1. Does there exist a quadratic trinomial $P(x)$ with integer coefficients, such that for any natural number $n$ in decimal notation consisting entirely of the digit 1, the value of the quadratic trinomial $P(n)$ is also a natural number consisting entirely of the digit 1.
1. According to the characteristics of the decimal numeral system, construct a quadratic trinomial $P(x)=90 x^{2}+20 x+1$. Let $n=\underbrace{11 \cdots 1}_{k \uparrow 1}$, then $P(n)=\underbrace{11 \cdots 1}_{2 k+1 \uparrow 1}$, i.e., $P(x)=90 x^{2}+20 x+1$. Satisfying the given condition, such a polynomial exists.