2003 IMO Shortlist Problems/A3

A3. Consider pairs of sequences of positive real numbers a1 ≥ a2 ≥ a3 ≥ ... , b1 ≥ b2 ≥ b3 ≥ ... and the sums An = a1 + .. + an, Bn = b1 + ... + bn; n = 1, 2, . . . . For any pair define ci = min{ai , bi} and Cn = c1 + ...+ cn, n = 1, 2, . . . . (1) Does there exist a pair (ai)i≥1, (bi)i≥1 such that the sequences (An)n≥1 and (Bn)n≥1 are unbounded while the sequence (Cn)n≥1 is bounded? (2) Does the answer to question (1) change by assuming additionally that bi = 1/i, i = 1, 2, . . . ? Justify your answer.