tự vẽ hình

nối MP

Xét t/g MNP có: AM=AN(gt),BN=BP(gt)

=>AB là đường tb của t/g MNP

=>AB//MP và  AB=1/2MP (1)

Xét t/g MQP có: MD=DQ(gt),QC=CP(gt)

=>CD là đường tb của t/g MQP

=.CD//MP và  CD=1/2MP(2)

Từ  (1) và  (2) => AB=CD (3)

Lại có:AB//MP, CD//MP

=>AB//CD (4)

Từ (3)và (4) => tứ giác ABCD là HBH

a/ Từ M dựng đường thẳng // AB cắt AD tại H ta có

\(AB\perp AD;\)MH//AB \(\Rightarrow MH\perp AD\)

Mà BC//AD

=> ABMH là hình bình hành => AB=MH

\(\Rightarrow S_{AMD}=\frac{AD.MH}{2}=\frac{AD.AB}{2}=\frac{S_{ABCD}}{2}\left(dpcm\right)\)

b/

\(\frac{S_{ABM}}{S_{DCM}}=\frac{\frac{1}{2}.BM.AB}{\frac{1}{2}.CM.CD}=\frac{BM}{CM}=\frac{1}{3}\) (do ABCD là HCN nên AB=CD)

\(\left(\frac{x}{xy-y^2}+\frac{2x-y}{xy-x^2}\right):\left(\frac{1}{x}+\frac{1}{y}\right)\)

\(=\left(\frac{x}{y\left(x-y\right)}-\frac{2x-y}{x\left(y-x\right)}\right):\left(\frac{y}{xy}-\frac{x}{xy}\right)\)

\(=\left(\frac{x}{y\left(x-y\right)}+\frac{2x-y}{x\left(x-y\right)}\right):\left(\frac{y-x}{xy}\right)\)

\(=\left(\frac{x^2}{xy\left(x-y\right)}+\frac{\left(2x-y\right)y}{xy\left(x-y\right)}\right):\left(\frac{y-x}{xy}\right)\)

\(=\frac{x^2+2xy-y^2}{xy\left(x-y\right)}.\frac{xy}{-\left(x-y\right)}=\frac{x^2+2xy-y^2}{-\left(x-y\right)}\)

\(x^3+5x^2-4x-20=0\)

\(\Leftrightarrow x^2\left(x+5\right)-4\left(x+5\right)=0\)

\(\Leftrightarrow\left(x^2-4\right)\left(x+5\right)=0\)

\(\Leftrightarrow\left(x-2\right)\left(x+2\right)\left(x+5\right)=0\Leftrightarrow x=\pm2;-5\)

Sửa đề : \(\left(\frac{2x}{2x+y}-\frac{4x^2}{4x^2+4xy+y^2}\right):\left(\frac{2x}{4x^2-y^2}+\frac{1}{y-2x}\right)\)

\(=\left(\frac{2x}{2x+y}-\frac{4x^2}{\left(2x+y\right)^2}\right):\left(\frac{2x}{\left(2x-y\right)\left(2x+y\right)}-\frac{1}{2x-y}\right)\)

\(=\left(\frac{2x\left(2x+y\right)-4x^2}{\left(2x+y\right)^2}\right):\left(\frac{2x-2x-y}{\left(2x-y\right)\left(2x+y\right)}\right)\)

\(=\frac{2xy}{\left(2x+y\right)^2}.\frac{\left(2x-y\right)\left(2x+y\right)}{-y}=-\frac{2xy\left(2x-y\right)}{\left(2x+y\right)y}\)

\(\frac{1}{1-x}+\frac{1}{1+x}+\frac{2}{1+x^2}+\frac{4}{1+x^4}+\frac{8}{1+x^8}+\frac{16}{1+x^{16}}=\frac{1+x+1-x}{\left(1-x\right)\left(1+x\right)}+\frac{2}{1+x^2}+\frac{4}{1+x^4}+\frac{8}{1+x^8}+\frac{16}{1+x^{16}}\)\(=\frac{2}{1-x^2}+\frac{2}{1+x^2}+\frac{4}{1+x^4}+\frac{8}{1+x^8}+\frac{16}{1+x^{16}}=\frac{2\left(1+x^2\right)+2\left(1-x^2\right)}{\left(1-x^2\right)\left(1+x^2\right)}+\frac{4}{1+x^4}+\frac{8}{1+x^8}+\frac{16}{1+x^{16}}\)\(=\frac{4}{1-x^4}+\frac{4}{1+x^4}+\frac{8}{1+x^8}+\frac{16}{1+x^{16}}=\frac{4\left(1+x^4\right)+4\left(1-x^4\right)}{\left(1-x^4\right)\left(1+x^4\right)}+\frac{8}{1+x^8}+\frac{16}{1+x^{16}}\)

\(=\frac{8}{1-x^8}+\frac{8}{1+x^8}+\frac{16}{1+x^{16}}=\frac{8\left(1+x^8\right)+8\left(1-x^8\right)}{\left(1-x^8\right)\left(1+x^8\right)}+\frac{16}{1+x^{16}}\)

\(=\frac{16}{1-x^{16}}+\frac{16}{1+x^{16}}=\frac{16\left(1+x^{16}\right)+16\left(1-x^{16}\right)}{\left(1-x^{16}\right)\left(1+x^{16}\right)}=\frac{32}{1-x^{32}}\)

a) x^2 - 2xy + y^2 - xz + yz 

= (x^2 - 2xy + y^2 ) - (xz + yz)

= (x - y)^2 - z(x + y)

= (x - y)(x - x + y)

Đặt biểu thức cần tính là A

Đặt B=1+2²+3²+4²+...+100²=1+2(1+1)+3(2+1)+4(3+1)+...+100(99+1)

B=1+1.2+2+2.3+3+3.4+4+...+99.100+100=(1+2+3+4+...+100)+(1.2+2.3+3.4+...+99.100)

Đặt C=1.2+2.3+3.4+...+99.100 => 3.C=1.2.3+2.3.3+3.4.3+...+99.100.3=1.2.3+2.3.(4-1)+3.4.(5-2)+...+99.100.(101-98)

3.C=1.2.3-1.2.3+2.3.4-2.3.4-2.3.4+3.4.5-...-98.99.100+99.100.101=99.100.101 => C=33.100.101

Đặt \(D=1+2+3+4+...+100=\frac{100\left(1+100\right)}{2}=5050.\)

=> B=D+C=5050+33.100.101

A=(2²+4²+6²++8²+...+100²)-(1+3²+5²+7²+...+99²)

Đặt E=2²+4²+6²+8²+...+100²=2².(1+2²+3²+4²+...+50²)=2².[1+2.(1+1)+3(2+1)+4(3+1)+...+50(49+1)]

E=2².(1+1.2+2+2.3+3+3.4+4+...+49.50+50)=2².[(1+2+3+...+50)+(1.2+2.3+3.4+...+49.50] Tính tương tự như C và D

=> \(E=2^2.\left(\frac{50.\left(1+50\right)}{2}+\frac{49.50.51}{3}\right)=2^2.\left(1275+17.49.50\right)\)

Mặt khác ta có

B=(1+3²+5²+7²+...+99²)+(2²+4²+6²+8²+...+100²)=(1+3²+5²+7²+...+99²)+E => 1+3²+5²+7²+...+99²=B-E

=> A=E-(B-E)=2.E-B

\(\Rightarrow A=2^3\left(1275+17.49.50\right)-\left(5050+33.100.101\right)\)