\section*{Problem 2 - 191032}

Determine all pairs \((x ; y)\) of real numbers for which, firstly, the term on the left side (as a real number) in the equation

\[
2^{\sqrt{1+x-3 y}}+3^{\sqrt{2 x-4 y+1}}=2
\]

is defined, and secondly, the equation is satisfied!
}

The term mentioned is defined if and only if the inequalities

\[
1+x-3 y \geq 0 \quad(2) \quad \text { and } \quad 2 x-4 y+1 \geq 0
\]

hold. If this is the case, then the inequalities

\[
2^{\sqrt{1+x-3 y}} \geq 1 \quad \text { (4) } \quad ; \quad 3^{\sqrt{2 x-4 y+1}} \geq 1
\]

also hold. Therefore, (1) can only be satisfied if the equality sign holds in both (4) and (5). This is only true if

\[
1+x-3 y=0 \quad(6) \quad \text { and } \quad 2 x-4 y+1=0
\]

hold. The system of equations (6), (7) has exactly the pair

\[
(x ; y)=\left(\frac{1}{2} ; \frac{1}{2}\right)
\]

as a solution. Conversely, for this pair, (2), (3), and (1) also follow from (6), (7). Therefore, this pair has all the required properties.