Consider a triangle $ABC$ and two congruent triangles $A_1B_1C_1$ and $A_2B_2C_2$ which are respectively similar to $ABC$ and inscribed in it: $A_i,B_i,C_i$ are located on the sides of $ABC$ in such a way that the points $A_i$ are on the side opposite to $A$, the points $B_i$ are on the side opposite to $B$, and the points $C_i$ are on the side opposite to $C$ (and the angle at A are equal to angles at $A_i$ etc.). The circumcircles of $A_1B_1C_1$ and $A_2B_2C_2$ intersect at points $P$ and $Q$. Prove that the line $PQ$ passes through the orthocenter of $ABC$.

Let $a_n$ and $b_n$ to be two sequences defined as below: $i)$ $a_1 = 1$ $ii)$ $a_n + b_n = 6n - 1$ $iii)$ $a_{n+1}$ is the least positive integer different of $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$. Determine $a_{2009}$.

Let $H$ be a complex Hilbert space. The bounded linear operator $A$ is called [i]positive[/i] if $\langle Ax, x\rangle \geq 0$ for all $x\in H$. Let $\sqrt A$ be the positive square root of $A$, i.e. the uniquely determined positive operator satisfying $(\sqrt{A})^2=A$. On the set of positive operators we introduce the $$A\circ B=\sqrt A B\sqrt B$$ operation. Prove that for a given pair $A, B$ of positive operators the identity $$(A\circ B)\circ C=A\circ (B\circ C)$$ holds for all positive operator $C$ if and only if $AB=BA$.

How many ways can you fill a $3 \times 3$ square grid with nonnegative integers such that no [i]nonzero[/i] integer appears more than once in the same row or column and the sum of the numbers in every row and column equals 7?

Vertex $ E$ of equilateral $ \triangle{ABE}$ is in the interior of unit square $ ABCD$. Let $ R$ be the region consisting of all points inside $ ABCD$ and outside $ \triangle{ABE}$ whose distance from $ \overline{AD}$ is between $ \frac{1}{3}$ and $ \frac{2}{3}$. What is the area of $ R$? $ \textbf{(A)}\ \frac{12\minus{}5\sqrt3}{72} \qquad \textbf{(B)}\ \frac{12\minus{}5\sqrt3}{36} \qquad \textbf{(C)}\ \frac{\sqrt3}{18} \qquad \textbf{(D)}\ \frac{3\minus{}\sqrt3}{9} \qquad \textbf{(E)}\ \frac{\sqrt3}{12}$

Points $M,\ N,\ K,\ L$ are taken on the sides $AB,\ BC,\ CD,\ DA$ of a rhombus $ABCD,$ respectively, in such a way that $MN\parallel LK$ and the distance between $MN$ and $KL$ is equal to the height of $ABCD.$ Show that the circumcircles of the triangles $ALM$ and $NCK$ intersect each other, while those of $LDK$ and $MBN$ do not.

A regular octagon $ ABCDEFGH$ has an area of one square unit. What is the area of the rectangle $ ABEF$? [asy]unitsize(8mm); defaultpen(linewidth(.8pt)+fontsize(6pt)); pair C=dir(22.5), B=dir(67.5), A=dir(112.5), H=dir(157.5), G=dir(202.5), F=dir(247.5), E=dir(292.5), D=dir(337.5); draw(A--B--C--D--E--F--G--H--cycle); label("$A$",A,NNW); label("$B$",B,NNE); label("$C$",C,ENE); label("$D$",D,ESE); label("$E$",E,SSE); label("$F$",F,SSW); label("$G$",G,WSW); label("$H$",H,WNW);[/asy]$ \textbf{(A)}\ 1\minus{}\frac{\sqrt2}{2} \qquad \textbf{(B)}\ \frac{\sqrt2}{4} \qquad \textbf{(C)}\ \sqrt2\minus{}1 \qquad \textbf{(D)}\ \frac12 \qquad \textbf{(E)}\ \frac{1\plus{}\sqrt2}{4}$

Sophia writes an algorithm to solve the graph isomorphism problem. Given a graph $G=(V,E)$, her algorithm iterates through all permutations of the set $\{v_1, \dots, v_{|V|}\}$, each time examining all ordered pairs $(v_i,v_j)\in V\times V$ to see if an edge exists. When $|V|=8$, her algorithm makes $N$ such examinations. What is the largest power of two that divides $N$?

Find the maximum possible value of $a + b + c ,$ if $a,b,c$ are positive real numbers such that $a^2 + b^2 + c^2 = a^3 + b^3 + c^3 .$

We only know that the password of a safe consists of $7$ different digits. The safe will open if we enter $7$ different digits, and one of them matches the corresponding digit of the password. Can we open this safe in less than $7$ attempts? [i](5 points for Juniors and 4 points for Seniors)[/i]

Let $AB$ be the diameter of a circle $k$ and let $E$ be a point in the interior of $k$. The line $AE$ intersects $k$ a second time in $C \ne A$ and the line $BE$ intersects $k$ a second time in $D \ne B$. Show that the value of $AC \cdot AE+BD\cdot BE$ is independent of the choice of $E$.

Find $2 \times (2 + 3)$ $\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }10\qquad\textbf{(D) }11\qquad\textbf{(E) }30$

The sequence $\{x_n\}$ is defined by $x_1=5$ and $x_{k+1}=x_k^2-3x_k+3$ for $k=1,2,3\cdots$. Prove that $x_k>3^{2^{k-1}}$ for any positive integer $k$.

The products of the opposite sidelengths of a cyclic quadrilateral $ABCD$ are equal. Let $B'$ be the reflection of $B$ about $AC$. Prove that the circle passing through $A,B', D$ touches $AC$

Prove that there is no function $f:\mathbb{R}^{+} \to \mathbb{R}^{+}$ such that $f(x)^2 \geq f(x+y)(f(x)+y)$ for all $x,y \in \mathbb{R}^{+}$.

The number of solutions in positive integers of $2x+3y=763$ is: $\textbf{(A)}\ 255 \qquad \textbf{(B)}\ 254\qquad \textbf{(C)}\ 128 \qquad \textbf{(D)}\ 127 \qquad \textbf{(E)}\ 0$

Given an acute angled triangle $ABC$ with $M$ being the mid-point of $AB$ and $P$ and $Q$ are the feet of heights from $A$ to $BC$ and $B$ to $AC$ respectively. Show that if the line $AC$ is tangent to the circumcircle of $BMP$ then the line $BC$ is tangent to the circumcircle of $AMQ$.

Positive numbers $ x_1, x_2, . . ., x_{2009}$ satisfy the equalities $$x^2_1 - x_1x_2 +x^2_2 =x^2_2 -x_2x_3+x^2_3=x^2_3 -x_3x_4+x^2_4= ...= x^2_{2008}- x_{2008}x_{2009}+x^2_{2009}= x^2_{2009}-x_{2009}x_1+x^2_1$$. Prove that the numbers $ x_1, x_2, . . ., x_{2009}$ are equal.

Consider a polygon with $m + n$ sides where $m, n$ are positive integers. Colour $m$ of its vertices red and the remaining $n$ vertices blue. A side is given the number $2$ if both its end vertices are red, the number $1/2.$ if both its end vertices are blue and the number $1$ otherwise. Let the product of these numbers be $P$. Find the largest possible value of $P$.

A circle centered at $ A$ with a radius of 1 and a circle centered at $ B$ with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. The radius of the third circle is [asy] size(220); real r1 = 1; real r2 = 3; real r = (r1*r2)/((sqrt(r1)+sqrt(r2))**2); pair A=(0,r1), B=(2*sqrt(r1*r2),r2); dot(A); dot(B); draw( circle(A,r1) ); draw( circle(B,r2) ); draw( (-1.5,0)--(7.5,0) ); draw( A -- (A+dir(210)*r1) ); label("$1$", A -- (A+dir(210)*r1), N ); draw( B -- (B+r2*dir(330)) ); label("$4$", B -- (B+r2*dir(330)), N ); label("$A$",A,dir(330)); label("$B$",B, dir(140)); draw( circle( (2*sqrt(r1*r),r), r )); [/asy] $ \displaystyle \textbf{(A)} \ \frac {1}{3} \qquad \textbf{(B)} \ \frac {2}{5} \qquad \textbf{(C)} \ \frac {5}{12} \qquad \textbf{(D)} \ \frac {4}{9} \qquad \textbf{(E)} \ \frac {1}{2}$

Let $a \spadesuit b = \frac{a^2-b^2}{2b-2a}$ . Given that $3 \spadesuit x = -10$, compute $x$.

An acute angle between the diagonals of a cyclic quadrilateral is equal to $\phi$. Prove that an acute angle between the diagonals of any other quadrilateral having the same sidelengths is smaller than $\phi$.

Prove that the equation $x^3+y^3+z^3=t^4$ has infinitely many solutions in positive integers such that $\gcd(x,y,z,t)=1$. [i]Mihai Pitticari & Sorin Rǎdulescu[/i]

Let $\omega=e^{\frac{2\pi i}{2017}}$ and $\zeta = e^{\frac{2\pi i}{2019}}$. Let $S=\{(a,b)\in\mathbb{Z}\,|\,0\leq a \leq 2016, 0 \leq b \leq 2018, (a,b)\neq (0,0)\}$. Compute $$\prod_{(a,b)\in S}(\omega^a-\zeta^b).$$

A magical castle has $n$ identical rooms, each of which contains $k$ doors arranged in a line. In room $i, 1 \leq i \leq n - 1$ there is one door that will take you to room $i + 1$, and in room $n$ there is one door that takes you out of the castle. All other doors take you back to room $1$. When you go through a door and enter a room, you are unable to tell what room you are entering and you are unable to see which doors you have gone through before. You begin by standing in room $1$ and know the values of $n$ and $k$. Determine for which values of $n$ and $k$ there exists a strategy that is guaranteed to get you out of the castle and explain the strategy. For such values of $n$ and $k$, exhibit such a strategy and prove that it will work.