Let F n be a free group of rank n generated by x 1 , … , x n . In this paper we discuss three algorithmic problems related to automorphisms of F 2 . A word u = u ( x 1 , … , x n ) of F n is called positive if no negative exponents of x i occur in u. A word u in F n is called potentially positive if ϕ ( u ) is positive for some automorphism ϕ of F n . We prove that there is an algorithm to decide whether or not a given word in F 2 is potentially positive, which gives an affirmative solution to problem F34a in [G. Baumslag, A.G. Myasnikov, V. Shpilrain, Open problems in combinatorial group theory, second ed., in: Contemp. Math., vol. 296, 2002, pp. 1–38, online version: http://www.grouptheory.info] for the case of F 2 . Two elements u and v in F n are said to be boundedly translation equivalent if the ratio of the cyclic lengths of ϕ ( u ) and ϕ ( v ) is bounded away from 0 and from ∞ for every automorphism ϕ of F n . We provide an algorithm to determine whether or not two given elements of F 2 are boundedly translation equivalent, thus answering question F38c in the online version of [G. Baumslag, A.G. Myasnikov, V. Shpilrain, Open problems in combinatorial group theory, second ed., in: Contemp. Math., vol. 296, 2002, pp. 1–38, online version: http://www.grouptheory.info] for the case of F 2 . We also provide an algorithm to decide whether or not a given finitely generated subgroup of F 2 is the fixed point group of some automorphism of F 2 , which settles problem F1b in [G. Baumslag, A.G. Myasnikov, V. Shpilrain, Open problems in combinatorial group theory, second ed., in: Contemp. Math., vol. 296, 2002, pp. 1–38, online version: http://www.grouptheory.info] in the affirmative for the case of F 2 .