Subject 1. Solve in real numbers the system: $\left\{\begin{array}{c}{[x]+y=4.2} \\ 2 x+[y]=5.3\end{array}\right.$.

Mathematical Gazette 2013
Subject 1. $\{y\}=0.2(\mathbf{1} \mathbf{p}) ;[x]+[y]=4(\mathbf{1} \mathbf{p}) ;[2 x]+[y]=5(\mathbf{1} \mathbf{p}) ;$ $\left.\left[x+\frac{1}{2}\right]=1 \mathbf{( 1 p}\right)$, so $\frac{1}{2} \leq x<\frac{3}{2}(\mathbf{1} \mathbf{p})$. Case 1. $[x]=0 \Rightarrow[y]=4 \Rightarrow$ $y=4.2$, then $x=0.65(\mathbf{1 p})$; Case 2. $[x]=1 \Rightarrow[y]=3 \Rightarrow y=3.2$, then $x=1.15(\mathbf{1} \mathbf{p})$.