Problem 6

Let

m\ge2

be an integer,

A

a finite set of (not necessarily positive) integers, and

B_1,B_2,B_3,\ldots,B_m

subsets of

A

. Suppose that for every

k=1,2,\ldots,m

the sum of the elements of

B_k

m^k

. Prove that

A

contains at least

m/2

elements.

Step 2 of 6: Reindex the sum over the elements of

A

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X=\sum_{a\in A}f_a(X)\,a,\quad f_a(X)=\sum_{i:\,a\in B_i}c_i

Detailed analysis

Substituting $m^i=\sum_{b\in B_i}b$ and swapping the order of summation, $X=\sum_{i=1}^m c_i\sum_{b\in B_i}b=\sum_{a\in A}\Big(\sum_{i:\,a\in B_i}c_i\Big)a=\sum_{a\in A}f_a(X)\,a$ , where $f_a(X):=\sum_{i:\,a\in B_i}c_i$ .