Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 6.1 National olympiad Prove it

Example 2: Let the three sides of $\triangle A B C$ be $a, b, c$ , and the area be $\Delta$ , then
$\min \left\{a^{4}+b^{4}, b^{4}+c^{4}, c^{4}+a^{4}\right\} \geq 8 \Delta^{2}$

Solution

Proof: Without loss of generality, let $a \geq b \geq c$ . Then, by the two-variable mean inequality and the boundedness of the sine function, we get
$\begin{aligned} \min \left\{a^{4}+b^{4}, b^{4}+c^{4}, c^{4}\right. & \left.+a^{4}\right\} \geq b^{4}+c^{4} \\ & \geq 2 b^{2} c^{2} \geq 8\left(\frac{1}{2} b c \sin A\right)^{2}=8 \Delta^{2} \end{aligned}$

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.