For positive numbers $a, b, c$, $a < b + c$ and $b < c + a$. What is the relationship between $a^{4} + b^{4} + c^{4}$ and $2\left(a^{2} b^{2} + b^{2} c^{2} + c^{2} a^{2}\right)$?
The difference of the two expressions can be easily transformed into a product:

$$
\begin{aligned}
K= & \left(a^{4}+b^{4}+c^{4}\right)-2\left(a^{2} b^{2}+b^{2} c^{2}+c^{2} a^{2}\right)=\left(a^{2}+b^{2}-c^{2}\right)^{2}-4 a^{2} b^{2}= \\
& =\left(a^{2}+b^{2}-c^{2}-2 a b\right)\left(a^{2}+b^{2}-c^{2}+2 a b\right)=\left\{(a-b)^{2}-c^{2}\right\}\left\{(a+b)^{2}-c^{2}\right\}= \\
& =(a-b-c)(a-b+c)(a+b-c)(a+b+c)
\end{aligned}
$$

According to the assumptions, the first factor is negative, the second and the fourth are positive, so the product of these three factors is negative, hence the sign of the difference is opposite to the sign of the third factor, if $a+b-c \neq 0$, and $K=0$, if $a+b-c=0$. Summarizing and rearranging:

$$
a^{4}+b^{4}+c^{4} \gtreqless 2\left(a^{2} b^{2}+b^{2} c^{2}+c^{2} a^{2}\right)
$$

depending on whether

$$
c \gtreqless a+b
$$

Zoltán Benkő (Miskolc, Földes F. Gymn., 1st year student)