Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.8 AIME, harder Prove it

3. In $\triangle A B C$ , prove:
(1) When $n$ is an integer, we have
$\cot ^{2} n A+\cot ^{2} n B+\cot ^{2} n C \geqslant 1 \text {; }$
(2) When $n$ is an odd number, we have
$\tan ^{2} \frac{n A}{2}+\tan ^{2} \frac{n B}{2}+\tan ^{2} \frac{n C}{2} \geqslant 1$

Solution

3. (1) Left side $\geqslant \cot n A \cot n B+\cot n B \cot n C+\cot n C \cot n A$ $=1$ .
$\text { (2) } \begin{aligned} \text { Left side } & \geqslant \tan \frac{n A}{2} \tan \frac{n B}{2}+\tan \frac{n B}{2} \tan \frac{n C}{2}+\tan \frac{n C}{2} \tan \frac{n A}{2} \\ & =1 . \end{aligned}$

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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.