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BMAT 2022 · Question 25 · Optimisation

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A village hall runs a small lending library which opens only on Saturdays. There are 80 members, each of whom may borrow up to four books at a time, which they may keep for three weeks. Members never overrun the three week time limit. After the annual stock-take, for which all books were returned, the library has 2000 books. On the next three Saturdays 120, 80, and 60 books are borrowed respectively.

What is the smallest number of books that could be left in the library after the fourth Saturday?