, 2020

The 2.020 in the original problem can be changed to any $k > 2$, so we will prove the general statement for $k$ below.
Let us label the monsters so that the weight $x_i$ of the $i$-th monster satisfies $x_1 \le x_2 \le \dots \le x_N$. We define a monster $i$ to be dangerous if and only if $x_i > k \sum_{j<i} x_j$.
First, note that the initial number of dangerous monsters is a lower bound for the dangerous level. Suppose $x_i$ is dangerous, and let $M = \{i-1\}$. Whenever monster number $i-1$ merges with monster number $j$, add $j$ to the set $M$. Then it is easy to see that in this process, the first time $M \cap \{j: j \ge i\}$ becomes nonempty, that mergence must be a dangerous mergence. This proves that the dangerous level is at least as large as the number of initially dangerous monsters.

Next, we prove that merging the lightest two monsters each time achieves this lower bound. Let us merge $x_1$ and $x_2$; this step is dangerous if and only if $x_2$ is dangerous. Next, let us consider the danger status of the $N-1$ monsters after the merge. Note that the danger status of a monster with weight greater than $x_1 + x_2$ does not change before and after the merge, so we only need to consider monsters with weight $\ge x_1 + x_2$; note that these monsters were not dangerous before the first two monsters were merged. Let one of these monsters have weight $y$, with $y \le x_1 + x_2$.
- If $y$ is the lightest after the merge, then it must not be dangerous after the merge.
- If $y$ is not the lightest after the merge, then by the initial ordering of weights, the lightest monster after the merge has weight at least $x_2$, and hence $kx_2 > 2x_2 \ge x_1 + x_2 \ge y$, so $y$ is also not dangerous.
The above two points together also prove that the merged weight $x_1 + x_2$ is not dangerous either.

Combining the above, when we merge the lightest two monsters, if $x_2$ is not dangerous, then this mergence is not dangerous, and the number of dangerous monsters among the new $N-1$ monsters is the same as before. Conversely, if $x_2$ is dangerous, then this mergence is dangerous, and the number of dangerous monsters among the new $N-1$ monsters is the original number of dangerous monsters minus 1. Hence, by induction, this sequence achieves the lower bound of the dangerous level.