Let $z_1, z_2, z_3$ be pairwise distinct complex numbers satisfying $|z_1| = |z_2| = |z_3| = 1$ and \[\frac{1}{2 + |z_1 + z_2|}+\frac{1}{2 + |z_2 + z_3|}+\frac{1}{2 + |z_3 + z_1|} =1.\] If the points $A(z_1),B(z_2),C(z_3)$ are vertices of an acute-angled triangle, prove that this triangle is equilateral.

Prove the following inequality. \[ \frac{\pi}{4}\sqrt{\frac{3}{2}\plus{}\sqrt{2}}<\int_0^{\frac{\pi}{2}} \sqrt{1\minus{}\frac 12\sin ^ 2 x}\ dx<\frac{\sqrt{3}}{4}\pi\]

A triangle $ABC$ is inscribed in a circle $\omega$. A variable line $\ell$ chosen parallel to $BC$ meets segments $AB$, $AC$ at points $D$, $E$ respectively, and meets $\omega$ at points $K$, $L$ (where $D$ lies between $K$ and $E$). Circle $\gamma_1$ is tangent to the segments $KD$ and $BD$ and also tangent to $\omega$, while circle $\gamma_2$ is tangent to the segments $LE$ and $CE$ and also tangent to $\omega$. Determine the locus, as $\ell$ varies, of the meeting point of the common inner tangents to $\gamma_1$ and $\gamma_2$. [i](Russia) Vasily Mokin and Fedor Ivlev[/i]

In the diagram below, angle $ABC$ is a right angle. Point $D$ is on $\overline{BC}$, and $\overline{AD}$ bisects angle $CAB$. Points $E$ and $F$ are on $\overline{AB}$ and $\overline{AC}$, respectively, so that $AE=3$ and $AF=10.$ Given that $EB=9$ and $FC=27$, find the integer closest to the area of quadrilateral $DCFG.$ [asy] size(250); pair A=(0,12), E=(0,8), B=origin, C=(24*sqrt(2),0), D=(6*sqrt(2),0), F=A+10*dir(A--C), G=intersectionpoint(E--F, A--D); draw(A--B--C--A--D^^E--F); pair point=G+1*dir(250); 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)); label("$F$", F, dir(point--F)); label("$G$", G, dir(point--G)); markscalefactor=0.1; draw(rightanglemark(A,B,C)); label("10", A--F, dir(90)*dir(A--F)); label("27", F--C, dir(90)*dir(F--C)); label("3", (0,10), W); label("9", (0,4), W);[/asy]

Given an acute triangle $ ABC$. The incircle of triangle $ ABC$ touches $ BC,CA,AB$ respectively at $ D,E,F$. The angle bisector of $ \angle A$ cuts $ DE$ and $ DF$ respectively at $ K$ and $ L$. Suppose $ AA_1$ is one of the altitudes of triangle $ ABC$, and $ M$ be the midpoint of $ BC$. (a) Prove that $ BK$ and $ CL$ are perpendicular with the angle bisector of $ \angle BAC$. (b) Show that $ A_1KML$ is a cyclic quadrilateral.

For a complex number $z=1+2\sqrt{6}i$ and natural number $n=1,\ 2,\ 3,\ \cdots$, express the complex number $z^n$ in using real numbers $a_n,\ b_n$ as $z^n=a_n+b_ni$. Answer the following questions. (1) Show that $a_n^2+b_n^2=5^{2n}\ (n=1,\ 2,\ 3,\ \cdots).$ (2) Find the constants $p,\ q$ such that $a_{n+2}=pa_{n+1}+qa_n$ holds for all $n$. (3) Show that $a_n$ is not a multiple of $5$ for any $n$. (4) Show that $z^n\ (n=1,\ 2,\ 3,\ \cdots)$ is not a real number.

Let $n$ be positive integers and $t$ be a positive real number. Evaluate $\int_0^{\frac{2n}{t}\pi} |x\sin\ tx|\ dx.$

Let $p, q, r, s, t \in \mathbb{R}^{*}_{+}$ satisfying: i) $p^2 + pq + q^2 = s^2$ ii) $q^2 + qr + r^2 = t^2$ iii) $r^2 + rp + p^2 = s^2 - st + t^2$ Prove that \[\frac{s^2 - st + t^2}{s^2t^2} = \frac{r^2}{q^2t^2} + \frac{p^2}{q^2s^2} - \frac{pr}{q^2ts}\]

Evaluate $$\int_{0}^{ \pi \slash 2} \frac{ dx}{1+( \tan x)^{\sqrt{2}} }\;.$$

Evaluate $\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{2+\sin x}{1+\cos x}\ dx.$

For $i=1,2,$ let $T_i$ be a triangle with side length $a_i,b_i,c_i,$ and area $A_i.$ Suppose that $a_1\le a_2, b_1\le b_2, c_1\le c_2,$ and that $T_2$ is an acute triangle. Does it follow that $A_1\le A_2$?

Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic. [i]Proposed by John Cuya, Peru[/i]

Prove that for every natural number $n$, and for every real number $x \neq \frac{k\pi}{2^t}$ ($t=0,1, \dots, n$; $k$ any integer) \[ \frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx} \]

Let $ ABC$ be a triangle with $ AB \equal{} 5$, $ BC \equal{} 4$ and $ AC \equal{} 3$. Let $ \mathcal P$ and $ \mathcal Q$ be squares inside $ ABC$ with disjoint interiors such that they both have one side lying on $ AB$. Also, the two squares each have an edge lying on a common line perpendicular to $ AB$, and $ \mathcal P$ has one vertex on $ AC$ and $ \mathcal Q$ has one vertex on $ BC$. Determine the minimum value of the sum of the areas of the two squares. [asy]import olympiad; import math; import graph; unitsize(1.5cm); pair A, B, C; A = origin; B = A + 5 * right; C = (9/5, 12/5); pair X = .7 * A + .3 * B; pair Xa = X + dir(135); pair Xb = X + dir(45); pair Ya = extension(X, Xa, A, C); pair Yb = extension(X, Xb, B, C); pair Oa = (X + Ya)/2; pair Ob = (X + Yb)/2; pair Ya1 = (X.x, Ya.y); pair Ya2 = (Ya.x, X.y); pair Yb1 = (Yb.x, X.y); pair Yb2 = (X.x, Yb.y); draw(A--B--C--cycle); draw(Ya--Ya1--X--Ya2--cycle); draw(Yb--Yb1--X--Yb2--cycle); label("$A$", A, W); label("$B$", B, E); label("$C$", C, N); label("$\mathcal P$", Oa, origin); label("$\mathcal Q$", Ob, origin);[/asy]

Let the circumcenter of triangle $ABC$ be $O$. $H_A$ is the projection of $A$ onto $BC$. The extension of $AO$ intersects the circumcircle of $BOC$ at $A'$. The projections of $A'$ onto $AB, AC$ are $D,E$, and $O_A$ is the circumcentre of triangle $DH_AE$. Define $H_B, O_B, H_C, O_C$ similarly. Prove: $H_AO_A, H_BO_B, H_CO_C$ are concurrent

Prove that for every natural $n$ the following inequality is valid $$|\sin 1| + |\sin 2| + |\sin (3n-1)| + |\sin 3n| > \frac{8n}{5}$$

On the coordinate plane, find the area of the part enclosed by the curve $C: (a+x)y^2=(a-x)x^2\ (x\geq 0)$ for $a>0$.

Consider $\omega$ is circumcircle of an acute triangle $ABC$. $D$ is midpoint of arc $BAC$ and $I$ is incenter of triangle $ABC$. Let $DI$ intersect $BC$ in $E$ and $\omega$ for second time in $F$. Let $P$ be a point on line $AF$ such that $PE$ is parallel to $AI$. Prove that $PE$ is bisector of angle $BPC$. [i]Proposed by Mr.Etesami[/i]

A thin, uniform rod has mass $m$ and length $L$. Let the acceleration due to gravity be $g$. Let the rotational inertia of the rod about its center be $md^2$. The rod is suspended from a distance $kd$ from the center, and undergoes small oscillations with an angular frequency $\beta \sqrt{\frac{g}{d}}$. Find an expression for $\beta$ in terms of $k$. $ \textbf{(A)}\ 1+k^2$ $ \textbf{(B)}\ \sqrt{1+k^2}$ $ \textbf{(C)}\ \sqrt{\frac{k}{1+k}}$ $ \textbf{(D)}\ \sqrt{\frac{k^2}{1+k}}$ $ \textbf{(E)}\ \text{none of the above}$

Prove that for any triangle $ABC$, the inequality $\displaystyle\sum_{\text{cyclic}}\cos A\le\sum_{\text{cyclic}}\sin (A/2)$ holds.

A circle has center on the side $AB$ of the cyclic quadrilateral $ABCD$. The other three sides are tangent to the circle. Prove that $AD+BC=AB$.

Let $ P$ be a point in the interior of the equilateral triangle $ \triangle{}ABC$ such that $ PA \equal{} 5$, $ PB \equal{} 7$, $ PC \equal{} 8$. Find the length of the side of the triangle $ ABC$.

Let $ABC$ be an acute-angled triangle with $AB<AC$ and let $O$ be the centre of its circumcircle $\omega$. Let $D$ be a point on the line segment $BC$ such that $\angle BAD = \angle CAO$. Let $E$ be the second point of intersection of $\omega$ and the line $AD$. If $M$, $N$ and $P$ are the midpoints of the line segments $BE$, $OD$ and $AC$, respectively, show that the points $M$, $N$ and $P$ are collinear.

Two beads, each of mass $m$, are free to slide on a rigid, vertical hoop of mass $m_h$. The beads are threaded on the hoop so that they cannot fall off of the hoop. They are released with negligible velocity at the top of the hoop and slide down to the bottom in opposite directions. The hoop remains vertical at all times. What is the maximum value of the ratio $m/m_h$ such that the hoop always remains in contact with the ground? Neglect friction. [asy] pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((0,0)--(3,0)); draw(circle((1.5,1),1)); filldraw(circle((1.4,1.99499),0.1), gray(.3)); filldraw(circle((1.6,1.99499),0.1), gray(.3)); [/asy]

A circle $ \omega$ with center $ O$ is tangent to the rays of an angle $ BAC$ at $ B$ and $ C$. Point $ Q$ is taken inside the angle $ BAC$. Assume that point $ P$ on the segment $ AQ$ is such that $ AQ\perp OP$. The line $ OP$ intersects the circumcircles $ \omega_{1}$ and $ \omega_{2}$ of triangles $ BPQ$ and $ CPQ$ again at points $ M$ and $ N$. Prove that $ OM \equal{} ON$.