CGMO 2014

Let $x_1,x_2,\ldots,x_n$ be real numbers, where $n\ge 2$ is a given integer, and let $\lfloor{x_1}\rfloor,\lfloor{x_2}\rfloor,\ldots,\lfloor{x_n}\rfloor$ be a permutation of $1,2,\ldots,n$ . Find the maximum and minimum of $\sum\limits_{i=1}^{n-1}\lfloor{x_{i+1}-x_i}\rfloor$ (here $\lfloor x\rfloor$ is the largest integer not greater than $x$ ).

There are $n$ students; each student knows exactly $d$ girl students and $d$ boy students ("knowing" is a symmetric relation). Find all pairs $(n,d)$ of integers .

Let $a$ be a positive integer, but not a perfect square; $r$ is a real root of the equation $x^3-2ax+1=0$ . Prove that $r+\sqrt{a}$ is an irrational number.