IR-TST 2019 Problem 10

Let $1<t<2$ be a real number. Prove that for all sufficiently large positive integers like $d$ , there is a monic polynomial $P(x)$ of degree $d$ , such that all of its coefficients are either $+1$ or $-1$ and $$\left|P(t)-2019\right| <1.$$ Proposed by Navid Safaei