Olympiad problems

1974 IMO Longlists, 8

Tags: trigonometry , algebra , Trigonometric Identities , trigonometric substitution , equation , IMO Shortlist

Let $x, y, z$ be real numbers each of whose absolute value is different from $\frac{1}{\sqrt 3}$ such that $x + y + z = xyz$. Prove that \[\frac{3x - x^3}{1-3x^2} + \frac{3y - y^3}{1-3y^2} + \frac{3z -z^3}{1-3z^2} = \frac{3x - x^3}{1-3x^2} \cdot \frac{3y - y^3}{1-3y^2} \cdot \frac{3z - z^3}{1-3z^2}\]

1966 IMO Shortlist, 25

Tags: trigonometry , algebra , Trigonometric Identities , equation , IMO Shortlist , IMO Longlist

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

1979 IMO Longlists, 33

Tags: trigonometry , Inequality , geometric inequality , Trigonometric Identities , IMO Shortlist , IMO Longlist , inequalities

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

1967 IMO Shortlist, 4

Tags: parameterization , calculus , trigonometry , algebra , Trigonometric Identities , IMO Longlist

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 $x.$

1979 IMO Shortlist, 13

Tags: trigonometry , Inequality , geometric inequality , Trigonometric Identities , IMO Shortlist , IMO Longlist , inequalities

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