Problem:
For t=1,2,3,4, define St​=∑i=1350​ait​, 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