Let $b_1, ... ,b_n$ be given real numbers and $max_{1 \leq i \leq j\leq n}\mid b_i-b_j\mid \leq 1$.
Then, I need to minimize $\sum_{i=1}^{n}(b_i+k_i)^2$ which is subject to the constraint that $k_i$'s are integers and $\sum_{i=1}^{n}k_i=0$.
Also, I have to show that there is a unique minimizing $(k_1,...,k_n)$ if $max_{1 \leq i \leq j \leq n}\mid b_i-b_j\mid < 1$ and more than one minimizing such tuples if $max_{1 \leq i \leq j \leq n}\mid b_i-b_j\mid = 1$.
The hint given says me to choose $q$ such that $\mid b_i-q \mid \leq 1$ and notice that $\sum_{i=1}^{n}b_ik_i=\sum_{i=1}^{n}(b_i-q)k_i$. However I cannot find a way to approach...I tried to switch the $k_i's$ but to no avail. Could anyone please help me?