Let $\mathcal L$ be the set of all lines in the plane and let $f$ be a function that assigns to each line $\ell\in\mathcal L$ a point $f(\ell)$ on $\ell$. Suppose that for any point $X$, and for any three lines $\ell_1,\ell_2,\ell_3$ passing through $X$, the points $f(\ell_1),f(\ell_2),f(\ell_3)$, and $X$ lie on a circle. Prove that there is a unique point $P$ such that $f(\ell)=P$ for any line $\ell$ passing through $P$. [i]Australia[/i]

The numbers $1,2,\dots, 2013$ are written on $2013$ stones weighing $1,2,\dots, 2013$ grams such that each number is used exactly once. We have a two-pan balance that shows the difference between the weights at the left and the right pans. No matter how the numbers are written, if it is possible to determine in $k$ weighings whether the weight of each stone is equal to the number that is written on the stone, what is the least possible value of $k$? $ \textbf{(A)}\ 15 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ \text{None of above} $

Let $ y_1(x)$ be an arbitrary, continuous, positive function on $ [0,A]$, where $ A$ is an arbitrary positive number. Let \[ y_{n+1}=2 \int_0^x \sqrt{y_n(t)}dt \;(n=1,2,...)\ .\] Prove that the functions $ y_n(x)$ converge to the function $ y=x^2$ uniformly on $ [0,A]$.

A convex polygon $P$ is placed inside a unit square $Q$. Prove that the perimeter of $P$ does not exceed $4$.

Let $n$ be a natural number. The graph $G$ has $10n$ vertices. They are separated into $10$ groups with $n$ vertices and we know that there is an edge between two of them if and only if they belong to two different groups. What’s the greatest number of edges a subgraph of $G$ can have, so that there are no 4-cliques in it?

Let $a, b, c$ be positive real numbers such that $a^2+b^2+c^2+(a+b+c)^2\leq4$. Prove that \[\frac{ab+1}{(a+b)^2}+\frac{bc+1}{(b+c)^2}+\frac{ca+1}{(c+a)^2}\geq 3.\]

Calculate $ \lim_{n\to\infty }\sum_{t=1}^n\frac{1}{n+t+\sqrt{n^2+nt}} . $ [i]D.M. Bătinețu[/i] and [i]Neculai Stanciu[/i]

Three positive integers are written on the board. In every minute, instead of the numbers $a, b, c$, Elbek writes $a+\gcd(b,c), b+\gcd(a,c), c+\gcd(a,b)$ . Prove that there will be two numbers on the board after some minutes, such that one is divisible by the other. Note. $\gcd(x,y)$ - Greatest common divisor of numbers $x$ and $y$ [i]Proposed by Sergey Berlov, Russia[/i]

Let AOB and COD be angles which can be identified by a rotation of the plane (such that rays OA and OC are identified). A circle is inscribed in each of these angles; these circles intersect at points E and F. Show that angles AOE and DOF are equal.

Prove that the equation $$1320x^3=(y_1+y_2+y_3+y_4)(z_1+z_2+z_3+z_4)(t_1+t_2+t_3+t_4+t_5)$$ has infinitely many solutions in the set of Fibonacci numbers. [i]Proposed by Mihály Bencze[/i]

i) Show that a regular hexagon, six squares, and six equilateral triangles can be assembled without overlapping to form a regular dodecagon. ii) Let $P_1 , P_2 ,\ldots, P_{12}$ be the vertices of a regular dodecagon. Prove that the three diagonals $P_{1}P_{9}, P_{2}P_{11}$ and $P_{4}P_{12}$ intersect.

Let $F$ be a field in which $1+1 \neq 0$. Show that the set of solutions to the equation $x^2+y^2=1$ with $x$ and $y$ in $F$ is given by $(x,y)=(1,0)$ and \[ (x,y) = \left( \frac{r^2-1}{r^2+1}, \frac{2r}{r^2+1} \right) \] where $r$ runs through the elements of $F$ such that $r^2\neq -1$.

Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$. Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$

A multiple choice examination consists of $20$ questions. The scoring is $+5$ for each correct answer, $-2$ for each incorrect answer, and $0$ for each unanswered question. John's score on the examination is $48$. What is the maximum number of questions he could have answered correctly? $\text{(A)}\ 9 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 11 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 16$

Consider the real number axis (i. e. the $x$-axis of a Cartesian coordinate system). We mark the points $1$, $2$, ..., $2n$ on this axis. A flea starts at the point $1$. Now it jumps along the real number axis; it can jump only from a marked point to another marked point, and it doesn't visit any point twice. After the ($2n-1$)-th jump, it arrives at a point from where it cannot jump any more after this rule, since all other points are already visited. Hence, with its $2n$-th jump, the flea breaks this rule and gets back to the point $1$. Assume that the sum of the (non-directed) lengths of the first $2n-1$ jumps of the flea was $n\left(2n-1\right)$. Show that the length of the last ($2n$-th) jump is $n$.

Let $\triangle ABC$ be a triangle with $AB=65$, $BC=70$, and $CA=75$. A semicircle $\Gamma$ with diameter $\overline{BC}$ is constructed outside the triangle. Suppose there exists a circle $\omega$ tangent to $AB$ and $AC$ and furthermore internally tangent to $\Gamma$ at a point $X$. The length $AX$ can be written in the form $m\sqrt{n}$ where $m$ and $n$ are positive integers with $n$ not divisible by the square of any prime. Find $m+n$.

How many sequences of $0$s and $1$s of length $19$ are there that begin with a $0$, end with a $0$, contain no two consecutive $0$s, and contain no three consecutive $1$s? $\textbf{(A) }55\qquad\textbf{(B) }60\qquad\textbf{(C) }65\qquad\textbf{(D) }70\qquad\textbf{(E) }75$

Let $t$ be a positive constant. Given two points $A(2t,\ 2t,\ 0),\ B(0,\ 0,\ t)$ in a space with the origin $O$. Suppose mobile points $P$ in such way that $\overrightarrow{OP}\cdot \overrightarrow{AP}+\overrightarrow{OP}\cdot \overrightarrow{BP}+\overrightarrow{AP}\cdot \overrightarrow{BP}=3.$ Find the value of $t$ such that the maximum value of $OP$ is 3.

Triangle $ABC$ with $AB = AC$ is inscribed in circle $o$. Circles $o_1$ and $o_2$ are internally tangent to circle $o$ in points $P$ and $Q$, respectively, and they are tangent to segments $AB$ and $AC$, respectively, and they are disjoint with the interior of triangle $ABC$. Let $m$ be a line tangent to circles $o_1$ and $o_2$, such that points $P$ and $Q$ lie on the opposite side than point $A$. Line $m$ cuts segments $AB$ and $AC$ in points $K$ and $L$, respectively. Prove, that intersection point of lines $PK$ and $QL$ lies on bisector of angle $BAC$.

Given $n \ge 4$ points in the plane such that any four of them are the vertices of a convex quadrilateral, prove that these points are the vertices of a convex polygon.

Let $a$ be an integer and $n$ a positive integer . Show that the sum : $$\sum_{k=1}^{n} a^{(k,n)}$$ is divisible by $n$ , where $(x,y)$ is the greatest common divisor of the numbers $x$ and $y$ .

Determine if there exist positive real numbers $x, \alpha$, so that for any non-empty finite set of positive integers $S$, the inequality \[\left|x-\sum_{s\in S}\frac{1}{s}\right|>\frac{1}{\max(S)^\alpha}\] holds, where $\max(S)$ is defined as the maximum element of the finite set $S$.

Define a [i]T-grid[/i] to be a $ 3\times3$ matrix which satisfies the following two properties: (1) Exactly five of the entries are $ 1$'s, and the remaining four entries are $ 0$'s. (2) Among the eight rows, columns, and long diagonals (the long diagonals are $ \{a_{13},a_{22},a_{31}\}$ and $ \{a_{11},a_{22},a_{33}\}$, no more than one of the eight has all three entries equal. Find the number of distinct T-grids.

Let $A,B$ be two points on a give circle, and $M$ be the midpoint of one of the arcs $AB$ . Point $C$ is the orthogonal projection of $B$ onto the tangent $l$ to the circle at $A$. The tangent at $M$ to the circle meets $AC,BC$ at $A',B'$ respectively. Prove that if $\hat{BAC}<\frac{\pi}{8}$ then $S_{ABC}<2S_{A'B'C'}$.

Prove that for all prime $p \ge 5$, there exist an odd prime $q \not= p$ such that $q$ divides $(p-1)^p + 1$