$\text{ If }x=(99{)}^{50}+(100{)}^{50}\text{ and }y=(101{)}^{50},\text{ then }$

$$\begin{gathered} (101{)}^{50}-(99{)}^{50}=(100+1{)}^{50}-(100-1{)}^{50} \\ =2\left[{}^{50}C_1{100}^{49}+{}^{50}C_3{100}^{47}+....\right] \\ ={100}^{50}+2{}^{50}C_3{100}^{47}+..... \\ >{100}^{50} \\ \therefore (101{)}^{50}-(99{)}^{50}>{100}^{50} \\ \Rightarrow (99{)}^{50}+{100}^{50}<(101{)}^{50} \end{gathered}$$