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\(3x^2+2xy+3y^2=\left(x+y\right)^2+2\left(x^2+y^2\right)\ge\left(x+y\right)^2+\left(x+y\right)^2=2\left(x+y\right)^2\)
\(\Rightarrow A\ge\sqrt{2}\left(a+b\right)+\sqrt{2}\left(b+c\right)+\sqrt{2}\left(c+a\right)\)
\(A\ge2\sqrt{2}\left(a+b+c\right)\ge\frac{2\sqrt{2}}{3}\left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)^2=6\sqrt{2}\)
\(A_{min}=6\sqrt{2}\) khi \(a=b=c=1\)
Áp dụng tính chất dãy tỉ số bằng nhau, ta có :
\(\frac{a}{3b}=\frac{b}{3c}=\frac{c}{3d}=\frac{d}{3a}=\frac{a+b+c+d}{3b+3c+3d+3a}=\frac{a+b+c+d}{3\cdot\left(b+c+d+a\right)}=\frac{1}{3}\)
Do đó :
\(\frac{a}{3b}=\frac{1}{3}\Rightarrow\frac{a}{b}.\frac{1}{3}=\frac{1}{3}\Rightarrow\frac{a}{b}=\frac{1}{3}:\frac{1}{3}=1\Rightarrow a=b\)
\(\frac{b}{3c}=\frac{1}{3}\Rightarrow\frac{b}{c}\cdot\frac{1}{3}=\frac{1}{3}\Rightarrow\frac{b}{c}=\frac{1}{3}:\frac{1}{3}=1\Rightarrow b=c\)
\(\frac{c}{3d}=\frac{1}{3}\Rightarrow\frac{c}{d}\cdot\frac{1}{3}=\frac{1}{3}\Rightarrow\frac{c}{d}=\frac{1}{3}:\frac{1}{3}=1\Rightarrow c=d\)
\(\frac{d}{3a}=\frac{1}{3}\Rightarrow\frac{d}{a}\cdot\frac{1}{3}=\frac{1}{3}\Rightarrow\frac{d}{a}=\frac{1}{3}:\frac{1}{3}=1\Rightarrow d=a\)
\(\Rightarrow a=b=c=d\)
\(\frac{3a+b+c}{a}=\frac{a+3b+c}{b}=\frac{a+b+3c}{c}=\frac{\left(3a+b+c\right)+\left(a+3b+c\right)+\left(a+b+3c\right)}{a+b+c}\)
\(=\frac{5\left(a+b+c\right)}{a+b+c}=5\)
\(\Rightarrow\frac{3a+b+c}{a}=5\Rightarrow3a+b+c=5a\Rightarrow b+c=2a\)
Tương tự ta có : \(a+c=2b;a+b=2c\)
\(\Rightarrow B=\left(1+\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right)\)
\(=\frac{a+b}{b}.\frac{b+c}{c}.\frac{a+c}{a}=\frac{2c}{b}.\frac{2a}{c}.\frac{2b}{a}\)
\(=\frac{8abc}{abc}=8\)
Ta có:
\(\frac{3a+b+c}{a}=\frac{a+3b+c}{b}=\frac{3c+b+a}{c}\)
\(\Rightarrow3+\frac{b+c}{a}=3+\frac{a+c}{b}=3+\frac{b+a}{c}\)
\(\Rightarrow\frac{b+c}{a}=\frac{a+c}{b}=\frac{b+a}{c}=\frac{2\left(a+b+c\right)}{a+b+c}\)
TH1:\(a+b+c=0\)\(\Rightarrow\left\{{}\begin{matrix}a+b=-c\\b+c=-a\\a+c=-b\end{matrix}\right.\)
Thay vào B, ta có:
\(B=\left(1+\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right)=-1\)
TH2:\(a+b+c\ne0\)
\(\Rightarrow\frac{b+c}{a}=\frac{a+c}{b}=\frac{b+a}{c}=\frac{2\left(a+b+c\right)}{a+b+c}=2\)
\(\Rightarrow\left\{{}\begin{matrix}b+c=2a\\a+c=2b\\b+a=2c\end{matrix}\right.\)
Thay vào B, ta có:
\(B=\left(1+\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right)=8\)
Vậy \(\left[{}\begin{matrix}B=-1\\B=8\end{matrix}\right.\)
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\(\frac{2a+a+b+c}{a}=\frac{2b+a+b+c}{b}=\frac{2c+a+b+c}{c}\)
\(\Rightarrow2+\frac{a+b+c}{a}=2+\frac{a+b+c}{b}=2+\frac{a+b+c}{c}\)
\(\Rightarrow\frac{a+b+c}{a}=\frac{a+b+c}{b}=\frac{a+b+c}{c}\Rightarrow a=b=c\)
\(\Rightarrow B=\left(1+1\right)\left(1+1\right)\left(1+1\right)\)
Bài 2:
a) \(\left|x+1\right|+\left|x+2\right|+\left|x+4\right|+\left|x+5\right|-6x=0\)
\(\Rightarrow\left|x+1\right|+\left|x+2\right|+\left|x+4\right|+\left|x+5\right|=6x\)
Ta có: \(\left|x+1\right|\ge0;\left|x+2\right|\ge0;\left|x+4\right|\ge0;\left|x+5\right|\ge0\)
\(\Rightarrow\left|x+1\right|+\left|x+2\right|+\left|x+4\right|+\left|x+5\right|\ge0\)
\(\Rightarrow6x\ge0\)
\(\Rightarrow x\ge0\)
\(\Rightarrow\left|x+1\right|+\left|x+2\right|+\left|x+4\right|+\left|x+5\right|=x+1+x+2+x+4+x+5=6x\)
\(\Rightarrow4x+12=6x\)
\(\Rightarrow2x=12\)
\(\Rightarrow x=6\)
Vậy x = 6
b) Giải:
Áp dụng tính chất dãy tỉ số bằng nhau ta có:
\(\frac{x-2}{2}=\frac{y-3}{3}=\frac{z-3}{4}=\frac{2y-6}{6}=\frac{3z-9}{12}=\frac{x-2-2y+6+3z-9}{2-6+12}=\frac{\left(x-2y+3z\right)-\left(2-6+9\right)}{8}\)
\(=\frac{14-5}{8}=\frac{9}{8}\)
+) \(\frac{x-2}{2}=\frac{9}{8}\Rightarrow x-2=\frac{9}{4}\Rightarrow x=\frac{17}{4}\)
+) \(\frac{y-3}{3}=\frac{9}{8}\Rightarrow y-3=\frac{27}{8}\Rightarrow y=\frac{51}{8}\)
+) \(\frac{z-3}{4}=\frac{9}{8}\Rightarrow z-3=\frac{9}{2}\Rightarrow z=\frac{15}{2}\)
Vậy ...
c) \(5^x+5^{x+1}+5^{x+2}=3875\)
\(\Rightarrow5^x+5^x.5+5^x.5^2=3875\)
\(\Rightarrow5^x.\left(1+5+5^2\right)=3875\)
\(\Rightarrow5^x.31=3875\)
\(\Rightarrow5^x=125\)
\(\Rightarrow5^x=5^3\)
\(\Rightarrow x=3\)
Vậy x = 3
Bài 1:
\(\frac{3a+b+c}{a}=\frac{a+3b+c}{b}=\frac{a+b+3c}{c}.\)
\(\Rightarrow\frac{3a}{a}+\frac{b+c}{a}=\frac{3b}{b}+\frac{a+c}{b}=\frac{3c}{c}+\frac{a+b}{c}\)
\(\Rightarrow3+\frac{b+c}{a}=3+\frac{a+c}{b}=3+\frac{a+b}{c}\)
\(\Rightarrow\frac{b+c}{a}=\frac{a+c}{b}=\frac{a+b}{c}.\)
+ TH1: \(a+b+c=0.\)
\(\Rightarrow\left\{{}\begin{matrix}a+b=-c\\a+c=-b\\b+c=-a\end{matrix}\right.\)
Ta có: \(P=\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}\)
\(\Rightarrow P=\frac{-c}{c}+\frac{-a}{a}+\frac{-b}{b}\)
\(\Rightarrow P=\left(-1\right)+\left(-1\right)+\left(-1\right)\)
\(\Rightarrow P=-3.\)
+ TH2: \(a+b+c\ne0.\)
Áp dụng tính chất dãy tỉ số bằng nhau ta được:
\(\frac{b+c}{a}=\frac{a+c}{b}=\frac{a+b}{c}=\frac{b+c+a+c+a+b}{a+b+c}=\frac{2a+2b+2c}{a+b+c}=\frac{2.\left(a+b+c\right)}{a+b+c}=2.\)
\(\Rightarrow\left\{{}\begin{matrix}\frac{b+c}{a}=2\\\frac{a+c}{b}=2\\\frac{a+b}{c}=2\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}b+c=2a\\a+c=2b\\a+b=2c\end{matrix}\right.\)
Lại có: \(P=\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}\)
\(\Rightarrow P=\frac{2c}{c}+\frac{2a}{a}+\frac{2b}{b}\)
\(\Rightarrow P=2+2+2\)
\(\Rightarrow P=6.\)
Vậy \(P=-3\) hoặc \(P=6.\)
Chúc bạn học tốt!
Bài 1:
\(\left(x-1\right).\left(xy-5\right)=5\)
Vì \(x,y\in Z\Rightarrow\left\{{}\begin{matrix}x-1\in Z\\xy-5\in Z\end{matrix}\right.\)
\(\Rightarrow x-1\inƯC\left(5\right);xy-5\inƯC\left(5\right)\)
\(\Rightarrow x-1\in\left\{\pm1;\pm5\right\};xy-5\in\left\{\pm1;\pm5\right\}.\)
\(\Rightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x-1=1\\xy-5=5\end{matrix}\right.\\\left\{{}\begin{matrix}x-1=5\\xy-5=1\end{matrix}\right.\\\left\{{}\begin{matrix}x-1=-1\\xy-5=-5\end{matrix}\right.\\\left\{{}\begin{matrix}x-1=-5\\xy-5=-1\end{matrix}\right.\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x=2\\2y=10\end{matrix}\right.\\\left\{{}\begin{matrix}x=6\\6y=6\end{matrix}\right.\\\left\{{}\begin{matrix}x=0\\0y=0\end{matrix}\right.\\\left\{{}\begin{matrix}x=-4\\-4y=4\end{matrix}\right.\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x=2\\y=5\end{matrix}\right.\left(TM\right)\\\left\{{}\begin{matrix}x=6\\y=1\end{matrix}\right.\left(TM\right)\\\left\{{}\begin{matrix}x=0\\y=0\end{matrix}\right.\left(TM\right)\\\left\{{}\begin{matrix}x=-4\\y=-1\end{matrix}\right.\left(TM\right)\end{matrix}\right.\)
Vậy cặp số nguyên \(\left(x;y\right)\) thỏa mãn đề bài là: \(\left(2;5\right),\left(6;1\right),\left(0;0\right),\left(-4;-1\right).\)
Chúc bạn học tốt!
Bài 1:
Ta có bảng sau:
| \(x-1\) | -5 | -1 | 1 | 5 |
| \(xy-5\) | -1 | -5 | 5 | 1 |
| x | -4 | 0 | 2 | 6 |
| y | -1 | mọi y∈Z | 5 | 1 |
\(\frac{2b+c-a}{a}=\frac{2c-b+a}{b}=\frac{2a+b-c}{c}=\frac{2b+c-a+2c-b+a+2a+b-c}{a+b+c}=\)
\(=\frac{2a+2b+2c}{a+b+c}=2\)
+ Từ \(\frac{2b+c-a}{a}=2\Rightarrow2b+c-a=2a\Rightarrow3a-2b=c\) và \(3a-c=2b\)
+ Tương tự ta cũng có \(3b-2c=a\) và \(3b-a=2c\)
Và \(3c-2a=b\); \(3c-b=2a\)
Thay vào P
\(P=\frac{c.a.b}{2.b.2.c.2.a}=\frac{1}{8}\)
Làm hộ tui đi à,đây là Sol của thầy Sỹ,đọc là 1 chuyện nhưng hiểu mới là vấn đề.
Sol 2:Phạm Kim Hùng