Gold, Silver, Bronze
I have 3 gold coins, 4 silver coins, and 5 bronze coins. These 3 types of coins have different monetary values: $5, $4, and $3, respectively. I distribute all these coins to Albert, Bernard, and Caleb such that each of them receives a different total value.
Now, each of them donates all but 1 coin. Albert is left with a gold coin, Bernard a silver coin, and Caleb a bronze coin.
If Albert, Bernard, and Caleb each donated at least $11, who donated the most money?
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1 solution
We know 1 gold, 1 silver, 1 bronze were not used, so the used ones were 2 golds, 3 silvers, 4 bronzes, all summing up to 34 dollars. We know each needed to pay at least 11 dollars. So one of them paid 12, for 3 × 1 1 = 3 3 . The owner of 12 dollars can't have neither silver nor bronze because, for instance, he has 1 bronze left, he will have 15 dollars. The other two equally have 11 dollars, and if one of them got a silver coin, he would also have 15 dollars, contradicting the condition.
Thus, since Albert had a gold coin left, he donated the most.