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We study two extremal problems for the product of powers of conformal radii of symmetric disjoint domains.Extremal problems for disjoint domains were posed for the first time by Lavrent'ev [1], who solved the problem of the product of conformal radii of a pair of disjoint domains. Later, these problems were extensively studied and generalized by numerous authors (see, e. g., [2][3][4][5][6]).In the present work, we study the problem of the product of powers of conformal radii in the case where disjoint domains are symmetric with respect to the unit circle. This problem belongs to the class of problems with free poles on the unit circle. For the first time, problems of the indicated type were formulated and studied by Bakhtina in [7]. Later, statements of problems of this type were generalized and strengthened by Dubinin [6,8] and Emel'yanov [9]. We especially mention a new method suggested by Dubinin based on the application of piecewiseseparating transformations.

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