Olympiad problems

1967 IMO Shortlist, 4

Tags: algebra , parameterization , calculus , trigonometry , trigonometric identities

Find values of the parameter $u$ for which the expression \[y = \frac{ \tan(x-u) + \tan(x) + \tan(x+u)}{ \tan(x-u)\tan(x)\tan(x+u)}\] does not depend on $x.$

1969 IMO Longlists, 38

Tags: trigonometry , algebra , series summation , trigonometric identities

$(HUN 5)$ Let $r$ and $m (r \le m)$ be natural numbers and $Ak =\frac{2k-1}{2m}\pi$. Evaluate $\frac{1}{m^2}\displaystyle\sum_{k=1}^{m}\displaystyle\sum_{l=1}^{m}\sin(rA_k)\sin(rA_l)\cos(rA_k-rA_l)$

1979 IMO Shortlist, 13

Tags: inequalities , geometric inequality , trigonometry , inequality , trigonometric identities

Show that $\frac{20}{60} <\sin 20^{\circ} < \frac{21}{60}.$

1966 IMO Longlists, 25

Tags: trigonometry , trigonometric identities , equation , algebra

Prove that \[\tan 7 30^{\prime }=\sqrt{6}+\sqrt{2}-\sqrt{3}-2.\]

1966 IMO Shortlist, 61

Tags: trigonometry , algebra , trigonometric identities , induction

Prove that for every natural number $n$, and for every real number $x \neq \frac{k\pi}{2^t}$ ($t=0,1, \dots, n$; $k$ any integer) \[ \frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx} \]