AT-OEMO-REG 2003

Find the minimum value of the expression $\frac{a+1}{a(a+2)}+\frac{b+1}{b(b+2)}+\frac{c+1}{c(c+2)}$ , where $a,b,c$ are positive real numbers with $a+b+c \le 3$ .

Find all prime numbers $p$ with $5^p+4p^4$ is the square of an integer.

Given are two parallel lines $g$ and $h$ and a point $P$ , that lies outside of the corridor bounded by $g$ and $h$ . Construct three lines $g_1$ , $g_2$ and $g_3$ through the point $P$ . These lines intersect $g$ in $A_1,A_2, A_3$ and $h$ in $B_1, B_2, B_3$ respectively. Let $C_1$ be the intersection of the lines $A_1B_2$ and $A_2B_1$ , $C_2$ be the intersection of the lines $A_1B_3$ and $A_3B_1$ and let $C_3$ be the intersection of the lines $A_2B_3$ and $A_3B_2$ . Show that there exists exactly one line $n$ , that contains the points $C_1,C_2,C_3$ and that $n$ is parallel to $g$ and $h$ .

For every real number $b$ determine all real numbers $x$ satisfying $x-b= \sum_{k=0}^{\infty}x^k$ .