There are many solutions. Notice that we have the following identity: \[ xy(x+y-10)-3x^2-2y^2+21x+16y-30=(x-2)(y-3)(x+y-5) \] We let $ a=x-2 \ge -1, b=y-3 \ge -2 $ and it's suffice to find out the solutions of the equation \[ ab(a+b)=30 \] After some calculus, we get \[ (a,b)=(1,5),(2,3),(3,2),(5,1) \Leftrightarrow (x,y)=(3,8),(4,6),(5,5),(7,4) \]

How did you got that identity?

No need for such guesswork. Write the equation as $(y-3)x^2 + (y^2-10y+21)x - (2y^2-16y+60)=0$. Its discriminant is $\Delta = (y^2-10y+21)^2 + 4(y-3)(2y^2-16y+60) = (y-3)^4 + 120(y-3)$. We thus have $((y-3)^2)^2 < \Delta < ((y-3)^2+1)^2$ for $y\geq 11$, so it is enough to manually check the cases $1\leq y \leq 10$.

@mavropnevma, Hi! I think your solution is as great as the first one, and I understand that if $y$ satisfies the inequality $((y-3)^2)^2 < \Delta < ((y-3)^2+1)^2$ , then there is no way the discriminant can be a perfect square, but, I just don't understand why it holds for $y\geq 11$... I tried it out numerically and I got: At $y=11$: $64^2 <8^4+120(8)=5056 < 65^2=4225$ Could you please enlighten me with this? Many thanks in advance...

He could not, he is no more.

I am so sorry to hear this...he was such a great person and helper and he will be missed.

Why? WhatsApp hapened?

jaspion wrote: Why? WhatsApp hapened? https://www.artofproblemsolving.com/community/q2h1099628p5067889 Rest In Peace mavropnevma...

The link does not open

It opens for me. What browser are you on? If it continues to not work you can make a bug report on site support. @below Try this link. http://www.artofproblemsolving.com/community/q2h1099628p5067889 I changed it from https to http. Did it work?

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