Arithmetic and harmonic mean of two numbers.

Question

What is the maximum value of arithmetic mean of two integer number if their harmonic mean is 40?

$$\frac{1}{x}+\frac{1}{y} = \frac{2}{40}$$ $$xy=20x+20y$$ $$xy−20x−20y+20⋅20=400$$ $$(x−20)(y−20)=400$$

Answer 1

$$\frac{1}{x}+\frac{1}{y} = \frac{2}{40}$$ $$xy=20x+20y$$ $$xy−20x−20y+20⋅20=400$$ $$(x−20)(y−20)=400$$ $400=1\cdot 400 =2\cdot 200 =4\cdot 100=5\cdot 80 = 8\cdot 50= 10\cdot 40=16\cdot 25=20\cdot 20$
$$ x+y = \{21+420; \;22+220;\;24+120;\;25+100;\;28+70;\;30+60;\;36+45;\;40+40\} $$ the maximum value of summ is 441 and arithmetic mean is 220.5