Alice and Bob are in a hardware store. The store sells coloured sleeves that fit over keys to distinguish them. The following conversation takes place: [color=#0000FF]Alice:[/color] Are you going to cover your keys? [color=#FF0000]Bob:[/color] I would like to, but there are only $7$ colours and I have $8$ keys. [color=#0000FF]Alice:[/color] Yes, but you could always distinguish a key by noticing that the red key next to the green key was different from the red key next to the blue key. [color=#FF0000]Bob:[/color] You must be careful what you mean by "[i]next to[/i]" or "[i]three keys over from[/i]" since you can turn the key ring over and the keys are arranged in a circle. [color=#0000FF]Alice:[/color] Even so, you don't need $8$ colours. [b]Problem:[/b] What is the smallest number of colours needed to distinguish $n$ keys if all the keys are to be covered.

A person enters the social network facebook. He befriends at least one person a day for the first $30$ days. At the end of those $30$ days, it has been exactly $45$ friends. Prove that there is a sequence of consecutive days where made exactly $14$ friends.

Let $E$ be the ellipse lying in the $x, y$ plane centered at $(0, 0)$ with semi-major axis of length $2$ along the $x$-axis and semi-minor axis of length $1$ along the $y$-axis. Let $C$ be a cone created by revolving two perpendicular lines about an angle bisector of the perpendicular angle. There are some points $(x, y, z)$ where the vertex of $C$ could be so that $E$ is the intersection of $C$ with the $x, y$ plane. These points define a convex polygon in the $x, z$ plane. The area of this polygon can be expressed as $\sqrt{n}$ for some positive integer $n.$ Find $n.$ (Some definitions: the semi-major axis is the longest distance from the center of the ellipse to the boundary, and the semi-minor axis is the shortest distance from the center of the ellipse to the boundary.)

Let $p$ and $q$ be complex polynomials with $\deg p>\deg q$ and let $f(z)=\frac{p(z)}{q(z)}$. Suppose that all roots of $p$ lie inside the unit circle $|z|=1$ and that all roots of $q$ lie outside the unit circle. Prove that $$\max_{|z|=1}|f'(z)|>\frac{\deg p-\deg q}2\max_{|z|=1}|f(z)|.$$

Find the number of dierent positive integer triples (x; y; z) satsfying the equations x+y-z=1 and $x^2+y^2-z^2=1$.

The number $B=\overline{a_1a_2\dots a_na_1a_2\dots a_n}$ it is called $repetition$ of the natural positive number $A = \overline{a_1a_2\dots a_n}$.Prove that there is an infinity of natural numbers ,whose $repetition$ is a perfect square .

Let $N$ and $K$ be positive integers satisfying $1 \leq K \leq N$. A deck of $N$ different playing cards is shuffled by repeating the operation of reversing the order of $K$ topmost cards and moving these to the bottom of the deck. Prove that the deck will be back in its initial order after a number of operations not greater than $(2N/K)^2$.

Let the area and the perimeter of a cyclic quadrilateral $C$ be $A_C$ and $P_C$, respectively. If the area and the perimeter of the quadrilateral which is tangent to the circumcircle of $C$ at the vertices of $C$ are $A_T$ and $P_T$ , respectively, prove that $\frac{A_C}{A_T} \geq \left (\frac{P_C}{P_T}\right )^2$.

A $ 3\times3\times3$ cube composed of $ 27$ unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes from a $ 3\times3\times1$ block (the order is irrelevant) with the property that the line joining the centers of the two cubes makes a $ 45^\circ$ angle with the horizontal plane.

Tyler rolls two $ 4025 $ sided fair dice with sides numbered $ 1, \dots , 4025 $. Given that the number on the first die is greater than or equal to the number on the second die, what is the probability that the number on the first die is less than or equal to $ 2012 $?

Minneapolis-St. Paul International Airport is $ 8$ miles southwest of downtown St. Paul and $ 10$ miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis? $ \textbf{(A)}\ 13\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 15\qquad \textbf{(D)}\ 16\qquad \textbf{(E)}\ 17$

A positive integer $n$ is called a $\textit{supersquare}$ if there exists a positive integer $m$, such that $10 \nmid m$ and the decimal representation of $n=m^2$ consists only of digits among $\{0, 4, 9\}$. Are there infinitely many $\textit{supersquares}$?

Let $u$ be the positive root of the equation $x^2+x-4=0$. The polynomial $$P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0$$ where $n$ is positive integer has non-negative integer coefficients and $P(u)=2017$. 1) Prove that $a_0+a_1+...+a_n\equiv 1\mod 2$. 2) Find the minimum possible value of $a_0+a_1+...+a_n$.

Let $\mathbb{R}^{+}$ denote the set of all positive real numbers. Find all functions $f:\mathbb{R}^{+}\longrightarrow \mathbb{R}$ satisfying \[f(x)+f(y)\le \frac{f(x+y)}{2}, \frac{f(x)}{x}+\frac{f(y)}{y}\ge \frac{f(x+y)}{x+y},\] for all $x, y\in \mathbb{R}^{+}$.

Let $ABC$ be a triangle with incenter $I$. Let segment $AI$ intersect the incircle of triangle $ABC$ at point $D$. Suppose that line $BD$ is perpendicular to line $AC$. Let $P$ be a point such that $\angle BPA = \angle PAI = 90^\circ$. Point $Q$ lies on segment $BD$ such that the circumcircle of triangle $ABQ$ is tangent to line $BI$. Point $X$ lies on line $PQ$ such that $\angle IAX = \angle XAC$. Prove that $\angle AXP = 45^\circ$. [i]Luke Robitaille[/i]

$\triangle{ABC}$ is equailateral. $E$ is any point on $\overline{AC}$ produced and the equilateral $\triangle{ECD}$ is drawn. If $M$ and $N$ are the midpoints of $\overline{AD}$ and $\overline{EB}$ respectively then show that $\triangle{CMN}$ is equilateral.

An $7\times 7$ array is divided in $49$ unit squares. Find all integers $n \in N^*$ for which $n$ checkers can be placed on the unit squares so that each row and each line have an even number of checkers. ($0$ is an even number, so there may exist empty rows or columns. A square may be occupied by at most $1$ checker).

Determine all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying \[f(x + yf(x)) + f(xy) = f(x) + f(2019y),\] for all real numbers $x$ and $y$.

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]

For $n \geq 3$, a [i]special n-triangle[/i] is a triangle with $n$ distinct numbers on each side such that the sum of the numbers on a side is the same for all sides. For instance, because $41 + 23 + 43 = 43 + 17 + 47 = 47 + 19 + 41$, the following is a special $3$-triangle: $$41$$ $$23\text{ }\text{ }\text{ }\text{ }\text{ }19$$ $$43\text{ }\text{ }\text{ }\text{ }\text{ }17\text{ }\text{ }\text{ }\text{ }\text{ }47$$ Note that a special $n$-triangle contains $3(n - 1)$ numbers. An infinite set $A$ of positive integers is a [i]special set[/i] if, for each $n \geq 3$, the smallest $3(n - 1)$ numbers of $A$ can be used to form a special $n$-triangle. Show that the set of positive integers that are not multiples of $2023$ is a special set.

The Euclidean Algorithm on inputs $a$ and $b$ is a way to find the greatest common divisor $\gcd(a,b)$. Suppose WLOG that $a>b$. On each step of the Euclidan Algorithm, we solve the equation $a=bq+r$ for integers $q,r$ such that $0\leq r<b$, and repeat on $b$ and $r$. Thus $\gcd(a,b)=\gcd(b,r)$, and we repeat. If $r=0$, we are done. For example, $\gcd(100,15)=\gcd(15,10)=\gcd(10,5)=5$, because $100=15\cdot6+10$, $15=10\cdot1+5$, and $10=5\cdot2+0$. Thus, the Euclidean Algorithm here takes $3$ steps. What is the largest number of steps that the Euclidean Algorithm can take on some integer inputs $a,b$ where $0<a,b<10^{2016}$? Let $C$ be the actual answer and $A$ be the answer you submit. If $\tfrac{|A-C|}{C}>\tfrac{1}{2}$, then your score will be $0$. Otherwise, your score will be given by $\max\{0,\lceil25-2(\tfrac{|A-C|}{20})^{1/2.2}\rceil\}$.

Solve the equation $$\frac{10}{x+10}+\frac{10\cdot 9}{(x+10)(x+9)}+\frac{10\cdot 9\cdot 8}{(x+10)(x+9)(x+8)}+ ...+\frac{10\cdot 9\cdot ... \cdot 2 \cdot 1}{(x+10)(x+9)\cdot ... \cdot(x+1)}=11$$

There are three flies of negligible size that start at the same position on a circular track with circumference 1000 meters. They fly clockwise at speeds of 2, 6, and $k$ meters per second, respectively, where $k$ is some positive integer with $7\le k \le 2013$. Suppose that at some point in time, all three flies meet at a location different from their starting point. How many possible values of $k$ are there? [i]Ray Li[/i]

Find all functions $f : \mathbb{R} \to \mathbb{R}$, such that for any real $x, y$ holds the following: $$f(x+yf(x+y)) = f(y^2) + xf(y) + f(x)$$ [i]Proposed by Vadym Koval[/i]

What is the maximum number of colours that can be used to paint an $8 \times 8$ chessboard so that every square is painted in a single colour, and is adjacent , horizontally, vertically but not diagonally, to at least two other squares of its own colour? (A Shapovalov)