That isn't possible. Clearly if we can do it for the number x,x+1,...x+9 we can do it for 1,2,3...,9 because we can subtract 1 from each vertex and then each midpoint will be decreased by 1 as well clearly. lets name the numbers we put on the vertices a,b,c,d. the sum of all of the numbers is 2.5(a+b+c+d)=1+2+..9=45 Therefore a+b+c+d=18. we must place the number 1 on a vertex, because it can't be a mean of two others (all others are bigger). All numbers on the vertices must be odd because otherwise the mean won't be an integer. Therefore we have to choose 4 numbers of 1,3,5,7,9 which sums to 18, and clearly 1,3,5,9 is the only option for that. but then 1+5/2=3, which means 3 is on a vertex as well as on a midpoint of an edge, which means we can't do that.

You can see this topic https://artofproblemsolving.com/community/c6h1502369p8868819