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2012 UNCO Math Contest II Problems/Problem 9

Revision as of 03:22, 17 October 2014 by Timneh (talk | contribs) (Created page with "== Problem == Treasure Chest . You have a long row of boxes. The 1st box contains no coin. The next <math>2</math> boxes each contain <math>1</math> coin. The next <math>4</math...")
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Problem

Treasure Chest . You have a long row of boxes. The 1st box contains no coin. The next $2$ boxes each contain $1$ coin. The next $4$ boxes each contain $2$ coins. The next $8$ boxes each contain $3$ coins. And so on, so that there are $2^N$ boxes containing exactly $N$ coins.

(a) If you combine the coins from all the boxes that contain $1, 2, 3$, or $4$ coins you get $98$ coins. How many coins do you get when you combine the coins from all the boxes that contain $1, 2, 3,\ldots,$ or $N$ coins? Give a closed formula in terms of $N$. That is, give a formula that does not use ellipsis $(\ldots)$ or summation notation.

(b) Combine the coins from the first $K$ boxes. What is the smallest value of $K$ for which the total number of coins exceeds $20120$? (Remember to count the first box.)


Solution

See Also

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