Olympiad problems

1994 Tuymaada Olympiad, 5

Find the smallest natural number $n$ for which $sin \Big(\frac{1}{n+1934}\Big)<\frac{1}{1994}$ .

1975 IMO Shortlist, 12

Tags: trigonometry , trigonometric inequality , trigonometric identities , inequality , optimization

Consider on the first quadrant of the trigonometric circle the arcs $AM_1 = x_1,AM_2 = x_2,AM_3 = x_3, \ldots , AM_v = x_v$ , such that $x_1 < x_2 < x_3 < \cdots < x_v$. Prove that \[\sum_{i=0}^{v-1} \sin 2x_i - \sum_{i=0}^{v-1} \sin (x_i- x_{i+1}) < \frac{\pi}{2} + \sum_{i=0}^{v-1} \sin (x_i + x_{i+1})\]

1974 IMO Shortlist, 10

Tags: trigonometry , circumcircle , 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]

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} $.

2010 Laurențiu Panaitopol, Tulcea, 4

Tags: trigonometric inequality , inequalities , trigonometry

Let be a natural number $ n, $ and $ n $ real numbers $ a_1,a_2,\ldots ,a_n . $ Then, $$ \sum_{1\le i<j\le n} \cos\left( a_i-a_j \right)\ge -n/2. $$

2007 Peru MO (ONEM), 1

Tags: inequalities , algebra , trigonometric inequality , trigonometry

Find all values of $A$ such that $0^o < A < 360^o$ and also $\frac{\sin A}{\cos A - 1} \ge 1$ and $\frac{3\cos A - 1}{\sin A} \ge 1.$

Indonesia Regional MO OSP SMA - geometry, 2012.4

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

Given an acute triangle $ABC$. Point $H$ denotes the foot of the altitude drawn from $A$. Prove that $$AB + AC \ge BC cos \angle BAC + 2AH sin \angle BAC$$

2002 USA Team Selection Test, 1

Tags: inequalities , trigonometry , trigonometric inequality , function

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

1974 IMO, 2

Tags: trigonometry , circumcircle , 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]

1967 IMO Shortlist, 2

Tags: trigonometry , inequality , trigonometric inequality , minimum

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

1977 IMO Longlists, 20

Tags: function , trigonometry , trigonometric inequality , inequality

Let $a,b,A,B$ be given reals. We consider the function defined by \[ f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x). \] Prove that if for any real number $x$ we have $f(x) \geq 0$ then $a^2 + b^2 \leq 2$ and $A^2 + B^2 \leq 1.$