Fundamental properties of the helices induced on a polymer chain due to hydrogen bonding of chiral and achiral molecules on the chain side are studied by a new statistical-mechanical theory. Unlike optically active random copolymers, the selection process of the helical sequences on a polymer chain is thermoreversible because of the reversible nature of association. The molecular origin for nonlinear amplification of chiral order by small enantiomer excess (majority rule) in such annealed randomness is clarified. Enhancement of the chirality by added achiral molecules (sergeants-and-soldiers rule) is also derived. Theoretical calculations of the helix content theta and the helical order parameter psi are compared with recent experimental measurements of circular dichroism in solutions of poly((4-carboxyphenyl)acetylene) bearing optically active amino groups that are hydrogen-bonded to carboxylic groups on the polymer chain.