A purse contains a finite number of coins, each with distinct positive integer values. Is it possible that there are exactly 2020 ways to use coins from the purse to make the value 2020?
It is possible.

Consider a coin purse with coins of values 2, 4, 8, 2014, 2016, 2018, 2020, and every odd number between 503 and 1517. Call such a coin big if its value is between 503 and 1517. Call a coin small if its value is 2, 4, or 8, and huge if its value is 2014, 2016, 2018, or 2020. Suppose some subset of these coins contains no huge coins and sums to 2020. If it contains at least four big coins, then its value must be at least \(503 + 505 + 507 + 509 > 2020\). Furthermore, since all of the small coins are even in value, if the subset contains exactly one or three big coins, then its value must be odd. Thus, the subset must contain exactly two big coins. The eight possible subsets of the small coins have values \(0, 2, 4, 6, 8, 10, 12, 14\). Therefore, the ways to make the value 2020 using no huge coins correspond to the pairs of big coins with sums 2006, 2008, 2010, 2012, 2014, 2016, 2018, and 2020. The numbers of such pairs are 250, 251, 251, 252, 252, 253, 253, 254, respectively. Thus, there are exactly 2016 subsets of this coin purse with value 2020 using no huge coins. There are exactly four ways to make a value of 2020 using huge coins; these are \(\{2020\}, \{2, 2018\}, \{4, 2016\}\), and \(\{2, 4, 2014\}\). Thus, there are exactly 2020 ways to make the value 2020.

Alternate construction: Take the coins \(1, 2, \ldots, 11, 1954, 1955, \ldots, 2019\). The only way to get 2020 is a non-empty subset of \(1, \ldots, 11\) and a single large coin. There are 2047 non-empty such subsets of sums between 1 and 66. Thus, they each correspond to a unique large coin making 2020, so we have 2047 ways. Thus, we only need to remove some large coins, so that we remove exactly 27 small sums. This can be done, for example, by removing coins \(2020 - n\) for \(n = 1, 5, 6, 7, 8, 9\), as these correspond to \(1 + 3 + 4 + 5 + 6 + 8 = 27\) partitions into distinct numbers that are at most 11.