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\(2020\left|x-1\right|+2021\ge2021\)
\(\Rightarrow\dfrac{2021}{2020\left|x-1\right|+2021}\le\dfrac{2021}{2021}=1\)
\(\Rightarrow A=\dfrac{-2021}{2020\left|x-1\right|+2021}\ge-1\)
\(minA=-1\Leftrightarrow x=1\)
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\(\text{f(1)=}2.1^2+1=3\)
\(\text{f(-1)=}2.\left(-1\right)^2+1=3\)
\(\text{f(2)=}2.2^2+1=9\)
\(\text{f(0)=}2.0^2+1=1\)
\(\text{f(-3)=}=2.\left(-3\right)^2+1=19\)
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Đặt : A = 1 + 2 + 2^2 + 2^3 + ... + 2^2016
=> 2A = 2 + 2^2 + 2^3 + 2^4 + ... + 2^2017
=> 2A - A = ( 2 + 2^2 + 2^3 + 2^4 + ... + 2^2017 ) - ( 1 + 2 + 2^2 + 2^3 + ... + 2^2016 )
=> A = 2^2017 - 1
=> A < 2^2017
Vậy A < 2^2017
Ta đặt A = 1 + 2 + 22 + 23 + ....+ 22016
=> 2A = 2 + 22 + 23 + ...+22017
=> 2A - A = (2+22+23+...+22017) - (1+2+22+...+22016 )
=> A = 22017 - 1
Mà 22017 - 1 < 22017
=> A < 22017
Vậy 1 + 2 + 22 + ...+ 22016 < 22017
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\(a,2\dfrac{1}{3}.5\dfrac{4}{7}=\dfrac{7}{3}.\dfrac{39}{7}=13\)
\(b,\dfrac{-1}{2}+\dfrac{4}{9}+\dfrac{3}{2}+\dfrac{5}{9}=-1+1=0\)
\(c,0,75+\left(-1,25\right)=-1,50\)
\(d,\dfrac{2}{5}\cdot\dfrac{3}{8}-\dfrac{2}{5}\cdot\dfrac{6}{11}=\dfrac{2}{5}\cdot\left(\dfrac{3}{8}-\dfrac{6}{11}\right)=\dfrac{2}{5}.\dfrac{-15}{88}=\dfrac{-3}{44}\)
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a) Xét \(\Delta ABC:\)
\(BC^2=10^2=100.\\ AB^2+AC^2=6^2+8^2=100.\\ \Rightarrow BC^2=AB^2+AC^2.\)
\(\Rightarrow\Delta ABC\) vuông tại A (Pytago đảo).
\(\Rightarrow\widehat{BAC}=90^o.\)
b) Ta có: \(\widehat{B}+\widehat{D}=90^o.\\ \widehat{B}+\widehat{C}=90^o.\)
\(\Rightarrow\widehat{D}=\widehat{C}.\)
Xét \(\Delta ABC\) và \(\Delta AED:\)
\(\widehat{D}=\widehat{C}\left(cmt\right).\)
\(AC=AD\left(=8cm\right).\)
\(\widehat{BAC}=\widehat{EAD}\left(=90^o\right).\)
\(\Rightarrow\) \(\Delta ABC\) \(=\Delta AED\left(g-c-g\right).\)
c) Xét \(\Delta BDC:\)
DK là đường cao \(\left(DK\perp BC\right).\)
CA là đường cao \(\left(CA\perp AB\right).\)
Mà E là giao điểm của DK; CA (gt).
\(\Rightarrow\) E là trực tâm.\(\Rightarrow\) BE là đường cao.\(\Rightarrow\) \(BE\perp CD.\)