Problem

Source: 2020 China Southeast 10.5/11.5

Tags: number theory



Consider the set $I=\{ 1,2, \cdots, 2020 \}$. Let $W= \{w(a,b)=(a+b)+ab | a,b \in I \} \cap I$, $Y=\{y(a,b)=(a+b) \cdot ab | a,b \in I \} \cap I$ be its two subsets. Further, let $X= W \cap Y$. (1) Find the sum of maximal and minimal elements in $X$. (2) An element $n=y(a,b) (a \le b)$ in $Y$ is called excellent, if its representation is not unique (for instance, $20=y(1,5)=y(2,3)$). Find the number of excellent elements in $Y$.

HIDE: Note (2) is only for Grade 11.