We have to thank who write problems, as they have How to solve them, but we may have how to solve Different from writer which it may be true or fault And important we can share each other. 1 ∀a,b,c∈R^+, prove that (I much interest it) 2( (a+b)/2-√ab)≤3( (a+b+b)/3-∛abc) 2 Give a,b,c,d≥0 and ab+bc+cd+da=1 Prove that a^3/(b+c+d)+b^3/(c+d+a)+c^3/(d+a+b)+d^3/(a+b+c)≤1/3 3 ∀a,b,c∈R^+, prove that 1/a+1/b+1/c≤(a^8+b^8+c^8)/(a^3 b^3 c^3 ) 4 ∀a,b,c∈R^+, where (b+c)/a+(c+a)/b+(a+b)/c=2(1/ab+1/bc+1/ca) Prove that a^2+b^2+c^2+3≥2(ab+bc+ca) 5 Give a,b,c≥0,and a^2+b^2+c^2=3 Prove that a/(b+2)+b/(c+2)+C/(a+2)≤1 All 6 points, if there are someone can Show how to solve, we want to see and Next, they are my hints 1 we must show that a+b+c≥3∛abc and ∛(a^2 b)+∛(ab^2 )≥2√ab 2 we must separate to be 3 cases 2.1 a≠0,b≠0,c=d=0 2.2 a≠0,b≠0,c≠0,d=0 and 2.3 a≠0,b≠0,c≠0,and d≠0 Remark: a=1/3,b=1/2,c=2/3,and d=1/2 We have ab+bc+cd+da=1, a+b+c+d=2,a^2+b^2+c^2+d^2=38/36≥1 a=1/4,b=2/3,c=1/2,and d=2/3 , we have ab+bc+cd+da=1, a+b+c+d=25/12≥2,a^2+b^2+c^2+d^2=173/144≥1 3 by using a^2+b^2+c^2≥ab+bc+ca 4 from defined we must show that ab+bc+ca≤3 5 we must separate to be 3 cases 5.1 a≠0,b=0,and c=0 5.2 a≠0,b≠0,and c=0 and 5.3 a≠0,b≠0,and c≠0
Posted on: Thu, 14 Aug 2014 06:03:15 +0000
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