imagine copying from 1991 aime #1

Let $a=xy$ and $b=x+y$. We have $a+b=71$ and $ab=880$, so by Vieta $a,b$ are the roots of $t^2-71t+880$, which are $16$ and $55$. Then $x,y$ are the roots of $t^2-16t+55$ or $t^2-55t+16$, so $(x,y)=(5,11),(11,5)$ since the other quadratic doesn't have integer roots.

asdf334 wrote: imagine copying from 1991 aime #1 This is 1991 AIME 1?

jhu08 wrote: asdf334 wrote: imagine copying from 1991 aime #1 This is 1991 AIME 1? Yes, see here.

Lol ig puerto rico copied the problem. Would be an unfair advantage to those who did the problem, am I right?

parmenides51 wrote: Find the solutions of positive integers for the system $xy + x + y = 71$ and $x^2y + xy^2 = 880$. \((x+1)(y+1)=72\),\(xy(x+y)=880\) the rest is just a number check analysis lol