55. There are 2019 plates arranged in a circle, each with one pastry on it. Petya and Vasya are playing a game. In one move, Petya points to a pastry and names a number from 1 to 16, and Vasya moves the indicated pastry to the specified number of plates clockwise or counterclockwise (Vasya chooses the direction each time). Petya wants to have at least $k$ pastries on one of the plates at some point, while Vasya wants to prevent this. For what largest $k$ can Petya achieve his goal?
55. Answer: 32 pastries.

Let's show how Petya can make it so that one of the plates ends up with no fewer than 32 pastries. Number the plates in a circle from 0 to 2018. Paint 32 plates in red with numbers $0, 32, 2 \cdot 32, \ldots, 31 \cdot 32$. Petya will only point to plates with numbers from 0 to $31 \cdot 32$, and forget about the rest. Initially, he will point to all pastries on plates with odd numbers (from this range) and ask to move them one position. As a result, the odd-numbered plates will be empty, and all the pastries from them will accumulate on even-numbered plates (and will not go beyond this range). Then he will point to all pastries on plates of the form $4k+2$ and ask to move them by 2. As a result, all these pastries will accumulate on plates (from our range) with numbers divisible by four. Next, he will give similar instructions for plates of the form $8k+4$ (move by 4), and of the form $16k+8$ (move by 8), and finally, of the form $32k+16$ (move by 16). As a result of this activity, all $31 \cdot 32 + 1$ pastries will end up on 32 red plates. By the pigeonhole principle, one of them will have no fewer than 32 pastries.

Now let's teach Vasya how to prevent more than 32 pastries from ending up on any plate. Let Vasya associate each plate with a block of 32 plates, containing this plate and the next 31 plates clockwise. Vasya will ensure that the original pastry from each plate moves only within the corresponding block. (Obviously, this is possible: no matter which of the 32 plates in this block the pastry is on, and no matter what shift from 1 to 16 Petya names, Vasya can choose the direction so that the pastry does not leave this block.) But each plate is contained in exactly 32 such blocks, and only pastries moving within these blocks can end up on it, that is, no more than 32 pastries!