$A, B, C$, and $D$ are the four different points on the circle $O$ in the order. Let the centre of the scribed circle of triangle $ABC$, which is tangent to $BC$, be $O_1$. Let the centre of the scribed circle of triangle $ACD$, which is tangent to $CD$, be $O_2$. (1) Show that the circumcentre of triangle $ABO_1$ is on the circle $O$. (2) Show that the circumcircle of triangle $CO_1O_2$ always pass through a fixed point on the circle $O$, when $C$ is moving along arc $BD$.

The graph of $y=e^{x+1}+e^{-x}-2$ has an axis of symmetry. What is the reflection of the point $(-1,\tfrac{1}{2})$ over this axis? $\textbf{(A) }\left(-1,-\frac{3}{2}\right)\qquad\textbf{(B) }(-1,0)\qquad\textbf{(C) }\left(-1,\tfrac{1}{2}\right)\qquad\textbf{(D) }\left(0,\frac{1}{2}\right)\qquad\textbf{(E) }\left(3,\frac{1}{2}\right)$

A train, an hour after starting, meets with an accident which detains it a half hour, after which it proceeds at $ \frac{3}{4}$ of its former rate and arrives $ 3 \frac{1}{2}$ hours late. Had the accident happened $ 90$ miles farther along the line, it would have arrived only $ 3$ hours late. The length of the trip in miles was: $ \textbf{(A)}\ 400 \qquad \textbf{(B)}\ 465 \qquad \textbf{(C)}\ 600 \qquad \textbf{(D)}\ 640 \qquad \textbf{(E)}\ 550$

Two players are writting in turn natural numbers not exceeding $p$. The rules forbid to write the divisors of the numbers already having been written. Those who cannot make his move looses. a) Who, and how, can win if $p=10$? b) Who wins if $p=1000$?

Ivan and Kolya play a game, Ivan starts. Initially, the polynomial $x-1$ is written of the blackboard. On one move, the player deletes the current polynomial $f(x)$ and replaces it with $ax^{n+1}-f(-x)-2$, where $\deg(f)=n$ and $a$ is a real root of $f$. The player who writes a polynomial which does not have real roots loses. Can Ivan beat Kolya?

We consider the real numbers $a _ 1, a _ 2, a _ 3, a _ 4, a _ 5$ with the zero sum and the property that $| a _ i - a _ j | \le 1$ , whatever it may be $i,j \in \{1, 2, 3, 4, 5 \} $. Show that $a _ 1 ^ 2 + a _ 2 ^ 2 + a _ 3 ^ 2 + a _ 4 ^ 2 + a _ 5 ^ 2 \le \frac {6} {5}$ .

Let $p > 5$ be a prime number. Show that there exists a prime number $q < p$ and a positive integer $n$ such that $p$ divides $n^2-q$. [i]Proposed by Andrew Gu.[/i]

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 $y$.