In an acute triangle $ ABC$ segments $ BE$ and $ CF$ are altitudes. Two circles passing through the point $ A$ and $ F$ and tangent to the line $ BC$ at the points $ P$ and $ Q$ so that $ B$ lies between $ C$ and $ Q$. Prove that lines $ PE$ and $ QF$ intersect on the circumcircle of triangle $ AEF$. [i]Proposed by Davood Vakili, Iran[/i]

Let $f:\mathbb{R}\backslash0 \rightarrow \mathbb{R}\backslash0$ be a non-constant, continuous function defined such that $f(3^x2^y)=\frac{y}{x}f(3^y)$ for any $x,y \neq 0.$ Compute $\frac{f(1296)}{f(6)}.$ [i]Proposed by Richard Chen and Zachary Perry[/i]

The square $ABCD$ has side length $1$. The point $E$ lies on the side $CD$. The line through $A$ and $E$ intersects the line through $B$ and $C$ at the point $F$. Prove that $$\frac{1}{|AE|^2}+\frac{1}{|AF|^2}= 1.$$ [img]https://cdn.artofproblemsolving.com/attachments/5/8/4e803eb7748f7a72783065717044cfc06f565f.png[/img]

Simon draws some line segments on the face of a regular polygon, dissecting it into exactly $2021$ triangles, such that no two drawn line segments are collinear, and no two triangles share a pair of vertices. Simon then assigns each drawn line segment and each side of the polygon with one of three colours. Prove that there is some triangle in the dissection with a pair of identically-coloured sides.

Let $a,b\in\mathbb{N}$ such that : \[ ab(a-b)\mid a^3+b^3+ab \] Then show that $\operatorname{lcm}(a,b)$ is a perfect square.

Let $O_n$ be the $n$-dimensional vector $(0,0,\cdots, 0)$. Let $M$ be a $2n \times n$ matrix of complex numbers such that whenever $(z_1, z_2, \dots, z_{2n})M = O_n$, with complex $z_i$, not all zero, then at least one of the $z_i$ is not real. Prove that for arbitrary real numbers $r_1, r_2, \dots, r_{2n}$, there are complex numbers $w_1, w_2, \dots, w_n$ such that \[ \mathrm{re}\left[ M \left( \begin{array}{c} w_1 \\ \vdots \\ w_n \end{array} \right) \right] = \left( \begin{array}{c} r_1 \\ \vdots \\ r_n \end{array} \right). \] (Note: if $C$ is a matrix of complex numbers, $\mathrm{re}(C)$ is the matrix whose entries are the real parts of the entries of $C$.)

Let $V=[(x,y,z)|0\le x,y,z\le 2008]$ be a set of points in a 3-D space. If the distance between two points is either $1, \sqrt{2}, 2$, we color the two points differently. How many colors are needed to color all points in $V$?

Let a, b, c be the sidelengths and S the area of a triangle ABC. Denote $x=a+\frac{b}{2}$, $y=b+\frac{c}{2}$ and $z=c+\frac{a}{2}$. Prove that there exists a triangle with sidelengths x, y, z, and the area of this triangle is $\geq\frac94 S$.

Consider the set of numbers $\{1,10,10^2,10^3, ... 10^{10} \}$. The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integer? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 11 \qquad \textbf{(E)}\ 101 $

Given an integer $n\geq 3$, determine the smallest positive number $k$ such that any two points in any $n$-gon (or at its boundary) in the plane can be connected by a polygonal path consisting of $k$ line segments contained in the $n$-gon (including its boundary).

Find the maximum number of subsets from $\left \{ 1,...,n \right \}$ such that for any two of them like $A,B$ if $A\subset B$ then $\left | B-A \right |\geq 3$. (Here $\left | X \right |$ is the number of elements of the set $X$.)

Determine the minimum possible amount of distinct prime divisors of $19^{4n}+4$, for a positive integer $n$.

A [i]Benelux n-square[/i] (with $n\geq 2$) is an $n\times n$ grid consisting of $n^2$ cells, each of them containing a positive integer, satisfying the following conditions: $\bullet$ the $n^2$ positive integers are pairwise distinct. $\bullet$ if for each row and each column we compute the greatest common divisor of the $n$ numbers in that row/column, then we obtain $2n$ different outcomes. (a) Prove that, in each Benelux n-square (with $n \geq 2$), there exists a cell containing a number which is at least $2n^2.$ (b) Call a Benelux n-square [i]minimal[/i] if all $n^2$ numbers in the cells are at most $2n^2.$ Determine all $n\geq 2$ for which there exists a minimal Benelux n-square.

Given a sequence $a_n$: \[ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \dots \] (one '1', two '2' and so on) and another sequence $b_n$ such that $a_{b_n}=b_{a_n}$ for all positive integers $n$. It is known that $b_k=1$ for some $k>100$. Prove that $b_m=1$ for all $m>k$.

Determine all polynomials $P(x)$ with real coefficients such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial.

A gas diffuses one-third as fast as $\ce{O2}$ at $100^{\circ}\text{C}$. This gas could be: $ \textbf{(A)}\hspace{.05in}\text{He (M=4)}\qquad\textbf{(B)}\hspace{.05in}\ce{C2H5F}(\text{M=48})$ $\qquad\textbf{(C)}\hspace{.05in}\ce{C7H12}\text{(M=96)}\qquad\textbf{(D)}\hspace{.05in}\ce{C5F12}\text{(M=288)}\qquad$

If $x$ and $y$ are positive integers, and $x^4+y^4=4721$, find all possible values of $x+y$

How many solutions does the equation system \[\dfrac{x-1}{xy-3}=\dfrac{3-x-y}{7-x^2-y^2} = \dfrac{y-2}{xy-4}\] have? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 $

In a $n\times n$ table ($n>1$) $k$ unit squares are marked.One wants to rearrange rows and columns so that all the marked unit squares are above the main diagonal or on it.For what maximum $k$ is it always possible?

In the plane the points with integer coordinates are the vertices of unit squares. The squares are coloured alternately black and white (as on a chessboard). For any pair of positive integers $ m$ and $ n$, consider a right-angled triangle whose vertices have integer coordinates and whose legs, of lengths $ m$ and $ n$, lie along edges of the squares. Let $ S_1$ be the total area of the black part of the triangle and $ S_2$ be the total area of the white part. Let $ f(m,n) \equal{} | S_1 \minus{} S_2 |$. a) Calculate $ f(m,n)$ for all positive integers $ m$ and $ n$ which are either both even or both odd. b) Prove that $ f(m,n) \leq \frac 12 \max \{m,n \}$ for all $ m$ and $ n$. c) Show that there is no constant $ C\in\mathbb{R}$ such that $ f(m,n) < C$ for all $ m$ and $ n$.

On the plane is drawn a triangle $ABC$ and a circle $\omega$ passing through the vertex $C$, the midpoints of the sides $AC$ and $BC$ and the point of intersection of the medians of the triangle $ABC$. The point $K$ lies on the circle $\omega$ such that $\angle AKB=90^o$. Using only with a ruler, draw a tangent to the circle $\omega$ at point $K$.

Determine all the integers $n > 1$ such that $$\sum_{i=1}^{n}x_i^2 \ge x_n \sum_{i=1}^{n-1}x_i$$ for all real numbers $x_1, x_2, ... , x_n$.

In the space given three segments $A_1A_2, B_1B_2$ and $C_1C_2$, do not lie in one plane and intersect at a point $P$. Let $O_{ijk}$ be center of sphere that passes through the points $A_i, B_j, C_k$ and $P$. Prove that $O_{111}O_{222}, O_{112}O_{221}, O_{121}O_{212}$ and$O_{211}O_{122}$ intersect at one point. (P.Kozhevnikov)

In the plane we have $16$ lines(not parallel and not concurrents), we have $120$ point(s) of intersections of this lines. Sebastian has to paint this $120$ points such that in each line all the painted points are with colour differents, find the minimum(quantity) of colour(s) that Sebastian needs to paint this points. If we have have $15$ lines(in this situation we have $105$ points), what's the minimum(quantity) of colour(s)?

Let $ABCDEF$ be a convex hexagon and denote by $A',B',C',D',E',F'$ the middle points of the sides $AB$, $BC$, $CD$, $DE$, $EF$ and $FA$ respectively. Given are the areas of the triangles $ABC'$, $BCD'$, $CDE'$, $DEF'$, $EFA'$ and $FAB'$. Find the area of the hexagon. [i]Kvant Magazine[/i]