Toán [toán 9] bất đẳng thức

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riverflowsinyou1

a+ba+b \geq cc \geq b+1b+1 \Leftrightarrow aa \geq 11

bb \geq aa \Leftrightarrow bb\geq1

\Rightarrow (a1)(b1)(a-1)(b-1) \geq 0 \Leftrightarrow ab+1ab+1\geqa+ba+b\geqc \Leftrightarrow abab\geqc1c-1

QQ\geq2ab+abc(a+1)(b+1)(c1)=ab(c+2)(ab+a+b+1)(c+1)\dfrac{2ab+abc}{(a+1)(b+1)(c-1)}=\dfrac{ab(c+2)}{(ab+a+b+1)(c+1)}

\Rightarrow QQ \geq ab(c+2)2(ab+1)(c+1)=c+22(1+1ab)(c+1)\dfrac{ab(c+2)}{2(ab+1)(c+1)}=\dfrac{c+2}{2(1+ \dfrac{1}{ab} )(c+1)}

\Rightarrow QQ \geq c+22(1+1c1)(c+1)=(c+2)(c1)2c(c+1)\dfrac{c+2}{2(1+\dfrac{1}{c-1})(c+1)}=\dfrac{(c+2)(c-1)}{2c(c+1)}

\Leftrightarrow QQ \geq c2+c22(c2+c)=121c2+c\dfrac{c^2+c-2}{2(c^2+c)}=\dfrac{1}{2}-\dfrac{1}{c^2+c}

\Leftrightarrow QQ \geq 12132+3=512\dfrac{1}{2}-\dfrac{1}{3^2+3}=\dfrac{5}{12}
 
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