2005 AMC 12A Problems/Problem 13
Problem
The regular 5-point star is drawn and in each vertex, there is a number. Each
and
are chosen such that all 5 of them came from set
. Each letter is a different number (so one possible way is
). Let
be the sum of the numbers on
and
, and so forth. If
and
form an arithmetic sequence (not necessarily in increasing order), find the value of
.
Solution
Solution I
. The sum
will always be
, so the arithmetic sequence has a sum of
. Since
is the middle term, it must be the average of the five numbers, of
.
Solution II
Assume AB is the smallest term (this assumption will drop out by the end). If AB is the smallest the sequence goes A+B,B+C,C+D,D+E,E+A. If (A+B)<(B+C) then A must be less than C. Using this method we eventually get A<C<E and B<D<A so we know B<D<A<C<E. Therefore D is 5 and C is 7 and CD is 12. If AB is the biggest it just works out to D being 7 and C being 5. D
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 |
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