The correct option is B competitive
The given sentence requires an adjective. The words "very ___ spirit" comprise of an adverb 'very' and a noun 'spirit'. Adverbs modify verbs, adjectives and other adverbs. But, only adjectives can modify nouns, like 'spirit'. Hence, the required word is an adjective.
Option C: 'Competitive' is an adjective. It relates to the given sentence, as the word means 'displaying a strong desire to be better than others'. Hence option C is correct.
Option A: 'Compete' is a verb form of 'competitive' and verbs cannot modify nouns. ('a very compete spirit' sounds strange) Hence option A is incorrect.
Options B and D: 'Competition' and 'competitor' are both nouns. Adverbs do not modify nouns. Hence B and D are also incorrect.