In a computer game, we can earn an integer score each time we play. On the list that keeps track of the best 30 scores, the game's creator has displayed the scores $30,29,28, \ldots, 1$ next to fantasy names. If our score in a game - under our own name - is higher than the smallest score currently on the list, it will be added to the list, and the smallest score will logically be removed from the list. In the case of equal scores, the score that was added last will be removed from the list. The ranking is done "from the bottom," meaning we will only surpass the results on the list where we have achieved a higher score. Assuming that we get on the list after each game, what is the minimum number of games we need to play to ensure that our name appears next to every score on the list?
I. solution. Let's examine how many times we can achieve a given score during the game to bypass the list with it. We claim that if $1 \leq p \leq 30$, then we can achieve $p$ points at most $(p-1)$ times. Indeed, during the game, the number of scores higher than a given score on the list does not decrease, and initially, $(31-p)$ better scores (i.e., $(p-1)$ better scores) appear on the table. The game program works in such a way that these all precede our $p$ points.

Since we cannot get on the list with a score less than 1, and a score greater than 30 can be achieved with 30 games, we have

$$
30+\sum_{p=1}^{30}(p-1)=465
$$

games that we must definitely play to ensure that only our name appears on the list.

Now we show that we might need this many games, i.e., the sequence of games can develop in such a way that we only occupy the entire list with our 465th game. If, according to the above, for every $p \leq 30$, we achieve the $p$ points exactly $(p-1)$ times, and in increasing order, and then we score more than 30 points on thirty consecutive occasions, then we have played $(1+2+3+\ldots+29+30)=465$ games, and only after our last game will all the aliases disappear from the list.

We need to play at least 465 games to be sure, and this is enough. This completes the proof.

II. solution. We get on the list exactly when we surpass the 30th, last place. This implies that if the list changes, the sum of the scores increases by at least 1. Initially, this sum is 465. If we consider our results that surpass the 30-point pseudonym as 31 points, then as long as a pseudonym appears on the list, the total score will increase. After 465 successful games, the total score will increase to at least 930, but $930=30 \cdot 31$, so at that point, a 30-point score cannot appear on the list anymore.

That we might need 465 games can be proven as shown in the previous solution.

Remarks. 1. From the line of thought in the second solution, it follows that we might need 465 games only if the sequence of game results is as specified in the second part, since if we do not increase the score sum by 1 in any game, we will reach the 930 limit sooner.

2. According to the statistics of the problem, a disproportionately large number of incomplete solutions were submitted. The authors of many 3-point solutions essentially tried to justify the first part of their correct solution with the statement that "it is enough to examine the worst case": "when we get on the list with the worst possible result". Indeed, this is the "worst" case, but why, is precisely illuminated by the 2nd solution. It is not enough that it "seems obvious", the proof of this is an essential part of the problem.