Inegalitate in 5 variabile

[Mateescu Constantin]

Fie \( a,\ b,\ c,\ d,\ e \in \mathbb{R} \) astfel incat \( a+b+c+d+e=0 \) si \( a^2+b^2+c^2+d^2+e^2=4 \).
Aratati ca \( a+2b+3c+4d+5e<7 \).

[Marius Mainea]

Folosim CBS :

\( LHS= a+2b-3(a+b+d+e)+4d+5e=-2a-b+d+2e\le\sqrt{((-2)^2+(-1)^2+1^1+2^2)(a^2+b^2+d^2+e^2)}\le\sqrt{40}<7 \)