I spent a lot of time trying to sovle this so I would really appreciate it if someone could just give me some hints what I am doing wrong.
There are positive intergers k, n, m such that
19/20 < 1/k + 1/n + 1/m < 1
What is the smallest possible value of k+n+m?
19 < 20 ( 1/k + 1/n + 1/m ) < 20
19 < 20 ( nm + km + kn/ knm ) < 20
nm + km + kn = try to make the value about 19.01
knm = 20
I did a lot of guessing and checking but now I am starting to see my solution doesn't make any sense..
since the both right side and the left side are integers then I just start to guess numbers first take 2,2,2 that dose not work cuz the sum is 30
it is clear that if we choose two numbers (2) the sum of them will be more than 20 so try
2,3,3 dose not work too >20
2,3,4 dose not work too >20
2,4,4 =20 the same
3,3,3 < 19 so hold 2
2,3,6 = 20 as you see
you can see that if we take bigger number the sum become smalle