Find the greatest integer less than $10.3^3$ without using a calculator.
To solve this problem, I need to estimate $10.3^3$ and find the largest integer that is smaller than that. I know that $10^3 = 1000$, so $10.3^3$ must be a bit larger than that. To get a better estimate, I can use the binomial formula: $(10 + 0.3)^3 = 10^3 + 3 \cdot 10^2 \cdot 0.3 + 3 \cdot 10 \cdot 0.3^2 + 0.3^3$. The first term is $1000$, the second term is $3 \cdot 100 \cdot 0.3 = 90$, the third term is $3 \cdot 10 \cdot 0.09 = 2.7$, and the last term is $0.027$. The greatest integer less than this is $1092$. 