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Prove the inequality and find the limit

Source: 2019 Jozsef Wildt International Math Competition

May 19, 2020
inequalitieslimittrigonometry

Problem Statement

[*] If aa, bb, cc, d>0d > 0, show inequality:(tan⁡−1(ad−bcac+bd))2≥2(1−ac+bd(a2+b2)(c2+d2))\left(\tan^{-1}\left(\frac{ad-bc}{ac+bd}\right)\right)^2\geq 2\left(1-\frac{ac+bd}{\sqrt{\left(a^2+b^2\right)\left(c^2+d^2\right)}}\right) [*] Calculate lim⁡n→∞nα(n−∑k=1nn+k2−k(n2+k2)(n2+(k−1)2))\lim \limits_{n \to \infty}n^{\alpha}\left(n- \sum \limits_{k=1}^n\frac{n^+k^2-k}{\sqrt{\left(n^2+k^2\right)\left(n^2+(k-1)^2\right)}}\right)where α∈R\alpha \in \mathbb{R}
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