Nicolae Paun problema 1

DrAGos Calinescu · Post by **DrAGos Calinescu** » Sat Dec 13, 2008 7:48 pm

Fie \( a,b,c\in\mathbb{R}^*. \) Stiind ca \( abc(a^3+b^3+c^3)<\sum_{cyc}a^3b^3 \) aratati ca exact doua dintre ecuatiile\( ax^2+2bx+c=0;bx^2+2cx+a=0;cx^2+2ax+b=0 \) au solutii reale.

Claudiu Mindrila · Post by **Claudiu Mindrila** » Sat Dec 13, 2008 8:02 pm

Cred ca ai o greseala in enunt. In loc de\( 2b \) trebuia \( 2bx \) si la fel pentru \( 2c \) si \( 2a \).

DrAGos Calinescu · Post by **DrAGos Calinescu** » Sat Dec 13, 2008 8:12 pm

gata

DrAGos Calinescu · Post by **DrAGos Calinescu** » Wed Dec 17, 2008 12:41 am

Ma rog, sa nu ramana problema nerezolvata

Obtinem \( \Delta_1=4(b^2-ac) , \Delta_2=4(c^2-ab) , \Delta_3=4(a^2-bc) \)
Inmultindu-le obtinem \( (b^2-ac)(c^2-ab)(a^2-bc)=abc(a+b+c)-a^3b^3-b^3c^3-c^3a^3<0\Rightarrow \) toti 3 discriminantii sunt negativi sau numai unul.
Daca sunt toti 3 relatia e echivalenta cu \( a^2+b^2+c^2-ab-bc-ca<0 \)(contradictie)\( \Rightarrow \) un discriminant negativ, deci exact doua ecuatii cu solutii reale.[/tex][/code]