Found problems: 36
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.
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
\[a\le\cos A+\sqrt3\sin A.\]
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]
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
\[a\le\cos A+\sqrt3\sin A.\]
Prove that for every $ x $ such that $ \sin x \neq 0 $, exists natural $ n $ such that $ | \sin nx | \geq \frac {\sqrt {3}} {2} $.
Let $n$ and $k$ be positive integers such that $1 \leq n \leq N+1$, $1 \leq k \leq N+1$. Show that: \[ \min_{n \neq k} |\sin n - \sin k| < \frac{2}{N}. \]
Prove that for an arbitrary triangle $ABC$ : $sin \frac{A}{2} sin \frac{B}{2} sin \frac{C}{2} < \frac{1}{4}$.
Prove that for all $x \in \left( 0, \frac{\pi}{3} \right)$ inequality $sin2x+cosx>1$ holds.
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]
Find all $x$ for which, for all $n,$ \[\sum^n_{k=1} \sin {k x} \leq \frac{\sqrt{3}}{2}.\]
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]
Prove the following inequality: $tan \, 1>\frac{3}{2}$.
Let $n$ and $k$ be positive integers such that $1 \leq n \leq N+1$, $1 \leq k \leq N+1$. Show that: \[ \min_{n \neq k} |\sin n - \sin k| < \frac{2}{N}. \]
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
\[a\le\cos A+\sqrt3\sin A.\]
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
\[\sum_{i=0}^{v-1} \sin 2x_i - \sum_{i=0}^{v-1} \sin (x_i- x_{i+1}) < \frac{\pi}{2} + \sum_{i=0}^{v-1} \sin (x_i + x_{i+1})\]
$ \sin\frac{\pi }{4n}\ge \frac{\sqrt 2 }{2n} ,\quad \forall n\in\mathbb{N} $
Find all values of $A$ such that $0^o < A < 360^o$ and also
$\frac{\sin A}{\cos A - 1} \ge 1$ and $\frac{3\cos A - 1}{\sin A} \ge 1.$
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]
Given an acute triangle $ABC$. Point $H$ denotes the foot of the altitude drawn from $A$. Prove that $$AB + AC \ge BC cos \angle BAC + 2AH sin \angle BAC$$
Prove the trigonometric inequality $\cos x < 1 - \frac{x^2}{2} + \frac{x^4}{16},$ when $x \in \left(0, \frac{\pi}{2} \right).$
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 $A^2 + B^2 \leq 1.$
Let be a natural number $ n, $ and $ n $ real numbers $ a_1,a_2,\ldots ,a_n . $ Then,
$$ \sum_{1\le i<j\le n} \cos\left( a_i-a_j \right)\ge -n/2. $$
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 $A^2 + B^2 \leq 1.$
If reals $x, y, z $ satises $tan x + tan y + tan z = 2$ and $0 < x, y,z < \frac{\pi}{2}.$ Prove that
$$sin^2 x + sin^2 y + sin^2 z < 1.$$
Let $ ABC$ be a triangle, and $ A$, $ B$, $ C$ its angles. Prove that
\[ \sin\frac{3A}{2}+\sin\frac{3B}{2}+\sin\frac{3C}{2}\leq \cos\frac{A-B}{2}+\cos\frac{B-C}{2}+\cos\frac{C-A}{2}. \]