Quadrilateral $ABCD$ is circumscribed around the circle with centre $I$. Let points $M$ and $N$ be the midpoints of sides $AB$ and $CD$ respectively and let $\frac{IM}{AB} = \frac{IN}{CD}$. Prove that $ABCD$ is either a trapezoid or a parallelogram.

Let $ABC$ be an equilateral triangle. On the small arc $AB{}$ of its circumcircle $\Omega$, consider the point $N{}$ such that the small arc $NB$ measures $30^\circ{}$. The perpendiculars from $N{}$ onto $AC$ and $AB$ intersect $\Omega$ again at $P{}$ and $Q{}$ respectively. Let $H_1,H_2$ and $H_3$ be the orthocenters of the triangles $NAB, QBC$ and $CAP$ respectively. [list=a] [*]Prove that the triangle $NPQ$ is equilateral. [*]Prove that the triangle $H_1H_2H_3$ is equilateral. [/list]

Let $f\in\mathbb{Z}[X]$ be an irreducible polynomial over the ring of integer polynomials, such that $|f(0)|$ is not a perfect square. Prove that if the leading coefficient of $f$ is 1 (the coefficient of the term having the highest degree in $f$) then $f(X^2)$ is also irreducible in the ring of integer polynomials. [i]Mihai Piticari[/i]

Let $ABC$ be a triangle inscribed in circle $\omega$, and let the medians from $B$ and $C$ intersect $\omega$ at $D$ and $E$ respectively. Let $O_1$ be the center of the circle through $D$ tangent to $AC$ at $C$, and let $O_2$ be the center of the circle through $E$ tangent to $AB$ at $B$. Prove that $O_1$, $O_2$, and the nine-point center of $ABC$ are collinear. [i]Proposed by Michael Kural[/i]

Let $w_1, w_2, \dots, w_n$ be complex numbers. A line $L$ in the complex plane is called a mean line for the points $w_1, w_2, \dots, w_n$ if $L$ contains points (complex numbers) $z_1, z_2, \dots, z_n$ such that \[ \sum_{k = 1}^n (z_k - w_k) = 0. \] For the numbers $w_1 = 32 + 170i$, $w_2 = -7 + 64i$, $w_3 = -9 +200i$, $w_4 = 1 + 27i$, and $w_5 = -14 + 43i$, there is a unique mean line with $y$-intercept 3. Find the slope of this mean line.

For some complex number $\omega$ with $|\omega| = 2016$, there is some real $\lambda>1$ such that $\omega, \omega^{2},$ and $\lambda \omega$ form an equilateral triangle in the complex plane. Then, $\lambda$ can be written in the form $\tfrac{a + \sqrt{b}}{c}$, where $a,b,$ and $c$ are positive integers and $b$ is squarefree. Compute $\sqrt{a+b+c}$.

$z=x+i y$ where $x$ and $y$ are two real numbers. Find the locus of the point $(x, y)$ in the plane, for which $\frac{z+i}{z-i}$ is purely imaginary (that is, it is of the form $i b$ where $b$ is a real number). [Here, $i=\sqrt{-1}$ [list=1] [*] A straight line [*] A circle [*] A parabole [*] None of these [/list]

Prove that if $n$ can be written as $n=a^2+ab+b^2$, then also $7n$ can be written that way.

Let $\zeta= e^{2\pi i/99}$ and $\omega e^{2\pi i/101}$. The polynomial $$x^{9999} + a_{9998}x^{9998} + ...+ a_1x + a_0$$ has roots $\zeta^m + \omega^n$ for all pairs of integers $(m, n)$ with $0 \le m < 99$ and $0 \le n < 101$. Compute $a_{9799} + a_{9800} + ...+ a_{9998}$.

Let $D, E$ and $F{}$ be the midpoints of the sides $BC, CA$ and $AB{}$ of the acute-angled triangle $ABC$ and let $H_a, H_b$ and $H_c{}$ be the orthocenters of the triangles $ABD, BCE$ and $CAF{}$ respectively. Prove that the triangles $H_aH_bH_c$ and $DEF$ have equal areas. [i]Proposed by Tran Quang Hung[/i]

Let $ABC$ be a triangle with $AB < AC$. As shown below, $T$ is the point on $\overline{BC}$ such that $\overline{AT}$ is tangent to the circumcircle of $\triangle{}ABC$. Additionally, $H$ and $O$ are the orthocenter and circumcenter of $\triangle{}ABC$, respectively. Suppose that $\overline{CH}$ passes through the midpoint of $\overline{AT}$. Prove that $\overline{AO}$ bisects $\overline{CH}$. [asy] size(8cm); pair A = dir(132.5); pair B = dir(200); pair C = dir(340); draw(A--B--C--cycle, black); draw(circumcircle(A, B, C), black); pair O = circumcenter(A, B, C); pair U = 2*C*A/(C+A); pair V = 2*A*B/(A+B);pair T = extension(U, V, B, C); draw(A--T); draw(T--B);pair X = (T+A)/2; pair H = (A+B+C); draw(A--H); pair Y = (H+C)/2; draw(H--X, dashed); draw(C--H); draw(O--Y, dashed); draw(A--O); dot("$A$", A, dir(A)); dot("$B$", B, SW); dot("$C$", C, dir(C)); dot("$O$", O, NE); dot("$T$", T, dir(T)); dot("$H$", H, SW); dot("$X$", X, NW); dot("$Y$", Y, SW); [/asy]

Denote $A_{DE}$ by the foot of perpendicular line from $A$ to line $DE$. Given concyclic points $A,B,C,D,E,F$, show that the three points determined by the lines $A_{FD}A_{DE}$ , $B_{DE}B_{EF}$ , $C_{EF}C_{FD}$, and the three points determined by the lines $D_{CA}D_{AB}$ , $E_{AB}E_{BC}$ , $F_{BC}F_{CA}$ are concyclic.

Solve the following system for real $x,y,z$ \[ \{ \begin{array}{ccc} x+ y -z & =& 4 \\ x^2 - y^2 + z^2 & = & -4 \\ xyz & =& 6. \end{array} \]

A regular $n$-gon is inscribed in a unit circle. Compute the product from a fixed vertex to all the other vertices.