Азарт is a temporary feeling or permanent personal feature that makes someone to perform risky activities that can probably or even mostly probably lead to failure. Азарт makes people to play lottery, casino, sweepstakes. If a person is an addict, he can't resist the Азарт and wastes all his money and even property for the game. "Азарт взял свое" - she tried to struggle to overcome the feeling, but finally the feeling took over and she finally decided to participate (though she didn't believe she wins). She says "во что бы то ни стало" - it seems there were some difficulties on a way to participating, no matter external or internal (e.g. she was too shy?). But it didn't stop her, because азарт взял свое.

Paola, "взял своё" is literally "took what-is-his", "took what-belongs-to-it". It is a metaphor. E.g. "усталость взяла своё и он заснул". Азарт is about a _competition_, _game_, argument etc. From French 'hasard', as you could guess:) (and further from Arabic az-zahr - as I understand you are an Arabic learner)

I would translate it as: "Excitement took over me, and I decided to take part in the selection, no matter what." "брать/взять своё" is a fixed expression and means that the feeling or state expressed by the noun makes things happen and act in a way that is inherent to this feeling or state, very often becoming dominant even though you're trying to resist it. It can also be used with abstract nouns, for example in expressions like "возраст берет свое" or "годы берут свое", simply meaning that we are getting older, with all the consequences that it implies.

do you know what "Азарт" is?

Азарт взял своё - у меня появился азарт! мне захотелось сделать что-то!