There are $n{}$ wise men in a hall and everyone sees each other. Each man will wear a black or white hat. The wise men should simultaneously write down on their piece of paper a guess about the color of their hat. If at least one does not guess, they will all be executed. The wise men can discuss a strategy before the test and they know that the layout of the hats will be chosen randomly from the set of all $2^n$ layouts. They want to choose their strategy so that the number of layouts for which everyone guesses correctly is as high as possible. What is this number equal to?

The excircles of triangle $ABC$ touch its sides $BC$, $CA$, and $AB$ at points $A_1$, $B_1$, and $C_1$, respectively. Let $B_2$ and $C_2$ be the midpoints of segments $BB_1$ and $CC_1$, respectively. Line $B_2C_2$ intersects line $BC$ at point $W$. Prove that $AW = A_1W$.

Let $n>1$ be a natural number and $x_k{}$ be the residue of $n^2$ modulo $\lfloor n^2/k\rfloor+1$ for all natural $k{}$. Compute the sum \[\bigg\lfloor\frac{x_2}{1}\bigg\rfloor+\bigg\lfloor\frac{x_3}{2}\bigg\rfloor+\cdots+\left\lfloor\frac{x_n}{n-1}\right\rfloor.\]

We call a natural number almost a square if it can be represented as a product of two numbers that differ by no more than one percent of the larger of them. Prove that there are infinitely many consecutive quadruples of almost squares.

On a table, cards numbered $1, 2, \ldots , 200$ are laid out in a row in some order, and a line is drawn on the table between some two of them. It is allowed to swap two adjacent cards if the number on the left is greater than the number on the right. After a few such moves, the cards were arranged in ascending order. Prove we have swapped pairs of cards separated by the line no more than 1884 times.

The symmetric difference of two homothetic triangles $T_1$ and $T_2$ consists of six triangles $t_1, \ldots, t_6$ with circumcircles $\omega_1, \omega_2, \ldots, \omega_6$ (counterclockwise, no two intersect). Circle $\Omega_1$ with center $O_1$ is externally tangent to $\omega_1, \omega_3,$ and $\omega_5$; circle $\Omega_2$ with center $O_2$ is externally tangent to $\omega_2, \omega_4,$ and $\omega_6$; circle $\Omega_3$ with center $O_3$ is internally tangent to $\omega_1, \omega_3,$ and $\omega_5$; circle $\Omega_4$ with center $O_4$ is internally tangent to $\omega_2, \omega_4,$ and $\omega_6$. Prove that $O_1O_3 = O_2O_4$. Proposed by Ilya Zamotorin

Each student at a school divided 18 subjects into six disjoint triples. Could it happen that every triple of subjects is among the triples of exactly one student?

Let $r\geqslant 0$ be a real number and define $f(x)=1/(1+x^2)^r$. Prove that \[|f^{(k)}(x)|\leqslant\frac{2r\cdot(2r+1)\cdots(2r+k-1)}{(1+x^2)^{r+k/2}},\]for every natural number $k{}$. Here, $f^{(k)}(x)$ denotes the $k^{\text{th}}$ derivative of $f$.

Each cell of an $100\times 100$ board is divided into two triangles by drawing some diagonal. What is the smallest number of colors in which it is always possible to paint these triangles so that any two triangles having a common side or vertex have different colors?

Let $1 < a_1 < a_2 < \cdots < a_N$ be natural numbers. It is known that for any $1\leqslant i\leqslant N$ the product of all these numbers except $a_i$ increased by one, is a multiple of $a_i$. Prove that $a_1\leqslant N$.

In quadrilateral $ABCD$, a circle $\omega$ is inscribed. A point $K$ is chosen on diagonal $AC$. Segment $BK$ intersects $\omega$ at a unique point $X$, and segment $DK$ intersects $\omega$ at a unique point $Y$. It turns out that $XY$ is the diameter of $\omega$. Prove that it is perpendicular to $AC$. Proposed by Tseren Frantsuzov

There are a million numbered chairs at a large round table. The Sultan has seated a million wise men on them. Each of them sees the thousand people following him in clockwise order. Each of them was given a cap of black or white color, and they must simultaneously write down on their own piece of paper a guess about the color of their cap. Those who do not guess will be executed. The wise men had the opportunity to agree on a strategy before the test. What is the largest number of survivors that they can guarantee?

Let $a{}$ and $b>1$ be natural numbers. Prove that there exists a natural number $n < b^2$ such that the number $a^n + n$ is divisible by $b{}$.

An arrangement of 12 real numbers in a row is called good if for any four consecutive numbers the arithmetic mean of the first and last numbers is equal to the product of the two middle numbers. How many good arrangements are there in which the first and last numbers are 1, and the second number is the same as the third?

The diagonals of convex quadrilateral $ABCD$ intersect at point $E$. Triangles $ABE$ and $CED$ have a common excircle $\Omega$, tangent to segments $AE$ and $DE$ at points $B_1$ and $C_1$, respectively. Denote by $I$ and $J$ the centers of the incircles of these triangles, respectively. Segments $IC_1$ and $JB_1$ intersect at point $S$. It is known that $S$ lies on $\Omega$. Prove that the circumcircle of triangle $AED$ is tangent to $\Omega$. Proposed by David Brodsky

Let $n{}$ and $k{}$ be natural numbers, with $n > 2k$. In the deck of cards, each card contains a subset of the set $\{1, 2, \ldots , n\}$ consisting of at least $k+1$, but no more than $n-k$ elements. Each $m$-element set is written exactly on $m-k$ cards. Is it possible to split these cards into $n- 2k$ stacks so that in each stack all subsets on the cards are different, and any two of them intersect?