Y by Adventure10, Mango247, and 1 other user
Consider pairs of the sequences of positive real numbers
and the sums
For any pair define
and
,
.
(1) Does there exist a pair
,
such that the sequences
and
are unbounded while the sequence
is bounded?
(2) Does the answer to question (1) change by assuming additionally that
,
?
Justify your answer.
![\[a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots\]](http://latex.artofproblemsolving.com/7/9/e/79e388eba20412905964dc2b4ebb491f3c79143b.png)
![\[A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.\]](http://latex.artofproblemsolving.com/3/c/4/3c44766edcc9e5269f0d3396386e41676cbdf42c.png)



(1) Does there exist a pair





(2) Does the answer to question (1) change by assuming additionally that


Justify your answer.
This post has been edited 2 times. Last edited by djmathman, May 27, 2018, 3:38 PM
Reason: overhauled problem wording to fit https://anhngq.files.wordpress.com/2010/07/imo-2003-shortlist.pdf
Reason: overhauled problem wording to fit https://anhngq.files.wordpress.com/2010/07/imo-2003-shortlist.pdf