2026-338

As a resistance leader, you have obtained a schematic map of the dictatorship communication cables. Because of the technology used, the signal travels one-way only, as depicted in the map. To launch your attack, you must completely cut all paths from the Capital (s) to the Military Base (t).

Each one-way cable is guarded. The cost to destroy a cable (in C4 charges and personnel) is shown in the map. You have a total budget of B = 10.

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Problem Statement

Which of the following statements is true regarding your sabotage plan?

a) The cheapest way to disconnect the base costs 11, so your budget of B = 10 is not enough to go through with the plan.

b) You can successfully implement the plan for a cost of 9 by destroying the cables (a, t) and (b, t).

c) The most efficient plan is to hit cables (s, b) and (b, a), which only costs 10 and leaves the base isolated.

d) You would need to destroy every cable connected to the Capital, costing you exactly 18.

e) None of the above

Original idea by: Luis Alberto Vásquez Vargas

2026-337

On Easter Sunday, due to high demand, airlines had to limit the number of planes that could fly over certain regions. Help Zula Airlines maximize the number of planes that could travel on April 4, 2026, at 2 PM, through the areas shown in the figure below, where SP is the source node and AM is the sink node. In this network flow problem, what would be the residual network formed from the flows and capacities shown?

 

a.) 

b.) 

c.) 

d.) 

e) None of the above

Original idea by: Julia de Pietro Bigi

2026-336

A research data center needs to transfer data to a remote backup facility during a 4‑hour maintenance window. The data transfer process is modeled as a directed network, where nodes represent logical components (functional stages) of the data transfer pipeline, and edges represent logical communication channels, with limited capacity.

In this model, node \(A\) represents the point where data is generated in the primary data center. The intermediate nodes (\(B, C, D\)) represent functional stages of the transfer process, such as internal processing, aggregation, or interfaces to external networks. And node \(E\) represents the logical destination where data is finally stored at the backup site.

All edges have constant transmission capacity, except for one critical link, whose capacity varies over time due to shared usage with other services. The network links are as follows, with capacities measured in GB/hour:

\( A \rightarrow B : 8 \)

\( B \rightarrow C : 4 + \pi \sin(\pi t/4) \)

\( C \rightarrow E : 9 \)

\( A \rightarrow D : 3 \)

\( D \rightarrow E : 6 \)

where \(t\) is the time in hours since the start of the maintenance window.

Which of the following alternatives represents approximately the maximum total amount of data (in GB) that can be transferred from \(A\) to \(E\) during the maintenance window?

a) 40

b) 36

c) 28

d) 18

e) None

 

Original idea by: Ingrid Barbosa