Five positive integers have mean 10 and median 12. What is the greatest possible value of the smallest of the five integers?

The five integers add up to $5\times 10=50$ and the middle one is 12. Write them in order as a ≤ $b$ ≤ 12 ≤ $d$ ≤ $e$. The two largest are each at least 12, so $d+e$ ≥ 24 and therefore $a+b$ ≤ 50 − 12 − 24 = 14. Since a ≤ $b$ this forces $2a$ ≤ 14, that is a ≤ 7, and $a=7$ really is reached, by 7, 7, 12, 12, 12. Pushing the smallest up to 8 would need $a+b$ ≥ 16 and leave under 24 for the top two, which is impossible.