1. Find all positive roots of the equation $x^{x}+x^{1-x}=x+1$.

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# Solution

Since $x>0$, then

$0=x^{2 x}+x-x^{x+1}-x^{x}=x^{x}\left(x^{x}-1\right)-x\left(x^{x}-1\right)=x\left(x^{x}-1\right)\left(x^{x-1}-1\right)$.

Thus, $x=1$.

## Answer $x=1$.