\(\text{Vì }a,b,c\inℕ^∗\Rightarrow\hept{\begin{cases}\frac{a}{a+b}>\frac{a}{a+b+c}\\\frac{b}{b+c}>\frac{b}{a+b+c}\\\frac{c}{c+a}>\frac{c}{a+b+c}\end{cases}\Rightarrow M>\frac{a+b+c}{a+b+c}=1}\)(1)

\(\hept{\begin{cases}\frac{a}{a+b}< \frac{a+c}{a+b+c}\\\frac{b}{b+c}< \frac{b+a}{a+b+c}\\\frac{c}{a+c}< \frac{c+b}{c+a+b}\end{cases}}\Rightarrow M< \frac{2.\left(a+b+c\right)}{a+b+c}=2\)(2) (chỉ áp dụng cho p/s có tử  bé hơn mẫu)

từ (1) và (2) => 1<M<2 => M không phải là STN

\(M=\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\)

Ta có

\(\frac{a}{a+b+c}< \frac{a}{a+b}< \frac{a+c}{a+b+c}\)

\(\frac{b}{b+c+a}< \frac{b}{b+c}< \frac{b+a}{b+c+a}\)

\(\frac{ c}{c+a+b}< \frac{c}{c+a}< \frac{c+b}{c+a+b}\)

\(\Rightarrow1< M< 2\Rightarrow\)M không phải là số tự nhiên

\(\left|x+1\right|+\left|x+2\right|+\left|x+3\right|+...+\left|x+99\right|=100x\)

\(\left|x+1\right|\ge0;\left|x+2\right|\ge0;...;\left|x+99\right|\ge0\)

\(\Rightarrow100x\ge0\)

\(\Rightarrow x\ge0\)

\(\Rightarrow x+1+x+2+x+3+...+x+99=100x\)

\(\Rightarrow99x+1+2+3+...+99=100x\)

\(\Rightarrow99x+4950=100x\)

\(\Rightarrow-x=-4950\)

\(\Rightarrow x=4950\)

\(\left|x+\frac{1}{1\cdot2}\right|+\left|x+\frac{1}{2\cdot3}\right|+\left|x+\frac{1}{3\cdot4}\right|+...+\left|x+\frac{1}{49\cdot50}\right|=50x\)

\(\left|x+\frac{1}{1\cdot2}\right|\ge0;\left|x+\frac{1}{2\cdot3}\right|\ge0;...;\left|x+\frac{1}{49\cdot50}\right|\ge0\)

\(\Rightarrow50x\ge0\)

\(\Rightarrow x\ge0\)

\(\Rightarrow x+\frac{1}{1\cdot2}+x+\frac{1}{2\cdot3}+...+x+\frac{1}{49\cdot50}\)

\(\Rightarrow49x+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{49}-\frac{1}{50}=50x\)

\(\Rightarrow49x+\frac{49}{50}=50x\)

tu lam 

\(a;\left|x+1\right|+\left|x+2\right|+\left|x+3\right|+..............+\left|x+99\right|=100x^{\left(1\right)}\)

Ta có \(\left|x+1\right|\ge0;\left|x+2\right|\ge0;\left|x+3\right|\ge0;.............;\left|x+99\right|\ge0\)

\(\Rightarrow VT\ge0\Rightarrow VP\ge0\Rightarrow100x\ge0\Rightarrow x\ge0\)

Với \(x\ge0\).Từ (1) \(\Rightarrow x+1+x+2+x+3+..................+x+99=100x\)

\(\Rightarrow\left(x+x+x+........+x\right)+\left(1+2+3+..........+99\right)=100x\)

\(\Rightarrow99x+4950=100x\)

\(\Rightarrow x=4950\)(t/m đk x > =  0)

\(\left|x+\frac{1}{1.2}\right|+\left|x+\frac{1}{2.3}\right|+.........+\left|x+\frac{1}{49.50}\right|=50x^{(∗)}\)

\(\left|x+\frac{1}{1.2}\right|\ge0;\left|x+\frac{1}{2.3}\right|\ge0;............;\left|x+\frac{1}{49.50}\right|\ge0\)

\(\Rightarrow VT\ge0\Rightarrow VP\ge0\Rightarrow50x\ge0\Rightarrow x\ge0\)

Với x > = 0 .Từ (*) \(\Rightarrow x+\frac{1}{1.2}+x+\frac{1}{2.3}+............+x+\frac{1}{49.50}=50x\)

\(\Rightarrow\left(x+x+x+.......+x\right)+\left(\frac{1}{1.2}+\frac{1}{2.3}+...........+\frac{1}{49.50}\right)=50x\)

\(\Rightarrow49x+\left(1-\frac{1}{50}\right)=50x\)

\(\Rightarrow49x+\frac{49}{50}=50x\)

\(\Rightarrow x=\frac{49}{50}\)(t/m đk \(x\ge0\))

Trước hết ta chứng minh bổ đề: \(|a|+|b|\ge|a+b|.\left(1\right)\)

CM: \(\left(1\right)\Leftrightarrow\left(|a|+|b|\right)^2\ge\left(|a+b\right)^2\)

                  \(\Leftrightarrow a^2+b^2+2|ab|\ge a^2+b^2+2ab\)

                  \(\Leftrightarrow2|ab|\ge2ab\)

                  \(\Leftrightarrow\left|ab\right|\ge ab\)(điều này đúng do tính chất của giá trị tuyệt đối).

Vậy ta có đpcm. Dấu bằng xảy ra \(\Leftrightarrow ab\ge0.\)

a) A = \(\left|x-1\right|+\left|x-2\right|+\left|x-3\right|=\left|x-1\right|+\left|3-x\right|+\left|x-2\right|.\)

Ta thấy rằng \(\left|x-2\right|\ge0\)với mọi x.

Áp dụng bổ đề trên ta có:

\(A\ge\left|x-1+3-x\right|+0=\left|2\right|+0=2+0=2.\)

Dấu bằng xảy ra \(\Leftrightarrow\hept{\begin{cases}\left(x-1\right)\left(3-x\right)\ge0\\x-2=0\end{cases}}\Leftrightarrow\hept{\begin{cases}1\le x\le3\\x=2\end{cases}}\Leftrightarrow x=2.\)

Vậy GTNN của A bằng 2 khi x = 2.

b) Áp dụng bổ đề trên ta có:\(B=\left|x-4\right|+\left|7-x\right|+\left|x-5\right|+\left|6-x\right|\ge\left|x-4+7-x\right|+\left|x-5+6-x\right|=\left|3\right|+\left|1\right|=3+1=4.\)

Dấu bằng xảy ra \(\Leftrightarrow\hept{\begin{cases}\left(x-4\right)\left(7-x\right)\ge0\\\left(x-5\right)\left(6-x\right)\ge0\end{cases}\Leftrightarrow\hept{\begin{cases}4\le x\le7\\5\le x\le6\end{cases}\Leftrightarrow}5\le x\le6}\)(vì với mọi x nằm giữa 5 và 6 thì cũng nằm giữa 4 và 7).

Vậy GTNN của B bằng 4 khi \(5\le x\le6.\)

a;\(A=\left|x-1\right|+\left|x-2\right|+\left|x-3\right|\)

\(\Rightarrow A=\left|x-1\right|+\left|x-2\right|+\left|3-x\right|\)

Ta có +) \(\left|x+1\right|+\left|3-x\right|\ge\left|x+1+3-x\right|=4\)

Dấu "=" xảy ra \(\Leftrightarrow\left(x-1\right)\left(3-x\right)\ge0\Leftrightarrow1\le x\le3\)

+)\(\left|x-2\right|\ge0\)Dấu "=" xảy ra \(\Leftrightarrow x-2=0\Leftrightarrow x=2\)

\(\Rightarrow A=\left|x-1\right|+\left|x-2\right|+\left|x-3\right|\ge2\)

\(\Rightarrow A_{min}=2\Leftrightarrow\hept{\begin{cases}1\le x\le3\\x=2\end{cases}\Leftrightarrow x=2}\)

b;\(B=\left|x-4\right|+\left|x-5\right|+\left|x-6\right|+\left|x-7\right|\)

\(\Rightarrow B=\left|x-4\right|+\left|x-5\right|+\left|6-x\right|+\left|7-x\right|\)

Ta có +) \(\left|x-4\right|+\left|7-x\right|\ge\left|x-4+7-x\right|=3\)

Dấu "=" xảy ra \(\Leftrightarrow\left(x-4\right)\left(7-x\right)\ge0\Leftrightarrow4\le x\le7\)

+) \(\left|x-5\right|+\left|6-x\right|\ge\left|x-5+6-x\right|=1\)

Dấu "=" xảy ra \(\Leftrightarrow\left(x-5\right)\left(6-x\right)\ge0\Leftrightarrow5\le x\le6\)

\(\Rightarrow B=\left|x-4\right|+\left|x-5\right|+\left|x-6\right|+\left|x-7\right|\ge4\)

\(\Rightarrow B_{min}=4\Leftrightarrow\hept{\begin{cases}4\le x\le7\\5\le x\le6\end{cases}\Leftrightarrow5\le x\le6}\)

\(C=\frac{x+5}{2x+\left(x-2y\right)}+\frac{y-5}{2y-\left(x-2y\right)}\)

\(=\frac{x+5}{2x+10}+\frac{y-5}{2y-10}=\frac{x+5}{2\left(x+5\right)}+\frac{y-5}{2\left(y-5\right)}=\frac{1}{2}+\frac{1}{2}=1\left(x\ne-5,y\ne5\right)\)

Trả lời :

Ta có :

C = \(\frac{x+5}{3x-2y}+\frac{y-5}{4y-x}\)

C = \(\frac{2\left(x+5\right)}{2\left(3x-2y\right)}+\frac{2\left(y-5\right)}{2\left(4y-x\right)}\)

C = \(\frac{2x+10}{6x-4y}+\frac{2y-10}{8y-2x}\)

Thay x - 2y = 10 . Ta được :

C = \(\frac{2x+x-2y}{6x-4y}+\frac{2y-x-2y}{8y-2x}\)

C = \(\frac{x\left(2+1\right)-2y}{6x-4y}+\frac{y\left(2+2\right)-x}{8y-2x}\)

C = \(\frac{3x-2y}{6x-4y}+\frac{4y-x}{8y-2x}\)

C = \(\frac{1}{2}+\frac{1}{2}\)

C = \(1\)

Vậy C = 1

Hok tốt

a, xy + 2x - y = 9

=> x(y + 2) - y - 2 = 7

=> (x - 1)(y + 2) = 7

lap bang 

b, xy - 5x - y = 8

=> x(y - 5) - y + 5 = 13

=> (x - 1)(y - 5) = 13

c, xy - 5x + y = 8

=> x(y - 5) + y - 5 = 3

=> (x + 1)(y - 5) = 3

Ta có: 104²⁰¹⁵ + 2 < 104²⁰¹⁶ + 2.

=> A = \(\frac{104^{2015}+2}{104^{2016}+2}\)< 1 (1).

Ta có: 104²⁰¹⁶ + 2 > 104²⁰ + 2 > 104²⁰.

=> B = \(\frac{104^{2016}+2}{104^{20}}\) > 1 (2).

Từ (1) và (2) => A < 1 < B => A < B.

Chúc bạn học tốt nhé!

bạn ơi giúp mình sửa mẫu của B thành 104^2017+2 nhé

Thanks bạn nhìu

tu ve hinh :

a, xet tamgiac OCB va tamgiac OCA co : OC chung

goc OBC = goc OAC = 90 do BC | Oy va AC | Ox (GT)

OB = OA (gt)

=> tamgiac OCB = tamgiac OCA (ch - cgv)

=> goc BOC = goc AOC  (dn) ma OC nam giac Ox va Oy 

=> OC la phan giac cua goc xOy (dn)

b, xet tamgiac OBD va tamgiac OAE co : OB = OA (gt)

goc BOD = goc AOE (doi dinh)

goc OBD = goc OAE = 90 do BC | Oy va AC | Ox (GT)

=>  tamgiac OBD = tamgiac OAE  (cgv - gnk)

=> OD = OE (dn)

=> tamgiac ODE can tai O (dn)

c, tu nghi di cau c-g-c

tu  ve hinh :

cau b la vuong goc phai k

a, tamgiac ABC can tai A(gt) => AB = AC va goc ABC = goc ACB (dn)

goc ADB = goc ADC do AD | BC (GT)

=> tamgiac ADB = tamgiac ADC (ch - gn)

=> BD = DC (dn)

b, xet tamgiac BHD va tamgiac CKD co :  BD = DC (Cau a)

goc ABC = goc ACB (cau a)

goc BHD = goc DKC = 90 do HD | AB va HK | AC (gt)

=> tamgiac BHD = tamgiac CKD (ch - gn)

=> HD = DK (dn)

c, xet tamgiac AHD va tamgiac AKD co : AD chung

HD = DK (cau b) 

goc AHD = goc AKD = 90 do HD | AB va HK | AC (gt) 

=> tamgiac AHD = tamgiac AKD  (ch - cgv)

=> tamgiac AHK can tai A (dn)

=> goc AHK = (180 - goc BAC) : 2

tamgiac ABC can tai A (gt) => goc ABC = (180 - goc BAC) : 2

=> goc AHK = goc ABC  2 goc nay dong vi

=> HK // BC (tc)

d, tu ap dung py-ta-go 

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