Question 20
(ROM 3) Let $c_1,\ldots,c_n\in\mathbb R$ ($n\ge2$) such that $0\le\sum_{i=1}^n c_i\le n$. Show that we can find integers $k_1,\ldots,k_n$ such that $\sum_{i=1}^n k_i=0$ and
$$1-n\le c_i+nk_i\le n$$
for every $i=1,\ldots,n$.
$$1-n\le c_i+nk_i\le n$$
for every $i=1,\ldots,n$.