CEMC Cayley

Suppose that the three integers are $x, y$ and $z$ where $x+y=998$, $x+z=1050$, and $y+z=1234$. From the first two equations, $(x+z)-(x+y)=1050-998$ or $z-y=52$. Since $z+y=1234$ and $z-y=52$, then $(z+y)+(z-y)=1234+52$ or $2z=1286$ and so $z=643$. Since $z=643$ and $z-y=52$, then $y=z-52=643-52=591$. Since $x+y=998$ and $y=591$, then $x=998-y=998-591=407$. The three original numbers are 407, 591, and 643. The difference between the largest and smallest of these integers is $643-407=236$.