mateforum.ro

a) Sa se arate ca pentru orice \( x \in \mathbb{R} \) are loc inegalitatea \( 3(x^4+1)\ge 2x(x^2+x+1) \).
b) Aratati ca orice \( a,b,c>0 \) verifica egalitatea \( \frac{a^{4}}{\left(a+b\right)\left(a^{2}+b^{2}\right)}+\frac{b^{4}}{\left(b+c\right)\left(b^{2}+c^{2}\right)}+\frac{c^{4}}{\left(c+a\right)\left(c^{2}+a^{2}\right)}=\frac{b^{4}}{\left(a+b\right)\left(a^{2}+b^{2}\right)}+\frac{c^{4}}{\left(b+c\right)\left(b^{2}+c^{2}\right)}+\frac{a^{4}}{\left(c+a\right)\left(c^{2}+a^{2}\right)} \).

c) Dedueti ca orice \( a,b,c>0 \) satisfac inegalitatea \( \frac{a^{4}}{\left(a+b\right)\left(a^{2}+b^{2}\right)}+\frac{b^{4}}{\left(b+c\right)\left(b^{2}+c^{2}\right)}+\frac{c^{4}}{\left(c+a\right)\left(c^{2}+a^{2}\right)}\ge\frac{a+b+c}{4}. \)

Concursul "TMMATE", 2009

b)Notam cu \( S_1 \) membrul stang si \( S_2 \) membrul drept .
\( S_1=\sum_{cyc}\frac{a^4}{(a+b)(a^2+b^2)}=\sum_{cyc}\frac{a^4-b^4+b^4}{(a+b)(a^2+b^2)}=\sum_{cyc}\frac{(a^2-b^2)(a^2+b^2)}{(a+b)(a^2+b^2)}+S_2=\sum_{cyc}\frac{(a-b)(a+b)(a^2+b^2)}{(a+b)(a^2+b^2)}+S_2=\sum_{cyc}(a-b)+S_2=S_2 \)