CAPS 2020 Problem 5

Let $n$ be a positive integer and let $d(n)$ denote the number of ordered pairs of positive integers $(x,y)$ such that $(x+1)^2-xy(2x-xy+2y)+(y+1)^2=n$ . Find the smallest positive integer $n$ satisfying $d(n) = 61$ .

(Patrik Bak, Slovakia)