Question 80
Prove there are coprime polynomials $P(x)$ and $Q(x)$ with integer coefficients and a real number $u \gt 0$ such that if
$$
\left| \frac{a}{c} - 1 \right|^{2021} \leq \frac{u}{d c^{1010}}, \quad \left| \left( \frac{a}{c} \right)^{2020} - \frac{b}{d} \right| \leq \frac{u}{d c^{1010}},
$$
for some positive integers $a, b, c, d$, then we have $bP\left(\frac{a}{c}\right) = dQ\left(\frac{a}{c}\right)$.
$$
\left| \frac{a}{c} - 1 \right|^{2021} \leq \frac{u}{d c^{1010}}, \quad \left| \left( \frac{a}{c} \right)^{2020} - \frac{b}{d} \right| \leq \frac{u}{d c^{1010}},
$$
for some positive integers $a, b, c, d$, then we have $bP\left(\frac{a}{c}\right) = dQ\left(\frac{a}{c}\right)$.