Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

(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...")
 
(→‎Solution)
 
(2 intermediate revisions by the same user not shown)
Line 11: Line 11:
  
 
(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
−
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.)
  
  
 
== Solution ==
 
== Solution ==
−
 
+
(a) <math>{N-1}2^{N+1}+2</math> (b) <math>2201</math>
  
 
== See Also ==
 
== See Also ==
−
{{UNC Math Contest box|n=II|year=2012|num-b=8|num-a=10}}
+
{{UNCO Math Contest box|n=II|year=2012|num-b=8|num-a=10}}
  
 
[[Category:Intermediate Combinatorics Problems]]
 
[[Category:Intermediate Combinatorics Problems]]

Latest revision as of 02:26, 13 January 2019

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

(a) ${N-1}2^{N+1}+2$ (b) $2201$

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
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2012_UNCO_Math_Contest_II_Problems/Problem_9&oldid=100372"