Let
![c_1, \ldots, c_n \in \mathbb{R}](/media/m/d/b/f/dbfd68284e93c05bc97ef840776637dd.png)
with
![n \geq 2](/media/m/2/1/f/21fe2458de6d1580c44fd06e0fac11bb.png)
such that
![0 \leq \sum^n_{i=1} c_i \leq n.](/media/m/6/b/a/6ba7cd2dc6944c32a612365866ecb5e4.png)
Show that we can find integers
![k_1, \ldots, k_n](/media/m/6/a/c/6aca44ce58a5f892883f635bed32c665.png)
such that
![\sum^n_{i=1} k_i = 0](/media/m/c/0/6/c06bc5e99b255be814bbffb2323e2616.png)
and
![1-n \leq c_i + n \cdot k_i \leq n](/media/m/5/8/1/5815eb7819c82235256a3acac8330e64.png)
for every
Another formulation:Let
![x_1, \ldots, x_n,](/media/m/9/9/0/990aa4ab09e4fb4f57969107d3bcb108.png)
with
![n \geq 2](/media/m/2/1/f/21fe2458de6d1580c44fd06e0fac11bb.png)
be real numbers such that
![|x_1 + \ldots + x_n| \leq n.](/media/m/5/1/b/51ba29deebcb2810d5bd953959cf4ff8.png)
Show that there exist integers
![k_1, \ldots, k_n](/media/m/6/a/c/6aca44ce58a5f892883f635bed32c665.png)
such that
![|k_1 + \ldots + k_n| = 0.](/media/m/5/1/e/51e254cda7bd2af6c6c09e931d813130.png)
and
![|x_i + 2 \cdot n \cdot k_i| \leq 2 \cdot n -1](/media/m/6/0/1/601940cb7296369dc7532c689ea8e1e5.png)
for every
![i = 1, \ldots, n.](/media/m/c/2/8/c28b11873e25483161db20fcbc6e5ca2.png)
In order to prove this, denote
![c_i = \frac{1+x_i}{2}](/media/m/4/f/c/4fc5898ab2675a8d3afa8beabbfc86a8.png)
for
![i = 1, \ldots, n,](/media/m/6/9/b/69b8bbc2b5c8fb038490ffcf486023d0.png)
etc.
%V0
Let $c_1, \ldots, c_n \in \mathbb{R}$ with $n \geq 2$ such that $$0 \leq \sum^n_{i=1} c_i \leq n.$$ Show that we can find integers $k_1, \ldots, k_n$ such that $$\sum^n_{i=1} k_i = 0$$ and $$1-n \leq c_i + n \cdot k_i \leq n$$ for every $i = 1, \ldots, n.$
Another formulation:Let $x_1, \ldots, x_n,$ with $n \geq 2$ be real numbers such that $$|x_1 + \ldots + x_n| \leq n.$$ Show that there exist integers $k_1, \ldots, k_n$ such that $$|k_1 + \ldots + k_n| = 0.$$ and $$|x_i + 2 \cdot n \cdot k_i| \leq 2 \cdot n -1$$ for every $i = 1, \ldots, n.$ In order to prove this, denote $c_i = \frac{1+x_i}{2}$ for $i = 1, \ldots, n,$ etc.