Problem:
For t=1,2,3,4, define St=∑i=1350ait, where ai∈{1,2,3,4}. If S1=513 and S4=4745, find the minimum possible value for S2.
Solution:
For j=1,2,3,4, let mj be the number of ai 's that are equal to j. Then
m1+m2+m3+m4=350,S1=m1+2m2+3m3+4m4=513, and S4=m1+24m2+34m3+44m4=4745.
Subtracting the first equation from the second, then the first from the third yields
m2+2m3+3m4=163, and 15m2+80m3+255m4=4395.
Now subtracting 15 times the first of these equations from the second yields 50m3+210m4=1950 or 5m3+21m4=195. Thus m4 must be a nonnegative multiple of 5, and so m4 must be either 0 or 5 . If m4=0, then the mj 's must be (226,85,39,0), and if m4=5, then the mj 's must be (215,112,18,5). The first set of values results in S2= 12⋅226+22⋅85+32⋅39+42⋅0=917, and the second set of values results in S2=12⋅215+22⋅112+32⋅18+42⋅5=905. Thus the minimum value is 905 .
The problems on this page are the property of the MAA's American Mathematics Competitions