AT-OEMO-REG 2008

Show: For all real numbers $a,b,c$ with $0<a,b,c<1$ is: $$\sqrt{a^2bc+ab^2c+abc^2}+\sqrt{(1-a)^2(1-b)(1-c)+(1-a)(1-b)^2(1-c)+(1-a)(1-b)(1-c)^2}<\sqrt{3}.$$

For a real number $x$ is $[x]$ the next smaller integer to $x$ , that is the integer $g$ with $g\leqq<g+1$ , and $\{x\}=x-[x]$ is the "decimal part" of $x$ . Determine all triples $(a,b,c)$ of real numbers, which fulfil the following system of equations: $$\{a\}+[b]+\{c\}=2,9$$$$\{b\}+[c]+\{a\}=5,3$$$$\{c\}+[a]+\{b\}=4,0$$

Given is an acute angled triangle $ABC$ . Determine all points $P$ inside the triangle with $$1\leq\frac{\angle APB}{\angle ACB},\frac{\angle BPC}{\angle BAC},\frac{\angle CPA}{\angle CBA}\leq2$$

For every positive integer $n$ let $$a_n=\sum_{k=n}^{2n}\frac{(2k+1)^n}{k}$$ Show that there exists no $n$ , for which $a_n$ is a non-negative integer.