The subject of the sentence is: 'eleven-year-old boy'. This noun phrase does not refer to a specific person. Hence, an indefinite article ('a' or 'an') is to be used before this phrase. The word, 'eleven' starts with a vowel sound. Hence, the article to be used before the said noun phrase is 'an'.
Options A and D are incorrect as they indicate the use of the article, 'the' before the noun phrase, 'eleven-year-old boy'. Hence, these options are incorrect.
The correct answer has to be either option B and C.
'National Essay-writing competition' is the name of a particular national level competition which is well known. Hence, this noun phrase refers to a specific competition and the article to be used before it is: 'the'.
Of the two options B and C, option C uses the article, 'the before the phrase, 'National Essay-writing competition. Hence, option C is correct.
Option B is incorrect as it indicates the use of the article 'a' before the noun phrase, 'National Essay-writing competition.