Cho `a,b,c > 0` Chứng minh rằng : `P = a^2/(a^2 + (b+c)^2) + b^2/(b^2 + (c+a)^2) + c^2/(c^2 + (a+b)^2) ≥ 3/5`

Question

Cho `a,b,c > 0` Chứng minh rằng :
`P = a^2/(a^2 + (b+c)^2) + b^2/(b^2 + (c+a)^2) + c^2/(c^2 + (a+b)^2) ≥ 3/5`

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Kennedy 2 tháng 2021-07-18T19:35:40+00:00 1 Answers 2 views 0

Answers ( )

    0
    2021-07-18T19:37:02+00:00

    ;-; sai thì góp ý 

    ta có :

    `2(a^2+b^2)≥(a+b)^2`

    `⇔1/(2(a^2+b^2))≤1/((a+b)^2)`

    `P=(a^2)/(a^2+(b+c)^2)+(b^2)/(a^2+(b+c)^2)+(a^2)/(a^2+(b+c)^2)`

    `⇒P≥(a^2)/(a^2+2(b^2+c^2))+(b^2)/(b^2+2(a^3+c^2))+(c^2)/(c^2+2(a^2+b^2))`

    `⇔P+3≥(2(a^2+b^2+c^2))/(a^2+2(b^2+c^2))+(2(a^2+b^2+c^2))/(b^2+2(a^3+c^2))+(2(a^2+b^2+c^2))/(c^2+2(a^2+b^2))`

    `⇔P+3≥2(a^2+b^2+c^2) . 1/(a^2+2(b^2+c^2))+1/(b^2+2(a^3+c^2))+1/(c^2+2(a^2+b^2))`

    `⇔P+3≥2(a^2+b^2+c^2).(1+1+1)^2/(a^2+2b^2+c^2+b^2+2a^3+c^2+c^2+2a^2+b^2)`

    `⇔P+3≥2(a^2+b^2+c^2).9/(5(a^2+b^2+c^2))`

    `⇔P+3≥(18)/5`

    `⇔P≥(18)/5-3=3/5`

    `”=”`xẩy ra khi :

    `a=b=c=1`

     

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