The trick is to first think about the number of coins, rather than the amount of money on the table - at first, anyway. This game involves one hundred coins - an even number, in other words, and that's something that Rainbow, who's going first, can exploit.
First, label all coins from 1-100. Once Rainbow has taken the first coin at one end, all her opponent would be left with to choose from are either even-numbered coins on both ends (if she picks 1, the opponent's left with 2-100) or odd-numbered coins if she picks 100, with a 1-99 split. For example, if Rainbow wants all the odd ones, she'll pick coin 1. Whichever coin they take leaves an odd coin on one end for her to take, and then this pattern just repeats until there are no coins left. Therefore, in order to assure herself of collecting the most money, she needs to add the total value of the odd-numbered coins against the even-numbered coins, and then take ONLY those, depending on which ones have more value.
However, the strategy only works if the total value of the odd coins and the total value of the even coins are different. Even if they are the same though, the system will just reset and you can just repeat the game until Rainbow eventually wins.