Difference between revisions of "2010 AMC 12B Problems/Problem 14"

Latest revision as of 14:38, 17 April 2024

Problem

Let $a$ , $b$ , $c$ , $d$ , and $e$ be positive integers with $a+b+c+d+e=2010$ and let $M$ be the largest of the sum $a+b$ , $b+c$ , $c+d$ and $d+e$ . What is the smallest possible value of $M$ ?

$\textbf{(A)}\ 670 \qquad \textbf{(B)}\ 671 \qquad \textbf{(C)}\ 802 \qquad \textbf{(D)}\ 803 \qquad \textbf{(E)}\ 804$

Solution 1

We want to try make $a+b$ , $b+c$ , $c+d$ , and $d+e$ as close as possible so that $M$ , the maximum of these, is smallest.

Notice that $2010=670+670+670$ . In order to express $2010$ as a sum of $5$ 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): $2010=670+1+670+1+668$ or $2010=670+1+669+1+669$ . We see that in both cases, the value of $M$ is $671$ , so the answer is $671 \Rightarrow \boxed{B}$ .

Solution 2

Since $a + b \le M$ , $d + e \le M$ , and $c < b + c \le M$ , we have that $2010 = a + b + c + d + e < 3M$ . Hence, $M > 670$ , or $M \ge 671$ .

For the values $(a,b,c,d,e) = (669,1,670,1,669)$ , $M = 671$ , so the smallest possible value of $M$ is $\boxed{671}$ . The answer is (B).

~ math31415926535

Solution 3

Notice that only the sums of adjacent numbers matter. (For example, a & c could be extremely high, as long as b is relatively low.) Therefore creating "mountains" and "valleys" is the best way to lower the sum of adjacent numbers. We can do

1. (high, low, high, low, high)

or

2. (low, high, low, high, low)

In the extreme case that each "low" = $0$ , $2010$ will be divided into either $3$ or $2$ numbers for cases 1 and 2, respectively. Obviously dividing by $3$ will yield a lower number, so we consider case 1.

Dividing $2010$ by $3$ yields $670$ , or

( $670, 0, 670, 0, 670$ )

However, all five numbers must be positive, and the closest we can get to this is

( $668, 3, 668, 3, 668$ )

The lowest possible sum of two adjacent numbers then becomes $671$ , or $\boxed{B}$ .

@@ Line 1: / Line 1: @@
-== Problem 14 ==
+== Problem ==
 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>
-== Solution ==
+== Solution 1 ==
-We want to try make <math>a+b</math>, <math>b+c</math>, <math>c+d</math>, and <math>d+e</math> as close as possible so that <math>M</math>, the maximum of these, if smallest.
+We want to try make <math>a+b</math>, <math>b+c</math>, <math>c+d</math>, and <math>d+e</math> as close as possible so that <math>M</math>, the maximum of these, is smallest.
 Notice that <math>2010=670+670+670</math>. In order to express <math>2010</math> as a sum of <math>5</math> 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): <math>2010=670+1+670+1+668</math> or <math>2010=670+1+669+1+669</math>. We see that in both cases, the value of <math>M</math> is <math>671</math>, so the answer is <math>671 \Rightarrow \boxed{B}</math>.
+== Solution 2 ==
+Since <math>a + b \le M</math>, <math>d + e \le M</math>, and <math>c < b + c \le M</math>, we have that <math>2010 = a + b + c + d + e < 3M</math>. Hence, <math>M > 670</math>, or <math>M \ge 671</math>.
+For the values <math>(a,b,c,d,e) = (669,1,670,1,669)</math>, <math>M = 671</math>, so the smallest possible value of <math>M</math> is <math>\boxed{671}</math>. The answer is (B).
+~ math31415926535
+==Solution 3==
+Notice that only the sums of adjacent numbers matter. (For example, a & c could be extremely high, as long as b is relatively low.)
+Therefore creating "mountains" and "valleys" is the best way to lower the sum of adjacent numbers. We can do
+. (high, low, high, low, high)
+or
+. (low, high, low, high, low)
+In the extreme case that each "low" = <math>0</math>, <math>2010</math> will be divided into either <math>3</math> or <math>2</math> numbers for cases 1 and 2, respectively. Obviously dividing by <math>3</math> will yield a lower number, so we consider case 1.
+Dividing <math>2010</math> by <math>3</math> yields <math>670</math>, or
+(<math>670, 0, 670, 0, 670</math>)
+However, all five numbers must be positive, and the closest we can get to this is
+(<math>668, 3, 668, 3, 668</math>)
+The lowest possible sum of two adjacent numbers then becomes <math>671</math>, or <math>\boxed{B}</math>.
 == See also ==
 {{AMC12 box|year=2010|num-b=13|num-a=15|ab=B}}
+[[Category:Introductory Algebra Problems]]
 {{MAA Notice}}

2010 AMC 12B (Problems • Answer Key • Resources)
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