3. Find all three-digit numbers $\overline{L O M}$, composed of distinct digits $L, O$ and $M$, for which the following equality holds:

$$
\overline{L O M}=(L+O+M)^{2}+L+O+M
$$
Answer: 156. Instructions. Let $x=L+O+M$. Then $\overline{L O M}=x(x+1)$. In this case, $x \geq 10$ (otherwise $x(x+1)<100)$ and $x \leq 24$ (the sum of digits does not exceed $9+8+7=24$). Therefore, $x \in[10 ; 24]$. From the relation $100 \cdot L+10 \cdot O+M=x^{2}+L+O+M$ it follows that $x^{2}=99 \cdot L+9 \cdot O$, which means $x$ is divisible by 3. It remains to substitute the values $12,15,18,21$ and 24 into $x(x+1)$ and check the sum of the digits of the resulting number. It matches $x$ only when $x=12$.