. . . . . . . . eleven year old boy won . . . . . . . . National Essay-writing competition.

Here, the sentence talks about the 'boy' as general class of things. Since we are referring to general things, we need to use the indefinite article 'a/an'. The word 'eleven' starts with a vowel and therefore 'an' will be used.
In the second blank a specific competition is being talked about. Definite article 'the' will be used.