MathDB
Problems
Contests
International Contests
IMO Shortlist
2023 ISL
A1
A1
Part of
2023 ISL
Problems
(1)
Feeding Pokémon is hard
Source: IMO Shortlist 2023 A1
7/17/2024
Professor Oak is feeding his
100
100
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
100
100
kilograms. Professor Oak distributes
100
100
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 the bowl). The dissatisfaction level of a Pokémon who received
N
N
N
kilograms of food and whose bowl has a capacity of
C
C
C
kilograms is equal to
∣
N
−
C
∣
\lvert N-C\rvert
∣
N
−
C
∣
.Find the smallest real number
D
D
D
such that, regardless of the capacities of the bowls, Professor Oak can distribute food in a way that the sum of the dissatisfaction levels over all the
100
100
100
Pokémon is at most
D
D
D
.Oleksii Masalitin, Ukraine
IMO Shortlist
algebra