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222-9+11+12:2*14+14 = ? ( )

Toán Lớp 8: Cho `a, b, c \ne 0; a+b+c\ne0` thỏa mãn ĐK `1/a+1/b+1/c=1/(a+b+c)` CMR `1/a^(2021)+1/b^(2021)+1/c^(2021)=1/(a^(2021)+b^(2021)+c^(2021)

Home/ Toán Lớp 8: Cho `a, b, c \ne 0; a+b+c\ne0` thỏa mãn ĐK `1/a+1/b+1/c=1/(a+b+c)` CMR `1/a^(2021)+1/b^(2021)+1/c^(2021)=1/(a^(2021)+b^(2021)+c^(2021)

Toán Lớp 8: Cho `a, b, c \ne 0; a+b+c\ne0` thỏa mãn ĐK `1/a+1/b+1/c=1/(a+b+c)` CMR `1/a^(2021)+1/b^(2021)+1/c^(2021)=1/(a^(2021)+b^(2021)+c^(2021)

Tháng Bảy 10, 2022 Toán học 1 Comment 0 views

Toán Lớp 8: Cho a, b, c \ne 0; a+b+c\ne0 thỏa mãn ĐK 1/a+1/b+1/c=1/(a+b+c)
CMR 1/a^(2021)+1/b^(2021)+1/c^(2021)=1/(a^(2021)+b^(2021)+c^(2021))

Comments ( 1 )

  1. maianhnhiquynh
    maianhnhiquynh
    10 Tháng Bảy, 2022 at 11:46 sáng
    Reply
    $\\$
    1/a+1/b+1/c=1/(a+b+c)
    <=> (ab+bc+ca)/(abc) – 1/(a+b+c)=0
    <=> ((ab+bc+ca)(a+b+c))/(abc(a+b+c)) – (abc)/(abc(a+b+c)) = 0
    <=> a^2b + abc + a^2c + ab^2 + b^2c + abc + abc + bc^2 + c^2a – abc=0
    <=> a^2b + 2abc +a^2c + ab^2 + b^2c + bc^2 +c^2a=0
    <=> (a^2b + a^2c) + (ab^2+abc) + (b^2c +bc^2)+(abc+c^2a)=0
    <=> a^2(b+c) + ab (b+c) + bc (b+c)+ac (b+c)=0
    <=>(b+c)(a^2+ab+bc+ac)=0
    <=>(b+c) [a(a+b)+c(a+b)]=0
    <=> (b+c)(a+c)(a+b)=0
    Không mất tính tổng quát giả sử : a+b=0
    <=>a=-b
    1/a^{2021}+1/b^{2021}+1/c^{2021}
    = 1/a^{2021} – 1/a^{2021}+1/c^{2021}
    =1/c^{2021}(1)
    1/(a^{2021}+b^{2021}+c^{2021})
    =1/(a^{2021}-a^{2021}+c^{2021})
    =1/c^{2021}(2)
    (1)(2)=>1/a^{2021}+1/b^{2021}+1/c^{2021}=1/(a^{2021}+b^{2021}+c^{2021})

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222-9+11+12:2*14+14 = ? ( )

An Kim

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