3A. Determine the smallest natural number $n$ such that the proportion

$$
h_{a}: h_{b}: h_{c}=n:(n+1):(n+2)
$$

holds, where $h_{a}, h_{b}, h_{c}$ are the lengths of the altitudes in triangle $ABC$.
Solution. From $2 P=a h_{a}=b h_{b}=c h_{c}$ we obtain

$$
a: b: c=\frac{1}{h_{a}}: \frac{1}{h_{b}}: \frac{1}{h_{c}}=\frac{1}{n}: \frac{1}{n+1}: \frac{1}{n+2}
$$

that is,

$$
a: b: c=(n+1)(n+2): n(n+2): n(n+1)=\left(n^{2}+3 n+2\right):\left(n^{2}+2 n\right):\left(n^{2}+n\right)
$$

Since $h_{a}b>c$, it is sufficient to find a natural number $n$ for which the inequality $a2$ holds. The required smallest natural number is 2.