\(ĐKXĐ:-1\le x\le1\)

\(pt\Leftrightarrow2\left[\sqrt{1-x}-\left(1-x\right)\right]+\left[\sqrt{1-x^2}-\left(1+x\right)\right]=0\)

\(\Leftrightarrow2\sqrt{1-x}\left(1-\sqrt{1-x}\right)+\sqrt{1+x}\)

\(\left(\sqrt{1-x}-\sqrt{1+x}\right)=0\)

\(\Leftrightarrow2\sqrt{1-x}.\frac{x}{1+\sqrt{1-x}}+\sqrt{1+x}.\frac{-2x}{\sqrt{1-x}+\sqrt{1+x}}=0\)

\(\Leftrightarrow x\left(\frac{\sqrt{1-x}}{1+\sqrt{1-x}}-\frac{\sqrt{1+x}}{\sqrt{1-x}+\sqrt{1+x}}\right)=0\)

\(\Leftrightarrow\orbr{\begin{cases}x=0\left(tm\right)\\\frac{\sqrt{1-x}}{1+\sqrt{1-x}}=\frac{\sqrt{1+x}}{\sqrt{1-x}+\sqrt{1+x}}\left(^∗\right)\end{cases}}\)

\(\left(^∗\right)\Leftrightarrow1-x=\sqrt{1+x}\)

\(\Leftrightarrow1+x^2-2x=1+x\Leftrightarrow x^2-3x=0\)

\(\Leftrightarrow x\left(x-3\right)=0\Leftrightarrow\orbr{\begin{cases}x=0\left(tmđk\right)\\x=3\left(ktmđk\right)\end{cases}}\)

Vậy phương trình có 1 nghiệm duy nhất là 0

ĐKXĐ: \(-1\le x\le1\)

Đặt \(\sqrt{1+x}=a\ge0;\sqrt{1-x}=b\ge0\)

Ta thu được hệ: \(\hept{\begin{cases}2b+ab=b^2+2\\a^2+b^2=2\end{cases}}\)

Cộng hai phương trình của hệ lại với nhau, vế với vế: \(a^2+b^2+2b+ab=b^2+4\)

\(\Leftrightarrow\left(a+2\right)\left(a+b-2\right)=0\)\(\Leftrightarrow\orbr{\begin{cases}a=-2\left(L\right)\\a=2-b\left(1\right)\end{cases}}\)

(1) \(\Leftrightarrow a+b=2\Leftrightarrow\sqrt{1+x}+\sqrt{1-x}=2\)

\(\Leftrightarrow2+2\sqrt{\left(1+x\right)\left(1-x\right)}=4\Leftrightarrow\left(1+x\right)\left(1-x\right)=1\) (chuyển vế, chia hai vế cho 2)

\(\Leftrightarrow-x^2=0\Rightarrow x=0\)

https://www.youtube.com/channel/UCU_DXbWfhapaSkAR7XsK5yQ?view_as=subscriber

Gọi OD cắt (O) tại E,F \(\left(E\in DF\right)\)ta có:

     \(\widehat{DAE}=\widehat{DFM}\)(cùng bù với \(\widehat{MAE}\))

     \(\widehat{ADE}=\widehat{FDM}\)(chung)

Do đó \(\Delta DAE\text{~}\Delta DFM\text{ }\left(g.g\right)\)

\(\Rightarrow\frac{DA}{DF}=\frac{DE}{DM}\)

\(\Rightarrow DA.DM=DE.DF\)

\(=\left(DO-OE\right)\left(DO+OF\right)=\left(DO-OM\right)\left(DO+OM\right)=DO^2-OM^2\)(đpcm)

Ta có: \(3\left(x^2+y^2+z^2\right)-\left(x+y+z\right)^2=\left(x-y\right)^2+\left(y-z\right)^2+\left(x-z\right)^2\ge0\)

\(\Rightarrow\left(x+y+z\right)^2\le3\left(x^2+y^2+z^2\right)\) nên với \(x,y,z>0\) ta có:

\(x+y+z\le\sqrt{3\left(x^2+y^2+z^2\right)}\) áp dụng ta có:

\(\frac{1}{\sqrt{ab+a+2}}+\frac{1}{\sqrt{bc+b+2}}+\frac{1}{\sqrt{ca+c+2}}\le\sqrt{3\left(\frac{1}{ab+a+2}+\frac{1}{bc+b+2}+\frac{1}{ca+c+2}\right)}\)

Với: \(x,y>0\) ta có: \(x+y\ge2\sqrt{xy}\Rightarrow\left(x+y\right)^2\ge4xy\Rightarrow\frac{1}{x+y}\le\frac{1}{4}\left(\frac{1}{x}+\frac{1}{y}\right)\)

Áp dụng ta được:

 \(\frac{1}{ab+a+2}=\frac{1}{ab+1+a+1}=\frac{1}{ab+abc+a+1}=\frac{1}{ab\left(c+1\right)+\left(a+1\right)}\)

\(\le\frac{1}{4}\left(\frac{1}{ab\left(c+1\right)}+\frac{1}{a+1}\right)=\frac{1}{4}\left(\frac{abc}{ab\left(c+1\right)}+\frac{1}{a+1}\right)=\frac{1}{4}\left(\frac{c}{c+1}+\frac{1}{a+1}\right)\)

Vậy ta có: \(\frac{1}{ab+a+2}\le\frac{1}{4}\left(\frac{c}{c+1}+\frac{1}{a+1}\right)\)

Tương tự như trên ta có: \(\frac{1}{bc+b+2}\le\frac{1}{4}\left(\frac{a}{a+1}+\frac{1}{b+1}\right)\) và \(\frac{1}{ca+c+2}\le\frac{1}{4}\left(\frac{b}{b+1}+\frac{1}{c+1}\right)\) nên:

\(\Rightarrow\sqrt{3\left(\frac{1}{ab+a+2}+\frac{1}{bc+b+2}+\frac{1}{ca+c+2}\right)}\)

\(\le\sqrt{3.\frac{1}{4}\left(\frac{c}{c+1}+\frac{1}{a+1}+\frac{a}{a+1}+\frac{1}{b+1}+\frac{b}{b+1}+\frac{1}{c+1}\right)}=\frac{3}{2}\)

Vậy \(\frac{1}{\sqrt{ab+a+2}}+\frac{1}{\sqrt{bc+b+2}}+\frac{1}{\sqrt{ca+c+2}}\le\frac{3}{2}\left(đpcm\right)\)

Dấu " = " xảy ra \(\Leftrightarrow a=b=c=1\)

Đặt \(\left(a;b;c\right)=\left(\frac{x}{y};\frac{y}{z};\frac{z}{x}\right)\). BĐT quy về:\(\Sigma_{cyc}\frac{\sqrt{yz}}{\sqrt{xy+xz+2yz}}\le\frac{3}{2}\)

Áp dụng liên hoàn BĐT Cô si:

\(VT=\Sigma_{cyc}\sqrt{\frac{yz}{\left(xy+yz\right)+\left(xz+yz\right)}}\le\Sigma_{cyc}\sqrt{\frac{yz}{4}\left(\frac{1}{xy+yz}+\frac{1}{xz+yz}\right)}\)

\(=\frac{1}{2}\Sigma_{cyc}\sqrt{1\left(\frac{yz}{xy+yz}+\frac{yz}{xz+yz}\right)}\le\frac{1}{4}\Sigma_{cyc}\left(1+\frac{yz}{xy+yz}+\frac{yz}{xz+yz}\right)=\frac{3}{2}\)

Áp dụng BĐT bunhiacopxki cho 2 bộ số \(\left(\sqrt{a}.\sqrt{b+c};\sqrt{b}.\sqrt{d+c};\sqrt{c}.\sqrt{d+a};\sqrt{d}.\sqrt{a+b}\right)\)

và \(\left(\frac{\sqrt{a}}{\sqrt{b+c}};\frac{\sqrt{b}}{\sqrt{d+c}};\frac{\sqrt{c}}{\sqrt{d+a}};\frac{\sqrt{d}}{\sqrt{a+b}}\right)\), ta được:

\(\left[a\left(b+c\right)+b\left(d+c\right)+c\left(d+a\right)+d\left(a+b\right)\right]\)\(\left(\frac{a}{b+c}+\frac{b}{d+c}+\frac{c}{a+d}+\frac{d}{a+b}\right)\)\(\ge\left(a+b+c+d\right)^2\)

\(\Leftrightarrow\frac{a}{b+c}+\frac{b}{d+c}+\frac{c}{a+d}+\frac{d}{a+b}\)\(\ge\frac{\left(a+b+c+d\right)^2}{ab+ac+bd+bc+cd+ac+ad+bd}\)(1)

Ta có \(\left(a+b+c+d\right)^2\ge2\left(ab+ac+bc+bd+cd+ac+ad+bd\right)\)

\(\Leftrightarrow\left(a-c\right)^2+\left(b-d\right)^2\ge0\)(luôn đúng)

Do đó: \(\left(a+b+c+d\right)^2\ge2\left(ab+ac+bc+bd+cd+ac+ad+bd\right)\)(2)

Từ (1) và (2) suy ra ĐPCM

Dấu "=" xảy ra khi và chỉ khi a=b=c=d

Áp dụng BĐT : \(\frac{1}{xy}\ge\frac{4}{\left(x+y\right)^2}\)với x,y > 0

Ta có : \(\frac{a}{b+c}+\frac{c}{d+a}=\frac{a^2+ad+bc+c^2}{\left(b+c\right)\left(a+d\right)}\ge\frac{4\left(a^2+ad+bc+c^2\right)}{\left(a+b+c+d\right)^2}\)

Tương tự : \(\frac{b}{c+d}+\frac{d}{a+b}\ge\frac{4\left(b^2+ab+cd+d^2\right)}{\left(a+b+c+d\right)^2}\)

\(\Rightarrow\frac{a}{b+c}+\frac{b}{c+d}+\frac{c}{d+a}+\frac{d}{a+b}\ge\frac{4\left(a^2+b^2+c^2+d^2+ad+bc+ab+cd\right)}{\left(a+b+c+d\right)^2}\)

Cần chứng minh : \(\frac{a^2+b^2+c^2+d^2+ad+bc+ab+cd}{\left(a+b+c+d\right)^2}\ge\frac{1}{2}\)

\(\Leftrightarrow2\left(a^2+b^2+c^2+d^2+ad+bc+ab+cd\right)\ge\left(a+b+c+d\right)^2\)

\(\Leftrightarrow\left(a-c\right)^2+\left(b-d\right)^2\ge0\)

Dấu "=" xảy ra khi a = c ; b = d

Vậy ....

1/ BĐT \(\Leftrightarrow\left(a+b+c\right)\left(a^2+b^2+c^2\right)+4abc\ge104=\frac{13}{27}\left(a+b+c\right)^3\)

Hay: \(27\left(a+b+c\right)\left(a^2+b^2+c^2\right)+108abc\ge13\left(a+b+c\right)^3\)

\(VT-VP=2\left[6\left\{\Sigma_{cyc}a^3+3abc-\Sigma_{cyc}ab\left(a+b\right)\right\}+\left(a^3+b^3+c^3-3abc\right)\right]\ge0\)

(đúng theo BĐT Schur bậc 3 và Cô si cho 3 số dương)

Đẳng thức xảy ra khi a = b = c = 2

tth_new trả lời nốt luôn đi 

đkxđ : \(x,y,z\ge\frac{1}{4}\)

Ta có : 

\(x-z=\sqrt{4z-1}-\sqrt{4x-1}=\frac{4\left(z-x\right)}{\sqrt{4z-1}+\sqrt{4x-1}}=-\frac{4\left(x-z\right)}{\sqrt{4z-1}+\sqrt{4x-1}}\)

\(\Rightarrow\left(x-z\right)\left(1+\frac{4}{\sqrt{4z-1}+\sqrt{4x-1}}\right)=0\)

Dễ thấy \(1+\frac{4}{\sqrt{4z-1}+\sqrt{4x-1}}>0\)nên x - z = 0 hay x = z

Tương tự : x = y

Suy ra : x = y = z

Thay vào đầu bài, ta có : \(2x=\sqrt{4x-1}\Rightarrow4x^2=4x-1\Rightarrow x=\frac{1}{2}\)

Vậy x = y = z = \(\frac{1}{2}\)