C. Have Your Cake and Eat It Too

Alice, Bob and Charlie want to share a rectangular cake cut into 𝑛 pieces. Each person considers every piece to be worth a different value. The 𝑖-th piece is considered to be of value 𝑎𝑖 by Alice, 𝑏𝑖 by Bob and 𝑐𝑖 by Charlie.

The sum over all 𝑎𝑖, all 𝑏𝑖 and all𝑐𝑖 individually is the same, equal to 𝑡𝑜𝑡.

Given the values of each piece of the cake for each person, you need to give each person a contiguous slice of cake. In other words, the indices at the left and right ends of these subarrays (the slices given to each person) can be represented as (𝑙𝑎,𝑟𝑎), (𝑙𝑏,𝑟𝑏) and (𝑙𝑐,𝑟𝑐) respectively for Alice, Bob and Charlie. The division needs to satisfy the following constraints:

• No piece is assigned to more than one person, i.e., no two subarrays among [𝑙𝑎,…,𝑟𝑎], [𝑙𝑏,…,𝑟𝑏] and [𝑙𝑐,…,𝑟𝑐] intersect.
• ∑𝑖=𝑙𝑎𝑟𝑎𝑎𝑖,∑𝑖=𝑙𝑏𝑟𝑏𝑏𝑖,∑𝑖=𝑙𝑐𝑟𝑐𝑐𝑖≥⌈𝑡𝑜𝑡3⌉.

Here, the notation ⌈𝑎𝑏⌉ represents ceiling division. It is defined as the smallest integer greater than or equal to the exact division of 𝑎 by 𝑏. In other words, it rounds up the division result to the nearest integer. For instance⌈103⌉=4 and ⌈153⌉=5.