The three rational numbers a, b, c satisfy a: b: c=2:3:5, and a2+b2+c2=abc, then a+b+c=___.

The three rational numbers a, b, c satisfy a: b: c=2:3:5, and a2+b2+c2=abc, then a+b+c=___.

Let a=2k, b=3k, c=5k,
A2+b2+c2=abc,
(2K)2+(3k)2+(5k)2=2k×3k×5k, i.e.38k2=30k2•k,
K=0,
K=19
15,
A+b+c=10k=38
3.
Therefore,38
3.

If the rational numbers a, b, c satisfy a+b+c=0, abc=2, c >0, it is proved that |a||b|is greater than or equal to 2. From abc=2>0, c >0, ab is greater than 0. If the rational numbers a, b, c satisfy a+b+c=0, abc=2, c >0, it is proved that |a||b|is greater than or equal to 2. From abc=2>0, c >0, ab is greater than 0. From a+b+c=0, a

Abc=2, c >0, c=2/ab
A+b+c=0, c=-a-b
C^3=2/ab*(-a-b)^2=2/ab*(a+b)^2
(A+b)^2-4ab=(a-b)^2>=0
(A+b)^2>=4ab
C^3>=2/ab*4ab >=8
C >=2, c >0
-A-b <=-2
|A b|>=2