Let $ABC$ be an equilateral triangle, and let $P$ be some point in its circumcircle. Determine all positive integers $n$, for which the value of the sum $S_n (P) = |PA|^n + |PB|^n + |PC|^n$ is independent of the choice of point $P$.

Determine the smallest integer $n$ for which there exist integers $x_1,\ldots,x_n$ and positive integers $a_1,\ldots,a_n$ so that \begin{align*} x_1+\cdots+x_n &=0,\\ a_1x_1+\cdots+a_nx_n&>0, \text{ and }\\ a_1^2x_1+\cdots+a_n^2x_n &<0. \end{align*}

A set $S$ of integers is Balearic, if there are two (not necessarily distinct) elements $s,s'\in S$ whose sum $s+s'$ is a power of two; otherwise it is called a non-Balearic set. Find an integer $n$ such that $\{1,2,\ldots,n\}$ contains a 99-element non-Balearic set, whereas all the 100-element subsets are Balearic.

Let $x,y,z$ and $a,b,c$ be positive real numbers with $a+b+c=1$. Prove that $$\left(x^2+y^2+z^2\right) \left( \frac{a^3}{x^2+2y^2} + \frac{b^3}{y^2+2z^2} + \frac{c^3}{z^2+2x^2} \right) \ge\frac19.$$