- 设a,b,c满足1/a+1/b+1/c=1/(a+b+c),求证?
- 设a,b,c满足1/a+1/b+1/c=1/(a+b+c),求证：当n为奇数时，1/(a^n+b^n+c^n)=1/a^n+1/b^n+1/c^n
- 证明:
∵ a,b,c满足1/a+1/b+1/c=1/(a+b+c), ∴ (a+b+c)(ab+bc+ca)=abc,
(b/a+a/b)+(c/b+b/c)+(c/a+a/c)+2=0, [(a-b)^/ab]+[(c+b)^/bc]+[(c+a)/ac]=0…(*), ∵ (a-b)^≥0, (c+b)^≥0, (c+a)^≥0, ∴ a,b,c中必有一个负数,不妨设a>≥b>0>c.
∴ 由(*)式知,当且仅当a=b=-c时,"="成立.
∴ 当n为奇数时，1/(a^n+b^n+c^n)=1/(a^n+1/a^n-1/a^n)=1/a^n, 1/a^n+1/b^n+1/c^n=1/a^n+1/a^n-1/a^n=1/a^n
∴ n为奇数时，1/(a^n+b^n+c^n)=1/a^n+1/b^n+1/c^n.