MT【287】余弦的线性组合

(2017北大特优)在$Delta ABC$中,$cos A+sqrt{2}cos B+sqrt{2}cos C$的最大值____

解答 :2
$cos A+sqrt{2}cos B+sqrt{2}cos C$

$=cos A+2sqrt{2}cosdfrac{B+C}{2}cosdfrac{B-C}{2}le1-2sin^2dfrac{A}{2}+2sqrt{2}sindfrac{A}{2}le2$

提示:$pcosalpha+qcoseta+rcosgammaledfrac{1}{2}sumlimits_{cyc}dfrac{qr}{p}$
嵌入不等式:$$2yzcosA+2zxcosB+2xycosCle x^2+y^2+z^2$$中令$(x,y,z)=(sqrt{dfrac{qr}{p}},sqrt{dfrac{rp}{q}},sqrt{dfrac{pq}{r}})$即得.