In the beginning, there is a circle with three points on it. The points are colored (clockwise): Green, blue, red. Jonathan may perform the following actions, as many times as he wants, in any order: [list] [*] Choose two adjacent points with [u]different[/u] colors, and add a point between them with one of the two colors only. [*] Choose two adjacent points with [u]the same[/u] color, and add a point between them with any of the three colors. [*] Choose three adjacent points, at least two of them having the same color, and delete the middle point. [/list] Can Jonathan reach a state where only three points remain on the circle, colored (clockwise): Blue, green, red?

LeBron James and Carmelo Anthony play a game of one-on-one basketball where the first player to $3$ points or more wins. LeBron James has a $20\%$ chance of making a $3$-point shot; Carmelo has a $10\%$ chance of making a $3$-pointer. LeBron has a $40\%$ chance of making a $2$-point shot from anywhere inside the $3$-point line (excluding dunks, which are also worth $2$ points); Carmelo has a $52\%$ chance of making a $ 2$-point shot from anywhere inside the 3-point line (excluding dunks). LeBron has a $90\%$ chance of dunking on Carmelo; Carmelo has a $95\%$ chance of dunking on LeBron. If each player has $3$ possessions to try to win, LeBron James goes first, and both players follow a rational strategy to try to win, what is the probability that Carmelo Anthony wins the game?

Let $P = \{(x, y) | x, y \in \{0, 1, 2,... , 2015\}\}$ be a set of points on the plane. Straight wires of unit length are placed to connect points in $P$ so that each piece of wire connects exactly two points in $P$, and each point in $P$ is an endpoint of exactly one wire. Prove that no matter how the wires are placed, it is always possible to draw a straight line parallel to either the horizontal or vertical axis passing through midpoints of at least $506$ pieces of wire.

Let $n>1$ be an integer and $S_n$ the set of all permutations $\pi : \{1,2,\cdots,n \} \to \{1,2,\cdots,n \}$ where $\pi$ is bijective function. For every permutation $\pi \in S_n$ we define: \[ F(\pi)= \sum_{k=1}^n |k-\pi(k)| \ \ \text{and} \ \ M_{n}=\frac{1}{n!}\sum_{\pi \in S_n} F(\pi) \] where $M_n$ is taken with all permutations $\pi \in S_n$. Calculate the sum $M_n$.

Let $n$ and $r$ two positive integers. It is wanted to make $r$ subsets $A_1,\ A_2,\dots,A_r$ from the set $\{0,1,\cdots,n-1\}$ such that all those subsets contain exactly $k$ elements and such that, for all integer $x$ with $0\leq{x}\leq{n-1}$ there exist $x_1\in{}A_1,\ x_2\in{}A_2 \dots,x_r\in{}A_r$ (an element of each set) with $x=x_1+x_2+\cdots+x_r$. Find the minimum value of $k$ in terms of $n$ and $r$.

Let $A$ be an infinite set of points in the plane such that inside each circle there are only a finite number of points of $A$, with the following properties: $\bullet$ $(0, 0)$ belongs to $A$. $\bullet$ If $(a, b)$ and $(c, d)$ belong to $A$, then $(a-c, b-d)$ belongs to $A$. $\bullet$ There is a value of $\alpha$ such that by rotating the set $A$ with center at $(0, 0)$ and angle $\alpha$, the set $A$ is obtained again. Prove that $\alpha$ must be equal to $\pm 60^{\circ}$ or $\pm 90^{\circ}$ or $\pm 120^{\circ}$ or $\pm 180^{\circ}$.

In how many ways can Eve fill each of the six squares of a $2 \times 3$ grid with either a $0$ or a $1$, such that Anne can then divide the grid into three congruent rectangles: one containing two $0$s, one containing two $1$s, and one containing a $0$ and a $1$? [i]Proposed by Michael Tang[/i]

In a group of $6$ people playing the card game Tractor, all $54$ cards from $3$ decks are dealt evenly to all the players at random. Each deck is dealt individually. Let the probability that no one has at least two of the same card be $X$. Find the largest integer $n$ such that the $n$th root of $X$ is rational. [i]Proposed by Sammy Charney[/i] [b]Due to the problem having infinitely many solutions, all teams who inputted answers received points.[/b]

Call a convex polyhedron a [i]footballoid [/i] if it has the following properties. (1) Any face is either a regular pentagon or a regular hexagon. (2) All neighbours of a pentagonal face are hexagonal (a [i]neighbour [/i] of a face is a face that has a common edge with it). Find all possibilities for the number of pentagonal and hexagonal faces of a footballoid.

Sofiya and Marquis are playing a game. Sofiya announces to Marquis that she's thinking of a polynomial of the form $f(x)=x^3+px+q$ with three integer roots that are not necessarily distinct. She also explains that all of the integer roots have absolute value less than (and not equal to) $N$, where $N$ is some fixed number which she tells Marquis. As a "move" in this game, Marquis can ask Sofiya about any number $x$ and Sofiya will tell him whether $f(x)$ is positive negative, or zero. Marquis's goal is to figure out Sofiya's polynomial. If $N=3\cdot 2^k$ for some positive integer $k$, prove that there is a strategy which allows Marquis to identify the polynomial after making at most $2k+1$ "moves".

Ana & Bruno decide to play a game with the following rules.: a) Ana has cards $1, 3, 5,7,..., 2n-1$ b) Bruno has cards $2, 4,6, 8,...,2n$ During the first turn and all odd turns afterwards, Bruno chooses one of his cards first and reveals it to Ana, and Ana chooses one of her cards second. Whoever's card is higher gains a point. During the second turn and all even turns afterwards, Ana chooses one of her cards first and reveals it to Bruno, and Bruno chooses one of his cards second. Similarly, whoever's card is higher gains a point. During each turn, neither player can use a card they have already used on a previous turn. The game ends when all cards have been used after $n$ turns. Determine the highest number of points Ana can earn, and how she manages to do this.

Let $ p \equiv 3 \pmod{4}$ be a prime. Define $T = \{ (i,j) \mid i, j \in \{ 0, 1, \cdots , p-1 \} \} \smallsetminus \{ (0,0) \} $. For arbitrary subset $ S ( \ne \emptyset ) \subset T $, prove that there exist subset $ A \subset S $ satisfying following conditions: (a) $ (x_i , y_i ) \in A ( 1 \le i \le 3) $ then $ p \not | x_1 + x_2 - y_3 $ or $ p \not | y_1 + y_2 + x_3 $. (b) $ 8 n(A) > n(S) $

A [i]discrete hexagon with center \((a,b,c)\) \emph{(where \(a\), \(b\), \(c\) are integers)[/i] and radius \(r\) [i](a nonnegative integer)[/i]} is the set of lattice points \((x,y,z)\) such that \(x+y+z=a+b+c\) and \(\max(|x-a|,|y-b|,|z-c|)\le r\). Let \(n\) be a nonnegative integer and \(S\) be the set of triples \((x,y,z)\) of nonnegative integers such that \(x+y+z=n\). If \(S\) is partitioned into discrete hexagons, show that at least \(n+1\) hexagons are needed. [i]Proposed by Linus Tang[/i]

A positive integer given in decimal representation $\overline{ a_na_{n-1} \ldots a_1a_0 }$ is called [i]monotone[/i] if $a_n\leq a_{n-1} \leq \cdots \leq a_0$. Determine the number of monotone positive integers with at most 1993 digits.

How many positive five-digit integers are there that have the product of their five digits equal to $900$? (Karl Czakler)

It is given positive integer $n$. Let $a_1, a_2,..., a_n$ be positive integers with sum $2S$, $S \in \mathbb{N}$. Positive integer $k$ is called separator if you can pick $k$ different indices $i_1, i_2,...,i_k$ from set $\{1,2,...,n\}$ such that $a_{i_1}+a_{i_2}+...+a_{i_k}=S$. Find, in terms of $n$, maximum number of separators

There is a frog in every vertex of a regular 2n-gon with circumcircle($n \geq 2$). At certain time, all frogs jump to the neighborhood vertices simultaneously (There can be more than one frog in one vertex). We call it as $\textsl{a way of jump}$. It turns out that there is $\textsl{a way of jump}$ with respect to 2n-gon, such that the line connecting any two distinct vertice having frogs on it after the jump, does not pass through the circumcentre of the 2n-gon. Find all possible values of $n$.

Two players play the following game: alternatively they write numbers $1$ or $0$ in the vertices of an $n$-gon. First player starts the game and wins if after any of his moves there exists a triangle, whose vertices are three consecutive vertices of the $n$-gon, such that the sum of numbers in it's vertices is divisible by $3$. Second player wins if he prevents this. Determine which player has a winning strategy if: a) $n=2019$ b) $n=2020$ c) $n=2021$

Let $\mathcal{P}$ be a regular polygon, and let $\mathcal{V}$ be its set of vertices. Each point in $\mathcal{V}$ is colored red, white, or blue. A subset of $\mathcal{V}$ is [i]patriotic[/i] if it contains an equal number of points of each color, and a side of $\mathcal{P}$ is [i]dazzling[/i] if its endpoints are of different colors. Suppose that $\mathcal{V}$ is patriotic and the number of dazzling edges of $\mathcal{P}$ is even. Prove that there exists a line, not passing through any point in $\mathcal{V}$, dividing $\mathcal{V}$ into two nonempty patriotic subsets. [i]Ankan Bhattacharya[/i]

On a board there are $4$ positive integers $a, b, c$ and $d$. Dan chooses three of them and writes their product on a paper. Then he substracts $1$ from the other number. He does this until $0$ appears on the board. What are the possible values of the sum of the numbers written on the paper?

Let $n$ be a positive integer. There are $n$ islands with $n-1$ bridges connecting them such that one can travel from any island to another. One afternoon, a fire breaks out in one of the islands. Every morning, it spreads to all neighbouring islands. (Two islands are neighbours if they are connected by a bridge.) To control the spread, one bridge is destroyed every night until the fire has nowhere to spread the next day. Let $X$ be the minimum possible number of bridges one has to destroy before the fire stops spreading. Find the maximum possible value of $X$ over all possible configurations of bridges and island where the fire starts at.

The numbers $1$ to $15$ are each coloured blue or red. Determine all possible colourings that satisfy the following rules: • The number $15$ is red. • If numbers $x$ and $y$ have different colours and $x + y \le 15$, then $x + y$ is blue. • If numbers $x$ and $y$ have different colours and $x \cdot y \le 15$, then $x \cdot y$ is red.

Consider a rectangular grid of $ 10 \times 10$ unit squares. We call a [i]ship[/i] a figure made up of unit squares connected by common edges. We call a [i]fleet[/i] a set of ships where no two ships contain squares that share a common vertex (i.e. all ships are vertex-disjoint). Find the greatest natural number that, for each its representation as a sum of positive integers, there exists a fleet such that the summands are exactly the numbers of squares contained in individual ships.

For which positive integer n can you color the numbers 1,2...2n with n colors, such that every color is used twice and the numbers 1,2,3...n occur as difference of two numbers of the same color exatly once.

Prove that a graph containing a copy of each possible tree on $n$ vertices as a subgraph has at least $n(\ln n - 2)$ edges.