Belorusija 2012

Answer: $(-10,-11),(-10,9),(-3,-5),(-3,3),(2,-5),(2,3),(9,-11),(9,9)$.

Multiplying the given equality by $4$, we obtain $$4n^2+4n=4m^2+8m-36\iff 4n^2+4n+1=4m^2+8m+4-39\iff$$ $$(2n+1{)}^2=(2m+2{)}^2-39\iff (2m+2{)}^2-(2n+1{)}^2=39\iff (2m+2-2n-1)(2m+2+2n+1)=39\iff (2m-2n+1)(2m+2n+3)=39.$$ Since $39=1\cdot 39=39\cdot 1=3\cdot 13=13\cdot 3=(-1)\cdot (-39)=(-39)\cdot (-1)=(-3)\cdot (-13)=(-13)\cdot (-3)$ are all possible factorizations of $39$, it suffices to consider the following cases. \begin{array}{l@{\quad}l@{\qquad}l} 1) \left\{ \begin{array}{l} 2m - 2n + 1 = 1, \\ 2m + 2n + 3 = 39 \end{array} \right. & \text{which gives } \left\{ \begin{array}{l} 4m + 4 = 40, \\ 4n + 2 = 38 \end{array} \right. & \text{so } \left\{ \begin{array}{l} m = 9, \\ n = 9. \end{array} \right. \\ 2) \left\{ \begin{array}{l} 2m - 2n + 1 = 3, \\ 2m + 2n + 3 = 13 \end{array} \right. & \Leftrightarrow & \left\{ \begin{array}{l} 4m + 4 = 16, \\ 4n + 2 = 10 \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} m = 3, \\ n = 2. \end{array} \right. \\ 3) \left\{ \begin{array}{l} 2m - 2n + 1 = 39, \\ 2m + 2n + 3 = 1 \end{array} \right. & \Leftrightarrow & \left\{ \begin{array}{l} 4m + 4 = 40, \\ 4n + 2 = -38 \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} m = 9, \\ n = -10. \end{array} \right. \\ 4) \left\{ \begin{array}{l} 2m - 2n + 1 = 13, \\ 2m + 2n + 3 = 3 \end{array} \right. & \Leftrightarrow & \left\{ \begin{array}{l} 4m + 4 = 16, \\ 4n + 2 = -10 \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} m = 3, \\ n = -3. \end{array} \right. \\ 5) \left\{ \begin{array}{l} 2m - 2n + 1 = -1, \\ 2m + 2n + 3 = -39 \end{array} \right. & \Leftrightarrow & \left\{ \begin{array}{l} 4m + 4 = -40, \\ 4n + 2 = -38 \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} m = -11, \\ n = -10. \end{array} \right. \\ 6) \left\{ \begin{array}{l} 2m - 2n + 1 = -3, \\ 2m + 2n + 3 = -13 \end{array} \right. & \Leftrightarrow & \left\{ \begin{array}{l} 4m + 4 = -16, \\ 4n + 2 = -10 \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} m = -5, \\ n = -3. \end{array} \right. \\ 7) \left\{ \begin{array}{l} 2m - 2n + 1 = -39, \\ 2m + 2n + 3 = -1 \end{array} \right. & \Leftrightarrow & \left\{ \begin{array}{l} 4m + 4 = -40, \\ 4n + 2 = 38 \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} m = -11, \\ n = 9. \end{array} \right. \\ 8) \left\{ \begin{array}{l} 2m - 2n + 1 = -13, \\ 2m + 2n + 3 = -3 \end{array} \right. & \Leftrightarrow & \left\{ \begin{array}{l} 4m + 4 = -16, \\ 4n + 2 = 10 \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} m = -5, \\ n = 2. \end{array} \right. \end{array}