Answer to Question #279376 in Discrete Mathematics for Jaishree

Question

In a mathematics contest with three problems, 80% of the participants solved

the first problem, 75% solved the second and 70% solved the third. Prove that

at least 25% of the participants solved all three problems.

Answer 1

Let the total number of participants be $n > 0$ (if $n = 0,$ the proof is trivial). Denote

the set of people who missed the first problem by $A,$

the set of people who missed the second by $B,$

and the set who missed the third by $C.$

We know that $|A| = n − 0.8n = 0.2n, |B| = n − 0.75n = 0.25n,$ and $|C| = n − 0.7n = 0.3n.$

We also know, that

$$|A ∪ B ∪ C| ≤ |A| + |B| + |C|$$

$$= 0.2n + 0.25n + 0.3n = 0.75n$$

The set of people who solved all three problems is the complement of $A ∪ B ∪ C$ (the set who missed at least one problem), so it has size

$n-|A ∪ B ∪ C|.$

$$n − |A ∪ B ∪ C| ≥ n − 0.75n = 0.25n$$

Therefore at least 25% of the participants solved all three problems.