ISL 1979 Problem 23

Consider the set $E$ consisting of pairs of integers $(a, b)$ , with $a \geq 1$ and $b \geq 1$ , that satisfy in the decimal system the following properties:(i) $b$ is written with three digits, as $\overline{\alpha_2\alpha_1\alpha_0}$ , $\alpha_2 \neq 0$ ;(ii) $a$ is written as $\overline{\beta_p \ldots \beta_1\beta_0}$ for some $p$ ;(iii) $(a + b)^2$ is written as $\overline{\beta_p\ldots \beta_1 \beta_0 \alpha_2 \alpha_1 \alpha_0}.$ Find the elements of $E$ .