Problem 18 · 2009 Math Kangaroo
Hard
Counting & Probability
complementary-countingsum-constraint
100 students take an exam with 4 questions. 90 solve the first question, 85 the second, 80 the third and 70 the fourth. Determine the smallest possible number of students that have solved all four questions.
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Answer: D — 25
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Hint 1 of 2
Count the ‘misses’ rather than the solves: how many failures are there in total?
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Hint 2 of 2
Spread the failures over as many different students as possible to minimise those who got everything.
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Approach: count failures and subtract from 100
- The numbers who missed each question are 10, 15, 20, 30, totalling 75 failures.
- At best each failure falls on a different student, so at most 75 students missed something.
- Then at least 100 − 75 = 25 students solved all four.
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