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# Exponents and Equations | AIME I, 2010 Question 3

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

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

AIME, 2010, Question 3.

Elementary Number Theory by Sierpinsky

## Try with Hints

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.