TA CÓ BỔ ĐỀ SAU:

\(a^3+b^3=a^3+3a^2b+3ab^2+b^3-3ab.\left(a+b\right)=\left(a+b\right)^3-3ab.\left(a+b\right)\)

ta có:

\(\left(x-2023\right)^3+\left(x-2021\right)^3=\left(2x-4044\right)^3\)

\(\Leftrightarrow\left(x-2023+x-2021\right)^3-3.\left(x-2023\right).\left(x-2021\right).\left(x-2023+x-2021\right)=\left(2x-4044\right)^3\)

\(\Leftrightarrow\left(2x-4044\right)^3-3.\left(x-2023\right).\left(x-2021\right).\left(2x-4044\right)=\left(2x-4044\right)^3\)

\(\Leftrightarrow\left(2x-4044\right)^3-\left(2x-4044\right)^3=3.\left(x-2023\right).\left(x-2021\right).\left(2x-4044\right)\)

\(\Leftrightarrow3.\left(x-2023\right).\left(x-2021\right).\left(2x-4044\right)=0\)

\(\Leftrightarrow x-2023=0;x-2021=0;2x-4044=0\)

\(\Rightarrow\) x=2023 hoạc x=2021 hoặc x=2022 là nghiệm của phương trình.

vậy................

KHÓ QUÉ                   

gọi độ dài quãng đường AB là \(x\) \(\left(km\right)\)\(\left(x>0\right)\)

thời gian dự định đi quãng đường AB là \(\frac{x}{15}\left(h\right)\)

thời gian thực tế đi quãng đường AB là: \(\frac{x}{15-3}=\frac{x}{12}\left(h\right)\)

theo đề bài người đó đến B chậm hơn dự tính 12 phút=1/5h nên ta có phương trình:

\(\frac{x}{12}-\frac{x}{15}=\frac{1}{5}\)

\(\Leftrightarrow\frac{5x}{60}-\frac{4x}{60}=\frac{12}{5}\)

\(\Leftrightarrow5x-4x=12\)

\(\Leftrightarrow x=12\left(TMđk\right)\)

vậy đọ dài AB là 12 km

\(S=\left(\frac{1}{16x}+\frac{1}{4y}+\frac{1}{z}\right)=\left(\frac{1}{16x}+\frac{1}{4y}+\frac{1}{z}\right).\left(x+y+z\right)\)   (do x+y+z=1 nên michf nhân vào kết quả sẽ ko bị  thay đổi)

\(S=\frac{21}{16}+\left(\frac{x}{4y}+\frac{y}{16x}\right)+\left(\frac{x}{z}+\frac{z}{16x}\right)+\left(\frac{y}{z}+\frac{z}{4y}\right)\)

AD BĐT cô si,ta có:

\(S\ge\frac{21}{16}+2.\sqrt{\frac{x}{4y}.\frac{y}{16x}}+2\sqrt{\frac{x}{z}.\frac{z}{16x}}+2.\sqrt{\frac{y}{z}.\frac{z}{4y}}=\frac{21}{16}+\frac{1}{4}+\frac{1}{2}+1=\frac{49}{16}\)

dấu bằng xảy ra \(\Leftrightarrow\hept{\begin{cases}4x=2y=z\\x+y+z=1\\x;y;z>0\end{cases}\Leftrightarrow\hept{\begin{cases}x=\frac{1}{7}\\y=\frac{2}{7}\\z=\frac{4}{7}\end{cases}}}\)

T=116x+14y+1zT=116x+14y+1z  ; x + y + z = 1

⇒T=x+y+z16x+x+y+z4y+x+y+zz⇒T=x+y+z16x+x+y+z4y+x+y+zz

=116+y16x+z16x+x4y+14+z4y+xz+yz+1=116+y16x+z16x+x4y+14+z4y+xz+yz+1

=(116+14+1)+(y16x+x4y)+(z16x+xz)+(z4y+yz)=(116+14+1)+(y16x+x4y)+(z16x+xz)+(z4y+yz)                    (1)

x;y;z>0⇒y16x;x4y;z16x;xz;z4y;yz>0x;y;z>0⇒y16x;x4y;z16x;xz;z4y;yz>0

áp dụng bđt cô si : 

y16x+x4y≥2√y16x⋅x4y=14y16x+x4y≥2y16x⋅x4y=14                             (2)

z16x+xz≥2√z16x⋅xz=12z16x+xz≥2z16x⋅xz=12                                 (3)

x4y+yz≥2√z4y⋅yz=1x4y+yz≥2z4y⋅yz=1                                        (4)

(1)(2)(3)(4) ⇒T≥116+14+1+14+12+1⇒T≥116+14+1+14+12+1

⇒T≥4916⇒T≥4916

dấu "=" xảy ra khi \hept⎧⎪ ⎪⎨⎪ ⎪⎩y16x=x4yz16x=xzz4y=yz⇔\hept⎧⎨⎩4y2=16x2z2=16x2z2=4y2\hept{y16x=x4yz16x=xzz4y=yz⇔\hept{4y2=16x2z2=16x2z2=4y2

⇔\hept⎧⎨⎩y=2xz=4xz=2y⇔\hept{y=2xz=4xz=2y có x+y+z = 1

=> x + 2x + 4x = 1

=> x = 1/7

xong tìm ra y = 2/7 và z = 4/7

\(\frac{3}{x-1}=\frac{3x+2}{1-x^2}-\frac{4}{x+1}\)\(\left(đkxđ:x\ne\pm1\right)\)

\(\Leftrightarrow\frac{3.\left(x+1\right)}{\left(x-1\right).\left(x+1\right)}=\frac{-3x-2}{\left(x-1\right).\left(x+1\right)}-\frac{4.\left(x-1\right)}{\left(x-1\right).\left(x+1\right)}\)

\(\Rightarrow3.\left(x+1\right)=-3x-2-4.\left(x-1\right)\)

\(\Leftrightarrow3x+3=-3x-2-4x+4\)

\(\Leftrightarrow3x+3=2-7x\)

\(\Leftrightarrow3x+7x=2-3\)

\(\Leftrightarrow10x=-1\)

\(\Leftrightarrow x=-\frac{1}{10}\left(TMđkxđ\right)\)

vậy phương trình có 1 nghiệm duy nhất là x=-1/10

đặt \(A=\left(\frac{m-n}{p}+\frac{n-p}{m}+\frac{p-m}{n}\right)\)

\(\Rightarrow S=A.\left(\frac{p}{m-n}+\frac{m}{n-p}+\frac{n}{p-m}\right)=A.\frac{p}{m-n}+A.\frac{m}{n-p}+A.\frac{n}{p-m}\)

giờ ta xét từng hạng tử 1 nhé:

\(A.\frac{p}{m-n}=\left(\frac{m-n}{p}+\frac{n-p}{m}+\frac{p-m}{n}\right).\frac{p}{m-n}\)

\(=1+\frac{p}{m-n}.\left(\frac{n-p}{m}+\frac{p-m}{n}\right)\)

\(=1+\frac{p}{m-n}.\left(\frac{\left(n-p\right).n+m.\left(p-m\right)}{m.n}\right)\)

\(=1+\frac{p}{m-n}.\left(\frac{n^2-pn+m.p-m^2}{m.n}\right)\)

\(=1+\frac{p}{m-n}.\left(\frac{\left(n-m\right).\left(n+m\right)+p.\left(m-n\right)}{m.n}\right)\)

\(=1+\frac{p}{m-n}.\left(\frac{\left(p-m-n\right).\left(m-n\right)}{m.n}\right)\)

\(=1+\frac{p.\left(p-m-n\right)}{m.n}\)

\(=1+\frac{p^2-p.\left(m+n\right)}{m.n}\)

bây h ta sẽ sử dụng giả thiết \(m+n+p=0\Rightarrow m+n=-p\)

\(\Rightarrow A.\frac{p}{m-n}=1+\frac{p^2+p^2}{m.n}=1+\frac{2p^3}{m.n.p}\)

CM tương tự ta có:  \(A.\frac{m}{n-p}=\frac{2m^3}{mnp}\)  ;    \(A.\frac{n}{p-m}=\frac{2n^3}{mnp}\)

\(\Rightarrow S=A.\left(\frac{p}{m-n}+\frac{m}{n-p}+\frac{n}{p-m}\right)=A.\frac{p}{m-n}+A.\frac{m}{n-p}+A.\frac{n}{p-m}=3+\frac{2\left(p^3+m^3+n^3\right)}{m.n.p}\)

\(m+n+p=0\Rightarrow\left(m+n+p\right).\left(m^2+p^2+n^2-mn-mp-np\right)=0\Leftrightarrow m^3+n^3+p^3-3mnp=0\)

\(\Leftrightarrow m^3+n^3+p^3=3mnp\)

\(S=3+\frac{2.3mnp}{mnp}=3+6=9\)

Vậy \(S=9\Leftrightarrow m+n+p=0\)