Cho a2+b2+c2=ab+bc+ca.Tính C=\(\dfrac{a^{2010}+b^{2010}}{c^{2010}}+\dfrac{b^{2010}+c^{2010}}{a^{2010}}+\dfrac{c^{2010}+a^{2010}}{b^{2010}}\)
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21 tháng 3 2018
Ta có:\(\dfrac{x^{2010}+y^{2010}+z^{2010}+t^{2010}}{a^2+b^2+c^2+d^2}=\dfrac{x^{2010}}{a^2}=\dfrac{y^{2010}}{b^2}=\dfrac{z^{2010}}{c^2}=\dfrac{t^{2010}}{d^2}\)
\(\Rightarrow\dfrac{x^{2010}}{a^2}+\dfrac{y^{2010}}{b^2}+\dfrac{z^{2010}}{c^2}+\dfrac{t^{2010}}{d^2}=\dfrac{x^{2010}}{a^2}\)
\(\Rightarrow\dfrac{y^{2010}}{b^2}+\dfrac{z^{2010}}{c^2}+\dfrac{t^{2010}}{d^2}=0\)
\(\Leftrightarrow3\cdot\dfrac{y^{2010}}{b^2}=0\)
\(\Leftrightarrow y^{2010}=0\)
\(\Leftrightarrow y=0\)
CMTT\(\Rightarrow x=z=t=0\)
\(\Rightarrow T=0\)
17 tháng 1 2023
A = \(\dfrac{2008}{2009+2010+2011}+\dfrac{2009}{2009+2010+2011}+\dfrac{2010}{2009+2010+2011}\)
Ta có:
\(\dfrac{2008}{2009}>\dfrac{2008}{2009+2010+2011}\)
\(\dfrac{2009}{2010}>\dfrac{2009}{2009+2010+2011}\)
\(\dfrac{2010}{2011}>\dfrac{2010}{2009+2010+2011}\)
Từ 3 điều trên suy ra : A < B
25 tháng 6 2022
\(2010A=\dfrac{2010^{2012}+2010}{2010^{2012}+1}=1+\dfrac{2009}{2010^{2012}+1}\)
\(2010B=\dfrac{2010^{2011}+2010}{2010^{2011}+1}=1+\dfrac{2009}{2010^{2011}+1}\)
mà \(2010^{2012}+1>2010^{2011}+1\)
nên A<B
ta có : \(a^2+b^2+c^2=ab+bc+ca\Leftrightarrow2a^2+2b^2+2c^2=2ab+2bc+2ca\)
\(\Leftrightarrow a^2-2ab+b^2+b^2-2bc+c^2+c^2-2ca+a^2=0\)
\(\Leftrightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\) \(\Leftrightarrow a=b=c\)
\(\Rightarrow C=\dfrac{a^{2010}+b^{2010}}{c^{2010}}+\dfrac{b^{2010}+c^{2010}}{a^{2010}}+\dfrac{c^{2010}+a^{2010}}{b^{2010}}=3\dfrac{a^{2010}+a^{2010}}{a^{2010}}\)
\(=3\dfrac{2a^{2010}}{a^{2010}}=3.2=6\)