Olympiad problems

2002 USA Team Selection Test, 1

Let $ ABC$ be a triangle, and $ A$, $ B$, $ C$ its angles. Prove that \[ \sin\frac{3A}{2}+\sin\frac{3B}{2}+\sin\frac{3C}{2}\leq \cos\frac{A-B}{2}+\cos\frac{B-C}{2}+\cos\frac{C-A}{2}. \]

1967 IMO Longlists, 3

Tags: trigonometry , calculus , Taylor series , Inequality , Trigonometric inequality , IMO Shortlist , IMO Longlist

Prove the trigonometric inequality $\cos x < 1 - \frac{x^2}{2} + \frac{x^4}{16},$ when $x \in \left(0, \frac{\pi}{2} \right).$

1967 IMO Longlists, 8

Tags: geometry , parallelogram , trigonometry , geometric inequality , Trigonometric inequality , IMO , IMO 1967

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 \[a\le\cos A+\sqrt3\sin A.\]

2008 IMAC Arhimede, 3

Tags: inequalities , trigonometry , Trigonometric inequality

Let $ 0 \leq x \leq 2\pi$. Prove the inequality $ \sqrt {\frac {\sin^{2}x}{1 + \cos^{2}x}} + \sqrt {\frac {\cos^{2}x}{1 + \sin^{2}x}}\geq 1 $

1967 IMO Longlists, 55

Tags: trigonometry , Trigonometric inequality , Inequality , algebra , IMO Shortlist , IMO Longlist

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: geometry , trigonometry , geometric inequality , Trigonometric inequality , IMO Shortlist

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]

2008 Mathcenter Contest, 3

Tags: geometry , trigonometry , Trigonometric inequality , inequalities , geometric inequality

Let $ABC$ be a triangle whose side lengths are opposite the angle $A,B,C$ are $a,b,c$ respectively. Prove that $$\frac{ab\sin{2C}+bc\sin{ 2A}+ca\sin{2B}}{ab+bc+ca}\leq\frac{\sqrt{3}}{2}$$. [i](nooonuii)[/i]

2017 Romania National Olympiad, 3

Tags: inequalities , Trigonometric inequality

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

1967 IMO Shortlist, 3

Tags: trigonometry , calculus , Taylor series , Inequality , Trigonometric inequality , IMO Shortlist , IMO Longlist

Prove the trigonometric inequality $\cos x < 1 - \frac{x^2}{2} + \frac{x^4}{16},$ when $x \in \left(0, \frac{\pi}{2} \right).$

2012 IFYM, Sozopol, 3

Tags: algebra , Trigonometric inequality

Prove the following inequality: $tan \, 1>\frac{3}{2}$.

1967 IMO Shortlist, 2

Tags: trigonometry , Inequality , Trigonometric inequality , minimum , IMO Shortlist , IMO Longlist

Let $n$ and $k$ be positive integers such that $1 \leq n \leq N+1$, $1 \leq k \leq N+1$. Show that: \[ \min_{n \neq k} |\sin n - \sin k| < \frac{2}{N}. \]