Question 7
For each integer $ k \geq 3 $, prove that there exists a unique tuple $ (x_1, x_2, \ldots, x_k) $ of positive real numbers such that $ x_1 \geq x_2 \geq \cdots \geq x_k $ and
$$ \left\lfloor nx_{1}+\frac{1}{2}\right\rfloor+\left\lfloor nx_{2}+\frac{1}{2}\right\rfloor+\cdots+\left\lfloor nx_{k}+\frac{1}{2}\right\rfloor=n $$
for every integer n.
Here $\lfloor z\rfloor$ denotes the greatest integer less than or equal to z. For example, $\lfloor-\pi\rfloor=-4$ and $\lfloor2\rfloor=\lfloor2.9\rfloor=2$.
$$ \left\lfloor nx_{1}+\frac{1}{2}\right\rfloor+\left\lfloor nx_{2}+\frac{1}{2}\right\rfloor+\cdots+\left\lfloor nx_{k}+\frac{1}{2}\right\rfloor=n $$
for every integer n.
Here $\lfloor z\rfloor$ denotes the greatest integer less than or equal to z. For example, $\lfloor-\pi\rfloor=-4$ and $\lfloor2\rfloor=\lfloor2.9\rfloor=2$.