Problem

Source: FKMO 2024 P2

Tags: combinatorics



For a positive integer $n(\geq 2)$, there are $2n$ candies. Alice distributes $2n$ candies into $4n$ boxes $B_1, B_2, \dots, B_{4n}.$ Bob checks the number of candies that Alice puts in each box. After this, Bob chooses exactly $2n$ boxes $B_{k_1}, B_{k_2}, \dots, B_{k_{2n}}$ out of $4n$ boxes that satisfy the following condition, and takes all the candies. (Condition) $k_i - k_{i - 1}$ is either $1$ or $3$ for each $i = 1, 2, \dots, 2n$, and $k_{2n} = 4n$. ($k_0 = 0$) Alice takes all the candies in the $2n$ boxes that Bob did not choose. If Alice and Bob both use their best strategy to take as many candies as possible, how many candies can Alice take?