a. Ta có: 

\(\Delta MNP\)vuông tại \(N\left(gt\right)\)

\(\Rightarrow\widehat{N}=90^0\)

\(HC\perp MN\left(gt\right)\)

\(\Rightarrow\widehat{C}=90^0\)

\(HD\perp NP\left(gt\right)\)

\(\Rightarrow\widehat{D}=90^0\)

Xét tứ giác HDNC, ta có:

\(\widehat{N}=\widehat{C}=\widehat{D}\left(=90^0\right)\left(cmt\right)\)

\(\Rightarrow HDNC\)là hình chữ nhật (dhnb)

b, xét ΔMHN và ΔMNP có : ^P chung

^PNM = ^NHM = 90

=> ΔMHN ~ ΔMNP (g-g)

=> NH/MN = NP/MP 

=> NH.MP = MN.NP

a) \(\frac{3x+5}{2\left(x-1\right)}+\frac{4}{x-2}=\frac{\left(3x+5\right)\left(x-2\right)+4\cdot2\left(x-1\right)}{2\left(x-1\right)\left(x-2\right)}=\frac{3x^2-6x+5x-10+8x-8}{2\left(x-1\right)\left(x-2\right)}\)

\(=\frac{3x^2+7x-18}{2\left(x-1\right)\left(x-2\right)}\)

b) \(\frac{2x^2+1}{4x^2-2x}+\frac{3-3x}{1-2x}+\frac{3}{2x}=\frac{2x^2+1+4x\left(3-3x\right)+2\cdot3\left(1-2x\right)}{4x\left(1-2x\right)}=\frac{2x^2+1+12-12x+6-12x}{4x\left(1-2x\right)}\)\(=\frac{2x^2-24x+19}{4x\left(1-2x\right)}\)

Đề này... bạn xem lại đi. Chứ thế này thì dùng máy tính cũng không làm nổi T-T

Ta có : \(x^4-7x^2+y^2+16=2xy\)

=> \(\left(x^2-8x^2+16\right)+\left(x^2-2xy+y^2\right)=0\)

=> \(\left(x-4\right)^2+\left(x-y\right)^2=0\)

Vì \(\left(x-4\right)^2\ge0 \forall x ,\left(x-y\right)^2 \ge0 \forall x,y \)

=> \(\left(x-4\right)^2+\left(x-y\right)^2\ge0 \forall x,y\)

=> \(\hept{\begin{cases}x-4=0\\x-y=0\end{cases}\Rightarrow\hept{\begin{cases}x=4\\x=y=4\end{cases}}}\)

Thay vào \(A=4^{2016}.4^{2017}-4^{2017}.4^{2016}+4+4=8\)

Vậy A=8

https://olm.vn/thanhvien/nguyentrangth8 bạn giỏi thế

a) \(A=\frac{3x^2+5}{x^2+1}=\frac{3\left(x^2+1\right)+2}{x^2+1}=3+\frac{2}{x^2+1}\)

để A nguyên =>\(x^2+1\inƯ\left(2\right)\)

\(\Leftrightarrow x^2\in\left\{0;1\right\}\)

\(\Leftrightarrow x\in\left\{0;\pm1\right\}\)

Để A nguyên thì \(3x^2+5⋮x^2+1\)

\(\Rightarrow3\left(x^2+1\right)+2⋮x^2+1\)

\(\Rightarrow2⋮x^2+1\)

\(\Rightarrow x^2+1\in\left\{1;2\right\}\Rightarrow x=0;x=1;x=-1\)

a) \(Q=\frac{x^4-x^2+2x+2}{x^4+x^3+x+1}\)

\(Q=\frac{x^2\left(x^2-1\right)+2\left(x+1\right)}{x^3\left(x+1\right)+\left(x+1\right)}\)

\(Q=\frac{x^2\left(x+1\right)\left(x-1\right)+2\left(x+1\right)}{\left(x+1\right)\left(x^3+1\right)}\)

\(Q=\frac{\left(x+1\right)\left[x^2\left(x-1\right)+2\right]}{\left(x+1\right)\left(x^3+1\right)}\)

\(Q=\frac{x^3-x^2+2}{x^3+1}\)

b) \(Q=\left|Q\right|=\frac{x^3-x^2+2}{x^3+1}\)

Ta có: \(x^3-x=\left(x-1\right)x\left(x+1\right)\)

Tích 3 số nguyên liên tiếp chia hết cho 3 nên \(\left(x-1\right)x\left(x+1\right)⋮3\)

hay \(x^3-x⋮3\)

Tương tự \(y^3-y⋮3\);\(z^3-z⋮3\)

\(\Rightarrow x^3+y^3+z^3-\left(x+y+z\right)⋮3\)

Mà \(\left(x+y+z\right)⋮3\left(gt\right)\Rightarrow a^3+b^3+c^3⋮3\left(đpcm\right)\)

\(x^2+5y^2+2y-4xy-3=0.\)

\(\Rightarrow x^2-4xy+4y^2+y^2+2y-3=0\)

\(\Rightarrow\left(x-2y\right)^2+\left(y+1\right)^2-4=0\)

Vậy cặp số x,y nhỏ nhất thỏa mãn là \(\hept{\begin{cases}x-2y=0\\y+1=0\end{cases}\Rightarrow\hept{\begin{cases}x-2y=0\\y=-1\end{cases}\Rightarrow}\hept{\begin{cases}x+2=0\\y=-1\end{cases}}}\)

\(\Rightarrow x=-2;y=-1\)

\(x^2+5y^2+2y-4xy-3=0\)

=> \(\left(x^2-4xy+4y^2\right)+\left(y^2+2y+1\right)-4=0\)

=> \(\left(x-2y\right)^2+\left(y+1\right)^2-2^2=0\)

=> \(\left(x-2y\right)^2+\left(y+1-2\right)\left(y+1+2\right)=0\)

=> \(\left(x-2y\right)^2+\left(y-1\right)\left(y+3\right)=0\)

Mà   \(\left(x-2y\right)^2 \ge 0 \forall x\) 

=> \(\left(y-1\right)\left(y+3\right)\le0\)   Mặt khác \(y-1 < y+3 \)

=> \(\hept{\begin{cases}y-1\le0\\y+3\ge0\end{cases}}\)=> \(-3\le y\le1\)  mà y nhỏ nhất 

=> \(y=-3\)

Thay vào biểu thức, ta có \(\left(x+6\right)^2+\left(-3-1\right)\left(-3+3\right)=0\) => \(\left(x+6\right)^2=0\)  => \(x+6=0\) => \(x=-6\)

    Vậy x=-6 , y=-3