Rarity needs 56 soldiers minimum to win the contest.
The total number of winnable points when we add the value of the towers (1+2+3...+9+10) is 55 points. Therefore, to secure a victory, Rarity needs to win 28 points in total at least.
Since Rarity is proceeding under the assumption she has all the knowledge in this contest, we need to first discern what the opponent's most optimal strategy would be, and then devise a countermeasure. Since the opponent won't want to yield 28 or more points, they (that is to say you) will naturally make the highest scoring towers the ones that are the most difficult to conquer - in other words, to make it so that every tower has an equally poor number of soldiers required to conquer it.
Luckily, 100 is the perfect number, and so you'll most likely set up each tower with a distribution of 1, 3, 5, 7, 9, 11, 13, 15, 17, and 19, since that works out to 2 soldiers spent per victory won. As this is the worst case scenario, Rarity can earn the 28 points EXACTLY she needs by conquering towers 1-7, by sending 2, 4, 6, 8, 10, 12 and 14 soldiers out to the first 7 towers, which in total works out to 56 soldiers.
Just like any good war, this one is decided on economics.