Difference between revisions of "2010 AMC 12B Problems/Problem 14"
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== Problem 14 == | == Problem 14 == | ||
| − | Let <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math>, and <math>e</math> be | + | Let <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math>, and <math>e</math> be positive integers with <math>a+b+c+d+e=2010</math> and let <math>M</math> be the largest of the sum <math>a+b</math>, <math>b+c</math>, <math>c+d</math> and <math>d+e</math>. What is the smallest possible value of <math>M</math>? |
<math>\textbf{(A)}\ 670 \qquad \textbf{(B)}\ 671 \qquad \textbf{(C)}\ 802 \qquad \textbf{(D)}\ 803 \qquad \textbf{(E)}\ 804</math> | <math>\textbf{(A)}\ 670 \qquad \textbf{(B)}\ 671 \qquad \textbf{(C)}\ 802 \qquad \textbf{(D)}\ 803 \qquad \textbf{(E)}\ 804</math> | ||
Revision as of 18:32, 1 January 2012
Problem 14
Let 
, 
, 
, 
, and 
be positive integers with 
and let 
be the largest of the sum 
, 
, 
and 
. What is the smallest possible value of ?
Solution
We want to try make , , , and as close as possible so that , the maximum of these, if smallest.
Notice that 
. In order to express 
as a sum of 
numbers, we must split up some of these numbers. There are two ways to do this (while keeping the sum of two numbers as close as possible): 
or 
. We see that in both cases, the value of is 
, so the answer is 
.
See also
| 2010 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 13 |
Followed by Problem 15 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
