3. Masha bought 2021 weights of pairwise distinct masses. Now Masha places one weight on each pan of a two-pan balance (weights placed on the balance previously are not removed). Each time the balance is in equilibrium, Masha rejoices. What is the maximum number of times she can find a reason to be happy
Solution. Answer: 673 times.

Example. Let Masha have bought 673 triples of the form $x, y, x+y$ (we will choose the weights of the new triples so that they do not duplicate the old ones). The last two weights are any. Masha puts $x$ on the left pan, then $-y$ on the same left pan, and finally $x+y$ on the right pan and rejoices.

Estimate. Suppose the scales are in balance. By placing the next two weights on the scales, Masha will be sad (the first weight will tip the scales out of balance, and the second, being different from the first, will not be able to return them to balance). Thus, the first two weighings Masha will not be able to rejoice, and in each subsequent triple, she will find no more than one reason for joy, since between two joyful weighings there will be at least two unjoyful ones.