2024 DMC Mock 10 Problems/Problem 9

If Nathan brings $n$ people with him, then he pays $n+1+\frac{100}{n+1}$ dollars for the gift and the drive. $n+1+\frac{100}{n+1}<n+2+\frac{100}{n+2}\rightarrow \frac{100}{n+1}-\frac{100}{n+2}<1\rightarrow\frac{100}{(n+1)(n+2)}>1\rightarrow (n+1)(n+2)>100$ . The least $n$ that satisfies this inequality is $n=\boxed{9}$ . Alternatively, use guess and check.