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\(abc\ge\left(a+b-c\right)\left(b+c-a\right)\left(c+a-b\right)\)
\(\Leftrightarrow abc\ge\left(3-2a\right)\left(3-2b\right)\left(3-2c\right)\)
\(\Leftrightarrow9abc\ge12\left(ab+bc+ca\right)-27\)
\(\Rightarrow abc\ge\dfrac{4}{3}\left(ab+bc+ca\right)-3\)
\(P\ge\dfrac{9}{a\left(b^2+bc+c^2\right)+b\left(c^2+ca+a^2\right)+c\left(a^2+ab+b^2\right)}+\dfrac{abc}{ab+bc+ca}=\dfrac{9}{\left(ab+bc+ca\right)\left(a+b+c\right)}+\dfrac{abc}{ab+bc+ca}\)
\(\Rightarrow P\ge\dfrac{3}{ab+bc+ca}+\dfrac{abc}{ab+bc+ca}=\dfrac{3+abc}{ab+bc+ca}\)
\(\Rightarrow P\ge\dfrac{3+\dfrac{4}{3}\left(ab+bc+ca\right)-3}{ab+bc+ca}=\dfrac{4}{3}\)
Dấu "=" xảy ra khi \(a=b=c=1\)
Ta có: \(a^3+b^3+c^3-3abc=\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ac\right)\)
\(\Rightarrow\frac{a^3+b^3+c^3}{4abc}=\frac{3}{4}+\frac{\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ac\right)}{4abc}\)
\(=\frac{3}{4}+\frac{1}{4}\left(\frac{1}{ab}+\frac{1}{ac}+\frac{1}{bc}\right)\left(a^2+b^2+c^2-ab-bc-ca\right)\)
\(\ge\frac{9\left(a^2+b^2+c^2\right)}{4\left(ab+bc+ca\right)}-\frac{3}{2}\left(\frac{1}{ab}+\frac{1}{ac}+\frac{1}{bc}\ge\frac{9}{ab+ac+bc}\right)\)
\(\Rightarrow\frac{a^3+b^3+c^3}{4abc}\ge\frac{9}{4}\left(\frac{a^2+b^2+c^2}{ab+bc+ac}\right)-\frac{3}{2}\left(1\right)\)
Lại có:\(\frac{\left(a+b+c\right)^2}{30\left(a^2+b^2+c^2\right)}=\frac{a^2+b^2+c^2+2\left(ab+bc+ac\right)}{30\left(a^2+b^2+c^2\right)}\)
\(=\frac{1}{30}+\frac{1}{15}\left(\frac{ab+bc+ca}{a^2+b^2+c^2}\right)\left(2\right)\).Từ (1);(2) có:
\(P=\frac{1}{30}-\frac{3}{2}+\frac{1}{5}\left(\frac{ab+bc+ca}{a^2+b^2+c^2}\right)+\frac{9}{4}\left(\frac{a^2+b^2+c^2}{ab+bc+ca}\right)-\frac{131\left(a^2+b^2+c^2\right)}{60\left(ab+bc+ca\right)}\)
\(=\frac{1}{15}\left(\frac{a^2+b^2+c^2}{ab+bc+ac}+\frac{ab+bc+ca}{a^2+b^2+c^2}-22\right)\ge-\frac{4}{3}\)
đề thi hsg toán lớp 9 tỉnh thanh hóa năm 2016-2017 mà
Ta có: \(a^3+b^3+c^3-3abc=\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ac\right)\)
\(\Rightarrow\frac{a^3+b^3+c^3}{4abc}=\frac{3}{4}+\frac{\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ac\right)}{4abc}\)
\(=\frac{3}{4}+\frac{1}{4}\left(\frac{1}{ab}+\frac{1}{ac}+\frac{1}{bc}\right)\left(a^2+b^2+c^2-ab-bc-ca\right)\)
\(\ge\frac{9\left(a^2+b^2+c^2\right)}{4\left(ab+bc+ca\right)}-\frac{3}{2}\left(\frac{1}{ab}+\frac{1}{ac}+\frac{1}{bc}\ge\frac{9}{ab+ac+bc}\right)\)
\(\Rightarrow\frac{a^3+b^3+c^3}{4abc}\ge\frac{9}{4}\left(\frac{a^2+b^2+c^2}{ab+bc+ac}\right)-\frac{3}{2}\left(1\right)\)
Lại có: \(\frac{\left(a+b+c\right)^2}{30\left(a^2+b^2+c^2\right)}=\frac{a^2+b^2+c^2+2\left(ab+bc+ac\right)}{30\left(a^2+b^2+c^2\right)}\)
\(=\frac{1}{30}+\frac{1}{15}\left(\frac{ab+bc+ca}{a^2+b^2+c^2}\right)\left(2\right)\)
Từ \(\left(1\right)\) và \(\left(2\right)\)\(\Rightarrow P=\frac{1}{30}-\frac{3}{2}+\frac{1}{5}\left(\frac{ab+bc+ca}{a^2+b^2+c^2}\right)+\frac{9}{4}\left(\frac{a^2+b^2+c^2}{ab+bc+ca}\right)-\frac{131\left(a^2+b^2+c^2\right)}{60\left(ab+bc+ca\right)}\)
\(=\frac{1}{15}\left(\frac{a^2+b^2+c^2}{ab+bc+ac}+\frac{ab+bc+ca}{a^2+b^2+c^2}-22\right)\ge-\frac{4}{3}\)
\(Q=\sum\dfrac{\left(a+b\right)^2}{\sqrt{2\left(b+c\right)^2+bc}}\ge\sum\dfrac{\left(a+b\right)^2}{\sqrt{2\left(b+c\right)^2+\dfrac{1}{4}\left(b+c\right)^2}}=\dfrac{2}{3}\sum\dfrac{\left(a+b\right)^2}{b+c}\)
\(Q\ge\dfrac{2}{3}.\dfrac{\left(a+b+b+c+c+a\right)^2}{a+b+b+c+c+a}=\dfrac{4}{3}\left(a+b+c\right)=\dfrac{4}{3}\)
Trước hết theo BĐT Schur bậc 3 ta có:
\(\left(a+b+c\right)\left(a^2+b^2+c^2\right)+9abc\ge2\left(a+b+c\right)\left(ab+bc+ca\right)\)
\(\Leftrightarrow a^2+b^2+c^2+3abc\ge2\left(ab+bc+ca\right)\) (do \(a+b+c=3\)) (1)
Đặt vế trái BĐT cần chứng minh là P, ta có:
\(P=\dfrac{\left(a^2+abc\right)^2}{a^2b^2+2abc^2}+\dfrac{\left(b^2+abc\right)^2}{b^2c^2+2a^2bc}+\dfrac{\left(c^2+abc\right)^2}{a^2c^2+2ab^2c}\)
\(\Rightarrow P\ge\dfrac{\left(a^2+b^2+c^2+3abc\right)^2}{a^2b^2+b^2c^2+c^2a^2+2abc\left(a+b+c\right)}=\dfrac{\left(a^2+b^2+c^2+3abc\right)^2}{\left(ab+bc+ca\right)^2}\)
Áp dụng (1):
\(\Rightarrow P\ge\dfrac{\left[2\left(ab+bc+ca\right)\right]^2}{\left(ab+bc+ca\right)^2}=4\) (đpcm)
Dấu "=" xảy ra khi \(a=b=c=1\)
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Ta chứng minh \(P\ge-\dfrac{4}{3}\) hay
\(\dfrac{\left(a+b+c\right)^2}{30\left(a^2+b^2+c^2\right)}-\dfrac{1}{10}+\dfrac{a^3+b^3+c^3}{4abc}-\dfrac{3}{4}-\dfrac{131\left(a^2+b^2+c^2\right)}{60\left(ab+bc+ca\right)}+\dfrac{131}{60}\ge0\)
\(\Leftrightarrow\dfrac{\left(a+b+c\right)^2-3\left(a^2+b^2+c^2\right)}{30\left(a^2+b^2+c^2\right)}+\dfrac{a^3+b^3+c^3-3abc}{4abc}-\dfrac{131\left(a^2+b^2+c^2-ab-bc-ca\right)}{60\left(ab+bc+ca\right)}\ge0\)
\(\LeftrightarrowΣ_{cyc}\dfrac{-\left(a-b\right)^2}{30\left(a^2+b^2+c^2\right)}+Σ_{cyc}\dfrac{\dfrac{a+b+c}{2}\left(a-b\right)^2}{4abc}-Σ_{cyc}\dfrac{\dfrac{131}{2}\left(a-b\right)^2}{60\left(ab+bc+ca\right)}\ge0\)
\(\LeftrightarrowΣ_{cyc}\left(a-b\right)^2\left(\dfrac{\dfrac{a+b+c}{2}}{4abc}-\dfrac{\dfrac{131}{2}}{60\left(ab+bc+ca\right)}-\dfrac{1}{30\left(a^2+b^2+c^2\right)}\right)\ge0\)
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