I submitted this problem to Brilliant but it got rejected so I decided to share it here. Anyone willing to solve the problem is welcome. Enjoy!
I am sorry but there was a mistake in the problem. Thanks to Gabriel W. for pointing it out. I had a different approach but after getting the answers I did not verify them by triangle inequality.
A scalene triangle has an in-radius of 1 cm. Given that the altitudes are positive integers when measured in centimeters, what is the only possible value of the sum of the altitudes?