Let $ABCD$ be a convex quadrilateral. A circle passing through the points $A$ and $D$ and a circle passing through the points $B$ and $C$ are externally tangent at a point $P$ inside the quadrilateral. Suppose that \[\angle{PAB}+\angle{PDC}\leq 90^\circ\qquad\text{and}\qquad\angle{PBA}+\angle{PCD}\leq 90^\circ.\] Prove that $AB+CD \geq BC+AD$. [i]Proposed by Waldemar Pompe, Poland[/i]

A square is drawn in the Cartesian coordinate plane with vertices at $(2,2)$, $(-2,2)$, $(-2,-2)$, and $(2,-2)$. A particle starts at $(0,0)$. Every second it moves with equal probability to one of the eight lattice points (points with integer coordinates) closest to its current position, independently of its previous moves. In other words, the probability is $\frac{1}{8}$ that the particle will move from $(x,y)$ to each of $(x,y+1)$, $(x+1,y+1)$, $(x+1,y)$, $(x+1,y-1)$, $(x,y-1)$, $(x-1,y-1)$, $(x-1,y)$, $(x-1,y+1)$. The particle will eventually hit the square for the first time, either at one of the $4$ corners of the square or one of the $12$ lattice points in the interior of one of the sides of the square. The probability that it will hit at a corner rather than at an interior point of a side is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? $\textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 39$

Suppose $f : \mathbb{N} \longrightarrow \mathbb{N}$ is a function that satisfies $f(1) = 1$ and $f(n + 1) =\{\begin{array}{cc} f(n)+2&\mbox{if}\ n=f(f(n)-n+1),\\f(n)+1& \mbox{Otherwise}\end {array}$ $(a)$ Prove that $f(f(n)-n+1)$ is either $n$ or $n+1$. $(b)$ Determine$f$.

The determinant \[\begin{vmatrix}3&-2&5\\ 7&1&-4\\ 5&2&3\end{vmatrix}\] has the same value as the determinant \[\begin{vmatrix}x&1+x&2+x\\ 3&0&1\\ 1&1&0\end{vmatrix}\] Find $x$.

If the margin made on an article costing $ C$ dollars and selling for $ S$ dollars is $ M\equal{}\frac{1}{n}C$, then the margin is given by: $ \textbf{(A)}\ M\equal{}\frac{1}{n\minus{}1}S \qquad \textbf{(B)}\ M\equal{}\frac{1}{n}S \qquad \textbf{(C)}\ M\equal{}\frac{n}{n\plus{}1}S \\ \textbf{(D)}\ M\equal{}\frac{1}{n\plus{}1}S \qquad \textbf{(E)}\ M\equal{}\frac{n}{n\minus{}1}S$

Let $n$ and $m$ be positive integers. The daycare nanny uses $n \times m$ square floor mats to construct an $n \times m$ rectangular area, with a baby on each of the mats. Each baby initially faces toward one side of the rectangle. When the nanny claps, all babies crawl one mat forward in the direction it is facing at, and then turn 90 degrees clockwise. If a baby crawls outside of the rectangle, it cries. If two babies simultaneously crawl onto the same mat, they bump into each other and cry. Suppose that it is possible for the nanny to arrange the initial direction of each baby so that, no matter how many times she claps, no baby would cry. Find all possible values of $n$ and $m$. [i]Proposed by Chu-Lan Kao[/i]

In a club with $30$ members, every member initially had a hat. One day each member sent his hat to a different member (a member could have received more than one hat). Prove that there exists a group of $10$ members such that no one in the group has received a hat from another one in the group.

Let $a$ be a non-zero real number. For each integer $n$, we define $S_n = a^n + a^{-n}$. Prove that if for some integer $k$, the sums $S_k$ and $S_{k+1}$ are integers, then the sums $S_n$ are integers for all integers $n$.

Let $k$ be a natural number. For each function $f : \mathbb{N}\to \mathbb{N}$ define the sequence of functions $(f_{m})_{m\geq 1}$ by $f_{1}= f$ and $f_{m+1}= f \circ f_{m}$ for $m \geq 1$ . Function $f$ is called $k$-[i]nice[/i] if for each $n \in\mathbb{N}: f_{k}(n) = f (n)^{k}$. (a) For which $k$ does there exist an injective $k$-nice function $f$ ? (b) For which $k$ does there exist a surjective $k$-nice function $f$ ?

In triangle $ABC$, $AB=3$, $AC=5$, and $BC=4$. Let $P$ be a point inside triangle $ABC$, and let $D$, $E$, and $F$ be the projections of $P$ onto sides $BC$, $AC$, and $AB$, respectively. If $PD:PE:PF=1:1:2$, then find the area of triangle $DEF$. (Express your answer as a reduced fraction.) (will insert image here later)

Functions $f,g:\mathbb{Z}\to\mathbb{Z}$ satisfy $$f(g(x)+y)=g(f(y)+x)$$ for any integers $x,y$. If $f$ is bounded, prove that $g$ is periodic.

Let $G{}$ be an arbitrary finite group, and let $t_n(G)$ be the number of functions of the form \[f:G^n\to G,\quad f(x_1,x_2,\ldots,x_n)=a_0x_1a_1\cdots x_na_n\quad(a_0,\ldots,a_n\in G).\]Determine the limit of $t_n(G)^{1/n}$ as $n{}$ tends to infinity.

Given a polyhedron $P$. Mikita claims that he can write one integer on each face of $P$ such that not all the written numbers are zeros, and for each vertex $V$ of $P$ the sum of numbers on faces containing $V$ is equals to 0. Matvei claims that he can write one integer in each vertex of $P$ such that not all the written numbers are zeros, and for each face $F$ of $P$ the sum of numbers in vertices belonging to $F$ is equals to 0. Show that if the number of edges of polyhedron $P$ is odd, then at least one of the boys is right.

Let $f : [0, 1] \rightarrow \mathbb{R}$ be a monotonically increasing function such that $$f\left(\frac{x}{3}\right) = \frac{f(x)}{2}$$ $$f(1 0 x) = 2018 - f(x).$$ If $f(1) = 2018$, find $f\left(\dfrac{12}{13}\right)$.

Two players play a game as follows: there $n > 1$ rounds and $d \geq 1$ is fixed. In the first round A picks a positive integer $m_1$, then B picks a positive integer $n_1 \not = m_1$. In round $k$ (for $k = 2, \ldots , n$), A picks an integer $m_k$ such that $m_{k-1} < m_k \leq m_{k-1} + d$. Then B picks an integer $n_k$ such that $n_{k-1} < n_k \leq n_{k-1} + d$. A gets $\gcd(m_k,n_{k-1})$ points and B gets $\gcd(m_k,n_k)$ points. After $n$ rounds, A wins if he has at least as many points as B, otherwise he loses. For each $(n, d)$ which player has a winning strategy?

Find the smallest possible distance of points $ P$ and $ Q$ on a $ xy$-plane, if $ P$ lies on the line $ y \equal{} x$ and $ Q$ lies on the curve $ y \equal{} 2^x$.