The first thing to do is stop thinking about winning! It only makes you nervous. I learned that in my first competition in Paris in 1996. My teacher said I will surely win after I played Transcendental Etude number 11 of Liszt. It was pretty good at home during my lesson, but in the competition i was very nervous and my left and right hands got on different pages quickly. I settled down and the director later told me I played at high level.
Since then I played in 15 competitions, including St. Petersburg last year. I was the only American there and I won a diploma for playing Taneyev Prelude and Fugue. I haven't won any money yet, but I had a lot of fun and made a lot of friends. The competitions have given me performance opportunities. Each time I play with less stage fright because I feel more comfortable playing in front of people. It is a psychological problem. It's about self esteem. When you try do do anything difficult, it's natural to think about failure. If you are trying to impress others, to do something their way, to fit in, to be better than them, to care about what they think, this will affect your self esteem, put you in a false sense of expectation. A better way to believe you are doing your best and that your love of art is worth all the work and money you put into it. If you're not the best, somebody will surely like something about your performance and will tell you. It will feel good and you will go home wanting to do more of what you love.
I think that pianists who start at a very young age have an advantage over me, who started when I was 18. I was already worried about earning money, paying bills, studying for engineering degree, having a girlfriend. Shutting them out to practice piano was stressful. Performing made me worry if I was doing the right thing. Five year old kids don't have to worry about all those things, so I think it's easier for them to get on the stage. Maybe not for everyone, but that's part of it.
I think the best thing is to be logical, play the music you feel comfortable with because you understand it, learned it well, played it for people, if only your teacher. Know what you do the best octaves, scales, etc, or different style like impressionist. Also develope a meticulous effort to discover every little detail in the music and have a concept how to play it with articulation, dynamics, phrasing, tempo, rubato, etc. The better you know the music, the less stage fright you will have. Why take pills for a headache, when the cause of the headache is you keep hitting your head on the wall? It's better to stop hitting the wall.
I'm practicing hard for the St. Petersburg comp. So I hope to see you there in two weeks. I'm playing Schumann - Aufshwung, My work "Tolko Raz", Lyapunov Transcendental Etude no 3, Mendelssohn Rondo Capriciosso, Chopin Polonaise op 40 no 2, and Griffes Sonata. Nobody who wins these amature competitions will get rich or famous. The prize money is not much either, but it's great to express myself on the stage for a few minutes, to make new friends who love what I love, and to see St. Petersburg and other cities.