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.

In the $xyz$-space take four points $P(0,\ 0,\ 2),\ A(0,\ 2,\ 0),\ B(\sqrt{3},-1,\ 0),\ C(-\sqrt{3},-1,\ 0)$. Find the volume of the part satifying $x^2+y^2\geq 1$ in the tetrahedron $PABC$. 50 points

Let $\triangle ABC$ be an acute-angled triangle with $AB \not= AC$. Let $H$ be the orthocenter of triangle $ABC$, and let $M$ be the midpoint of the side $BC$. Let $D$ be a point on the side $AB$ and $E$ a point on the side $AC$ such that $AE=AD$ and the points $D$, $H$, $E$ are on the same line. Prove that the line $HM$ is perpendicular to the common chord of the circumscribed circles of triangle $\triangle ABC$ and triangle $\triangle ADE$.

The first two terms of a sequence are $ a_1 \equal{} 1$ and $ a_2 \equal{} \frac {1}{\sqrt3}$. For $ n\ge1$, \[ a_{n \plus{} 2} \equal{} \frac {a_n \plus{} a_{n \plus{} 1}}{1 \minus{} a_na_{n \plus{} 1}}. \]What is $ |a_{2009}|$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2 \minus{} \sqrt3\qquad \textbf{(C)}\ \frac {1}{\sqrt3}\qquad \textbf{(D)}\ 1\qquad \textbf{(E)}\ 2 \plus{} \sqrt3$

For how many positive integers $n$ less than or equal to $1000$ is \[(\sin t + i \cos t)^n=\sin nt + i \cos nt\] true for all real $t$?

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

Show that if $a,b,c$ are complex numbers that such that \[ (a+b)(a+c)=b \qquad (b+c)(b+a)=c \qquad (c+a)(c+b)=a\] then $a,b,c$ are real numbers.

If a triangle has side lengths $a,b,c$ where $a\leq 2 \leq b \leq 3$, what is the largest possible value of its area? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ \text{None of above} $

Let $ABCD$ be a square inscribed in a circle $\omega$ and let $P$ be a point on the shorter arc $AB$ of $\omega$. Let $CP\cap BD = R$ and $DP \cap AC = S.$ Show that triangles $ARB$ and $DSR$ have equal areas.

MTRPia in $2044$ is highly advanced and a lot of the work is done by disc-shaped robots, each of radius $1$ unit. In order to not collide with each other, there robots have a smaller $360$-degree camera mounted on top, as shown in the figure (robot $r_1$ 'sees' robot $r_2$). Each of there cameras themselves are smaller discs of radius $c$. Suppose there are three robots $r_1, r_2, r_3$ placed 'consecutively' such that $r_2$ is roughly in the middle. The angle between the lines joining the centres of $r_1, r_2$ and $r_2, r_3$ is given to be $\theta$. The distance between the centres of $r_1,r_2 = $ distance between centres of $r_2,r_3 = d$. Show (with the aid of clear diagrams) that $r_1$ and $r_3$ can see each other iff $\sin{\theta} > \frac{1-c}{d}$. As a bonus, try to show that in a longer 'chain' of such robots (same $d$, $\theta$), if $\sin{\theta} > \frac{1-c}{d}$ then all robots can see each other.

The quadrilateral $ABCD$, where $\angle BAD+\angle ADC>\pi$, is inscribed a circle with centre $I$. A line through $I$ intersects $AB$ and $CD$ in points $X$ and $Y$ respectively such that $IX=IY$. Prove that $AX\cdot DY=BX\cdot CY$.

Let $P$ be a point in the interior of an acute triangle $ABC$, and let $Q$ be its isogonal conjugate. Denote by $\omega_P$ and $\omega_Q$ the circumcircles of triangles $BPC$ and $BQC$, respectively. Suppose the circle with diameter $\overline{AP}$ intersects $\omega_P$ again at $M$, and line $AM$ intersects $\omega_P$ again at $X$. Similarly, suppose the circle with diameter $\overline{AQ}$ intersects $\omega_Q$ again at $N$, and line $AN$ intersects $\omega_Q$ again at $Y$. Prove that lines $MN$ and $XY$ are parallel. (Here, the points $P$ and $Q$ are [i]isogonal conjugates[/i] with respect to $\triangle ABC$ if the internal angle bisectors of $\angle BAC$, $\angle CBA$, and $\angle ACB$ also bisect the angles $\angle PAQ$, $\angle PBQ$, and $\angle PCQ$, respectively. For example, the orthocenter is the isogonal conjugate of the circumcenter.) [i]Proposed by Sammy Luo[/i]

Determine the number of angles $\theta$ between $0$ and $2 \pi$, other than integer multiples of $\pi /2$, such that the quantities $\sin \theta, \cos \theta, $ and $\tan \theta$ form a geometric sequence in some order.

A particle moves in $3$-space according to the equations: $$ \frac{dx}{dt} =yz,\; \frac{dy}{dt} =xz,\; \frac{dz}{dt}= xy.$$ Show that: (a) If two of $x(0), y(0), z(0)$ equal $0,$ then the particle never moves. (b) If $x(0)=y(0)=1, z(0)=0,$ then the solution is $$ x(t)= \sec t ,\; y(t) =\sec t ,\; z(t)= \tan t;$$ whereas if $x(0)=y(0)=1, z(0)=-1,$ then $$ x(t) =\frac{1}{t+1} ,\; y(t)=\frac{1}{t+1}, z(t)=- \frac{1}{t+1}.$$ (c) If at least two of the values $x(0), y(0), z(0)$ are different from zero, then either the particle moves to infinity at some finite time in the future, or it came from infinity at some finite time in the past (a point $(x, y, z)$ in $3$-space "moves to infinity" if its distance from the origin approaches infinity).

Find $\int_0^x \frac{dt}{1+t^2}+\int_0^{\frac{1}{x}} \frac{dt}{1+t^2}\ (x>0).$

Let $f(x) = \sin \frac{x}{3}+ \cos \frac{3x}{10}$ for all real $x$. Find the least natural number $n$ such that $f(n\pi + x)= f(x)$ for all real $x$.

Given positive real numbers $a, b, c, d$ that satisfy equalities \[a^2 + d^2 - ad = b^2 + c^2 + bc \ \ \text{and} \ \ a^2 + b^2 = c^2 + d^2\] find all possible values of the expression $\frac{ab+cd}{ad+bc}.$

Radius of the circle $\omega_A$ with centre at vertex $A$ of a triangle $\triangle{ABC}$ is equal to the radius of the excircle tangent to $BC$. The circles $\omega_B$ and $\omega_C$ are defined similarly. Prove that if two of these circles are tangent then every two of them are tangent to each other. [i](L. Emelyanov)[/i]

Let $f(x) = x^2 + ax + b cos x$. Find all values of parameter$ a$ and $b$, for which the equations $f(x) = 0$ and $f(f(x)) = 0 $have the same non-empty sets of real roots.

Prove that $\sin^318^{\circ}+\sin^218^{\circ}=\frac18$.

Let $\alpha, \beta, \gamma, \delta$ be the interior angles of a convex quadrilateral, If $$ \cos\alpha + \cos\beta + \cos\gamma, + \cos\delta = 0 , $$ then this quadrilateral is cyclic or a trapezium. Prove it.

Let $N$ be the midpoint of side $BC$ of triangle $ABC$. Right isosceles triangles $ABM$ and $ACP$ are constructed outside the triangle, with bases $AB$ and $AC$. Prove that $\triangle MNP$ is also a right isosceles triangle.

Consider an octagon with equal angles and with rational sides. Prove that it has a center of symmetry.

In $\triangle ABC$ , then \[\frac{1}{1+\cos^2 A+\cos^2 B}+\frac{1}{1+\cos^2 B+\cos^2 C}+\frac{1}{1+\cos^2 C+\cos^2 A}\le 2\]

Consider the obtuse-angled triangle $\triangle ABC$ and its side lengths $a,b,c$. Prove that $a^3\cos\angle A +b^3\cos\angle B + c^3\cos\angle C < abc$.