Question 460423
None.

Let a, b, and c be the lengths of the sides of the triangle.

Then by the given, a + b + c = 20.

Now the altitudes being in the ratio 1:2:3 means that if x is the shortest altitude , then the longer altitude is 2x, and the longest altitude is 3x.  

Without loss of generality, let x be the altitude for a, 2x be the altitude for b, and 3x be the altitude for c.

Then by using the area formula, we get 
{{{(ax)/2 = (2bx)/2 = (3cx)/2}}} <==> a = 2b = 3c.
==> c = a/3 and b = a/2.

==> a + a/2 + a/3 = 20
==> {{{(11a)/6 = 20}}} ==> {{{a = 120/11}}}.

Since a is not an integer, neither will b and c be.