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AbstractThe main result of the paper, based on the Borsuk Antipodality Theorem, describes extremal functions of the Kolmogorov-Landau problem(*) $f^{\left(m\right)}\left(\xi \right)\to sup$, $f\in W^r{H}^{\omega }[\xi ,b]$, ${\left|\right|f\left|\right|}_{ℂ[a,b]}\le B$,for all 0 < m ≤ r, ξ ≤ a or ξ = (a+b)/2, all B > 0 and concave moduli of continuity ω on ℝ₊. It is shown that any extremal function ${=}_{B,r,m,\omega ,\xi }$ of the problem (*) enjoys the following two characteristic properties. First, the function ${}^{\left(r\right)}\left(·\right){-}^{\left(r\right)}\left(\xi \right)$ is extremal for the problem(**) ${\int }_{\xi }^{b}h\left(t\right)\psi \left(t\right)dt\to sup$, $h\in H^{\omega }[\xi ,b]$, h(ξ) = 0,for an appropriate choice of the kernel ψ with a finite number of sign...