Difference between revisions of "2005 AMC 12A Problems/Problem 13"
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===Solution 3=== | ===Solution 3=== | ||
− | Not too bad with some logic and the awesome guess and check. Let <math>A=6</math>. Then let <math>B=7,E=5</math> and <math>C=3,D=9</math>. Our arithmetic sequence is <math>10,11,12,13,14 | + | Not too bad with some logic and the awesome guess and check. Let <math>A=6</math>. Then let <math>B=7,E=5</math> and <math>C=3,D=9</math>. Our arithmetic sequence is <math>10,11,12,13,14</math> so our answer is <math>12 \Longrightarrow \mathrm{(D)}</math>. |
Solution by franzliszt | Solution by franzliszt |
Revision as of 18:52, 11 July 2020
Problem
In the five-sided star shown, the letters ,
,
,
and
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
,
,
,
, and
form an
arithmetic sequence, although not necessarily in that order. What is the middle
term of the arithmetic sequence?
Solutions
Solution 1
(i.e., each number is counted twice). The sum
will always be
, so the arithmetic sequence has a sum of
. The middle term must be the average of the five numbers, which is
.
Solution 2
Let the terms in the arithmetic sequence be ,
,
,
, and
. We seek the middle term
.
These five terms are ,
,
,
, and
, in some order. The numbers
,
,
,
, and
are equal to 3, 5, 6, 7, and 9, in some order, so
Hence, the sum of the five terms is
But adding all five numbers, we also get
, so
Dividing both sides by 5, we get
, which is the middle term. The answer is (D).
Solution 3
Not too bad with some logic and the awesome guess and check. Let . Then let
and
. Our arithmetic sequence is
so our answer is
.
Solution by franzliszt
See also
2005 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 12 |
Followed by Problem 14 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.