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题1(2013年浙江省高中数学竞赛附加题)设a、b、c∈R~+,满足ab+bc+ca≥3.证明:∑a~5+∑a~3(b~2+c~2)≥9,其中∑表示轮换对称和.题2(2009年全国高中数学联赛福建省预赛第15题)若a、b、c为正实数,满足a+b+c≥3.求证:(a+1)/(a(a+2))+(b+1)/(b(b+2))+(c+1)/(c(c+2))≥2.对于题1,命题者提供的方法是将欲证不等式等价转化为(a~3+b~3+c~3)(a~2+b~2+c~2)≥9,再利用幂平均不等式和已知条件,便可得结果.
Question 1 (additional title of the Zhejiang Provincial High School Mathematics Competition in 2013) Let a, b, c∈R~+ meet ab+bc+ca≥3. Proof: ∑a~5+∑a~3(b~2+c ~2) ≥ 9, where ∑ means rotational symmetry and Problem 2 (2009 Question 15 of the Fujian Provincial Preliminaries of the Senior High School Mathematics League) If a, b, and c are positive real numbers, satisfy a+b+c ≥ 3. To verify: (a+1)/(a(a+2))+(b+1)/(b(b+2))+(c+1)/(c(c+2))≥2. The proposition method is to convert the desire inequality equivalence into (a~3+b~3+c~3)(a~2+b~2+c~2)≥9, and then use the power mean inequality and Known conditions, you can get results.