Determine all values of $x$ in the interval $0 \leq x \leq 2\pi$ which satisfy the inequality \[ 2 \cos{x} \leq \sqrt{1+\sin{2x}}-\sqrt{1-\sin{2x}} \leq \sqrt{2}. \]

[asy] size(100); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((0,1)--(1,0)); draw((0,0)--(.5,sqrt(3)/2)--(1,0)); label("$A$",(0,0),SW); label("$B$",(1,0),SE); label("$C$",(1,1),NE); label("$D$",(0,1),NW); label("$E$",(.5,sqrt(3)/2),E); label("$F$",intersectionpoint((0,0)--(.5,sqrt(3)/2),(0,1)--(1,0)),2W); //Credit to chezbgone2 for the diagram[/asy] Vertex $E$ of equilateral triangle $ABE$ is in the interior of square $ABCD$, and $F$ is the point of intersection of diagonal $BD$ and line segment $AE$. If length $AB$ is $\sqrt{1+\sqrt{3}}$ then the area of $\triangle ABF$ is $\textbf{(A) }1\qquad\textbf{(B) }\frac{\sqrt{2}}{2}\qquad\textbf{(C) }\frac{\sqrt{3}}{2}$ $\qquad\textbf{(D) }4-2\sqrt{3}\qquad \textbf{(E) }\frac{1}{2}+\frac{\sqrt{3}}{4}$

Let $ABC$ be an acute angle triangle. Suppose that $D,E,F$ are the feet of perpendicluar lines from $A,B,C$ to $BC,CA,AB$. Let $P,Q,R$ be the feet of perpendicular lines from $A,B,C$ to $EF,FD,DE$. Prove that \[ 2(PQ+QR+RP)\geq DE+EF+FD \]

Let $ABC$ be an equilateral triangle and $D$, $E$ points on the sides $[AB]$ and $[AC]$ respectively. If $DF$, $EF$ (with $F\in AE$, $G\in AD$) are the interior angle bisectors of the angles of the triangle $ADE$, prove that the sum of the areas of the triangles $DEF$ and $DEG$ is at most equal with the area of the triangle $ABC$. When does the equality hold? [i]Greece[/i]

11.6 Construct for each vertex of the quadrilateral of area $S$ a symmetric point wrt to the diagonal, which doesn't contain this vertex. Let $S'$ be an area of the obtained quadrilateral. Prove that $\frac{S'}{S}<3$. ([i]L. Emel'yanov[/i])

The tangents at $B$ and $A$ to the circumcircle of an acute angled triangle $ABC$ meet the tangent at $C$ at $T$ and $U$ respectively. $AT$ meets $BC$ at $P$, and $Q$ is the midpoint of $AP$; $BU$ meets $CA$ at $R$, and $S$ is the midpoint of $BR$. Prove that $\angle ABQ=\angle BAS$. Determine, in terms of ratios of side lengths, the triangles for which this angle is a maximum.

$a, b$ and $c$ are sides of a triangle, and $\gamma$ is its angle opposite $c$. Prove that $c \ge (a + b) \sin \frac{\gamma}{2}$ (V. Prasolov )

Find the volume of the region of points $(x,y,z)$ such that \[\left(x^{2}+y^{2}+z^{2}+8\right)^{2}\le 36\left(x^{2}+y^{2}\right). \]

The cosines of the angles of one triangle are respectively equal to the sines of the angles of the other triangle. Find the largest of these six angles of triangles.

In a triangle $ABC$, let $I$ denote its incenter. Points $D, E, F$ are chosen on the segments $BC, CA, AB$, respectively, such that $BD + BF = AC$ and $CD + CE = AB$. The circumcircles of triangles $AEF, BFD, CDE$ intersect lines $AI, BI, CI$, respectively, at points $K, L, M$ (different from $A, B, C$), respectively. Prove that $K, L, M, I$ are concyclic.

Consider the function \[f(x)= \sin \biggl( \frac{\pi}{2} \lfloor x \rfloor \biggr).\] Find the period of $f$ and sketch diagram of $f$ in one period. Also prove that $\lim_{x \to 1} f(x)$ does not exist.

Let $AXYBZ$ be a convex pentagon inscribed in a circle with diameter $\overline{AB}$. The tangent to the circle at $Y$ intersects lines $BX$ and $BZ$ at $L$ and $K$, respectively. Suppose that $\overline{AY}$ bisects $\angle LAZ$ and $AY=YZ$. If the minimum possible value of \[ \frac{AK}{AX} + \left( \frac{AL}{AB} \right)^2 \] can be written as $\tfrac{m}{n} + \sqrt{k}$, where $m$, $n$ and $k$ are positive integers with $\gcd(m,n)=1$, compute $m+10n+100k$. [i]Proposed by Evan Chen[/i]

Consider triangle $ABC$ with $\angle A=2\angle B$. The angle bisectors from $A$ and $C$ intersect at $D$, and the angle bisector from $C$ intersects $\overline{AB}$ at $E$. If $\dfrac{DE}{DC}=\dfrac13$, compute $\dfrac{AB}{AC}$.

Let $-2 < x_1 < 2$ be a real number and define $x_2, x_3, \ldots$ by $x_{n+1} = x_n^2-2$ for $n \geq 1$. Assume that no $x_n$ is $0$ and define a number $A$, $0 \leq A \leq 1$ in the following way: The $n^{\text{th}}$ digit after the decimal point in the binary representation of $A$ is a $0$ if $x_1x_2\cdots x_n$ is positive and $1$ otherwise. Prove that $A = \frac{1}{\pi}\cos^{-1}\left(\frac{x_1}{2}\right)$. [i]Evan O' Dorney.[/i]

Let $ABC$ be an acute triangle and let $\Gamma$ be its circumcircle. The bisector of $\angle{A}$ intersects $BC$ at $D$, $\Gamma$ at $K$ (different from $A$), and the line through $B$ tangent to $\Gamma$ at $X$. Show that $K$ is the midpoint of $AX$ if and only if $\frac{AD}{DC}=\sqrt{2}$.

Given $a, \theta \in \mathbb R, m \in \mathbb N$, and $P(x) = x^{2m}- 2|a|^mx^m \cos \theta +a^{2m}$, factorize $P(x)$ as a product of $m$ real quadratic polynomials.

For real number $ a$ with $ |a|>1$, evaluate $ \int_0^{2\pi} \frac{d\theta}{(a\plus{}\cos \theta)^2}$.

Let $\alpha, \beta$ and $\gamma$ denote the angles of the triangle $ABC$. The perpendicular bisector of $AB$ intersects $BC$ at the point $X$, the perpendicular bisector of $AC$ intersects it at $Y$. Prove that $\tan(\beta) \cdot \tan(\gamma) = 3$ implies $BC= XY$ (or in other words: Prove that a sufficient condition for $BC = XY$ is $\tan(\beta) \cdot \tan(\gamma) = 3$). Show that this condition is not necessary, and give a necessary and sufficient condition for $BC = XY$.

Prove that, if $\tan (a/2)$ is rational (or else, if $ a$ is an odd multiple of $\pi$ so that $\tan (a/2)$ is not defined), then $\cos a$ and $\sin a$ are rational. And, conversely, if $\cos a$ and $\sin a$ are rational, then $\tan (a/2)$ is rational unless $a$ is an odd multiple of $\pi$ so that $\tan (a/2)$ is not defined.

Let $\triangle ABC$ be a triangle with $AB = 7$, $BC = 8$, and $AC = 9$. Point $D$ is on side $\overline{AC}$ such that $\angle CBD$ has measure $45^\circ$. What is the length of $\overline{BD}$?

Let $ABC$ be a triangle. Let $D,E$ be points on the segment $BC$ such that $BD=DE=EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AD$ in $P$ and $AE$ in $Q$ respectively. Determine the ratio of the area of the triangle $APQ$ to that of the quadrilateral $PDEQ$.

Find $\lim_{n\to\infty} \int_0^{\pi} x|\sin 2nx| dx\ (n=1,\ 2,\ \cdots)$. [i]1992 Japan Women's University entrance exam/Physics, Mathematics[/i]

Triangle $ABC$ is given. Prove that $\frac{R}{r} > \frac{a}{h}$, where $R$ is the radius of the circumscribed circle, $r$ is the radius of the inscribed circle, $a$ is the length of the longest side, $h$ is the length of the shortest altitude.

Provided a convex equilateral pentagon. On every side of the pentagon We construct equilateral triangles which run through the interior of the pentagon. Prove that at least one of the triangles does not protrude the pentagon's boundary.

An isosceles trapezoid has legs and shorter base of length $1$. Find the maximum possible value of its area