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Ta có: \(A=\frac{a-d}{d+b}+\frac{d-b}{b+c}+\frac{b-c}{c+a}+\frac{c-a}{a+d}\)
\(\Leftrightarrow A+4=\frac{a-d}{d+b}+1+\frac{d-b}{b+c}+1+\frac{b-c}{c+a}+1+\frac{c-a}{a+d}+1\)
\(\Leftrightarrow A+4=\frac{a+b}{d+b}+\frac{d+c}{b+c}+\frac{b+a}{c+a}+\frac{c+d}{a+d}\)
\(\Leftrightarrow A+4=\left(a+b\right)\left(\frac{1}{d+b}+\frac{1}{a+c}\right)+\left(c+d\right)\left(\frac{1}{b+c}+\frac{1}{d+a}\right)\)
Áp dụng BĐT \(\frac{1}{x}+\frac{1}{y}\ge\frac{4}{xy}\)với mọi x,y>0
Ta có: \(A+4\ge\frac{4\left(a+b\right)}{a+b+c+d}+\frac{4\left(d+c\right)}{a+b+c+d}\)
\(A+4\ge\frac{4\left(a+b+c+d\right)}{a+b+c+d}=4\)
\(A\ge0\)(dpcm)
a/ Biến đổi tương đương:
\(\frac{1}{a}+\frac{1}{b}\ge\frac{4}{a+b}\Leftrightarrow\frac{a+b}{ab}\ge\frac{4}{a+b}\)
\(\Leftrightarrow\left(a+b\right)^2\ge4ab\Leftrightarrow a^2+2ab+b^2\ge4ab\)
\(\Leftrightarrow a^2-2ab+b^2\ge0\Leftrightarrow\left(a-b\right)^2\ge0\) (luôn đúng)
Vậy BĐT được chứng minh
b/ \(VT=\frac{a-d}{b+d}+1+\frac{d-b}{b+c}+1+\frac{b-c}{a+c}+1+\frac{c-a}{a+d}+1-4\)
\(VT=\frac{a+b}{b+d}+\frac{c+d}{b+c}+\frac{a+b}{a+c}+\frac{c+d}{a+d}-4\)
\(VT=\left(a+b\right)\left(\frac{1}{b+d}+\frac{1}{a+c}\right)+\left(c+d\right)\left(\frac{1}{b+c}+\frac{1}{a+d}\right)-4\)
\(\Rightarrow VT\ge\left(a+b\right).\frac{4}{b+d+a+c}+\left(c+d\right).\frac{4}{b+c+a+d}-4\)
\(\Rightarrow VT\ge\frac{4}{\left(a+b+c+d\right)}\left(a+b+c+d\right)-4=4-4=0\) (đpcm)
Dấu "=" xảy ra khi \(a=b=c=d\)
\(Để\frac{a-b}{b+c}+\frac{b-c}{c+d}+\frac{c-d}{d+a}+\frac{d-a}{a+b}\ge0\)
Thì \(\frac{a-b}{b+c}+1+\frac{b-c}{c+d}+1+\frac{c-d}{d+a}+1+\frac{d-a}{a+b}+1\ge4\)
\(\Leftrightarrow\frac{a+c}{b+c}+\frac{b+d}{c+d}+\frac{c+a}{d+a}+\frac{d+b}{a+b}\ge4\)
\(\Leftrightarrow\left(a+c\right)\left(\frac{1}{b+c}+\frac{1}{d+a}\right)+\left(b+d\right)\left(\frac{1}{c+d}+\frac{1}{a+b}\right)\ge4\)(Cần phải chứng minh)
Ta có : \(\Leftrightarrow\left(a+c\right)\left(\frac{1}{b+c}+\frac{1}{d+a}\right)+\left(b+d\right)\left(\frac{1}{c+d}+\frac{1}{a+b}\right)\)
\(\ge\left(a+c\right)\left(\frac{4}{a+b+c+d}\right)+\left(b+d\right)\left(\frac{4}{a+b+c+d}\right)=4\)(Áp dụng Cô-si dạng phân thức)
\(\Rightarrow\frac{a-b}{b+c}+\frac{b-c}{c+d}+\frac{c-d}{d+a}+\frac{d-a}{a+b}\ge0\)(Đpcm)
Học tốt ~~
Ta có:
\(\frac{a}{b+c+d}+1=\frac{b}{a+c+d}+1=\frac{c}{a+b+d}+1=\frac{d}{a+b+c}+1\)
\(\Rightarrow\) \(\frac{a+b+c+d}{b+c+d}=\frac{a+b+c+d}{a+c+d}=\frac{a+b+c+d}{a+b+d}=\frac{a+b+c+d}{a+b+c}\)
Vì \(a+b+c+d\ne0\) nên \(a=b=c=d\)
Do đó: \(M=4\)
Ta có: \(\frac{a}{a+b+c}< 1\Rightarrow\frac{a}{a+b+c}< \frac{a+d}{a+b+c+d}\left(1\right)\)
Mặt khác: \(\frac{a}{a+b+c}>\frac{a}{a+b+c+d}\left(2\right)\)
Từ (1) và (2) ta có: \(\frac{a}{a+b+c+d}< \frac{a}{a+b+c}< \frac{a+d}{a+b+c+d}\left(3\right)\)
Tương tự: \(\frac{b}{a+b+c+d}< \frac{b}{b+c+d}< \frac{b+a}{a+b+c+d}\left(4\right)\)
\(\frac{c}{a+b+c+d}< \frac{c}{c+d+a}< \frac{b+c}{a+b+c+d}\left(5\right)\)
\(\frac{d}{a+b+c+d}< \frac{d}{b+d+a}< \frac{d+c}{a+b+c+d}\left(6\right)\)
Cộng vế với vế (3);(4);(5);(6) ta có:
\(1< \frac{a}{a+b+c}+\frac{b}{b+c+d}+\frac{c}{c+d+a}+\frac{d}{d+a+b}< 2\left(đpcm\right)\)
Đặt A = a/a+b+c + b/b+c+d + c/c+d+a + d/d+a+b
A > a/a+b+c+d + b/a+b+c+d + c/a+b+c+d + d+a+b+c+d
A > a+b+c+d/a+b+c+d = 1 (1)
Áp dụng a/b < 1 <=> a/b < a+m/b+m (a;b;m > 0) ta có:
A < a+d/a+b+c+d + a+b/a+b+c+d + b+c/a+b+c+d + c+d/a+b+c+d
A < 2.(a+b+c+d)/a+b+c+d
A < 2
Từ (1) và (2) => đpcm
nguồn:soyeon_Tiểubàng giải
áp dung bdt 1/x+1/y>=4/x+y ta co
\(\frac{a+c}{a+b}+\frac{b+d}{b+c}+...\)
=(a+c)(\(\frac{1}{a+b}+\frac{1}{c+d}\)) + (b+d)(\(\frac{1}{b+c}+\frac{1}{a+d}\))\(\ge\)\(\frac{4a+4c}{a+b+c+d}+\frac{4b+4d}{a+b+c+d}\)=4(dpcm)
= \(\left(a+c\right)\left(\frac{1}{a+b}+\frac{1}{c+d}\right)+\left(b+d\right)\left(\frac{1}{b+c}+\frac{1}{d+a}\right)\)
Áp dụng \(\frac{1}{x}+\frac{1}{y}\ge\frac{4}{x+y}\left(x,y>0\right)\)
\(\ge\left(a+c\right)\left(\frac{4}{a+b+c+d}\right)+\left(b+d\right)\left(\frac{4}{a+b+c+d}\right)\)
\(\ge\frac{4\left(a+b+c+d\right)}{a+b+c+d}\)
Áp dụng tính chất tỉ số ta có: \(\frac{a+b+d}{a+b+c+d}>\frac{a+b}{a+b+c}>\frac{a+b}{a+b+c+d}\left(1\right)\)
Tương tự: với b,c rồi cộng vế theo vế có ĐPCM