Let $ d_1,\ldots,d_n \in \mathbb{N}_{\ge 2} $ (not necessarily distinct) be given. Define $ D:=\operatorname{lcm}(d_1,\ldots,d_n) $ and $ d:=\sum_{i=1}^n d_i $.
(1) Alice claims that whenever $ \lbrace d/D\rbrace \ge \frac{1}{2} $ holds (where $ \lbrace x \rbrace $ denotes the fractional part of $ x $), she is able to find a partition of $ \lbrace T_j \rbrace_{j=1}^{\left\lfloor d/D\right\rfloor} $ of $ \lbrace 1,\ldots,n \rbrace $ such that $ \sum_{m \in T_j} d_m \ge D $ ($j=1,\ldots, \left\lfloor d/D \right\rfloor $) are satisfied.
(2) Bob claims that whenever $ \lbrace d/D\rbrace \le \frac{1}{2} $ holds (where $ \lbrace x \rbrace $ denotes the fractional part of $ x $), he is able to find a partition of $ \lbrace T_j \rbrace_{j=1}^{\left\lceil d/D \right\rceil} $ of $ \lbrace 1,\ldots,n \rbrace $ such that $ \sum_{m \in T_j} d_m \le D $ ($j=1,\ldots, \left\lceil d/D \right\rceil $) are satisfied.
Are both of them right?