Evaluate $\int_{-\pi}^{\pi} \left(\sum_{k=1}^{2013} \sin kx\right)^2dx$.

Investigate the existence and uniqueness of the solution of the problem $yu_x = xu_y, u|_{x=1} =\cos y$ in a neighbourhood of the point $(1, y_0)$.

Inside the triangle $ ABC $ a point $ M $ is given. The line $ BM $ meets the side $ AC $ at $ N $. The point $ K $ is symmetrical to $ M $ with respect to $ AC $. The line $ BK $ meets $ AC $ at $ P $. If $ \angle AMP = \angle CMN $, prove that $ \angle ABP=\angle CBN $.

[b]a)[/b] Let be real numbers $ s,t\ge 0 $ and $ a,b\ge 1. $ Show that for any real $ x, $ it holds: $$ a^{s\sin x+t\cos x}b^{s\cos x+t\sin x}\le 10^{(s+t)\sqrt{\text{tg}^2 a+\text{tg}^2 b}} $$ [b]b)[/b] For $ a,b>0 $ is the above inequality still true? [i]Ilie Diaconu[/i]

Points $E$ and $C$ are chosen on a semicircle with diameter $AB$ and center $O$ such that $OE \perp AB$ and the intersection point $D$ of $AC$ and $OE$ is inside the semicircle. Find all values of $\angle{CAB}$ for which the quadrilateral $OBCD$ is tangent.

Suppose $a$ is a complex number such that \[a^2+a+\frac{1}{a}+\frac{1}{a^2}+1=0\] If $m$ is a positive integer, find the value of \[a^{2m}+a^m+\frac{1}{a^m}+\frac{1}{a^{2m}}\]

Let $a,b,c$ be positive real numbers such that $abc(a+b+c)=3.$ Prove that we have \[(a+b)(b+c)(c+a)\geq 8.\] Also determine the case of equality.

$(BEL 6)$ Evaluate $\left(\cos\frac{\pi}{4} + i \sin\frac{\pi}{4}\right)^{10}$ in two different ways and prove that $\dbinom{10}{1}-\dbinom{10}{3}+\frac{1}{2}\dbinom{10}{5}=2^4$

Let $P(x, y)$ be a non-constant homogeneous polynomial with real coefficients such that $P(\sin t, \cos t) = 1$ for every real number $t$. Prove that there exists a positive integer $k$ such that $P(x, y) = (x^2 + y^2)^k$.

In the $ x \minus{} y$ plane, let $ l$ be the tangent line at the point $ A\left(\frac {a}{2},\ \frac {\sqrt {3}}{2}b\right)$ on the ellipse $ \frac {x^2}{a^2} \plus{} \frac {y^2}{b^2}\equal{}1\ (0 < b < 1 < a)$. Let denote $ S$ be the area of the figure bounded by $ l,$ the $ x$ axis and the ellipse. (1) Find the equation of $ l$. (2) Express $ S$ in terms of $ a,\ b$. (3) Find the maximum value of $ S$ with the constraint $ a^2 \plus{} 3b^2 \equal{} 4$.

Given an acute angles triangle $ABC$, and $H$ is its orthocentre. The external bisector of the angle $\angle BHC$ meets the sides $AB$ and $AC$ at the points $D$ and $E$ respectively. The internal bisector of the angle $\angle BAC$ meets the circumcircle of the triangle $ADE$ again at the point $K$. Prove that $HK$ is through the midpoint of the side $BC$.

Let $n$ be a positive integer, $n\geq 2$. For each $t\in \mathbb{R}$, $t\neq k\pi$, $k\in\mathbb{Z}$, we consider the numbers \[ x_n(t) = \sum_{k=1}^n k(n-k)\cos{(tk)} \textrm{ and } y_n(t) = \sum_{k=1}^n k(n-k)\sin{(tk)}. \] Prove that if $x_n(t) = y_n(t) =0$ if and only if $\tan {\frac {nt}2} = n \tan {\frac t2}$. [i]Constantin Buse[/i]

Let $A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8$ be a regular octagon. Let $M_1$, $M_3$, $M_5$, and $M_7$ be the midpoints of sides $\overline{A_1 A_2}$, $\overline{A_3 A_4}$, $\overline{A_5 A_6}$, and $\overline{A_7 A_8}$, respectively. For $i = 1, 3, 5, 7$, ray $R_i$ is constructed from $M_i$ towards the interior of the octagon such that $R_1 \perp R_3$, $R_3 \perp R_5$, $R_5 \perp R_7$, and $R_7 \perp R_1$. Pairs of rays $R_1$ and $R_3$, $R_3$ and $R_5$, $R_5$ and $R_7$, and $R_7$ and $R_1$ meet at $B_1$, $B_3$, $B_5$, $B_7$ respectively. If $B_1 B_3 = A_1 A_2$, then $\cos 2 \angle A_3 M_3 B_1$ can be written in the form $m - \sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.

Show that the triangle whose angles satisfy the equality \[ \frac{sin^2(A) + sin^2(B) + sin^2(C)}{cos^2(A) + cos^2(B) + cos^2(C)} = 2 \] is a rectangular triangle.

Let $x, y, z$ be real numbers each of whose absolute value is different from $\frac{1}{\sqrt 3}$ such that $x + y + z = xyz$. Prove that \[\frac{3x - x^3}{1-3x^2} + \frac{3y - y^3}{1-3y^2} + \frac{3z -z^3}{1-3z^2} = \frac{3x - x^3}{1-3x^2} \cdot \frac{3y - y^3}{1-3y^2} \cdot \frac{3z - z^3}{1-3z^2}\]

If $\tan{\alpha}$ and $\tan{\beta}$ are the roots of $x^2 - px + q = 0$, and $\cot{\alpha}$ and $\cot{\beta}$ are the roots of $x^2 - rx + s = 0$, then $rs$ is necessarily $\text{(A)} \ pq \qquad \text{(B)} \ \frac{1}{pq} \qquad \text{(C)} \ \frac{p}{q^2} \qquad \text{(D)} \ \frac{q}{p^2} \qquad \text{(E)} \ \frac{p}{q}$

The complex numbers $z$ and $w$ satisfy $z^{13} = w$, $w^{11} = z$, and the imaginary part of $z$ is $\sin\left(\frac{m\pi}n\right)$ for relatively prime positive integers $m$ and $n$ with $m < n$. Find $n$.

Let $0 < \alpha, \beta, \gamma < \frac{\pi}{2}$ and $\sin^{3} \alpha + \sin^{3} \beta + \sin^3 \gamma = 1$. Prove that \[ \tan^{2} \alpha + \tan^{2} \beta + \tan^{2} \gamma \geq \frac{3 \sqrt{3}}{2} . \]

Given a triangle $ABC$. Find the minimum of \[\frac{\cos^{2}\frac{A}{2}\cos^{2}\frac{B}{2}}{\cos^{2}\frac{C}{2}}+\frac{\cos^{2}\frac{B}{2}\cos^{2}\frac{C}{2}}{\cos^{2}\frac{A}{2}}+\frac{\cos^{2}\frac{C}{2}\cos^{2}\frac{A}{2}}{\cos^{2}\frac{B}{2}}. \]

Let $ ABC$ be a triangle, and let the angle bisectors of its angles $ CAB$ and $ ABC$ meet the sides $ BC$ and $ CA$ at the points $ D$ and $ F$, respectively. The lines $ AD$ and $ BF$ meet the line through the point $ C$ parallel to $ AB$ at the points $ E$ and $ G$ respectively, and we have $ FG \equal{} DE$. Prove that $ CA \equal{} CB$. [i]Original formulation:[/i] Let $ ABC$ be a triangle and $ L$ the line through $ C$ parallel to the side $ AB.$ Let the internal bisector of the angle at $ A$ meet the side $ BC$ at $ D$ and the line $ L$ at $ E$ and let the internal bisector of the angle at $ B$ meet the side $ AC$ at $ F$ and the line $ L$ at $ G.$ If $ GF \equal{} DE,$ prove that $ AC \equal{} BC.$

Convex quadrilateral $ ABCD$ is inscribed in a circle, $ \angle{A}\equal{}60^o$, $ BC\equal{}CD\equal{}1$, rays $ AB$ and $ DC$ intersect at point $ E$, rays $ BC$ and $ AD$ intersect each other at point $ F$. It is given that the perimeters of triangle $ BCE$ and triangle $ CDF$ are both integers. Find the perimeter of quadrilateral $ ABCD$.

Let $ABC$ be a right triangle with $\angle A=90^\circ.$ Let $A'$ be the midpoint of $BC,$ $M$ be the midpoint of the height $AD$ and $P$ be the intersection of $BM$ and $AA'.$ Prove that $\tan\angle PCB=\sin C\cdot\cos C.$ [i]Daniel Văcărețu[/i]

Let $ABC$ be a triangle. Find all points $P$ on segment $BC$ satisfying the following property: If $X$ and $Y$ are the intersections of line $PA$ with the common external tangent lines of the circumcircles of triangles $PAB$ and $PAC$, then \[\left(\frac{PA}{XY}\right)^2+\frac{PB\cdot PC}{AB\cdot AC}=1.\]

For real $\theta_i$, $i = 1, 2, \dots, 2011$, where $\theta_1 = \theta_{2012}$, find the maximum value of the expression \[ \sum_{i=1}^{2011} \sin^{2012} \theta_i \cos^{2012} \theta_{i+1}. \] [i]Proposed by Lewis Chen [/i]

Prove that $\int_{0}^{\frac{\pi}{2}} arctg (1 - \sin^2x\cos^2x)dx = \frac{\pi^2}{4} - \pi arctg\sqrt{\frac{\sqrt{2}-1}{2}}$