2003 IMO Shortlist S01

Consider pairs of the sequences of positive real numbers $$a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots$$ and the sums $$A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.$$ For any pair define $c_n = \min\{a_i,b_i\}$ and $C_n = c_1 + \cdots + c_n$ , $n=1,2,\ldots$ .

(1) Does there exist a pair $(a_i)_{i\geq 1}$ , $(b_i)_{i\geq 1}$ such that the sequences $(A_n)_{n\geq 1}$ and $(B_n)_{n\geq 1}$ are unbounded while the sequence $(C_n)_{n\geq 1}$ is bounded?

(2) Does the answer to question (1) change by assuming additionally that $b_i = 1/i$ , $i=1,2,\ldots$ ?

Justify your answer.

考虑正实数序列对 $$a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots$$ 和和 $$A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.$$ 对于任何对，定义 $c_n = \min\{a_i,b_i\}$ 和 $C_n = c_1 + \cdots + c_n$ ， $n=1,2,\ldots$ 。

(1) 是否存在一对 $(a_i)_{i\geq 1}$ 、 $(b_i)_{i\geq 1}$ 使得序列 $(A_n)_{n\geq 1}$ 和 $(B_n)_{n\geq 1}$ 无界，而序列 $(C_n)_{n\geq 1}$ 有界？

(2) 问题(1)的答案是否会因额外假设 $b_i = 1/i$ , $i=1,2,\ldots$ 而改变？

证明你的答案合理。