The restricted minimum phi-divergence estimator, [Pardo, J.A., Pardo, L. and Zografos, K., 2002, Minimum phi-divergence estimators with constraints in multinomial populations. Journal of Statistical Planning and Inference, 104, 221-237], is employed to obtain estimates of the cell frequencies of an 1 x 1 contingency table under hypotheses of symmetry, marginal homogeneity or quasi-symmetry. The associated phi-divergence statistics are distributed asymptotically as chi-squared distributions under the null hypothesis. The new estimators and test statistics contain, as particular cases, the classical estimators and test statistics previously presented in the literature for the cited problems. A simulation study is presented, for the symmetry problem, to choose the best function phi(2) for estimation and the best function phi(1) for testing.