In essence, this is problems 1, 2 in Chap.1, Vol.1 of Polya G. *Aufgaben und Lehrsätze aus der Analysis*, 1st edn. 1925.^{[11]} ("Problems and theorems in analysis“). Springer, Berlin 1975 (with Gábor Szegő). This *math* (not Maple) problem is solved through generating functions. The solution of problem 1 is the theme of a scientific article (see the reference in the cited book).

Here are some results obtained with Maple.

restart; eqn1 := 30*a+75*b+110*c+85*d+255*e+160*f+15*g+12*h+120*i-800000:

DirectSearch:-SolveEquations(eqn1, assume = posint, AllSolutions, method = quadratic, number = 800, evaluationlimit = 20000);

produces 219 solutions (see the attached file).

The result of

floor(800000*(1/30))*floor(800000*(1/75))*floor(800000*(1/110))*floor(800000*(1/85))*floor(800000*(1/255))*floor(800000*(1/160))*floor(800000*(1/15))*floor(800000*(1/12))*floor(800000*(1/120));

7236013054105619739485107639205760000

evalf(log[10](%));

36.85949934

clearly shows the futility of applying the enumeration of possibilities.

Next,

with(Optimization):

LPSolve(30*a+75*b+110*c+85*d+255*t+160*f+15*g+12*h+120*i, {30*a+75*b+110*c+85*d+255*t+160*f+15*g+12*h+120 <= 800000}, assume = {integer, nonnegative}, maximize);

produces an incorrect result

Warning, problem appears to be unbounded

[0, [a = 0, b = 0, c = 0, d = 0, f = 0, g = 0, h = 0, i = 0, t = 0] ].

An SCR was submitted by me.

solve.mw

Edit. The numbers of the problems.