To resolve the apparent contradiction between Mr. Mamun having the highest rate of unsuccessful collections but still being considered the best bill collector, we need to find the option that provides a plausible explanation for this situation. Let's analyze the options provided:

A. None of the bill collectors at the agency has been working longer than Mr. Mamun.
This option does not directly address the contradiction. The length of service does not necessarily explain why Mr. Mamun, despite having the highest rate of unsuccessful collections, is still considered the best bill collector.

B. The other four bill collectors at the agency consider Mr. Mamun to be a very capable bill collector.
This option provides a plausible explanation for the apparent contradiction. If the other bill collectors consider Mr. Mamun to be a very capable bill collector, it suggests that they value his skills and expertise despite his higher rate of unsuccessful collections.

C. Mr. Mamun's rate of collections per year has remained fairly steady in the last few years.
This option does not directly address the contradiction. The stability of Mr. Mamun's rate of collections does not explain why he is considered the best bill collector despite having the highest rate of unsuccessful collections.

D. Before joining the agency, Mr. Mamun was affiliated with the credit department of a large department store.
This option does not directly address the contradiction. Mr. Mamun's previous affiliation with a credit department does not explain why he is considered the best bill collector despite his higher rate of unsuccessful collections.

E. Mr. Mamun is assigned to the majority of the most difficult cases at the agency.
This option provides a plausible explanation for the apparent contradiction. If Mr. Mamun is consistently assigned the most difficult cases, it suggests that his higher rate of unsuccessful collections could be due to the challenging nature of the cases he handles, and his overall performance may still be considered exceptional.

Based on the analysis, the option that most helps to resolve the apparent contradiction is option E: Mr. Mamun is assigned to the majority of the most difficult cases at the agency.