Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$. Proposed by Dominik Burek and Tomasz Ciesla, Poland

Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.

Let $n$ and $k$ be two integers with $n>k\geqslant 1$. There are $2n+1$ students standing in a circle. Each student $S$ has $2k$ neighbors - namely, the $k$ students closest to $S$ on the left, and the $k$ students closest to $S$ on the right. Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbors. Proposed by Gurgen Asatryan, Armenia

Let $a_1,a_2,a_3,\ldots$ be an infinite sequence of positive integers such that $a_{n+2m}$ divides $a_{n}+a_{n+m}$ for all positive integers $n$ and $m.$ Prove that this sequence is eventually periodic, i.e. there exist positive integers $N$ and $d$ such that $a_n=a_{n+d}$ for all $n>N.$

Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.

Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\]true?

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.

Determine the largest integer $N$ for which there exists a table $T$ of integers with $N$ rows and $100$ columns that has the following properties: $\text{(i)}$ Every row contains the numbers $1$, $2$, $\ldots$, $100$ in some order. $\text{(ii)}$ For any two distinct rows $r$ and $s$, there is a column $c$ such that $|T(r,c) - T(s, c)|\geq 2$. (Here $T(r,c)$ is the entry in row $r$ and column $c$.)

Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible. Carl Schildkraut, USA

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\]over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ Proposed by Shahjalal Shohag, Bangladesh

Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers. Proposed by Dorlir Ahmeti, Albania

Let $ABCD$ be a cyclic quadrilateral whose sides have pairwise different lengths. Let $O$ be the circumcenter of $ABCD$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $B_1$ and $D_1$, respectively. Let $O_B$ be the center of the circle which passes through $B$ and is tangent to $\overline{AC}$ at $D_1$. Similarly, let $O_D$ be the center of the circle which passes through $D$ and is tangent to $\overline{AC}$ at $B_1$. Assume that $\overline{BD_1} \parallel \overline{DB_1}$. Prove that $O$ lies on the line $\overline{O_BO_D}$.

Let $a, b, c$ be positive real numbers. Show that $$a^5+b^5+c^5 \geq 5abc(b^2-ac)$$and determine when the equality occurs.

Let $ABC$ be a triangle with orthocenter $H$, $\Gamma$ its circumcircle, and $A' \neq A$, $B' \neq B$, $C' \neq C$ points on $\Gamma$. Define $l_a$ as the line that passes through the projections of $A'$ over $AB$ and $AC$. Define $l_b$ and $l_c$ similarly. Let $O$ be the circumcenter of the triangle determined by $l_a$, $l_b$ and $l_c$ and $H'$ the orthocenter of $A'B'C'$. Show that $O$ is midpoint of $HH'$.

Let $p$ be an odd prime number and suppose that $2^h \not \equiv 1 \text{ (mod } p\text{)}$ for all integer $1 \leq h \leq p-2$. Let $a$ be an even number such that $\frac{p}{2} < a < p$. Define the sequence $a_0, a_1, a_2, \ldots$ as $$a_0 = a, \qquad a_{n+1} = p -b_n, \qquad n = 0,1,2, \ldots,$$where $b_n$ is the greatest odd divisor of $a_n$. Show that the sequence is periodic and determine its period.

Let $d_1, d_2, \ldots, d_n$ be given integers. Show that there exists a graph whose sequence of degrees is $d_1, d_2, \ldots, d_n$ and which contains an perfect matching if, and only if, there exists a graph whose sequence of degrees is $d_2, d_2, \ldots, d_n$ and a graph whose sequence of degrees is $d_1-1, d_2-1, \ldots, d_n-1$.