IMO Shortlist 2000 problem A5
Kvaliteta:
Avg: 0,0Težina:
Avg: 8,0 Let
be a positive integer and
a positive real number. Initially there are
fleas on a horizontal line, not all at the same point. We define a move as choosing two fleas at some points
and
, with
to the left of
, and letting the flea from
jump over the flea from
to the point
so that
.
Determine all values of
such that, for any point
on the line and for any initial position of the
fleas, there exists a sequence of moves that will take them all to the position right of
.
![n \geq 2](/media/m/2/1/f/21fe2458de6d1580c44fd06e0fac11bb.png)
![\lambda](/media/m/9/b/e/9be7eeb58b67ec913359062c0122ee80.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
![\frac {BC}{AB} = \lambda](/media/m/4/0/0/4000444da974446b36dc1ed256d3f8dd.png)
Determine all values of
![\lambda](/media/m/9/b/e/9be7eeb58b67ec913359062c0122ee80.png)
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
Izvor: Međunarodna matematička olimpijada, shortlist 2000