Find the greatest integer less than $10.3^3$ without using a calculator.
In general, \[(a+b)^3=a^3+3a^2b+3ab^2+b^3\]

Consider the decimal part separately, so we're trying to find the greatest integer less than $(10+0.3)^3$. By the Binomial Theorem in the above expansion, this is equal to  \[10^3+3(10^2)(.3)+3(10)(.3^2)+.3^3\] Expanding these terms, we have that $(10.3)^3=1000+90+2.7+.027$. From this, we find that the greatest integer less than this quantity is $\boxed{1092}$.