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 4 of 6: The tuple

(f_a(X))_{a\in A}

determines

X

Previous step Next step

X\ \text{is determined by}\ (f_a(X))_{a\in A}

Detailed analysis

By the second step, $X=\sum_{a\in A}f_a(X)\,a$ is recovered exactly from the tuple of values $(f_a(X))_{a\in A}$ , so distinct multiples $X$ of $m$ below $m^{m+1}$ must give distinct tuples: the map $X\mapsto(f_a(X))_{a\in A}$ is injective.