两个竞赛试题的推广

Promotion of Two Competition Questions

本文由第二届陈省身杯全国高中数学奥林匹克试题与2016年福建省高中数学竞赛试题提出一般性问题:设p,q,r是不全为零的三个非负数,对任意不全为零的实数x,y,z,求2pyz+2qxz+2rxy/x^2+y^2+z^2的最大值,并运用嵌入不等式与一个三角形恒等式解决了这个问题,获得结论:设p,q,r是不全为零的三个非负数,k是一元三次方程k^3-(p^2+q^2+r^2)k-2pqr=0的正根,则对任意不全为零的实数x,y,z,恒有x^2+y^2+z^2≥1/k(2pyz+2qxz+2rxy),仅当x:y:z=√k^2-p^2:√k^2-q^2:√k^2-r^2时等号成立.

This paper puts forward general questions from the second Chenshengshen Cup National Senior High School Mathematics Olympic Test and the 2016 Fujian Senior High School Mathematics Competition:if p, q, r are three non-negative numbers that are not all zero, for any real number x, y, z that is not all zero, find the maximum value of 2pyz + 2qxz + 2rxy/ x^2 + y^2 + z^2 . The question is solved by using the embedded inequality and a triangular constant equation, and the conclusion is obtained:if p, q, r are three non-negative numbers that are not all zero, k is the positive root of k3 -( p^2 + q^2 + r^2 ) k - 2pqr = 0. Then for any real number x, y, z that is not all zero, there is always x^2 + y^2 + z^2 ≥ 1 k ( 2pyz + 2qxz + 2rxy ). Only x:y:z =√k^2 - p^2 :√k^2 - q^2 :√k^2 - r^2 , the equal sign holds.