Partition of fractional parts where each sum of them has to be at least 1 Let $ a_1,\ldots,a_t \in \mathbb{Q} \setminus \mathbb{Z} $ be with $ \sum_{i=1}^t \lbrace a_i \rbrace \in \left[k,k+1\right) $ for some $ k \in \mathbb{N} $ with $ k \ge 4 $. Here $ \lbrace x \rbrace $ stands for the fractional part of $ x $. 
Are there always $ k-2 $ disjoint subsets $ A_j \subseteq \lbrace 1,\ldots,t \rbrace $ ($j =1,\ldots,k-2 $) with $ \sum_{i \in A_j} \lbrace a_i \rbrace \ge 1 $ for all $ j $?

No. Let $a_i=0.99$ for all $i$ and $t=6$. Then $k=5$. But if $ A_j \subseteq \lbrace 1,\ldots,t \rbrace $ with $ \sum_{i \in A_j} \lbrace a_i \rbrace \ge 1 $ then $|A_j|\ge 2$, so there can be at most $3<4=k-2$ such mutually disjiont subsets.
PS. Instead of fractional parts we can simply consider the rational numbers $a_i\in (0,1).$
