Unbounded Sequences
by DVDTSB, May 13, 2025, 12:06 PM
Let 
be a sequence of strictly positive real numbers. For each nonzero positive integer 
, define
![\[
s_n = a_1 + a_2 + \cdots + a_n \quad \text{and} \quad
\sigma_n = \frac{a_1}{1 + a_1} + \frac{a_2}{1 + a_2} + \cdots + \frac{a_n}{1 + a_n}.
\]](//latex.artofproblemsolving.com/1/9/1/191a2ebd41e3d236c90e9022ed65323461083815.png)
Show that if the sequence 
is unbounded, then the sequence 
is also unbounded.
Proposed by The Problem Selection Committee
be a sequence of strictly positive real numbers. For each nonzero positive integer 
, define![\[
s_n = a_1 + a_2 + \cdots + a_n \quad \text{and} \quad
\sigma_n = \frac{a_1}{1 + a_1} + \frac{a_2}{1 + a_2} + \cdots + \frac{a_n}{1 + a_n}.
\]](http://latex.artofproblemsolving.com/1/9/1/191a2ebd41e3d236c90e9022ed65323461083815.png)
Show that if the sequence 
is unbounded, then the sequence 
is also unbounded.Proposed by The Problem Selection Committee
This post has been edited 1 time. Last edited by DVDTSB, 3 hours ago
and 
be touch points of the incenter of 
at 
and 
,
and 
be the circumcenter of triangles 
and 
,
and 
concurrent.
be positive real numbers such that 
.
knowing that the denominators are positive real numbers.
satisfying the following condition: 
and 
be the incircle and circumcircle of the acute triangle 
,
so that all of its sides are tangent to 
,
are both on 
.
and 
,
and 
,
and 
intersects on 
are such that 
,
be the circumcircle of 
be A-excircle .Let the two common tangents of 
cut 
.
towns, arranged along a road running from left to right. Each town has a left bulldozer (put to the left of the town and facing left) and a right bulldozer (put to the right of the town and facing right). The sizes of the 
bulldozers are distinct. Every time when a left and right bulldozer confront each other, the larger bulldozer pushes the smaller one off the road. On the other hand, bulldozers are quite unprotected at their rears; so, if a bulldozer reaches the rear-end of another one, the first one pushes the second one off the road, regardless of their sizes.
and 
be two towns, with 
.
from left to right and their left and right bulldozer 
respectively.
towns..
and let some 
collide with 
and 
towns live unaffected by the remaining towns. And hence by inducton we are done.
.
be variable points on 
is tangent to 
-- the circumcenter of triangle 
.
such that 
September 2017
February 2017