Olympiad problems

1967 IMO Shortlist, 2

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

2010 Korea Junior Math Olympiad, 5

Tags: trigonometry , trigonometric inequality , algebra , inequalities

If reals $x, y, z $ satises $tan x + tan y + tan z = 2$ and $0 < x, y,z < \frac{\pi}{2}.$ Prove that $$sin^2 x + sin^2 y + sin^2 z < 1.$$

1977 IMO, 1

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 $A^2 + B^2 \leq 1.$

1987 IMO Longlists, 68

Tags: geometry , trigonometry , 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]

2012 IFYM, Sozopol, 3

Tags: algebra , trigonometric inequality

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

1967 IMO, 1

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

2010 Korea Junior Math Olympiad, 5

1977 IMO, 1

1987 IMO Longlists, 68

2012 IFYM, Sozopol, 3

1967 IMO, 1

1974 IMO Longlists, 15

2017 Romania National Olympiad, 3

1967 IMO Longlists, 55

1994 All-Russian Olympiad Regional Round, 11.1

2010 Victor Vâlcovici, 2