Cho abc = 1. CMR:
\(\frac{a}{ab+a+1}+\frac{b}{bc+b+1}+\frac{c}{ac+c+1}=1\)
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Cho abc=1
CMR: \(\frac{a}{ab+a+1}+\frac{b}{bc+b+1}+\frac{c}{ac+a+1}=1\)
\(\frac{a}{ab+a+1}+\frac{b}{bc+b+1}+\frac{c}{ca+c+1}\)
\(=\frac{a}{ab+a+abc}+\frac{b}{bc+b+1}+\frac{bc}{abc+bc+b}\)
\(=\frac{1}{bc+b+1}+\frac{b}{bc+b+1}+\frac{bc}{bc+b+1}\)
\(=\frac{bc+b+1}{bc+b+1}\)
\(=1\)
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sao cậu đánh được dấu phân số hay vậy. Tớ bấm hoài mà không thấy
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CMR :
\(\frac{a-b}{1+ab}+\frac{b-c}{1+bc}+\frac{c-a}{1+ac}=\frac{a-b}{1+ab}-\frac{b-c}{1+bc}-\frac{c-a}{1+ac}\)
Là đương nhiên hai biểu thức trên bằng nhau , giống nhau y hệt
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Xem thêm câu trả lời
Cho a,b,c>0 và a+b+c\(\le\)6
CMR:
\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{ab}+\frac{1}{ac}+\frac{1}{bc}+\frac{1}{abc}\ge\frac{19}{8}\)
Cho abc=1
CMR \(\frac{1}{ab+a+1}+\frac{1}{bc+b+1}+\frac{1}{ac+c+1}=1\)
Đặt \(A=\frac{1}{ab+a+1}+\frac{1}{bc+b+1}+\frac{1}{ac+c+1}\)
\(\Leftrightarrow A=\frac{c}{abc+ac+c}+\frac{ac}{abc^2+abc+ac}+\frac{1}{ac+c+1}\)
\(\Leftrightarrow A=\frac{c}{1+ac+c}+\frac{ac}{c+1+ac}+\frac{1}{ac+c+1}\)
\(\Leftrightarrow A=\frac{c+ac+1}{1+ac+c}=1\)
Vậy \(\frac{1}{ab+a+1}+\frac{1}{bc+b+1}+\frac{1}{ac+c+1}=1\left(đpcm\right)\)
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Cho a;b;c>0 thỏa mãn abc=1. CMR:
\(\frac{a}{\left(ab+a+1\right)^2}+\frac{b}{\left(bc+b+1\right)^2}+\frac{c}{\left(ac+c+1\right)^2}\ge\frac{1}{a+b+c}\)
Áp dụng BĐT Bunhiacopxki, ta có:
\(\left(a+b+c\right)\left(\frac{a}{\left(ab+a+1\right)^2}+\frac{b}{\left(bc+b+1\right)^2}+\frac{c}{\left(ca+c+1\right)^2}\right)\ge\left(\frac{a}{ab+a+1}+\frac{b}{bc+b+1}+\frac{c}{ca+c+1}\right)^2\)
Mà \(\frac{a}{ab+a+1}+\frac{b}{bc+b+1}+\frac{c}{ca+c+1}=\frac{a}{ab+a+abc}+\frac{b}{bc+b+1}+\frac{bc}{abc+bc+b}=\frac{1}{b+1+bc}+\frac{b}{bc+b+1}+\frac{bc}{1+bc+1}=1\)
\(\Rightarrow\left(\frac{a}{\left(ab+a+1\right)^2}+\frac{b}{\left(bc+b+1\right)^2}+\frac{c}{\left(ca+c+1\right)^2}\right)\left(a+b+c\right)\ge1\)
\(\Rightarrow\frac{a}{\left(ab+b+1\right)^2}+\frac{b}{\left(bc+b+1\right)^2}+\frac{c}{\left(ac+c+1\right)^2}\ge\frac{1}{a+b+c}\)
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\(\frac{a}{\left(ab+a+1\right)^2}+\frac{b}{\left(bc+b+1\right)^2}+\frac{c}{\left(ac+c+1\right)^2}\ge\frac{1}{a+b+c}\)
ta có \(\frac{a}{ab+a+1}+\frac{b}{bc+b+1}+\frac{c}{ca+c+1}\)
\(=\frac{1}{bc+b+1}+\frac{b}{bc+b+1}+\frac{bc}{bc+b+1}=1\)
đặt \(H=\frac{a}{\left(ab+a+1\right)^2}+\frac{b}{\left(bc+b+1\right)^2}+\frac{c}{\left(ac+c+1\right)^2}\)
áp dụng bất đẳng thức bunhiacopxki ta có
\(H\left(a+b+c\right)\ge\left(\frac{a}{ab+a+1}+\frac{b}{bc+b+1}+\frac{c}{ac+c+1}\right)^2=1\)
\(\Rightarrow H\ge\frac{1}{a+b+c}\)
hay \(\frac{a}{\left(ab+a+1\right)^2}+\frac{b}{\left(bc+b+1\right)^2}+\frac{c}{\left(ac+c+1\right)^2}\ge\frac{1}{a+b+c}\)
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Cho a+b+c=1 ( a,b,c khác 1 và 2 ) CMR: \(\frac{c+ab}{a^2+b^2+abc-1}+\frac{a+bc}{b^2+c^2+abc-1}+\frac{b+ac}{a^2+c^2+acb-1}=\frac{bc+ac+ab+8}{(a-2)(b-2)(c-2)}\)
CMR: \(\frac{a^2+b^2+c^2}{ab+bc+ac} + \frac{1}{3} \geq \frac{8}{9}(\frac{a}{b+c} + \frac{b}{a+c} +\frac{c}{a+b})\)
CMR:\((1+a+b+c)(1+ab+bc+ac) \geq 4\sqrt{2(a+bc)(b+ac)(c+ab)}\)
Đề:Cho biết abc 1. Chứng minh rằng:frac{a}{ab+a+1}+frac{b}{bc+b+1}+frac{c}{ac+c+1} là hằng số.Giải:Thay 1 abc vào biểu thức trên, ta có:frac{a}{ab+a+abc}+frac{b}{bc+b+abc}+frac{c}{ac+c+abc}frac{a}{aleft(b+1+abright)}+frac{b}{bleft(c+1+acright)}+frac{c}{cleft(a+1+abright)}frac{1}{b+1+ab}+frac{1}{c+1+ac}+frac{1}{a+1+ab}frac{abc}{b+abc+ab}+frac{1}{c+1+ac}+frac{1}{a+1+ab}frac{abc}{bleft(1+ac+aright)}+frac{1}{c+1+ac}+frac{1}{a+1+ab}frac{ac}{1+ac+a}+frac{1}{c+1+ac}+frac{1}{a+1+ab}frac{ac+1}{c+1+ac}...
Đọc tiếp
Đề:
Cho biết abc = 1. Chứng minh rằng:\(\frac{a}{ab+a+1}+\frac{b}{bc+b+1}+\frac{c}{ac+c+1}\) là hằng số.
Giải:
Thay 1 = abc vào biểu thức trên, ta có:
\(\frac{a}{ab+a+abc}+\frac{b}{bc+b+abc}+\frac{c}{ac+c+abc}\)
\(=\frac{a}{a\left(b+1+ab\right)}+\frac{b}{b\left(c+1+ac\right)}+\frac{c}{c\left(a+1+ab\right)}\)
\(=\frac{1}{b+1+ab}+\frac{1}{c+1+ac}+\frac{1}{a+1+ab}\)
\(=\frac{abc}{b+abc+ab}+\frac{1}{c+1+ac}+\frac{1}{a+1+ab}\)
\(=\frac{abc}{b\left(1+ac+a\right)}+\frac{1}{c+1+ac}+\frac{1}{a+1+ab}\)
\(=\frac{ac}{1+ac+a}+\frac{1}{c+1+ac}+\frac{1}{a+1+ab}\)
\(=\frac{ac+1}{c+1+ac}+\frac{1}{a+1+ab}\)
\(=\frac{ac+1}{c+abc+ac}+\frac{1}{a+1+ab}\)
\(=\frac{ac+1}{c\left(1+ab+a\right)}+\frac{1}{a+1+ab}\)
\(=\frac{ac+1}{c\left(1+ab+a\right)}+\frac{c}{c\left(a+1+ab\right)}\) \(MTC:c\left(a+1+ab\right)\)
\(=\frac{ac+1+c}{c\left(1+ab+a\right)}\)
\(=\frac{ac+abc+c}{c+abc+ac}\)
\(=1\)
Vậy \(\frac{a}{ab+a+1}+\frac{b}{bc+b+1}+\frac{c}{ac+c+1}\) là hằng số khi abc = 1 (đpcm)
Trịnh Trân Trân <3
cho M =\(\frac{b-c}{a^2-ac-ab+bc}+\frac{c-a}{b^2-ab-cb+ca}+\frac{a-b}{c^2-bc-ac+ab}\) và N=\(\frac{1}{a-b}+\frac{1}{b-c}+\frac{1}{c-a}\) cmr M=2N
\(M=\frac{b-c}{\left(a-b\right)\left(a-c\right)}+\frac{c-a}{\left(b-c\right)\left(b-a\right)}+\frac{a-b}{\left(c-a\right)\left(c-a\right)}\)
Đánh giá đại diện: \(\frac{b-c}{\left(a-b\right)\left(a-c\right)}=\frac{\left(a-c\right)-\left(a-b\right)}{\left(a-b\right)\left(a-c\right)}=\frac{1}{a-b}-\frac{1}{a-c}\)
Tương tự: \(\frac{c-a}{\left(b-c\right)\left(b-a\right)}=\frac{1}{b-c}-\frac{1}{b-a}\)
\(\frac{a-b}{\left(c-a\right)\left(c-b\right)}=\frac{1}{c-a}-\frac{1}{c-b}\)
\(\Rightarrow M=\frac{1}{a-b}-\frac{1}{a-c}+\frac{1}{b-c}-\frac{1}{b-a}+\frac{1}{c-a}-\frac{1}{c-b}\)
\(\Rightarrow M=\frac{1}{a-b}+\frac{1}{c-a}+\frac{1}{b-c}+\frac{1}{a-b}+\frac{1}{c-a}+\frac{1}{b-c}\)
\(\Rightarrow M=2\left(\frac{1}{a-b}+\frac{1}{b-c}+\frac{1}{c-a}\right)=2N\left(đpcm\right)\)