One Martian, one Venusian and one Human reside on Pluto.
One day they make the following conversation:
Martian: I have spent 1/12 of my life in Pluto.
Human: I also have.
Venusian: Me too.
Martian: But Venusian and I have spent much more time here than you, Human.
Human: However Venusian and I are of the same age.
Venusian: I have lived 300 Earth years.
Martian: Venusian and I have been on Pluto for the last 13 years.

It is known that Human and Martian together have lived for 104 Earth years.

Find their respective ages.

HINT: The numbers are not necessarily expressed in base 10 .
Source: 1971 IMO longlisted problem.

Denote the Martian and his age = M
Denote the Human and his age = H
Denote the Venusian and his age = V
Denote the lines of the problem starting with the first 'Martian:' 1,2,...

From 1,7: M/(m+2) = m+3 where m is the Martian base.

From 8: M+H = 104 (base 10)

Therefore (m+3)(m+2)+12h=104, (12h=H): {m=2, H=84}, (but '3'>2, a contradiction) or {m=5, H=48} are the only qualifying solutions.  But if so, the M's age = 56 and he has spent 1/(5+3)= 1/8 of his life or 7 years on Pluto, and and H has spent 1/12 or 4 of 48 years on Pluto. 

From 6: V is 3v^2 years old where v is the Venusian base. H must be using base 10 not base v for v's age  since 4v+8=3v^2 is not true in any base. So v=4 and V =H= 48. We can also tell that V has spent 1/(4+2)= 1/6 of his life or 8 years on Pluto, implying that he arrived the year before M.

 

Edited on June 19, 2018, 10:36 am

Posted by broll on 2018-06-19 10:33:49