2021 AMC 12A Fall Problem 16
Attempt Problem 16 of the 2021 AMC 12A Fall below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 12A Fall solutions, or check the answer key.
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16.
An organization has employees, of whom have a brand A computer while the other have a brand B computer. For security, the computers can only be connected to each other and only by cables. The cables can only connect a brand A computer to a brand B computer. Employees can communicate with each other if their computers are directly connected by a cable or by relaying messages through a series of connected computers. Initially, no computer is connected to any other. A technician arbitrarily selects one computer of each brand and installs a cable between them, provided there is not already a cable between that pair. The technician stops once every employee can communicate with each other. What is the maximum possible number of cables used?
Answer: B
Small hint:
To delay connectivity, keep the graph disconnected as long as possible, then one final cable joins everything
Big hint:
The most edges while still disconnected is a complete bipartite graph on brand A and brand B, leaving one brand A computer isolated
Solution:
The technician keeps adding cables until the graph becomes connected. To maximize the count, keep the network disconnected for as long as possible: leave a single brand A computer isolated and fully connect the remaining brand A computers to all brand B computers.
That uses cables while still disconnected. The next cable connects the last brand A computer, joining everyone, for a total of
Thus, the correct answer is B.
Problem 16 in other years
- 1950 AMC 12
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