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 $n$, \[ k ~\Big|~ \sum_{d|k} f(d). \]

Let $S$ be a set of $ 2 \times 2 $ integer matrices whose entries $a_{ij}(1)$ are all squares of integers and, $(2)$ satisfy $a_{ij} \le 200$. Show that $S$ has more than $ 50387 (=15^4-15^2-15+2) $ elements, then it has two elements that commute.

$ABC$ is an isosceles right triangle with right angle $B$ and $AB = 1$. $ABC$ has an incenter at $E$. The excircle to $ABC$ at side $AC$ is drawn and has center $P$. Let this excircle be tangent to $AB$ at $R$. Draw $T$ on the excircle so that $RT$ is the diameter. Extend line $BC$ and draw point $D$ on $BC$ so that $DT$ is perpendicular to $RT$. Extend $AC$ and let it intersect with $DT$ at $G$. Let $F$ be the incenter of $CDG$. Find the area of $\vartriangle EFP$.

What is the smallest number of $3$-cell corners needed to be painted in a $6\times 6$ square so that it was impossible to paint more than one corner of it? (The painted corners should not overlap.)

We have 11 boxes. On a move, we can choose 10 of them and put one ball in each of the boxes chosen. Two players move alternately. The one who gets a box of 21 balls wins. Which of the two players has winning strategy?

Show that $ 2005^{2005}$ is a sum of two perfect squares, but not a sum of two perfect cubes.

The Matini company released a special album with the flags of the $ 12$ countries that compete in the CONCACAM Mathematics Cup. Each postcard envelope has two flags chosen randomly. Determine the minimum number of envelopes that need to be opened to that the probability of having a repeated flag is $50\%$.

Let $ABCDEF$ be a convex hexagon satisfying $AC = DF, CE = FB$ and $EA = BD$. Prove that the lines connecting the midpoints of opposite sides of the hexagon $ABCDEF$ intersect in one point.

Medians $ BD$ and $ CE$ of triangle $ ABC$ are perpendicular, $ BD \equal{} 8$, and $ CE \equal{} 12$. The area of triangle $ ABC$ is [asy]defaultpen(linewidth(.8pt)); dotfactor=4; pair A = origin; pair B = (1.25,1); pair C = (2,0); pair D = midpoint(A--C); pair E = midpoint(A--B); pair G = intersectionpoint(E--C,B--D); dot(A);dot(B);dot(C);dot(D);dot(E);dot(G); label("$A$",A,S);label("$B$",B,N);label("$C$",C,S);label("$D$",D,S);label("$E$",E,NW);label("$G$",G,NE); draw(A--B--C--cycle); draw(B--D); draw(E--C); draw(rightanglemark(C,G,D,3));[/asy]$ \textbf{(A)}\ 24\qquad \textbf{(B)}\ 32\qquad \textbf{(C)}\ 48\qquad \textbf{(D)}\ 64\qquad \textbf{(E)}\ 96$

The quadrilateral $ABCD$ is inscribed in the circle $\omega$. Lines $AD$ and $BC$ intersect at point $E$. Points $M$ and $N$ are selected on segments $AD$ and $BC$, respectively, so that $AM: MD = BN: NC$. The circumscribed circle of the triangle $EMN$ intersects the circle $\omega$ at points $X$ and $Y$. Prove that the lines $AB, CD$ and $XY$ intersect at the same point or are parallel.

Find all the sequences of natural $k_n$ with two properties: a) $k_n \le n \sqrt {n}$ for all $n$ b) $(k_n - k_m)$ is divisible by $(m-n)$ for all $m>n$

A quadrilateral is called $\textit{innovative}$ if its diagonals divide it into $4$ triangles, having the same sets of angle measures. Find the angle measures of an $\textit{innovative}$ quadrilateral, given that one of its angles has measure $13^{\circ}$.

Amandine and Brennon play a turn-based game, with Amadine starting. On their turn, a player must select a positive integer which cannot be represented as a sum of multiples of any of the previously selected numbers. For example, if $3, 5$ have been selected so far, only $1, 2, 4, 7$ are available to be picked; if only $3$ has been selected so far, all numbers not divisible by three are eligible. A player loses immediately if they select the integer $1$. Call a number $n$ [i]feminist[/i] if $\gcd(n, 6) = 1$ and if Amandine wins if she starts with $n$. Compute the sum of the [i]feminist[/i] numbers less than $40$. [i]Proposed by Ashwin Sah[/i]

Let $ABC$ be a triangle such that $AB=AC$ and $\angle BAC$ is obtuse. Point $O$ is the circumcenter of triangle $ABC$, and $M$ is the reflection of $A$ in $BC$. Let $D$ be an arbitrary point on line $BC$, such that $B$ is in between $D$ and $C$. Line $DM$ cuts the circumcircle of $ABC$ in $E,F$. Circumcircles of triangles $ADE$ and $ADF$ cut $BC$ in $P,Q$ respectively. Prove that $DA$ is tangent to the circumcircle of triangle $OPQ$.

The figure on the right shows the ellipse $\frac{(x-19)^2}{19}+\frac{(x-98)^2}{98}=1998$. Let $R_1,R_2,R_3,$ and $R_4$ denote those areas within the ellipse that are in the first, second, third, and fourth quadrants, respectively. Determine the value of $R_1-R_2+R_3-R_4$. [asy] defaultpen(linewidth(0.7)); pair c=(19,98); real dist = 30; real a = sqrt(1998*19),b=sqrt(1998*98); xaxis("x",c.x-a-dist,c.x+a+3*dist,EndArrow); yaxis("y",c.y-b-dist*2,c.y+b+3*dist,EndArrow); draw(ellipse(c,a,b)); label("$R_1$",(100,200)); label("$R_2$",(-80,200)); label("$R_3$",(-60,-150)); label("$R_4$",(70,-150));[/asy]

Let be an infinite sequence $ \left( c_n \right)_{n\ge 1} $ of positive real numbers, with $ c_1=1, $ and satisfying $$ c_{n+1}-\frac{1}{c_{n+1}} =c_n+\frac{1}{c_n} , $$ for all natural numbers $ n. $ Prove that: [b]a)[/b] there exists a natural number $ k $ such that the sequence $ \left( c_n^k+\frac{1}{c_n^k} \right)_{n\ge 1} $ is an arithmetic one. [b]b)[/b] there exist two sequences $ \left( u_n \right)_{n\ge 1} ,\left( v_n \right)_{n\ge 1} $ of nonegative integers such that $ c_n=\sqrt{u_n} +\sqrt{v_n} , $ for any natural number $ n. $