The doors that are open after the 100th pass are the perfect squares from 1 to 100, i.e., the first door, the fourth door, the ninth door, and so on.
To see why this is the case, consider any particular door i. On the ith pass, you will toggle every ith door, starting with the ith door. So, for example, on the second pass (i=2), you will toggle every second door, starting with the second door. This means that the second door will be opened (since it starts out closed) and then closed again on the second pass.
Now, if a door is toggled an odd number of times, it will end up open, and if it is toggled an even number of times, it will end up closed. For any door i, the number of times it is toggled is equal to the number of factors it has, since each factor corresponds to a pass when that door is toggled. For example, door 12 is toggled on passes 1, 2, 3, 4, 6, and 12, so it is toggled 6 times, which is an even number. Therefore, it ends up closed.
However, for a perfect square like door 16, the factors come in pairs (1 and 16, 2 and 8, 4 and 4), except when the factor is the square root itself (in this case, 4). This means that the door is toggled an odd number of times, so it ends up open.
Therefore, the doors that end up open are precisely the perfect squares from 1 to 100, which are doors 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.