  Sorokin gave in 1996 a new proof that pi is transcendental. It is based on a
simultaneous Pad\'e approximation problem involving certain multiple
polylogarithms, which evaluated at the point 1 are multiple zeta values equal
to powers of pi. In this paper we construct a Pad\'e approximation problem of
the same flavour, and prove that it has a unique solution up to
proportionality. At the point 1, this provides a rational linear combination of
1 and multiple zeta values in an extended sense that turn out to be values of
the Riemann zeta function at odd integers. As an application, we obtain a new
proof of Vasilyev's conjecture for any odd weight, concerning the explicit
evaluation of certain hypergeometric multiple integrals; it was first proved by
Zudilin in 2003.
