已知a,b是正整数,求证;a^5+b^5>=a^3b^2+a^2b^3

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已知a,b是正整数,求证;a^5+b^5>=a^3b^2+a^2b^3已知a,b是正整数,求证;a^5+b^5>=a^3b^2+a^2b^3已知a,b是正整数,求证;a^5+b^5>=a^3b^2+a

已知a,b是正整数,求证;a^5+b^5>=a^3b^2+a^2b^3
已知a,b是正整数,求证;a^5+b^5>=a^3b^2+a^2b^3

已知a,b是正整数,求证;a^5+b^5>=a^3b^2+a^2b^3
a^5+b^5-(a^3b^2+a^2b^3)
=a^3(a^2-b^2)+b^3(b^2-a^2)
=(a^2-b^2)(a^3-b^3)
=(a+b)(a-b)^2(a^2+ab+b^2)
因为a>0 b>0
所以a+b>0 (a-b)^2>=0 a^2+ab+b^2>0
所以a^5+b^5-(a^3b^2+a^2b^3)>=0
即a^5+b^5>=a^3b^2+a^2b^3