Given triangle $ABC$ variable points $X$ and $Y$ are chosen on segments $AB$ and $AC$, respectively. Point $Z$ on line $BC$ is chosen such that $ZX=ZY$. The circumcircle of $XYZ$ cuts the line $BC$ for the second time at $T$. Point $P$ is given on line $XY$ such that $\angle PTZ = 90^ \circ$. Point $Q$ is on the same side of line $XY$ with $A$ furthermore $\angle QXY = \angle ACP$ and $\angle QYX = \angle ABP$. Prove that the circumcircle of triangle $QXY$ passes through a fixed point (as $X$ and $Y$ vary).

Find the largest integer $k$ such that $$k\leq\sqrt{2}+\sqrt[3]{\frac{3}{2}}+\sqrt[4]{\frac{4}{3}}+\sqrt[5]{\frac{5}{4}}+\cdots+\sqrt[2015]{\frac{2015}{2014}}.$$

For every pair of positive integers $(x, y)$ we define $f(x,y)$ as follows: $f(x,1) = x$ $f(x,y) = 0$ if $y > x$ $f(x +1,y) = y[f(x,y)+ f(x, y-1)]$ Evaluate $f(5, 5)$.

For an integer $n \ge 2$, let $G_n$ be an $n \times n$ grid of unit cells. A subset of cells $H \subseteq G_n$ is considered \textit{quasi-complete} if and only if each row of $G_n$ has at least one cell in $H$ and each column of $G_n$ has at least one cell in $H$. A subset of cells $K \subseteq G_n$ is considered \textit{quasi-perfect} if and only if there is a proper subset $L \subset K$ such that $|L| = n$ and no two elements in $L$ are in the same row or column. Let $\vartheta(n)$ be the smallest positive integer such that every quasi-complete subset $H \subseteq G_n$ with $|H| \ge \vartheta(n)$ is also quasi-perfect. Moreover, let $\varrho(n)$ be the number of quasi-complete subsets $H \subseteq G_n$ with $|H| = \vartheta(n) - 1$ such that $H$ is not quasi-perfect. Compute $\vartheta(20) + \varrho(20)$.

In some squares of a $2012\times 2012$ grid there are some beetles, such that no square contain more than one beetle. At one moment, all the beetles fly off the grid and then land on the grid again, also satisfying the condition that there is at most one beetle standing in each square. The vector from the centre of the square from which a beetle $B$ flies to the centre of the square on which it lands is called the [i]translation vector[/i] of beetle $B$. For all possible starting and ending configurations, find the maximum length of the sum of the [i]translation vectors[/i] of all beetles.

Let $\omega$ be a circle. Let $E$ be on $\omega$ and $S$ outside $\omega$ such that line segment $SE$ is tangent to $\omega$. Let $R$ be on $\omega$. Let line $SR$ intersect $\omega$ at $B$ other than $R$, such that $R$ is between $S$ and $B$. Let $I$ be the intersection of the bisector of $\angle ESR$ with the line tangent to $\omega$ at $R$; let $A$ be the intersection of the bisector of $\angle ESR$ with $ER$. If the radius of the circumcircle of $\triangle EIA$ is $10$, the radius of the circumcircle of $\triangle SAB$ is $14$, and $SA = 18$, then $IA$ can be expressed in simplest form as $\frac{m}{n}$. Find $m + n$.

Turbo the snail plays a game on a board with $2024$ rows and $2023$ columns. There are hidden monsters in $2022$ of the cells. Initially, Turbo does not know where any of the monsters are, but he knows that there is exactly one monster in each row except the first row and the last row, and that each column contains at most one monster. Turbo makes a series of attempts to go from the first row to the last row. On each attempt, he chooses to start on any cell in the first row, then repeatedly moves to an adjacent cell sharing a common side. (He is allowed to return to a previously visited cell.) If he reaches a cell with a monster, his attempt ends and he is transported back to the first row to start a new attempt. The monsters do not move, and Turbo remembers whether or not each cell he has visited contains a monster. If he reaches any cell in the last row, his attempt ends and the game is over. Determine the minimum value of $n$ for which Turbo has a strategy that guarantees reaching the last row on the $n$-th attempt or earlier, regardless of the locations of the monsters. [i]Proposed by Cheuk Hei Chu, Hong Kong[/i]

There is a finite number of towns in a country. They are connected by one direction roads. It is known that, for any two towns, one of them can be reached from another one. Prove that there is a town such that all remaining towns can be reached from it.

Let $a_1,\ldots a_{11}$ be distinct positive integers, all at least $2$ and with sum $407$. Does there exist an integer $n$ such that the sum of the $22$ remainders after the division of $n$ by $a_1,a_2,\ldots ,a_{11},4a_1,4a_2,\ldots ,4a_{11}$ is $2012$?

Which species has the largest $\text{F-A-F}$ bond angle where $\text{A}$ is the central atom? $ \textbf{(A) }\ce{BF3} \qquad\textbf{(B) }\ce{CF4} \qquad\textbf{(C) }\ce{NF3}\qquad\textbf{(D) }\ce{OF2}\qquad$

Given is a triangle $ABC$, its circumcircle $\omega$, and a circle $k$ that touches $\omega$ from the outside, and also touches rays $AB$ and $AC$ in $P$ and $Q$, respectively. Prove that the $A$-excenter of $\triangle ABC$ is the midpoint of $\overline{PQ}$.

Prove that, for every pair $ n$, $r$ of positive integers, there can be found a polynomial $ f(x)$ of degree $ n$ with integer coefficients, so that every polynomial $ g(x)$ of degree at most $ n$, for which the coefficients of the polynomial $ f(x)\minus{}g(x)$ are integers with absolute value not greater than $ r$, is irreducible over the field of rational numbers.

Given [i]Fibonacci[/i] sequence $(F_n)$ a) Prove that: for all $u,v\in \mathbb{N}, u\ge 1$, we have:$$F_{u+v}=F_{u-1}F_{v}+F_{u}F_{v+1}.$$ b) Prove that: for all $n\in \mathbb{N}, n\ge 1$, we have:$$F_{2n}=F_n(F_{n-1}+F_{n+1}),$$$$F_{2n+1}=F_n^2+F_{n+1}^2.$$

There are $N$ houses in a city. Every Christmas, Santa visits these $N$ houses in some order. Show that if $N$ is large enough, then it holds that for three consecutive years there are always are $13$ houses such that they have been visited in the same order for two years (out of the three consecutive years). Determine the smallest $N$ for which this holds.

When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)

Prove that if $x,y,z >1$ and $\frac 1x +\frac 1y +\frac 1z = 2$, then \[\sqrt{x+y+z} \geq \sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\]

Let $f(x)$ and $g(x)$ be given by $f(x) = \frac{1}{x} + \frac{1}{x-2} + \frac{1}{x-4} + \cdots + \frac{1}{x-2018}$ $g(x) = \frac{1}{x-1} + \frac{1}{x-3} + \frac{1}{x-5} + \cdots + \frac{1}{x-2017}$. Prove that $|f(x)-g(x)| >2$ for any non-integer real number $x$ satisfying $0 < x < 2018$.

Evaluate $ \int_0^{\sqrt{2}\minus{}1} \frac{1\plus{}x^2}{1\minus{}x^2}\ln \left(\frac{1\plus{}x}{1\minus{}x}\right)\ dx$.

In the beginning there are $100$ integers in a row on the blackboard. Kain and Abel then play the following game: A [i]move[/i] consists in Kain choosing a chain of consecutive numbers; the length of the chain can be any of the numbers $1,2,\dots,100$ and in particular it is allowed that Kain only chooses a single number. After Kain has chosen his chain of numbers, Abel has to decide whether he wants to add $1$ to each of the chosen numbers or instead subtract $1$ from of the numbers. After that the next move begins, and so on. If there are at least $98$ numbers on the blackboard that are divisible by $4$ after a move, then Kain has won. Prove that Kain can force a win in a finite number of moves.

Let $ABCD$ be a convex quadrilateral. Extend line $CD$ past $D$ to meet line $AB$ at $P$ and extend line $CB$ past $B$ to meet line $AD$ at $Q$. Suppose that line $AC$ bisects $\angle BAD$. If $AD = \frac{7}{4}$, $AP = \frac{21}{2}$, and $AB = \frac{14}{11}$ , compute $AQ$.

$ABCD$ is a cyclic quadrilateral, with diagonals $AC,BD$ perpendicular to each other. Let point $F$ be on side $BC$, the parallel line $EF$ to $AC$ intersect $AB$ at point $E$, line $FG$ parallel to $BD$ intersect $CD$ at $G$. Let the projection of $E$ onto $CD$ be $P$, projection of $F$ onto $DA$ be $Q$, projection of $G$ onto $AB$ be $R$. Prove that $QF$ bisects $\angle PQR$.

Let be given a real number $\alpha\ne0$. Show that there is a unique point $P$ in the coordinate plane, such that for every line through $P$ which intersects the parabola $y=\alpha x^2$ in two distinct points $A$ and $B$, segments $OA$ and $OB$ are perpendicular (where $O$ is the origin).

On the AMC 8 contest Billy answers 13 questions correctly, answers 7 questions incorrectly and doesn't answer the last 5. What is his score? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 19 \qquad \textbf{(E)}\ 26$

For a positive integer $n$ not divisible by $211$, let $f(n)$ denote the smallest positive integer $k$ such that $n^k - 1$ is divisible by $211$. Find the remainder when $$\sum_{n=1}^{210} nf(n)$$ is divided by $211$. [i]Proposed by ApraTrip[/i]

Given a natural number $n$, for which we can find a prime number less than $\sqrt{n}$ that is not a divisor of $n$. The sequence $a_1, a_2,... ,a_n$ is the numbers $1, 2,... ,n$ arranged in some order. For this sequence, we will find the longest ascending subsequense $a_{i_1} < a_{i_2} < ... < a_{i_k}$, ($i_1 <...< i_k$) and the longest decreasing substring $a_{j_1} > ... > a_{j_l}$, ($j_1 < ... < j_l$) . Prove that at least one of these two subsequnsces $a_{i_1} , . . . , a_{i_k}$ and $a_{j_1} > ... > a_{j_l}$ contains a number that is not a divisor of $n$.