2011 AMC 10B Problems/Problem 21
Problem 21
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 for
?
Solution
The largest difference, must be between
and
The smallest difference, must be directly between two integers. This also means the differences directly between the other two should add up to
The only remaining differences that would make this possible are
and
However, those two differences can't be right next to each other because they would make a difference of
This means
must be the difference between
and
We can express the possible configurations as the lines.
![[asy] unitsize(14mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair Z1=(0,1), Y1=(1,1), X1=(2,1), W1=(3,1); pair Z4=(4,1), Y4=(5,1), X4=(6,1), W4=(7,1); draw(Z1--W1); draw(Z4--W4); pair[] ps={W1,W4,X1,X4,Y1,Y4,Z1,Z4}; dot(ps); label("$z$",Z1,N); label("$y$",Y1,N); label("$x$",X1,N); label("$w$",W1,N); label("$z$",Z4,N); label("$y$",Y4,N); label("$x$",X4,N); label("$w$",W4,N); label("$1$",(X1--Y1),N); label("$1$",(X4--Y4),N); label("$3$",(Y1--Z1),N); label("$3$",(W4--X4),N); label("$5$",(X1--W1),N); label("$5$",(Y4--Z4),N); [/asy]](http://latex.artofproblemsolving.com/c/e/1/ce15bff994b2e1387b4b56d1de02fde16bfbfc54.png)
If we look at the first number line, you can express as
as
and
as
Since the sum of all these integers equal
,
You can do something similar to this with the second number line to find the other possible value of
The sum of the possible values of
is