2011 AMC 12B Problems/Problem 13
Problem
Brian writes down four integers whose sum is
. The pairwise positive differences of these numbers are
and
. What is the sum of the possible values of
?
Solution
Assume that
results in the greatest pairwise difference, and thus it is
.
This means
.
must be in the set
.
The only way for 3 numbers in the set to add up to 9 is if they are
.
, and
then must be the remaining two numbers which are
and
.
The ordering of
must be either
or
.
Case 1
$(a,b,c)=(3,1,5)\\
x=w-5\\
y=w-5-1\\
x=w-5-1-3\\
w+x+y+z=4w-20=44\\
w=16\$ (Error compiling LaTeX. Unknown error_msg)
Case 2
The sum of the two w's is
See also
2011 AMC 12B (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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