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Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2010 based on Exponents and Equations.

## Exponents and Equations – AIME 2010

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.

.

• is 107
• is 529
• is 840
• cannot be determined from the given information

### Key Concepts

Algebra

Equations

Number Theory

But try the problem first…

Source

AIME, 2010, Question 3.

Elementary Number Theory by Sierpinsky

## Try with Hints

First hint

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 $x=\frac{256}{81}$.

Second Hint

y=$\frac{3x}{4}=\frac{192}{81}$.

Final Step

x+y=$\frac{448}{81}$ then 448+81=529.