Question 7
(GDR) Prove that from $x+y=1$ ($x,y\in\mathbb R$) it follows that
$$x^{m+1}\sum_{j=0}^n\binom{m+j}{j}y^j+y^{n+1}\sum_{i=0}^m\binom{n+i}{i}x^i=1\quad(m,n=0,1,2,\ldots).$$
$$x^{m+1}\sum_{j=0}^n\binom{m+j}{j}y^j+y^{n+1}\sum_{i=0}^m\binom{n+i}{i}x^i=1\quad(m,n=0,1,2,\ldots).$$