A test with 20 questions was applied to 300 people. We know that 8 questions had at least 100 hits and the rest at least 200 hits. Prove that some student got at least 11 questions right.
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OK assume exactly 8 questions got 100 hits and the remaining 12 got exactly 200 hits. That means there were exactly 8 * 100 + 12 * 200 = 800 + 2400 = 3200 hits. If no student had at least 11 questions right, then the most any student could have got right is 10. If 300 students each got 10 answers right, that's just 3000 hits. But we know there were at least 3200 hits. Therefore, it can't be that there's no student with at least 11 hits; some student must have at least that many.