Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as \begin{align*} (x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z) \end{align*} with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.

Let $\mathcal{S}$ be a finite set of at least two points in the plane. Assume that no three points of $\mathcal S$ are collinear. A [i]windmill[/i] is a process that starts with a line $\ell$ going through a single point $P \in \mathcal S$. The line rotates clockwise about the [i]pivot[/i] $P$ until the first time that the line meets some other point belonging to $\mathcal S$. This point, $Q$, takes over as the new pivot, and the line now rotates clockwise about $Q$, until it next meets a point of $\mathcal S$. This process continues indefinitely. Show that we can choose a point $P$ in $\mathcal S$ and a line $\ell$ going through $P$ such that the resulting windmill uses each point of $\mathcal S$ as a pivot infinitely many times. [i]Proposed by Geoffrey Smith, United Kingdom[/i]

Let $x$ and $y$ be positive integers. If ${x^{2^n}}-1$ is divisible by $2^ny+1$ for every positive integer $n$, prove that $x=1$.

A convex quadrilateral $ABCD$ satisfies $AB\cdot CD = BC\cdot DA$. Point $X$ lies inside $ABCD$ so that \[\angle{XAB} = \angle{XCD}\quad\,\,\text{and}\quad\,\,\angle{XBC} = \angle{XDA}.\] Prove that $\angle{BXA} + \angle{DXC} = 180^\circ$. [i]Proposed by Tomasz Ciesla, Poland[/i]

In a dance party initially there are $20$ girls and $22$ boys in the pool and infinitely many more girls and boys waiting outside. In each round, a participant is picked uniformly at random; if a girl is picked, then she invites a boy from the pool to dance and then both of them elave the party after the dance; while if a boy is picked, then he invites a girl and a boy from the waiting line and dance together. The three of them all stay after the dance. The party is over when there are only (two) boys left in the pool. (a) What is the probability that the party never ends? (b) Now the organizer of this party decides to reverse the rule, namely that if a girl is picked, then she invites a boy and a girl from the waiting line to dance and the three stay after the dance; while if a boy is picked, he invites a girl from the pool to dance and both leave after the dance. Still the party is over when there are only (two) boys left in the pool. What is the expected number of rounds until the party ends?

Suppose there are $n$ distinct points on plane. There is circle with radius $r$ and center $O$ on the plane. At least one of the points are in the circle. We do the following instructions. At each step we move $O$ to the baricenter of the point in the circle. Prove that location of $O$ is constant after some steps.

$(NET 4)$ A boy has a set of trains and pieces of railroad track. Each piece is a quarter of circle, and by concatenating these pieces, the boy obtained a closed railway. The railway does not intersect itself. In passing through this railway, the train sometimes goes in the clockwise direction, and sometimes in the opposite direction. Prove that the train passes an even number of times through the pieces in the clockwise direction and an even number of times in the counterclockwise direction. Also, prove that the number of pieces is divisible by $4.$

Let $n$ ($\ge 4$) be an even integer. We label $n$ pairwise distinct real numbers arbitrarily on the $n$ vertices of a regular $n$-gon, and label the $n$ sides clockwise as $e_1, e_2, \ldots, e_n$. A side is called [i]positive[/i] if the numbers on both endpoints are increasing in clockwise direction. An unordered pair of distinct sides $\left\{ e_i,e_j \right\}$ is called [i]alternating[/i] if it satisfies both conditions: (i) $2 \mid (i+j)$; and (ii) if one rearranges the four numbers on the vertices of these two sides $e_i$ and $e_j$ in increasing order $a < b < c < d$, then $a$ and $c$ are the numbers on the two endpoints of one of sides $e_i$ or $e_j$. Prove that the number of alternating pairs of sides and the number of positive sides are of different parity.

The two circles pictured have the same center $C$. Chord $\overline{AD}$ is tangent to the inner circle at $B$, $AC$ is $10$, and chord $\overline{AD}$ has length $16$. What is the area between the two circles? [asy] unitsize(45); import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen xdxdff = rgb(0.49,0.49,1); draw((2,0.15)--(1.85,0.15)--(1.85,0)--(2,0)--cycle); draw(circle((2,1),2.24)); draw(circle((2,1),1)); draw((0,0)--(4,0)); draw((0,0)--(2,1)); draw((2,1)--(2,0)); draw((2,1)--(4,0)); dot((0,0),ds); label("$A$", (-0.19,-0.23),NE*lsf); dot((2,0),ds); label("$B$", (1.97,-0.31),NE*lsf); dot((2,1),ds); label("$C$", (1.96,1.09),NE*lsf); dot((4,0),ds); label("$D$", (4.07,-0.24),NE*lsf); clip((-3.1,-7.72)--(-3.1,4.77)--(11.74,4.77)--(11.74,-7.72)--cycle); [/asy] $ \textbf{(A)}\ 36 \pi \qquad\textbf{(B)}\ 49 \pi\qquad\textbf{(C)}\ 64 \pi\qquad\textbf{(D)}\ 81 \pi\qquad\textbf{(E)}\ 100 \pi $

Consider solutions to the equation \[x^2-cx+1 = \dfrac{f(x)}{g(x)},\] where $f$ and $g$ are polynomials with nonnegative real coefficients. For each $c>0$, determine the minimum possible degree of $f$, or show that no such $f,g$ exist. [i]Proposed by Linus Hamilton and Calvin Deng[/i]

Suppose that $a_1 = 2$ and the sequence $(a_n)$ satisfies the recurrence relation \[\frac{a_n -1}{n-1}=\frac{a_{n-1}+1}{(n-1)+1}\] for all $n \ge 2.$ What is the greatest integer less than or equal to \[\sum^{100}_{n=1} a_n^2?\] $\textbf{(A) } 338{,}550 \qquad \textbf{(B) } 338{,}551 \qquad \textbf{(C) } 338{,}552 \qquad \textbf{(D) } 338{,}553 \qquad \textbf{(E) } 338{,}554$

Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively. Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes. [i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]

A deck of $ 2n\plus{}1$ cards consists of a joker and, for each number between 1 and $ n$ inclusive, two cards marked with that number. The $ 2n\plus{}1$ cards are placed in a row, with the joker in the middle. For each $ k$ with $ 1 \leq k \leq n,$ the two cards numbered $ k$ have exactly $ k\minus{}1$ cards between them. Determine all the values of $ n$ not exceeding 10 for which this arrangement is possible. For which values of $ n$ is it impossible?

Let $a,b,c,d,p$ and $q$ be positive integers satisfying $ad-bc=1$ and $\frac{a}{b}>\frac{p}{q}>\frac{c}{d}$. Prove that: $(a)$ $q\ge b+d$ $(b)$ If $q=b+d$, then $p=a+c$.