Let $ABCD$ be a cyclic quadrilateral such that $AD=BD=AC$. A point $P$ moves along the circumcircle $\omega$ of triangle $ABCD$. The lined $AP$ and $DP$ meet the lines $CD$ and $AB$ at points $E$ and $F$ respectively. The lines $BE$ and $CF$ meet point $Q$. Find the locus of $Q$.

Let $Oxy$ be a fixed rectangular coordinate system in the plane. Each ordered pair of points $A_1, A_2$ from the same plane which are different from O and have coordinates $x_1, y_1$ and $x_2, y_2$ respectively is associated with real number $f(A_1,A_2)$ in such a way that the following conditions are satisfied: (a) If $OA_1 = OB_1$, $OA_2 = OB_2$ and $A_1A_2 = B_1B_2$ then $f(A_1,A_2) = f(B_1,B_2)$. (b) There exists a polynomial of second degree $F(u,v,w,z)$ such that $f(A_1,A_2)=F(x_1,y_1,x_2,y_2)$. (c) There exists such a number $\phi \in (0,\pi)$ that for every two points $A_1, A_2$ for which $\angle A_1OA_2 = \phi$ is satisfied $f(A_1,A_2) = 0$. (d) If the points $A_1, A_2$ are such that the triangle $OA_1A_2$ is equilateral with side $1$ then$ f(A_1,A_2) = \frac12$. Prove that $f(A_1,A_2) = \overrightarrow{OA_1} \cdot \overrightarrow{OA_2}$ for each ordered pair of points $A_1, A_2$.

Let $M$ be the circumcenter of a triangle $ABC.$ The line through $M$ perpendicular to $CM$ meets the lines $CA$ and $CB$ at $Q$ and $P,$ respectively. Prove that \[\frac{\overline{CP}}{\overline{CM}} \cdot \frac{\overline{CQ}}{\overline{CM}}\cdot \frac{\overline{AB}}{\overline{PQ}}= 2.\]

Find all functions $f:\mathbb{R^+} \to \mathbb{R^+}$ such that for all positive real $x, y$ holds $$xf(x)+yf(y)=(x+y)f\left(\frac{x^2+y^2}{x+y}\right)$$. [i]Fedir Yudin[/i]

Let $C$ be a right circular cone with apex $A$. Let $P_1, P_2, P_3, P_4$ and $P_5$ be points placed evenly along the circular base in that order, so that $P_1P_2P_3P_4P_5$ is a regular pentagon. Suppose that the shortest path from $P_1$ to $P_3$ along the curved surface of the cone passes through the midpoint of $AP_2$. Let $h$ be the height of $C$, and $r$ be the radius of the circular base of $C$. If $\left(\frac{h}{r}\right)^2$ can be written in simplest form as $\frac{a}{b}$ , find $a + b$.

Two toads named Gamakichi and Gamatatsu are sitting at the points $(0,0)$ and $(2,0)$ respectively. Their goal is to reach $(5,5)$ and $(7,5)$ respectively by making one unit jumps in positive $x$ or $y$ direction at a time. How many ways can they do this while ensuring that there is no point on the plane where both Gamakichi And Gamatatsu land on?

Prove that there exists a set of infinitely many positive integers such that the elements of no finite subset of this set add up to a perfect square.

Prove that for any two linear subspaces $V, W \subset R^n$ the same dimension there is an orthogonal transformation $A:R^n\to R^n$, such that $A(V )=W$ and $A(W) = V$

Which terms must be removed from the sum \[ \frac12 \plus{} \frac14 \plus{} \frac16 \plus{} \frac18 \plus{} \frac1{10} \plus{} \frac1{12} \]if the sum of the remaining terms is equal to $ 1$? $ \textbf{(A)}\ \frac14\text{ and }\frac18 \qquad \textbf{(B)}\ \frac14\text{ and }\frac1{12} \qquad \textbf{(C)}\ \frac18\text{ and }\frac1{12} \qquad \textbf{(D)}\ \frac16\text{ and }\frac1{10} \qquad \textbf{(E)}\ \frac18\text{ and }\frac1{10}$

Define $S(N)$ to be the sum of the digits of $N$ when it is written in base $10$, and take $S^k(N) = S(S(\dots(N)\dots))$ with $k$ applications of $S$. The \textit{stability} of a number $N$ is defined to be the smallest positive integer $K$ where $S^K(N) = S^{K+1}(N) = S^{K+2}(N) = \dots$. Let $T_3$ be the set of all natural numbers with stability $3$. Compute the sum of the two least entries of $T_3$. Proposed by Albert Wang (awang11)

Let $(x,y)$ be the coordinates of a point chosen uniformly at random within the unit square with vertices at $(0,0), (0,1), (1,0),$ and $(1,1).$ The probability that $|x - \tfrac{1}{2}| + |y - \tfrac{1}{2}| < \tfrac{1}{2}$ is $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime integers. Find $p + q.$

Let $n$ be a given positive integer. Find the smallest positive integer $k$ for which the following statement is true: for any given simple connected graph $G$ and minimal cuts $V_1, V_2,\ldots, V_n$, at most $k$ vertices can be chosen with the property that picking any two of the chosen vertices there exists an integer $1\le i\le n$ such that $V_i$ separates the two vertices. A partition of the vertices of $G$ into two disjoint non-empty sets is called a [i]minimal cut[/i] if the number of edges crossing the partition is minimal. [i]Proposed by András Imolay, Budapest[/i]

Fill up each square of a $50$ by $50$ grid with an integer. Let $G$ be the configuration of $8$ squares obtained by taking a $3$ by $3$ grid and removing the central square. Given that for any such $G$ in the $50$ by $50$ grid, the sum of integers in its squares is positive, show there exist a $2$ by $2$ square such that the sum of its entries is also positive.

For any positive integer $d$, prove there are infinitely many positive integers $n$ such that $d(n!)-1$ is a composite number.

Let $N$ be a positive integer. Initially, a positive integer $A$ is written on the board. At each step, we can perform one of the following two operations with the number written on the board: (i) Add $N$ to the number written on the board and replace that number with the sum obtained; (ii) If the number on the board is greater than $1$ and has at least one digit $1$, then we can remove the digit $1$ from that number, and replace the number initially written with this one (with removal of possible leading zeros) For example, if $N = 63$ and $A = 25$, we can do the following sequence of operations: $$25 \rightarrow 88 \rightarrow 151 \rightarrow 51 \rightarrow 5$$ And if $N = 143$ and $A = 2$, we can do the following sequence of operations: $$2 \rightarrow 145 \rightarrow 288 \rightarrow 431 \rightarrow 574 \rightarrow 717 \rightarrow 860 \rightarrow 1003 \rightarrow 3$$ For what values of $N$ is it always possible, regardless of the initial value of $A$ on the blackboard, to obtain the number $1$ on the blackboard, through a finite number of operations?

Let $ABC$ be a triangle, the point $E$ is in the segment $AC$, the point $F$ is in the segment $AB$ and $P=BE\cap CF$. Let $D$ be a point such that $AEDF$ is a parallelogram, Prove that $D$ is in the side $BC$, if and only if, the triangle $BPC$ and the quadrilateral $AEPF$ have the same area.

Let $ z = \frac{1}{2}(\sqrt{2} + i\sqrt{2}) $. The sum $$ \sum_{k = 0}^{13} \dfrac{1}{1 - ze^{k \cdot \frac{i\pi}{7}}} $$ can be written in the form $ a - bi $. Find $ a + b $.

Baron Minchausen talked to a mathematician. Baron said that in his country from any town one can reach any other town by a road. Also, if one makes a circular trip from any town, one passes through an odd number of other towns. By this, as an answer to the mathematician’s question, Baron said that each town is counted as many times as it is passed through. Baron also added that the same number of roads start at each town in his country, except for the town where he was born, at which a smaller number of roads start. Then the mathematician said that baron lied. How did he conclude that?

$300$ parliament members are divided into $3$ chambers, each chamber consists of $100$ members. For every $2$ members, they either know each other or are strangers to each other.Show that no matter how they are divided into these $3$ chambers, it is always possible to choose $2$ members, each from different chamber such that there exist $17$ members from the third chamber so that all of them knows these two members, or all of them are strangers to these two members.

Find all $x$ that satisfy the following equation: \[ \sqrt {1 + \frac {20}x } = \sqrt {1 + 24x} + 2 \] [i]Proposed by Andria Gvaramia, Georgia [/i]

Compute the value of \[\int_{-\pi}^\pi\frac{e^{x^2}-e^{-x^2}}{e^{x^2}-x\sqrt{2}}|x|dx.\]

Let $A, B \in \mathbb{N}$. Consider a sequence $x_1, x_2, \ldots$ such that for all $n\geq 2$, $$x_{n+1}=A \cdot \gcd(x_n, x_{n-1})+B. $$ Show that the sequence attains only finitely many distinct values.

Find all ordered pairs of integers $x,y$ such that $$xy(x^2y^2 - 12xy- 12x- 12y+2) = (2x + 2y)^2.$$ [i]Proposed by Henry Jiang[/i]

Consider the following layouts of nine triangles with the letters $A, B, C, D, E, F, G, H, I$ in its interior. [asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(200); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = 1.740000000000003, xmax = 8.400000000000013, ymin = 3.500000000000005, ymax = 9.360000000000012; /* image dimensions */ draw((5.020000000000005,8.820000000000011)--(2.560000000000003,4.580000000000005)--(7.461947712046029,4.569577506690286)--cycle); /* draw figures */ draw((5.020000000000005,8.820000000000011)--(2.560000000000003,4.580000000000005)); draw((2.560000000000003,4.580000000000005)--(7.461947712046029,4.569577506690286)); draw((7.461947712046029,4.569577506690286)--(5.020000000000005,8.820000000000011)); draw((3.382989341689345,5.990838871467448)--(4.193333333333338,4.580000000000005)); draw((4.202511849578174,7.405966442513598)--(5.828619600041468,4.573707435672692)); draw((5.841878190157451,7.408513542990484)--(4.193333333333338,4.580000000000005)); draw((6.656214943659867,5.990342259816768)--(5.828619600041468,4.573707435672692)); draw((4.202511849578174,7.405966442513598)--(5.841878190157451,7.408513542990484)); draw((3.382989341689345,5.990838871467448)--(6.656214943659867,5.990342259816768)); label("\textbf{A}",(4.840000000000007,8.020000000000010),SE*labelscalefactor,fontsize(22)); label("\textbf{B}",(3.980000000000006,6.640000000000009),SE*labelscalefactor,fontsize(22)); label("\textbf{C}",(4.820000000000007,7.000000000000010),SE*labelscalefactor,fontsize(22)); label("\textbf{D}",(5.660000000000008,6.580000000000008),SE*labelscalefactor,fontsize(22)); label("\textbf{E}",(3.160000000000005,5.180000000000006),SE*labelscalefactor,fontsize(22)); label("\textbf{F}",(4.020000000000006,5.600000000000008),SE*labelscalefactor,fontsize(22)); label("\textbf{G}",(4.800000000000007,5.200000000000007),SE*labelscalefactor,fontsize(22)); label("\textbf{H}",(5.680000000000009,5.620000000000007),SE*labelscalefactor,fontsize(22)); label("\textbf{I}",(6.460000000000010,5.140000000000006),SE*labelscalefactor,fontsize(22)); /* dots and labels */ clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy] A sequence of letters, each letter chosen from$ A, B, C, D, E, F, G, H, I$ is said to be [i]triangle-friendly[/i] if the first and last letter of the sequence is $C$, and for every letter except the first letter, the triangle containing this letter shares an edge with the triangle containing the previous letter in the sequence. For example, the letter after $C$ must be either $A, B$, or $D$. For example, $CBF BC$ is triangle-friendly, but $CBF GH$ and $CBBHC$ are not. [list] [*] (a) Determine the number of triangle-friendly sequences with $2012$ letters. [*] (b) Determine the number of triangle-friendly sequences with exactly $2013$ letters.[/list]

Find all triples $(a, b, c)$ of positive integers such that: $$a + b + c = 24$$ $$a^2 + b^2 + c^2 = 210$$ $$abc = 440$$