<s>[INST] Arithmetic and harmonic mean of two numbers. What is the maximum value of arithmetic mean of two integer number if their harmonic mean is 40?
$$\frac{1}{x}+\frac{1}{y} = \frac{2}{40}$$
$$xy=20x+20y$$
$$xy−20x−20y+20⋅20=400$$
$$(x−20)(y−20)=400$$
 [/INST] $$\frac{1}{x}+\frac{1}{y} = \frac{2}{40}$$
$$xy=20x+20y$$
$$xy−20x−20y+20⋅20=400$$
$$(x−20)(y−20)=400$$
$400=1\cdot 400 =2\cdot 200 =4\cdot 100=5\cdot 80 = 8\cdot 50= 10\cdot 40=16\cdot 25=20\cdot 20$
$$ x+y = \{21+420; \;22+220;\;24+120;\;25+100;\;28+70;\;30+60;\;36+45;\;40+40\}
$$ 
the maximum value of summ is 441 and arithmetic mean is 220.5
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