Difference between revisions of "2005 AMC 10A Problems/Problem 17"

Revision as of 23:47, 8 July 2024

Problem

In the five-sided star shown, the letters $A, B, C, D,$ and $E$ are replaced by the numbers $3, 5, 6, 7,$ and $9$ , although not necessarily in this order. The sums of the numbers at the ends of the line segments $AB$ , $BC$ , $CD$ , $DE$ , and $EA$ form an arithmetic sequence, although not necessarily in that order. What is the middle term of the arithmetic sequence?

$\textbf{(A) } 9\qquad \textbf{(B) } 10\qquad \textbf{(C) } 11\qquad \textbf{(D) } 12\qquad \textbf{(E) } 13$

Solution 1

(meta-solving; answer choices imply solution exists)

Each corner $(A,B,C,D,E)$ goes to two sides/numbers. ( $A$ goes to $AE$ and $AB$ , $D$ goes to $DC$ and $DE$ ). The sum of every term is equal to $2(3+5+6+7+9)=60$

Since the middle term in an arithmetic sequence is the average of all the terms in the sequence, the middle number is $\frac{60}{5}=\boxed{\textbf{(D) }12}$

Solution 2 (Doesn't assume a solution exists)

We know that the smallest number in the arithmetic sequence must be $\geq 3 + 5 = 8$ , and the largest number must be $\leq 7 + 9 = 16$ .

Since there are $5$ terms in this sequence, the common difference $d \leq (16 - 8)/(5-1) = 2$ .

Since $d$ is an integer (difference of sums of integers), and since exactly 2 of the sums must be odd, $d$ must be odd. Therefore, $d=1$ .

The middle term must have the majority parity, so it must be odd. The 2 terms adjacent to the middle are odd, $6+a$ and $6+b$ . $a-b = 2d = 2$ . $a$ and $b$ can't be the smallest (or largest) 2 odd numbers, because that would make it impossible to construct the smallest (or largest) sum from one of the remaining two numbers and one of the odd numbers. Therefore, $\{a,b\} = \{5,7\}$ . The middle sum must then be $(6+5) + 1 = (6+7)-1 = \boxed{12}$ . The remaining edges are $\{9,3\}$ (because $\{5,7\}$ can't be an edge, as that would make a triangle with 6), $\{3,7\}$ , and $\{5,9\}$ .

~oinava

Video Solution by OmegaLearn

https://youtu.be/tKsYSBdeVuw?t=544

~ pi_is_3.14

@@ Line 6: / Line 6: @@
 <math> \textbf{(A) } 9\qquad \textbf{(B) } 10\qquad \textbf{(C) } 11\qquad \textbf{(D) } 12\qquad \textbf{(E) } 13 </math>
-==Solution 1==
+==Solution 1 ==
+(meta-solving; answer choices imply solution exists)
 Each corner <math>(A,B,C,D,E)</math> goes to two sides/numbers. (<math>A</math> goes to <math>AE</math> and <math>AB</math>, <math>D</math> goes to <math>DC</math> and <math>DE</math>). The sum of every term is equal to <math>2(3+5+6+7+9)=60</math>
 Since the middle term in an arithmetic sequence is the average of all the terms in the sequence, the middle number is <math>\frac{60}{5}=\boxed{\textbf{(D) }12}</math>
-==Solution 2==
+==Solution 2 (Doesn't assume a solution exists) ==
-We know that the smallest number in this sequence must be <math>3 + 5 = 8</math>, and the biggest number must be <math>7 + 9 = 16</math>. Since there are <math>5</math> terms in this sequence, we know that <math>8 + 4d = 16</math>, or that <math>d = 2</math>. Thus, we know that the middle term must be <math>8 + 2 \cdot 2 = \boxed{12}.</math> This doesn’t necessarily work since not every single number will be added to every other number, but it is a good way to start trying during the test.
+We know that the smallest number in the arithmetic sequence must be <math>\geq 3 + 5 = 8</math>, and the largest number must be <math>\leq 7 + 9 = 16</math>.
-~yk2007 (Daniel K.)~sravya_m18
+Since there are <math>5</math> terms in this sequence, the common difference <math>d \leq (16 - 8)/(5-1) = 2</math>.
+Since <math>d</math> is an integer (difference of sums of integers), and since exactly 2 of the sums must be odd, <math>d</math> must be odd. Therefore, <math>d=1</math>.
+The middle term must have the majority parity, so it must be odd.
+The 2 terms adjacent to the middle are odd, <math>6+a</math> and <math>6+b</math>. <math>a-b = 2d = 2</math>. <math>a</math> and <math>b</math> can't be the smallest (or largest) 2 odd numbers, because that would make it impossible to construct the smallest (or largest) sum from one of the remaining two numbers and one of the odd numbers. Therefore, <math>\{a,b\} = \{5,7\}</math>. The middle sum must then be <math>(6+5) + 1 = (6+7)-1 = \boxed{12}</math>. The remaining edges are <math>\{9,3\}</math> (because <math>\{5,7\}</math> can't be an edge, as that would make a triangle with 6), <math>\{3,7\}</math>, and <math>\{5,9\}</math>.
+~oinava
 == Video Solution by OmegaLearn ==

2005 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 16	Followed by Problem 18
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 10 Problems and Solutions

Page

Toolbox

Search