MS_Kekas wrote: Prove the following inequality: $$\sin{1} + \sin{3} + \ldots + \sin{2021} > \frac{2\sin{1011}^2}{\sqrt{3}}$$ Proposed by Oleksii Masalitin $$LHS=\frac{2\sin{1}.\sin{1} + 2\sin{3}.\sin{1} + \ldots +2 \sin{2021}.\sin{1}}{2\sin{1}}$$$$LHS=\frac{1-\cos{2} +\cos{2}-\cos{4} + \ldots +\cos{2020}-\cos{2022}}{2\sin{1}}=\frac{1-\cos{2022}}{2\sin{1}}=\frac{\sin^2{1011}}{\sin{1}}$$$$LHS=\frac{\sin^2{1011}}{\sin{1}} >\frac{\sin^2{1011}}{\sin{\dfrac{\pi}{3}}}=RHS.$$ .