SMO 2025 Junior (Problem 8)EquationsEasyIf xxx and yyy are positive integers such that xy−9x−9y=20xy-9x-9y=20xy−9x−9y=20, find the value of x2+y2x^2+y^2x2+y2.Related TopicsCoreFactorization TechniquesCheck AnswerYour answer:CheckHints (3)Hint 1By SFFT (Simon's Favorite Factoring Trick),xy−9x−9y=20xy-9x-9y=20xy−9x−9y=20⟹(x−9)(y−9)=101.\Longrightarrow (x-9)(y-9)=101.⟹(x−9)(y−9)=101.Hint 2Since 101101101 is prime,x−9y−9xy110110110101111010−1−1018−92−101−1−928\begin{array}{c|c|c|c}x-9&y-9&x&y\\ \hline1&101&10&110\\[4pt]101&1&110&10\\[4pt]-1&-101&8&-92\\[4pt]-101&-1&-92&8\end{array}x−91101−1−101y−91011−101−1x101108−92y11010−928Since xxx and yyy are positive integers,{x,y}={10,110}.\{x,y\}=\{10,110\}.{x,y}={10,110}.Hint 3Therefore,x2+y2=102+1102x^2+y^2=10^2+110^2x2+y2=102+1102=100+12100=100+12100=100+12100=12200.=12200.=12200.Final Answer122001220012200Related Problems (1)AIME I 2026 (Problem 4)