Step 1: Identify the true subject of the sentence. The sentence begins with "The number of students who participate in the science fair". The head noun (true subject) is "The number" — singular. The phrase "of students who participate in the science fair" is a modifier, not the subject. Step 2: Apply subject-verb agreement. "The number" is singular → requires a singular verb form → "has" (not "have"). Note: "A number of students" would be plural, but "The number of students" is always singular. Step 3: Choose the correct tense. The clue "over the past three years" signals present perfect tense (an action that started in the past and continues to the present). Present perfect = has/have + past participle → "has increased". Step 4: Combine both rules. Singular subject ("the number") + present perfect → "has increased". Answer: has