2008 iTest Problems/Problem 11

Problem

After moving his sticky toy spider one morning, Tony heads outside to play "pirates" with his pal Nick, who lives a few doors down the street from the Kubiks. Tony and Nick imagine themselves as pirates in a rough skirmish over a chest of gold. Victorious over their foes, Tony and Nick claim the prize. However, they must split some of the gold with their crew, which they imagine consists of eight other bloodthirsty pirates. Each of the pirates receives at least one gold coin, but none receive the same number of coins, then Tony and Nick split the remainder equally. If there are $2000$ gold coins in the chest, what is the greatest number of gold coins Tony could take as his share? (Assume each gold coin is equally valuable.)

Solution

To minimize the amount of gold received by the other eight pirates (to maximize the remaining gold coins), one pirate gets $1$ coin, another pirate gets $2$ coins, and so on. Altogether, the eight bloodthirsty pirates got \[\tfrac{9 \cdot 8}{2} = 36\] gold coins, so there are $2000 - 36 = 1964$ gold coins left to split up evenly between Tony and Nick. Thus, Tony can take a maximum of $\tfrac{1964}{2} = \boxed{982}$ gold coins.

See Also

2008 iTest (Problems)
Preceded by:
Problem 10
Followed by:
Problem 12
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