# For a set of positive integers, is this inequality always true?

The input consists of a set of positive integers $\{b_1,...,b_2\}$ such that $$\sum_{i=1}^nb_i=CK,$$ with $C$ and $K$ two positive integers.

The question is the following, is there $k\in\{1,...,n\}$ for which the following inequality is true $$\left \lfloor \sum_{i=1,i\neq k}^n\frac{b_i}{C}\right \rfloor \leq \sum_{i=1,i\neq k}^n \left \lfloor \frac{b_i}{x}\right \rfloor,$$ where $x=C-\left(C\left \lceil\frac{b_k}{C}\right \rceil-b_k\right)$, it also means we always have $C\geq x$.

Example with $b_1=7804, b_2=6080, b_3=3376$. We have $7804+6080+3376=3452\times5$. Let's take $C=3452$.

• For $k=1$, we have $x=3452-\left(3452\left\lceil\frac{7804}{3452}\right\rceil-7804\right)=900$. Then the inequality is true because: $$\left\lfloor \frac{6080}{3452}+\frac{3376}{3452}\right\rfloor < \left\lfloor \frac{6080}{900}\right\rfloor+\left\lfloor \frac{3376}{900}\right\rfloor.$$
• For $k=2$, we have $x=3452-\left(3452\left\lceil\frac{6080}{3452}\right\rceil-6080\right)=2628$. Then the inequality is true because: $$\left\lfloor \frac{7804}{3452}+\frac{3376}{3452}\right\rfloor = \left\lfloor \frac{7804}{2628}\right\rfloor+\left\lfloor \frac{3376}{2628}\right\rfloor.$$
• For $k=3$, we have $x=3452-\left(3452\left\lceil\frac{3376}{3452}\right\rceil-3376\right)=3376$. Then the inequality is not true because: $$\left\lfloor \frac{7804}{3452}+\frac{6080}{3452}\right\rfloor > \left\lfloor \frac{7804}{3376}\right\rfloor+\left\lfloor \frac{6080}{3376}\right\rfloor.$$

Is there a way to show that for a set $b$ of integers (with $C$ and $K$ as described above) we always have $k\in\{1,...,n\}$ for which the inequality is true?

Yes. At first, if $b_i>C$ for some $i$, replace $b_i$ to $b_i-C$. If new set is ok, then the old also was ok. If all $b_i$'s do not exceed $C$, just choose $k$ for which $b_k$ is minimal. RHS is not less then $n-1$, LHS is at most $n-1$.