Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2010 based on Exponents and Equations.
Suppose that y=\(\frac{3x}{4}\) and \(x^{y}=y^{x}\). The quantity x+y can be expressed as a rational number \(\frac{r}{s}\) , where r and s are relatively prime positive integers. Find r+s.
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Algebra
Equations
Number Theory
Answer: is 529.
AIME, 2010, Question 3.
Elementary Number Theory by Sierpinsky
y=\(\frac{3x}{4}\) into \(x^{y}=y^{x}\) and \(x^{\frac{3x}{4}}\)=\((\frac{3x}{4})^{x}\) implies \(x^{\frac{3x}{4}}\)=\((\frac{3}{4})^{x}x^{x}\) implies \(x^{-x}{4}\)=\((\frac{3}{4})^{x}\) implies \(x^{\frac{-1}{4}}=\frac{3}{4}\) implies \(x=\frac{256}{81}\).
y=\(\frac{3x}{4}=\frac{192}{81}\).
x+y=\(\frac{448}{81}\) then 448+81=529.