cho số thực a,b,c thỏa mãn 1<=a,b,c<=2 CMR a^2+b^2/ab + b^2+c^2/bc + c^2+a^2/ac <=7
[tex]GT\Leftrightarrow \sum \frac{a}{b}+\sum \frac{b}{a}\\G/s:a\geq b\geq c\\\rightarrow \left ( a-b \right )\left ( b-c \right )\geq 0\\\rightarrow ab+bc\geq b^{2}+ac\\\rightarrow \left\{\begin{matrix} & \frac{b}{c}+\frac{a}{b}\leq \frac{a}{c}+1 & \\ & \frac{b}{a}+\frac{c}{b}\leq \frac{c}{a}+1 & \end{matrix}\right.\\\rightarrow \sum \frac{a}{b}+\sum \frac{b}{a}\leq 2(\frac{a}{c}+\frac{c}{a})+2[/tex]
Đặt [tex]\frac{a}{c}=ok[/tex]
Do [tex]ok\epsilon [1;2]\rightarrow (ok-1)(ok-2)\leq 0\rightarrow ok^2-3.ok+2\leq 0\\\rightarrow ok^2+1\leq 3.ok-1[/tex]
Có [tex]2.(ok+\frac{1}{ok})+2=2.\frac{ok^2+1}{ok}+2\leq 2.\frac{3.ok-1}{ok}+2\leq 2.(3-\frac{1}{ok})+2 \leq 2.(3-\frac{1}{2})+2=2.\frac{5}{2}+2=7(Q.E.D)[/tex]