I am posting in order to see that LOA works, even when we get low and we think it doesn't and we are losing hope.
1. Some years back, at the first year of the university, the very first thing I saw in the study guide was the possibility to study one year abroad via an exchange program. You had to be at the 4th year or sometimes at 3rd. I wanted to go to a specific country, there was only one university there, I hadn't heard of that town before, I didn't know where exactly it was, but I decided I wanted to go there.
2nd year comes, it was the dates of the applications. I did my part, I applied for that position, ONLY for that position, I didn't care about other places or countries. The professors were telling me to apply for more but I didn't.
Day of the results... I wasn't selected... just the first substitute in case the other person wouldn't go. (they always go! ). I remember that moment very clearly till now. I was at school, saw the results and went upstairs where there were toilets and I started crying there alone. The moment I was sure to come and waited for years hadn't come.
The following days, I decided to apply for the empty places that were left, it wasn't what I initially wanted, but at least I would spend the year abroad, I didn't want to wait another year to apply again. I had the papers done, I was sad doing this though, and the professor in charge told me she wouldn't approve me anyway for the places left because she prefered 4rth year students. I went to the office and I see an announcement that the person selected for the place I wanted, she won't go, she decided to go to another emply position!
So, they took me!
2. After those studies

I decided I wanted to do a postgraduate. I applied for Masters a couple of times, had exams, no success. I didn't really believe that I'd make it since I know that in my country, things are not very 'clear' and objective about the candidates selection.
I leave the Master dream, and focus on exams to enter in another program that I always liked. It wasn't easier than the Master exams, in fact it was a lotmore difficult since it wasn't my subject, but I considered it to be easier, so I made it, and started studying. Next year, I apply for a postgraduate that the selection is based on random selection among the candidates that have the qualifications. So I needed 'luck' in order to be one of them, the rate was 1/15. I believed I would be one of them. I started making schedules about how I was going to make it with two study programs, the job etc. I even started complaining to people that I\d have not enough free time!
I was 'lucky', I am studying it now!