Inegalitate cu variabile mai mici decat 1/2 si respectiv 1
Posted: Wed Apr 08, 2009 11:16 am
Fie numerele reale \( a,\ b,\ c,\ d\in\left[0,\ 1\right] \) si \( x,\ y,\ z,\ t\in\left[0,\ \frac{1}{2}\right] \) astfel incat \( a+b+c+d=x+y+z+t=1 \). Sa se arate ca:
a) \( ax+by+cz+dt\ge\min\left\{ \frac{a+b}{2},\ \frac{b+c}{2},\ \frac{c+d}{2},\ \frac{d+a}{2},\ \frac{a+c}{2},\ \frac{b+d}{2}\right\} \ ; \)
b) \( ax+by+cz+dt\ge 54abcd. \)
Octavian Purcaru, O.N.M. 1996
a) \( ax+by+cz+dt\ge\min\left\{ \frac{a+b}{2},\ \frac{b+c}{2},\ \frac{c+d}{2},\ \frac{d+a}{2},\ \frac{a+c}{2},\ \frac{b+d}{2}\right\} \ ; \)
b) \( ax+by+cz+dt\ge 54abcd. \)
Octavian Purcaru, O.N.M. 1996