Subject (2). Given the sets of natural numbers

\[
\begin{aligned}
& A=\{x-1,2 x-3,3 x+1,4 x-1\} \text { and } \\
& B=\{4 x-2,3 x+2,3 x-6,2 x-4\}
\end{aligned}
\]

Determine the natural values of \( x \) for which \( A=B \).

G.M.-B. nr. 11/2014
Solution. For the two sets to be equal, they must have the same sum of their elements, which forces the equality $10 x-4=12 x-10$, i.e., $x=3$, and it only remains to verify that then $A=\{2,3,10,11\}=B$. As can be seen, this solution does not require any special assumption about $x$ (and now $x$ can be left unspecified, but precludes simpler approaches related to parity, etc., as in the official solution).[^0]