Olympiad problems

2017 Romania National Olympiad, 3

$ \sin\frac{\pi }{4n}\ge \frac{\sqrt 2 }{2n} ,\quad \forall n\in\mathbb{N} $

1994 All-Russian Olympiad Regional Round, 11.1

Tags: trigonometry , inequalities , inequality , trigonometric inequality

Prove that for all $x \in \left( 0, \frac{\pi}{3} \right)$ inequality $sin2x+cosx>1$ holds.

2013 Bogdan Stan, 3

Tags: inequalities , trigonometry , trigonometric inequality , algebra

Let $ a,b,c $ be three real numbers such that $ \cos a+\cos b+\cos c=\sin a+\sin b+\sin c=0. $ Prove that [b]i)[/b] $ \cos 6a+\cos 6b+\cos 6c=3\cos (2a+2b+2c) $ [b]ii)[/b] $ \sin 6a+\sin 6b+\sin 6c=3\sin (2a+2b+2c) $ [i]Vasile Pop[/i]

2006 Kazakhstan National Olympiad, 5

Tags: trigonometry , inequalities , trigonometric inequality

Prove that for every $ x $ such that $ \sin x \neq 0 $, exists natural $ n $ such that $ | \sin nx | \geq \frac {\sqrt {3}} {2} $.

1994 Korea National Olympiad, Problem 2

Tags: algebra , inequalities , trigonometry , trigonometric inequality , inequality , geometry

Let $ \alpha,\beta,\gamma$ be the angles of a triangle. Prove that $csc^2\frac{\alpha}{2}+csc^2\frac{\beta}{2}+csc^2\frac{\gamma}{2} \ge 12$ and find the conditions for equality.

1974 IMO, 2

Tags: circumcircle , trigonometry , geometry , triangle , trigonometric inequality

Let $ABC$ be a triangle. Prove that there exists a point $D$ on the side $AB$ of the triangle $ABC$, such that $CD$ is the geometric mean of $AD$ and $DB$, iff the triangle $ABC$ satisfies the inequality $\sin A\sin B\le\sin^2\frac{C}{2}$. [hide="Comment"][i]Alternative formulation, from IMO ShortList 1974, Finland 2:[/i] We consider a triangle $ABC$. Prove that: $\sin(A) \sin(B) \leq \sin^2 \left( \frac{C}{2} \right)$ is a necessary and sufficient condition for the existence of a point $D$ on the segment $AB$ so that $CD$ is the geometrical mean of $AD$ and $BD$.[/hide]

1974 IMO Shortlist, 10

Tags: circumcircle , geometry , trigonometry , triangle , trigonometric inequality

Let $ABC$ be a triangle. Prove that there exists a point $D$ on the side $AB$ of the triangle $ABC$, such that $CD$ is the geometric mean of $AD$ and $DB$, iff the triangle $ABC$ satisfies the inequality $\sin A\sin B\le\sin^2\frac{C}{2}$. [hide="Comment"][i]Alternative formulation, from IMO ShortList 1974, Finland 2:[/i] We consider a triangle $ABC$. Prove that: $\sin(A) \sin(B) \leq \sin^2 \left( \frac{C}{2} \right)$ is a necessary and sufficient condition for the existence of a point $D$ on the segment $AB$ so that $CD$ is the geometrical mean of $AD$ and $BD$.[/hide]

1967 IMO Longlists, 55

Tags: algebra , trigonometry , trigonometric inequality , inequality

Find all $x$ for which, for all $n,$ \[\sum^n_{k=1} \sin {k x} \leq \frac{\sqrt{3}}{2}.\]

1987 IMO Shortlist, 19

Tags: trigonometry , geometry , geometric inequality , trigonometric inequality

Let $\alpha,\beta,\gamma$ be positive real numbers such that $\alpha+\beta+\gamma < \pi$, $\alpha+\beta > \gamma$,$ \beta+\gamma > \alpha$, $\gamma + \alpha > \beta.$ Prove that with the segments of lengths $\sin \alpha, \sin \beta, \sin \gamma $ we can construct a triangle and that its area is not greater than \[A=\dfrac 18\left( \sin 2\alpha+\sin 2\beta+ \sin 2\gamma \right).\] [i]Proposed by Soviet Union[/i]

1967 IMO, 1

Tags: trigonometry , parallelogram , geometric inequality , geometry , trigonometric inequality

The parallelogram $ABCD$ has $AB=a,AD=1,$ $\angle BAD=A$, and the triangle $ABD$ has all angles acute. Prove that circles radius $1$ and center $A,B,C,D$ cover the parallelogram if and only \[a\le\cos A+\sqrt3\sin A.\]

1974 IMO Longlists, 15

Tags: circumcircle , geometry , trigonometry , triangle , trigonometric inequality

Let $ABC$ be a triangle. Prove that there exists a point $D$ on the side $AB$ of the triangle $ABC$, such that $CD$ is the geometric mean of $AD$ and $DB$, iff the triangle $ABC$ satisfies the inequality $\sin A\sin B\le\sin^2\frac{C}{2}$. [hide="Comment"][i]Alternative formulation, from IMO ShortList 1974, Finland 2:[/i] We consider a triangle $ABC$. Prove that: $\sin(A) \sin(B) \leq \sin^2 \left( \frac{C}{2} \right)$ is a necessary and sufficient condition for the existence of a point $D$ on the segment $AB$ so that $CD$ is the geometrical mean of $AD$ and $BD$.[/hide]