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Difference between revisions of "2012 UNCO Math Contest II Problems/Problem 9"

m (moved 2012 UNC Math Contest II Problems/Problem 9 to 2012 UNCO Math Contest II Problems/Problem 9: disambiguation of University of Northern Colorado with University of North Carolina)
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(b) Combine the coins from the first <math>K</math> boxes. What is the smallest value of <math>K</math> for which
 
(b) Combine the coins from the first <math>K</math> boxes. What is the smallest value of <math>K</math> for which
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the total number of coins exceeds <math>20120</math>? (Remember to count the first box.)
+
the total number of coins exceeds <math>20120</math> ? (Remember to count the first box.)
  
  
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== See Also ==
 
== See Also ==
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{{UNC Math Contest box|n=II|year=2012|num-b=8|num-a=10}}
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{{UNCO Math Contest box|n=II|year=2012|num-b=8|num-a=10}}
  
 
[[Category:Intermediate Combinatorics Problems]]
 
[[Category:Intermediate Combinatorics Problems]]

Revision as of 21:26, 19 October 2014

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

2012 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10
All UNCO Math Contest Problems and Solutions
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