AT-OEMO-REG 2006

Let $0 < x <y$ be real numbers. Let $H=\frac{2xy}{x+y}$ , $G=\sqrt{xy}$ , $A=\frac{x+y}{2}$ , $Q=\sqrt{\frac{x^2+y^2}{2}}$ be the harmonic, geometric, arithmetic and root mean square (quadratic mean) of $x$ and $y$ . As generally known $H<G<A<Q$ . Arrange the intervals $[H,G]$ , $[G,A]$ and $[A,Q]$ in ascending order by their length.

Let $n>1$ be a positive integer an $a$ a real number. Determine all real solutions $(x_1,x_2,\dots,x_n)$ to following system of equations: $x_1+ax_2=0$ $x_2+a^2x_3=0$ … $x_k+a^kx_{k+1}=0$ … $x_n+a^nx_1=0$

In a non isosceles triangle $ABC$ let $w$ be the angle bisector of the exterior angle at $C$ . Let $D$ be the point of intersection of $w$ with the extension of $AB$ . Let $k_A$ be the circumcircle of the triangle $ADC$ and analogy $k_B$ the circumcircle of the triangle $BDC$ . Let $t_A$ be the tangent line to $k_A$ in A and $t_B$ the tangent line to $k_B$ in B. Let $P$ be the point of intersection of $t_A$ and $t_B$ . Given are the points $A$ and $B$ . Determine the set of points $P=P(C )$ over all points $C$ , so that $ABC$ is a non isosceles, acute-angled triangle.

Let $<h_n>$ $n\in\mathbb N$ a harmonic sequence of positive real numbers (that means that every $h_n$ is the harmonic mean of its two neighbours $h_{n-1}$ and $h_{n+1}$ : $h_n=\frac{2h_{n-1}h_{n+1}}{h_{n-1}+h_{n+1}}$ ) Show that: if the sequence includes a member $h_j$ , which is the square of a rational number, it includes infinitely many members $h_k$ , which are squares of rational numbers.