Difference between revisions of "2005 AMC 12A Problems/Problem 13"

Latest revision as of 10:01, 22 August 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 that order. The sums of the numbers at the ends of the line segments $\overline{AB}$ , $\overline{BC}$ , $\overline{CD}$ , $\overline{DE}$ , and $\overline{EA}$ form an arithmetic sequence, although not necessarily in that order. What is the middle term of the arithmetic sequence?

$[asy] draw((0,0)--(0.5,1.54)--(1,0)--(-0.31,0.95)--(1.31,0.95)--cycle); label("$A$",(0.5,1.54),N); label("$B$",(1,0),SE); label("$C$",(-0.31,0.95),W); label("$D$",(1.31,0.95),E); label("$E$",(0,0),SW); [/asy]$

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

Solutions

Solution 1

$(A+B) + (B+C) + (C+D) + (D+E) + (E+A) = 2(A+B+C+D+E)$ (i.e., each number is counted twice). The sum $A + B + C + D + E$ will always be $3 + 5 + 6 + 7 + 9 = 30$ , so the arithmetic sequence has a sum of $2 \cdot 30 = 60$ . The middle term must be the average of the five numbers, which is $\frac{60}{5} = 12 \Longrightarrow \mathrm{(D)}$ .

Solution 2

Let the terms in the arithmetic sequence be $a$ , $a + d$ , $a + 2d$ , $a + 3d$ , and $a + 4d$ . We seek the middle term $a + 2d$ .

These five terms are $A + B$ , $B + C$ , $C + D$ , $D + E$ , and $E + A$ , in some order. The numbers $A$ , $B$ , $C$ , $D$ , and $E$ are equal to 3, 5, 6, 7, and 9, in some order, so $A + B + C + D + E = 3 + 5 + 6 + 7 + 9 = 30.$ Hence, the sum of the five terms is $(A + B) + (B + C) + (C + D) + (D + E) + (E + A) = 2A + 2B + 2C + 2D + 2E = 60.$ But adding all five numbers, we also get $a + (a + d) + (a + 2d) + (a + 3d) + (a + 4d) = 5a + 10d$ , so $5a + 10d = 60.$ Dividing both sides by 5, we get $a + 2d = \boxed{12}$ , which is the middle term. The answer is (D).

Solution 3

Not too bad with some logic and the awesome guess and check. Let $A=6$ . Then let $B=7,E=5$ and $C=3,D=9$ . Our arithmetic sequence is $10,11,12,13,14$ so our answer is $12 \Longrightarrow \mathrm{(D)}$ .

Solution by franzliszt

@@ Line 1: / Line 1: @@
 == Problem ==
 In the five-sided star shown, the letters <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math> and <math>E</math> are replaced by the
-numbers 3, 5, 6, 7 and 9, although not necessarily in that order. The sums of the
+numbers <math>3, 5, 6, 7</math> and <math>9,</math> although not necessarily in that order. The sums of the
 numbers at the ends of the line segments <math>\overline{AB}</math>, <math>\overline{BC}</math>, <math>\overline{CD}</math>, <math>\overline{DE}</math>, and <math>\overline{EA}</math> form an
 arithmetic sequence, although not necessarily in that order. What is the middle

2005 AMC 12A (Problems • Answer Key • Resources)
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