Keywords: primary, 55M20, 54H25; secondary, 47H10, 65H10, 91A44, 91B50Existence theorems; Roots; Fixed points; Nonlinear equations; Bolzano theorem; Bolzano–Poincaré–Miranda theorem; Brouwer fixed point theorem; Sperner lemma; Knaster–Kuratowski–Mazurkiewicz coveri