Let $ABC$ be a triangle with incentre $I$ and circumcircle $\omega$. Let $D$ and $E$ be the second intersection points of $\omega$ with $AI$ and $BI$, respectively. The chord $DE$ meets $AC$ at a point $F$, and $BC$ at a point $G$. Let $P$ be the intersection point of the line through $F$ parallel to $AD$ and the line through $G$ parallel to $BE$. Suppose that the tangents to $\omega$ at $A$ and $B$ meet at a point $K$. Prove that the three lines $AE,BD$ and $KP$ are either parallel or concurrent. [i]Proposed by Irena Majcen and Kris Stopar, Slovenia[/i]

Let $P(x)$ be a polynomial with rational coefficients such that when $P(x)$ is divided by the polynomial $x^2 + x + 1$, the remainder is $x + 2$, and when $P(x)$ is divided by the polynomial $x^2 + 1$, the remainder is $2x + 1$. There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficients of that polynomial? $\textbf{(A) } 10 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 20 \qquad \textbf{(E) } 23$

Show that there exists an equiangular hexagon in the plane, whose sides measure 5,8,11,14,23, and 29 units in some order.

Let $n, m, k$ and $l$ be positive integers with $n \neq 1$ such that $n^k + mn^l + 1$ divides $n^{k+l} - 1$. Prove that [list] [*]$m = 1$ and $l = 2k$; or [*]$l|k$ and $m = \frac{n^{k-l}-1}{n^l-1}$. [/list]

In a restaurant, a meal consists of one sandwich and one optional drink. In other words, a sandwich is necessary for a meal but a drink is not necessary. There are two types of sandwiches and two types of drinks. How many possible meals can be purchased? $\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }12\qquad\textbf{(E) }16$

Prove that there exist 1000 consecutive numbers such that none of them is divisible by its sum of the digits

Consider a pyramid whose base is an equilateral triangle $BCD$ and whose other faces are triangles isosceles, right at the common vertex $A$. An ant leaves the vertex $B$ arrives at a point $P$ of the $CD$ edge, from there goes to a point $Q$ of the edge $AC$ and returns to point $B$. If the path you made is minimal, how much is the angle $PQA$ ?

Find the smallest positive integer $n$ such that there are at least three distinct ordered pairs $(x,y)$ of positive integers such that \[x^2-y^2=n.\]

Let $ABC$ an acute triangle with $\angle BAC\geq 60^\circ$ and $\Gamma$ it´s circumcircule. Let $P$ the intersection of the tangents to $\Gamma$ from $B$ and $C$. Let $\Omega$ the circumcircle of the triangle $BPC$. The bisector of $\angle BAC$ intersect $\Gamma$ again in $E$ and $\Omega$ in $D$, in the way that $E$ is between $A$ and $D$. Prove that $\frac{AE}{ED}\leq 2$ and determine when equality holds.

Let $S=\{1,2,3,4,5,6\}$. Compute the number of functions $f:S\rightarrow S$ such that $f(f(f(s)))=2$ if $s$ is odd and $f(f(f(s)))=1$ if $s$ is even.

For distinct real numbers $x$ and $y$, let $M(x,y)$ be the larger of $x$ and $y$ and let $m(x,y)$ be the smaller of $x$ and $y$. If $a<b<c<d<e$, then \[ M(M(a,m(b,c)),m(d,m(a,e)))= \] $ \textbf{(A)}\ a \qquad\textbf{(B)}\ b \qquad\textbf{(C)}\ c \qquad\textbf{(D)}\ d \qquad\textbf{(E)}\ e $

Let $ABC$ be a scalene acute triangle with orthocenter $H$. The circle with center $A$ and radius $AH$ meets the circumcircle of $BHC$ at $T_{a} \neq H$. Define $T_{b}$ and $T_{c}$ similarly. Show that $H$ lies on the circumcircle of $T_{a}T_{b}T_{c}$. [i]Authored by Nikola Velov[/i]

Let $ n$ be a positive integer. Find the number of odd coefficients of the polynomial \[ u_n(x) \equal{} (x^2 \plus{} x \plus{} 1)^n. \]

Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$. [i]Proposed by Netherlands[/i]

A graph $G(V,E)$ is triangle-free, but adding any edges to the graph will form a triangle. It's given that $|V|=2019$, $|E|>2018$, find the minimum of $|E|$ .

Let $ABCDEF$ be a hexagon inscribed in a circle $\Omega$ such that triangles $ACE$ and $BDF$ have the same orthocenter. Suppose that segments $BD$ and $DF$ intersect $CE$ at $X$ and $Y$, respectively. Show that there is a point common to $\Omega$, the circumcircle of $DXY$, and the line through $A$ perpendicular to $CE$. [i]Proposed by Michael Ren and Vincent Huang[/i]

We label every edge of a simple graph with the difference of the degrees of its endpoints. If the number of vertices is $N$, what can be the largest value of the sum of the labels on the edges? [i]Proposed by Dániel Lenger and Gábor Szűcs, Budapest[/i]

Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.

Solve in integers the equation $x + y = x^2 - xy + y^2$.

Three distinct points $A$, $B$, and $N$ are marked on the line $l$, with $B$ lying between $A$ and $N$. For an arbitrary angle $\alpha \in (0,\frac{\pi}{2})$, points $C$ and $D$ are marked in the plane on the same side of $l$ such that $N$, $C$, and $D$ are collinear; $\angle NAD = \angle NBC = \alpha$; and $A$, $B$, $C$, and $D$ are concyclic. Find the locus of the intersection points of the diagonals of $ABCD$ as $\alpha$ varies between $0$ and $\frac{\pi}{2}$.

Given an isosceles trapezoid $ABCD$ and a point $P$. Prove that a quadrilateral can be constructed from segments $PA, PB, PC, PD$. Note: It is allowed that the vertices of a quadrilateral lie not only not only on the sides of the trapezoid, but also on their extensions.

The given numbers are real numbers $ q,t \in \langle \frac{1}{2}; 1) $, $ t \in (0; 1 \rangle $. Prove that there is an increasing sequence of natural numbers $ {n_k} $ ($ k = 1,2, \ldots $) such that $$ t = \lim_{N\to \infty} \sum_{j=1}^N q^{n_j}.$$

Let $0 <a <b$ be two rational numbers. Let $M$ be a set of positive real numbers with the properties: (i) $a \in M$ and $b \in M$; (ii) if $x$ $\in M$ and $y \in M$, then $\sqrt{xy} \in M$. Let $M^*$denote the set of all irrational numbers in $M$. prove that every $c,d$ such that $a <c <d<b$, $M^*$ contains an element $m$ with property $c<m<d$

Let $k$ be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows. Bernardo starts by writing the smallest perfect square with $k+1$ digits. Every time Bernardo writes a number, Silvia erases the last $k$ digits of it. Bernardo then writes the next perfect square, Silvia erases the last $k$ digits of it, and this process continues until the last two numbers that remain on the board differ by at least $2$. Let $f(k)$ be the smallest positive integer not written on the board. For example, if $k = 1$, then the numbers that Bernardo writes are $16$, $25$, $36$, $49$, and $64$, and the numbers showing on the board after Silvia erases are $1$, $2$, $3$, $4$, and $6$, and thus $f(1) = 5$. What is the sum of the digits of $f(2) + f(4) + f(6) + \cdots + f(2016)$? $\textbf{(A) } 7986 \qquad\textbf{(B) } 8002 \qquad\textbf{(C) } 8030 \qquad\textbf{(D) } 8048 \qquad\textbf{(E) } 8064$

Given a positive integer $n$, let $D$ is the set of positive divisors of $n$, and let $f: D \to \mathbb{Z}$ be a function. Prove that the following are equivalent: (a) For any positive divisor $m$ of $n$, \[ n ~\Big|~ \sum_{d|m} f(d) \binom{n/d}{m/d}. \] (b) For any positive divisor $k$ of $n$, \[ k ~\Big|~ \sum_{d|k} f(d). \]