2010 AMC 10B Problems/Problem 24

Represent the teams' scores as: (a, an, an^2, an^3) and (a, a+m, a+2m, a+3m)

We have a+an+an^2+an^3=4a+6m+1 Manipulating this, we can get a(1+n+n^2+n^3)=4a+6m+1, or a(n^4-1)/(n-1)=4a+6m+1

Since both are increasing sequences, n>1. We can check cases up to n=4 because when n=5, we get 156a>100. When n=2, a=[1,6]

         n=3, a=[1,2]
         n=4, a=1

Checking each of these cases individually back into the equation a+an+an^2+an^3=4a+6m+1, we see that only when a=5 and n=2, we get an integer value for m, which is 9. The original question asks for the first half scores summed, so we must find (a)+(an)+(a)+(a+m)=(5)+(10)+(5)+(5+9)=34