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Problem of the Month – January 2021

There are 2021 distinguishable boxes on the table. Starting Alice, Alice and Bob take turn
writing an unordered box pair to the table (each unordered box pair can be written at most
once). They stop when there are 4038 written pairs on the table. After that Bob numerates
all box pairs by numbers 1, 2, . . . , 4038 and for each k = 1, 2, . . . , 4038 puts k balls into each
box belonging to the pair numbered k. Can Bob guarantee that any two boxes will contain
different number of balls?