MathDB
Problems
Contests
National and Regional Contests
Argentina Contests
Argentina National Olympiad
2016 Argentina National Olympiad
6
6
Part of
2016 Argentina National Olympiad
Problems
(1)
particles moving between ends of a segment
Source: 2016 Argentina OMA Finals L3 p6
1/16/2023
Let
A
B
AB
A
B
be a segment of length
1
1
1
. Several particles start moving simultaneously at constant speeds from
A
A
A
up to
B
B
B
. As soon as a particle reaches
B
B
B
, turns around and goes to
A
A
A
; when it reaches
A
A
A
, begins to move again towards
B
B
B
, and so on indefinitely. Find all rational numbers
r
>
1
r>1
r
>
1
such that there exists an instant
t
t
t
with the following property: For each
n
≥
1
n\ge 1
n
≥
1
, if
n
+
1
n+1
n
+
1
particles with constant speeds
1
,
r
,
r
2
,
…
,
r
n
1,r,r^2,…,r^n
1
,
r
,
r
2
,
…
,
r
n
move as described, at instant
t
t
t
, they all lie at the same interior point of segment
A
B
AB
A
B
.
combinatorics
time