Next 10 problems aren’t mine, but they belong to my Friends who have magnificent wisdom, may I must Admire all, although I will not state names. May you read this writing from notebook for Convenience and completeness of a proof 1 For all a,b,c,d>0 and a+b+c+d=4 Prove that 1/a^2 +1/b^2 +1/c^2 +1/d^2 ≥a^2+b^2+c^2+d^2 2 For all a,b,c≥0 and a+b+c=3 prove that (ab^3+bc^3+ca^3 )(ab+bc+ca)≤16 3 For all a,b,c∈R^+, a+b+c=1, prove that a(1-a^2 )/bc+b(1-b^2 )/ca+c(1-c^2 )/ab≥8 4 For all x,y,z∈R^+, x+y+z=3, find the minimum Value of 3((x+y)^2+(y+z)^2+(z+x)^2 )+ 2(x+y)(y+z)(z+x) 5 For all x,y,z≥0 and x+y+z=1 prove that xyz+8/27≥xy+xz+zx 6 For all a,b,c,d∈R^+, prove that (a^2+a+1)(b^2+b+1)(c^2+c+1) (d^2+d+1)≥81abcd 7 Give x∈R,where x≠0, find the maximum value Of (√(x^4+x^2+1)-√(x^4+1))/x 8 For x,y ∈N,find the maximum value of f(x)=√(x-116)+√(x+100) 9 Give a,b,c,d>0 and be less than1 prove that (1-a)(1-b)(1-c)(1-d)>1-a-b-c-d 10 Give x,y,z>0, and x+y+z=xyz Prove that (x+y)/(1+z^2 )+(y+z)/(1+x^2 )+(z+x)/(1+y^2 )≥(6√3)/(3√3+1) Solution 1 For all a,b,c,d>0,a+b+c+d=4 Prove that 1/a^2 +1/b^2 +1/c^2 +1/d^2 ≥a^2+b^2+c^2+d^2 For all a,b,c,d>0 and a+b+c+d=4 We will get that 00, and since ab+bc+cd+da>(abc+bcd+cda+dab) So, ab+bc+cd+da-(abc+bcd+cda+dab)+ ac+bd+abcd>0, it implies that ab+bc+cd+da-(abc+bcd+cda+dab)+ ac+bd+abcd+1-a-b-c-d> 1-a-b-c-d That is, (1-a)(1-b)(1-c)(1-d)> 1-a-b-c-d OKK 10 Give x,y,z>0, and x+y+z=xyz Prove that (x+y)/(1+z^2 )+(y+z)/(1+x^2 )+(z+x)/(1+y^2 )≥(6√3)/(3√3+1) from x,y,z>0, and x+y+z=xyz It will get that1/xy+1/yz+1/zx=1, and then 3+3+3≤xy+yz+zx ≤x^2+y^2+z^2 3√3+3√3+3√3≤xyz+yzx+zxy x+y+z≥√3+√3+√3 And it follows (x+y)+(y+z)+(z+x)≥2√3+2√3+2√3 1/(z^2+1)+1/(x^2+1)+(z+x)/(y^2+1)≥1/(xyz+1)+1/(xyz+1)+1/(xyz+1) (x+y)/(z^2+1)+(y+z)/(x^2+1)+(z+x)/(y^2+1)≥(2√3)/(xyz+1)+(2√3)/(xyz+1)+(2√3)/(xyz+1) =(6√3)/(xyz+1) ≥(6√3)/(3√3+1) Because, 1/(xyz+1)≤1/(3√3+1) Acknowledgement This writing, if there is a mistake, and then it is mine But if there is some profit that can make wisdom, then I Assign this success with Pro.Dr. Narong Phannim Who be my great teacher. Remark: if we think that varied problems are magnificent Food, then we will be capable to solve them happily And important we will get new wisdom by ourselves
Posted on: Thu, 02 Oct 2014 05:50:13 +0000
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