2023 IMO Shortlist A1

Professor Oak is feeding his 100 Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is 100 kilograms. Professor Oak distributes 100 kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may be larger than the capacity of their bowl). The dissatisfaction level of a Pokémon who received $N$ kilograms of food and whose bowl has a capacity of $C$ kilograms is equal to $|N-C|$. Find the smallest real number $D$ such that, regardless of the capacities of the bowls, Professor Oak can distribute the food in a way that the sum of the dissatisfaction levels over all the 100 Pokémon is at most $D$. (Ukraine)

大木教授正在喂养他的 100 只神奇宝贝。每个神奇宝贝都有一个碗，其容量为正实数公斤。奥克教授知道这些能力。所有碗的总容量为100公斤。大木教授分配了 100 公斤的食物，每个神奇宝贝都会收到非负整数公斤的食物(可能大于其碗的容量)。收到 $N$ 公斤食物且碗容量为 $C$ 公斤的神奇宝贝的不满程度等于 $|N-C|$。找到最小的实数 $D$，这样，无论碗的容量如何，大木教授都可以以一种方式分配食物，使所有 100 只神奇宝贝的不满程度总和最多为 $D$。 (乌克兰)