View source for 2012 UNCO Math Contest II Problems/Problem 9

== 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> boxes each contain <math>2</math> coins. The next <math>8</math> boxes each
contain <math>3</math> coins. And so on, so that there are <math>2^N</math> boxes containing exactly <math>N</math> coins.

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

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


== Solution ==
(a) <math>{N-1}2^{N+1}+2</math> (b) <math>2201</math>

== See Also ==
{{UNCO Math Contest box|n=II|year=2012|num-b=8|num-a=10}}

[[Category:Intermediate Combinatorics Problems]]