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Câu 2/
\(\frac{a^2+bc}{a^2\left(b+c\right)}+\frac{b^2+ca}{b^2\left(c+a\right)}+\frac{c^2+ab}{c^2\left(a+b\right)}\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\)
\(\Leftrightarrow\frac{a^2+bc}{a^2\left(b+c\right)}-\frac{1}{a}+\frac{b^2+ca}{b^2\left(c+a\right)}-\frac{1}{b}+\frac{c^2+ab}{c^2\left(a+b\right)}-\frac{1}{c}\ge0\)
\(\Leftrightarrow\frac{\left(b-a\right)\left(c-a\right)}{a^2\left(b+c\right)}+\frac{\left(a-b\right)\left(c-b\right)}{b^2\left(c+a\right)}+\frac{\left(a-c\right)\left(b-c\right)}{c^2\left(a+b\right)}\ge0\)
\(\Leftrightarrow a^4b^4+b^4c^4+c^4a^4-a^4b^2c^2-a^2b^4c^2-a^2b^2c^4\ge0\)
\(\Leftrightarrow a^4b^4+b^4c^4+c^4a^4\ge a^4b^2c^2+a^2b^4c^2+a^2b^2c^4\left(1\right)\)
Ma ta có: \(\hept{\begin{cases}a^4b^4+b^4c^4\ge2a^2b^4c^2\left(2\right)\\b^4c^4+c^4a^4\ge2a^2b^2c^4\left(3\right)\\c^4a^4+a^4b^4\ge2a^4b^2c^2\left(4\right)\end{cases}}\)
Cộng (2), (3), (4) vế theo vế rồi rút gọn cho 2 ta được điều phải chứng minh là đúng.
PS: Nếu nghĩ được cách khác đơn giản hơn sẽ chép lên cho b sau. Tạm cách này đã.
\(P=\frac{\frac{1}{a^2}}{\frac{1}{b}+\frac{1}{c}}+\frac{\frac{1}{b^2}}{\frac{1}{a}+\frac{1}{c}}+\frac{\frac{1}{c^2}}{\frac{1}{a}+\frac{1}{b}}\)
Đặt \(\hept{\begin{cases}x=\frac{1}{a}\\y=\frac{1}{b}\\z=\frac{1}{c}\end{cases}}\Rightarrow xyz=1\Rightarrow P=\frac{x^2}{y+z}+\frac{y^2}{x+z}+\frac{z^2}{x+y}\)
Áp dụng BĐT Cauchy-Schwarz dạng Engel ta có:
\(P\ge\frac{\left(x+y+z\right)^2}{y+z+x+z+x+y}=\frac{x+y+z}{2}\ge\frac{3\sqrt[3]{xyz}}{2}=\frac{3}{2}\)
Dấu "=" xảy ra khi \(x=y=z\Leftrightarrow a=b=c=1\)
Cần cách khác thì nhắn cái
Ta có:
\(\frac{1}{a}+\frac{1}{b}=\frac{1}{c}\Leftrightarrow\left(a+b\right)c=ab\Leftrightarrow ab-bc-ab=0\)
Hay \(ab-bc-ab+c^2=c^2\Leftrightarrow\left(b-c\right)\left(a-c\right)=c^2\)
Nếu \(\left(b-c;a-c\right)=d\ne1\Rightarrow c^2=d^2\left(loai\right)\)
Vậy \(\left(b-c;a-c\right)=1\Rightarrow c-b;c-a\) là 2 số chính phương
Đặt \(b-c=n^2;a-c=m^2\)
\(\Rightarrow a+b=b-c+a-c+2c=m^2+n^2+2mn=\left(m+n\right)^2\) là số chính phương
a) A = \(\frac{3x^2+3x-3}{x^2+x-2}-\frac{x+1}{x+2}+\frac{x-2}{x}\cdot\left(\frac{1}{1-x}-1\right)\)
A = \(\frac{3x^2+3x-3}{x^2+2x-x-2}-\frac{x+1}{x+2}+\frac{x-2}{x}\cdot\left(\frac{1-1+x}{1-x}\right)\)
A = \(\frac{3x^2+3x-3}{\left(x-1\right)\left(x+2\right)}-\frac{x+1}{x+2}+\frac{x-2}{x}\cdot\frac{x}{1-x}\)
A = \(\frac{3x^2+3x-3}{\left(x-1\right)\left(x+2\right)}-\frac{x+1}{x+2}-\frac{x-2}{x-1}\)
A = \(\frac{3x^2+3x-3}{\left(x-1\right)\left(x+2\right)}-\frac{\left(x+1\right)\left(x-1\right)}{\left(x-1\right)\left(x+2\right)}-\frac{\left(x-2\right)\left(x+2\right)}{\left(x-1\right)\left(x+2\right)}\)
A = \(\frac{3x^2+3x-3-x^2+1-x^2+4}{\left(x-1\right)\left(x+2\right)}\)
A = \(\frac{x^2+3x+2}{\left(x-1\right)\left(x+2\right)}\)
A = \(\frac{x^2+2x+x+2}{\left(x-1\right)\left(x+2\right)}\)
A = \(\frac{\left(x+1\right)\left(x+2\right)}{\left(x-1\right)\left(x+2\right)}\)
A = \(\frac{x+1}{x-1}\) (Đk: \(x-1\ge0\) => x \(\ge\)1)
b) Ta có: A = \(\frac{x+1}{x-1}=\frac{\left(x-1\right)+2}{x-1}=1+\frac{2}{x-1}\)
Để A \(\in\)Z <=> 2 \(⋮\)x - 1
<=> x - 1 \(\in\)Ư(2) = {1; -1; 2; -2}
<=> x \(\in\){2; 0; 3; -1}
c) Ta có: A < 0
=> \(\frac{x+1}{x-1}< 0\)
=> \(\hept{\begin{cases}x+1< 0\\x-1>0\end{cases}}\) hoặc \(\hept{\begin{cases}x+1>0\\x-1< 0\end{cases}}\)
=> \(\hept{\begin{cases}x< -1\\x>1\end{cases}}\)(loại) hoặc \(\hept{\begin{cases}x>-1\\x< 1\end{cases}}\)
=> -1 < x < 1
Edogawa Conan
Thiếu dòng đầu \(ĐKXĐ:\hept{\begin{cases}x\ne1\\x\ne-2\\x\ne0\end{cases}}\)
a, \(\left(\frac{x^3+1}{x^2-1}-\frac{x^2-1}{x-1}\right):\left(x+\frac{x}{x-1}\right)\)
\(=\left(\frac{x^3+1}{\left(x-1\right)\left(x+1\right)}-\frac{\left(x-1\right)\left(x+1\right)^2}{\left(x-1\right)\left(x+1\right)}\right):\left(\frac{x\left(x-1\right)}{x-1}+\frac{x}{x-1}\right)\)
\(=\left(\frac{\left(x+1\right)\left(x^2-x+1\right)-\left(x-1\right)\left(x+1\right)^2}{\left(x-1\right)\left(x+1\right)}\right):\left(\frac{x\left(x-1\right)+x}{x-1}\right)\)
\(=\left(\frac{\left(x+1\right)\left[x^2-x+1-x^2+1\right]}{\left(x-1\right)\left(x+1\right)}\right):\left(\frac{x^2}{x-1}\right)\)
\(=\frac{\left(x+1\right)\left(2-x\right)}{\left(x-1\right)\left(x+1\right)}.\frac{x-1}{x^2}=\frac{2-x}{x^2}\)
b, Ta có : A = 3 hay \(\frac{2-x}{x^2}=3\)
\(3x^2=2-x\Leftrightarrow3x^2+x-2=0\)
\(\Leftrightarrow3x^2+3x-2x-2=0\Leftrightarrow\left(x+1\right)\left(3x-2\right)=0\Leftrightarrow x=-1;\frac{2}{3}\)
a,\(A=\left(\frac{x^3+1}{x^2-1}-\frac{x^2-1}{x-1}\right)\div\left(x+\frac{x}{x-1}\right)\)
\(=\left(\frac{x^3+1}{\left(x+1\right)\left(x-1\right)}-\frac{\left(x^2-1\right)\left(x+1\right)}{\left(x+1\right)\left(x-1\right)}\right)\div\left(\frac{x\left(x-1\right)}{x-1}+\frac{x}{x-1}\right)\)
\(=\left(\frac{\left(x+1\right)\left(x^2-x+1\right)-\left(x-1\right)\left(x+1\right)^2}{\left(x-1\right)\left(x+1\right)}\right)\div\left(\frac{x\left(x-1\right)+x}{\left(x-1\right)}\right)\)
\(=\left(\frac{\left(x+1\right)\left(x^2-x+1-x^2+1\right)}{\left(x-1\right)\left(x+1\right)}\right)\div\left(\frac{x^2}{x-1}\right)\)
\(=\left(\frac{\left(x+1\right)\left(2-x\right)}{\left(x-1\right)\left(x+1\right)}\right)\div\frac{x^2}{x-1}\)
\(=\frac{\left(x+1\right)\left(2-x\right)}{\left(x-1\right)\left(x+1\right)}\times\frac{x-1}{x^2}\)
\(=\frac{\left(x+1\right)\left(2-x\right)\left(x-1\right)}{\left(x-1\right)\left(x+1\right)x^2}=\frac{2-x}{x^2}\)
Dễ dàng chứng minh được:
\(\frac{1}{x}+\frac{1}{y}\ge\frac{4}{x+y}\)với \(x,y>0\)(1)
Dấu bằng xảy ra \(\Leftrightarrow x=y>0\)
Ta có:
\(\frac{a}{bc\left(a+1\right)}=\frac{a}{abc+bc}=\frac{a}{ab+bc+ca+bc}=\frac{a}{\left(ab+bc\right)+\left(bc+ca\right)}\)
Áp dụng (1), ta được:
\(\frac{1}{ab+bc}+\frac{1}{bc+ca}\ge\frac{4}{\left(ab+bc\right)+\left(bc+ca\right)}\)
\(\Leftrightarrow\frac{1}{4\left(ab+bc\right)}+\frac{1}{4\left(bc+ca\right)}\ge\frac{1}{ab+bc+bc+ca}\)
\(\Leftrightarrow\frac{a}{4}\left(\frac{1}{ab+bc}+\frac{1}{bc+ca}\right)\ge\frac{a}{ab+bc+bc+ca}\)
\(\Leftrightarrow\frac{a}{4}\left(\frac{1}{ab+bc}+\frac{1}{bc+ca}\right)\ge\frac{a}{bc\left(a+1\right)}\left(2\right)\)
Dấu bằng xảy ra \(\Leftrightarrow b=c>0\)
Chúng minh tương tự, ta được:
\(\frac{b}{4}\left(\frac{1}{ab+ca}+\frac{1}{bc+ca}\right)\ge\frac{b}{ca\left(b+1\right)}\left(3\right)\)
Dấu bằng xảu ra \(\Leftrightarrow a=c>0\).
\(\frac{c}{4}\left(\frac{1}{ac+ab}+\frac{1}{ab+bc}\right)\ge\frac{c}{ab\left(c+1\right)}\left(4\right)\)
Từ (2), (3) và (4), ta được:
\(\frac{a}{bc\left(a+1\right)}+\frac{b}{ca\left(b+1\right)}+\frac{c}{ab\left(c+1\right)}\le\)\(\frac{a}{4}\left(\frac{1}{ab+bc}+\frac{1}{bc+ac}\right)+\frac{b}{4}\left(\frac{1}{ac+bc}+\frac{1}{ac+ab}\right)\)\(+\frac{c}{4}\left(\frac{1}{ab+bc}+\frac{1}{ab+ac}\right)\)
\(\Leftrightarrow P\le\frac{1}{4}.\left(\frac{a}{ab+bc}+\frac{c}{ab+bc}\right)+\frac{1}{4}\left(\frac{a}{bc+ac}+\frac{b}{bc+ac}\right)\)\(+\frac{1}{4}\left(\frac{b}{ab+ac}+\frac{c}{ab+ac}\right)\)
\(\Leftrightarrow P\le\frac{a+c}{4\left(ab+bc\right)}+\frac{a+b}{4\left(bc+ac\right)}+\frac{b+c}{4\left(ab+ac\right)}\)
\(\Leftrightarrow P\le\frac{a+c}{4b\left(a+c\right)}+\frac{a+b}{4c\left(a+b\right)}+\frac{b+c}{4a\left(b+c\right)}\)
\(\Leftrightarrow P\le\frac{1}{4b}+\frac{1}{4c}+\frac{1}{4a}\)
\(\Leftrightarrow P\le\frac{1}{4}\left(\frac{ab+bc+ca}{abc}\right)\)
\(\Leftrightarrow P\le\frac{1}{4}.\frac{abc}{abc}=\frac{1}{4}.1=\frac{1}{4}\)( vì \(ab+bc+ca=abc\))
Dấu bằng xảy ra
\(\Leftrightarrow\hept{\begin{cases}a=b=c>0\\ab+bc+ca=abc\end{cases}}\Leftrightarrow a=b=c=3\)
Vậy \(minP=\frac{1}{4}\Leftrightarrow a=b=c=3\)
\(ĐKXĐ:x\ne\pm1\)
a) \(A=\left(\frac{1}{1-x}+\frac{2}{1+x}-\frac{5-x}{1-x^2}\right):\frac{1-2x}{x^2-1}\)
\(=\left(\frac{\left(1+x\right)}{\left(1+x\right)\left(1-x\right)}+\frac{2\left(1-x\right)}{\left(1+x\right)\left(1-x\right)}-\frac{5-x}{1-x^2}\right):\frac{1-2x}{x^2-1}\)
\(=\frac{1+x+2-2x-5+x}{1-x^2}:\frac{2x-1}{1-x^2}\)
\(=\frac{8}{1-x^2}.\frac{1-x^2}{2x-1}=\frac{8}{2x-1}\)
b) Để A nguyên thì \(\frac{8}{2x-1}\inℤ\)
\(\Leftrightarrow8⋮2x-1\Rightarrow2x-1\inƯ\left(8\right)=\left\{\pm1;\pm2;\pm4;\pm8\right\}\)
Mà dễ thấy 2x - 1 lẻ nên\(2x-1\in\left\{\pm1\right\}\)
+) \(2x-1=1\Rightarrow x=1\left(ktmđkxđ\right)\)
+) \(2x-1=-1\Rightarrow x=0\left(tmđkxđ\right)\)
Vậy x nguyên bằng 0 thì A nguyên
c) \(\left|A\right|=A\Leftrightarrow A\ge0\)
\(\Rightarrow\frac{8}{2x-1}\ge0\Rightarrow2x-1>0\Leftrightarrow x>\frac{1}{2}\)
Vậy \(x>\frac{1}{2}\)thì |A| = A
a, \(A=\left(\frac{1}{1-x}+\frac{2}{1+x}-\frac{5-x}{1-x^2}\right):\frac{1-2x}{x^2-1}\left(x\ne\frac{1}{2};x\ne\pm1\right)\)
\(\Leftrightarrow A=\left(\frac{1+x}{\left(1-x\right)\left(1+x\right)}+\frac{2-2x}{\left(1-x\right)\left(1+x\right)}-\frac{5-x}{\left(1-x\right)\left(1+x\right)}\right):\frac{\left(x+1\right)\left(x-1\right)}{2x-1}\)
\(\Leftrightarrow A=\frac{1+x+2-2x-5+x}{\left(1-x\right)\left(1+x\right)}\cdot\frac{\left(x-1\right)\left(x+1\right)}{2x-1}\)
\(\Leftrightarrow A=\frac{-2\left(1-x^2\right)}{\left(1-x^2\right)\left(2x-1\right)}=\frac{2}{2x-1}\)
Vậy \(A=\frac{2}{2x-1}\left(x\ne\frac{1}{2};x\ne\pm1\right)\)
b) \(A=\frac{2}{2x-1}\left(x\ne\frac{1}{2};x\ne\pm1\right)\)
Để A nhận giá trị nguyên thì 2 chia hết cho 2x-1
Mà x nguyên => 2x-1 nguyên
=> 2x-1 thuộc Ư (2)={-2;-1;1;2}
Ta có bảng
| 2x-1 | -2 | -1 | 1 | 2 |
| 2x | -1 | 0 | 2 | 3 |
| x | -1/2 | 0 | 1 | 3/2 |
Đối chiếu điều kiện
=> x=0
Ez còn nhờ :
Để \(A=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}\) nghuyên \(\Leftrightarrow\hept{\begin{cases}1⋮a\text{ };\text{ }1⋮b\text{ };\text{ }1⋮c\\1⋮ab\text{ };\text{ }1⋮bc\text{ };\text{ }1⋮ac\end{cases}}\)
\(\Rightarrow\left(a;b;c\right)\inƯ\left(1\right)=\left\{\pm1\right\}\)Mà a;b;c nguyên dương \(\Rightarrow\left(a;b;c\right)=\left(1;1;1\right)\)
Vậy \(\left(a;b;c\right)=\left(1;1;1\right)\)
Có dễ như thế không vậy?