Prove that if $x,y,z >1$ and $\frac 1x +\frac 1y +\frac 1z = 2$, then \[\sqrt{x+y+z} \geq \sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\]

The incircle $I$ of $\triangle ABC$ is tangent to sides $AB,BC,CA$ at $D,E,F$ respectively. Line $EF$ intersects lines $AI,BI,DI$ at $M,N,K$ respectively. Prove that $DM\cdot KE=DN\cdot KF$.

Let $a$ be a real number. Evaluate \[\int _{-\pi+a}^{3\pi+a} |x-a-\pi|\sin \left(\frac{x}{2}\right)dx\]

Determine the absolute value of the sum \[ \lfloor 2013\sin{0^\circ} \rfloor + \lfloor 2013\sin{1^\circ} \rfloor + \cdots + \lfloor 2013\sin{359^\circ} \rfloor, \] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. (You may use the fact that $\sin{n^\circ}$ is irrational for positive integers $n$ not divisible by $30$.) [i]Ray Li[/i]

Find the smallest natural number $n$ for which the equality $\sin n^o= \sin (1997n)^o$ holds.

In a triangle $ABC$ with $B = 90^\circ$, $D$ is a point on the segment $BC$ such that the inradii of triangles $ABD$ and $ADC$ are equal. If $\widehat{ADB} = \varphi$ then prove that $\tan^2 (\varphi/2) = \tan (C/2)$.

Square $ ABCD$ has side length $ 2$. A semicircle with diameter $ \overline{AB}$ is constructed inside the square, and the tangent to the semicricle from $ C$ intersects side $ \overline{AD}$ at $ E$. What is the length of $ \overline{CE}$? [asy] defaultpen(linewidth(0.8)); pair A=origin, B=(1,0), C=(1,1), D=(0,1), X=tangent(C, (0.5,0), 0.5, 1), F=C+2*dir(C--X), E=intersectionpoint(C--F, A--D); draw(C--D--A--B--C--E); draw(Arc((0.5,0), 0.5, 0, 180)); pair point=(0.5,0.5); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E));[/asy] $ \textbf{(A)}\ \frac {2 \plus{} \sqrt5}{2} \qquad \textbf{(B)}\ \sqrt 5 \qquad \textbf{(C)}\ \sqrt 6 \qquad \textbf{(D)}\ \frac52 \qquad \textbf{(E)}\ 5 \minus{} \sqrt5$

Let $ABCDEF$ be a convex hexagon of area $1$, whose opposite sides are parallel. The lines $AB$, $CD$ and $EF$ meet in pairs to determine the vertices of a triangle. Similarly, the lines $BC$, $DE$ and $FA$ meet in pairs to determine the vertices of another triangle. Show that the area of at least one of these two triangles is at least $3/2$.

Eight spheres of radius 100 are placed on a flat surface so that each sphere is tangent to two others and their centers are the vertices of a regular octagon. A ninth sphere is placed on the flat surface so that it is tangent to each of the other eight spheres. The radius of this last sphere is $a+b\sqrt{c},$ where $a, b,$ and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a+b+c.$

When $4 \cos \theta - 3 \sin \theta = \tfrac{13}{3},$ it follows that $7 \cos 2\theta - 24 \sin 2\theta = \tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

In $\triangle ABC$ with circumcenter $O$, $\measuredangle A = 45^\circ$. Denote by $X$ the second intersection of $\overrightarrow{AO}$ with the circumcircle of $\triangle BOC$. Compute the area of quadrilateral $ABXC$ if $BX = 8$ and $CX = 15$. [i]Proposed by Aaron Lin[/i]

Let $ABCD$ be convex quadrilateral, such that exist $M,N$ inside $ABCD$ for which $\angle NAD= \angle MAB; \angle NBC= \angle MBA; \angle MCB=\angle NCD; \angle NDA=\angle MDC$ Prove, that $S_{ABM}+S_{ABN}+S_{CDM}+S_{CDN}=S_{BCM}+S_{BCN}+S_{ADM}+S_{ADN}$, where $S_{XYZ}$-area of triangle $XYZ$

In $ ABC$ we have $ AB \equal{} 25$, $ BC \equal{} 39$, and $ AC \equal{} 42$. Points $ D$ and $ E$ are on $ AB$ and $ AC$ respectively, with $ AD \equal{} 19$ and $ AE \equal{} 14$. What is the ratio of the area of triangle $ ADE$ to the area of quadrilateral $ BCED$? $ \textbf{(A)}\ \frac{266}{1521}\qquad \textbf{(B)}\ \frac{19}{75}\qquad \textbf{(C)}\ \frac{1}{3}\qquad \textbf{(D)}\ \frac{19}{56}\qquad \textbf{(E)}\ 1$

Let $ ABC$ be an arbitrary acute angled triangle. For any point $ P$ lying within the triangle, let $ D$, $ E$, $ F$ denote the feet of the perpendiculars from $ P$ onto the sides $ AB$, $ BC$, $ CA$ respectively. Determine the set of all possible positions of the point $ P$ for which the triangle $ DEF$ is isosceles. For which position of $ P$ will the triangle $ DEF$ become equilateral?

Evaluate the integrals as follows. (1) $\int \frac{x^2}{2-x}\ dx$ (2) $\int \sqrt[3]{x^5+x^3}\ dx$ (3) $\int_0^1 (1-x)\cos \pi x\ dx$

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 with sides $a$, $b$, $c$ and (corresponding) angles $A$, $B$, $C$. Prove that if $3A + 2B = 180^{\circ}$, then $a^2+bc=c^2$. [b]Additional problem:[/b] Prove that the converse also holds, i. e. prove the following: Let $ABC$ be an arbitrary triangle. Then, $3A + 2B = 180^{\circ}$ if and only if $a^2+bc=c^2$. [b]Similar problem:[/b] Let $ABC$ be an arbitrary triangle. Then, $3A + 2B = 360^{\circ}$ if and only if $a^2-bc=c^2$.

In triangle $ ABC$, $ AB \equal{} AC \equal{} 100$, and $ BC \equal{} 56$. Circle $ P$ has radius $ 16$ and is tangent to $ \overline{AC}$ and $ \overline{BC}$. Circle $ Q$ is externally tangent to $ P$ and is tangent to $ \overline{AB}$ and $ \overline{BC}$. No point of circle $ Q$ lies outside of $ \triangle ABC$. The radius of circle $ Q$ can be expressed in the form $ m \minus{} n\sqrt {k}$, where $ m$, $ n$, and $ k$ are positive integers and $ k$ is the product of distinct primes. Find $ m \plus{} nk$.

Let $ H$ be the orthocenter of an acute-angled triangle $ ABC$. The circle $ \Gamma_{A}$ centered at the midpoint of $ BC$ and passing through $ H$ intersects the sideline $ BC$ at points $ A_{1}$ and $ A_{2}$. Similarly, define the points $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$. Prove that the six points $ A_{1}$, $ A_{2}$, $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$ are concyclic. [i]Author: Andrey Gavrilyuk, Russia[/i]

Convex quadrilateral $ ABCD$ has $ AB\equal{}9$ and $ CD\equal{}12$. Diagonals $ AC$ and $ BD$ intersect at $ E$, $ AC\equal{}14$, and $ \triangle AED$ and $ \triangle BEC$ have equal areas. What is $ AE$? $ \textbf{(A)}\ \frac{9}{2}\qquad \textbf{(B)}\ \frac{50}{11}\qquad \textbf{(C)}\ \frac{21}{4}\qquad \textbf{(D)}\ \frac{17}{3}\qquad \textbf{(E)}\ 6$

Given a triangle $ ABC$ with $ \angle A \equal{} 90^{\circ}$ and $ AB \neq AC$. The points $ D$, $ E$, $ F$ lie on the sides $ BC$, $ CA$, $ AB$, respectively, in such a way that $ AFDE$ is a square. Prove that the line $ BC$, the line $ FE$ and the line tangent at the point $ A$ to the circumcircle of the triangle $ ABC$ intersect in one point.

Let triangle $ ABC$ acutangle, with $ m \angle ACB\leq\ m \angle ABC$. $ M$ the midpoint of side $ BC$ and $ P$ a point over the side $ MC$. Let $ C_{1}$ the circunference with center $ C$. Let $ C_{2}$ the circunference with center $ B$. $ P$ is a point of $ C_{1}$ and $ C_{2}$. Let $ X$ a point on the opposite semiplane than $ B$ respecting with the straight line $ AP$; Let $ Y$ the intersection of side $ XB$ with $ C_{2}$ and $ Z$ the intersection of side $ XC$ with $ C_{1}$. Let $ m\angle PAX \equal{} \alpha$ and $ m\angle ABC \equal{} \beta$. Find the geometric place of $ X$ if it satisfies the following conditions: $ (a) \frac {XY}{XZ} \equal{} \frac {XC \plus{} CP}{XB \plus{} BP}$ $ (b) \cos(\alpha) \equal{} AB\cdot \frac {\sin(\beta )}{AP}$

If $\tan x + \tan y = 4$ and $\cot x + \cot y = 5$, compute $\tan(x + y)$.

Let $ABC$ be an equilateral triangle, and $P$ be an arbitrary point within the triangle. Perpendiculars $PD,PE,PF$ are drawn to the three sides of the triangle. Show that, no matter where $P$ is chosen, \[ \frac{PD+PE+PF}{AB+BC+CA}=\frac{1}{2\sqrt{3}}. \]

Let $ ABC$ be a triangle with $ \angle A = 60^{\circ}$.Prove that if $ T$ is point of contact of Incircle And Nine-Point Circle, Then $ AT = r$, $ r$ being inradius.