  For a pair of conjugate trigonometrical polynomials $C (t) = \sum_ { j = 1 }
^N { { a_j}\cos jt }, S(t) = \sum_ { j = 1 } ^N { { a_j}\sin jt }$ with real
coefficients and normalization ${a_1} = 1 $ we solve the extremal problem \[
  \sup_ {a_2,...,a_N} \left ({ \min_t \left\{ {\Re \left ({ F\left ({ { e^ {it}
} } \right) } \right): \Im \left ({ F\left ({ { e^ {it} } } \right) } \right) =
0 } \right\} } \right) = -\frac14 \sec ^2\frac\pi{N + 2}. \] We show that the
solution is unique and is given by \[ a_j^ {(0)} = \frac {1} { { { U'_N}\left
({\cos \frac{\pi } { { N + 2 } } } \right) } } { U' _ { N - j + 1 } }\left
({\cos \frac{\pi } { { N + 2 } } } \right) { U_ { j - 1 } }\left ({\cos
\frac{\pi }
  { { N + 2 } } } \right), \] where the $U_j(x)$ are the Chebyshev polynomials
of the second kind, and the $U'_j(x)$ are their derivatives, $j = 1, \ldots,
N.$ As a consequence, we obtain some theorems on covering of intervals by
polynomial images of the unit disc. We formulate several conjectures on a
number of extremal problems on classes of polynomials.
