The paper is devoted to the explicit construction of optimal designs for discrimination between two polynomial regression models of degree n2 and n. In a fundamental paper Atkinson and Fedorov (1975a) proposed the T-optimality criterion for this purpose. Recently Atkinson (2010) determined T-optimal designs for polynomials up to degree 6 numerically and based on these results he conjectured that the support points of the optimal design are cosines of the angles that divide a half of the circle into equal parts if the coefficient of x^(n1) in the polynomial of larger degree vanishes. In the present paper we give a strong justification of the conjecture and determine all T-optimal designs explicitly for any degree nN. In particular, we show that there exists a one-dimensional class of T-optimal designs. Moreover, we also present a generalization to the case when the ratio between the coefficients of x^(n1) and x^n is smaller than a certain critical value. Because of the complexity of the optimization problem T-optimal designs have only been determined numerically so far and this paper provides the first explicit solution of the T-optimal design problem since its introduction by Atkinson and Fedorov (1975a). Finally, for the remaining cases (where the ratio of coefficients is larger than the critical value) we propose a numerical procedure to calculate the T-optimal designs. The results are also illustrated in an example.