Given is a triangle $ABC$ with $\angle BAC=45$; $AD, BE, CF$ are altitudes and $EF \cap BC=X$. If $AX \parallel DE$, find the angles of the triangle.

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 will say that a set of real numbers $A = (a_1,... , a_{17})$ is stronger than the set of real numbers $B = (b_1, . . . , b_{17})$, and write $A >B$ if among all inequalities $a_i > b_j$ the number of true inequalities is at least $3$ times greater than the number of false. Prove that there is no chain of sets $A_1, A_2, . . . , A_N$ such that $A_1>A_2> \cdots A_N>A_1$. Remark: For 11.4, the constant $3$ is changed to $2$ and $N=3$ and $17$ is changed to $m$ and $n$ in the definition (the number of elements don't have to be equal).

Altitudes $AA_1, BB_1, CC_1$ of acute triangle $ABC$ intersect at point $H$. On the tangent drawn from point $C$ to the circle $(AB_1C_1)$, the perpendicular $HQ$ is drawn (the point $Q$ lies inside the triangle $ABC$). Prove that the circle passing through the point $B_1$ and touching the line $AB$ at point $A$ is also tangent to line $A_1Q$.

Find all pairs of nonzero rational numbers $x, y$, such that every positive rational number can be written as $\frac{\{rx\}} {\{ry\}}$ for some positive rational $r$.

Given is a set of $2n$ cards numbered $1,2, \cdots, n$, each number appears twice. The cards are put on a table with the face down. A set of cards is called good if no card appears twice. Baron Munchausen claims that he can specify $80$ sets of $n$ cards, of which at least one is sure to be good. What is the maximal $n$ for which the Baron's words could be true?

Given is a triangle $ABC$ with altitude $AH$, diameter of the circumcircle $AD$ and incenter $I$. Prove that $\angle BIH = \angle DIC$.

There are two piles of stones: $1703$ stones in one pile and $2022$ in the other. Sasha and Olya play the game, making moves in turn, Sasha starts. Let before the player's move the heaps contain $a$ and $b$ stones, with $a \geq b$. Then, on his own move, the player is allowed take from the pile with $a$ stones any number of stones from $1$ to $b$. A player loses if he can't make a move. Who wins? Remark: For 10.4, the initial numbers are $(444,999)$

Let $n$ be a positive integer and let $a_1, a_2, \cdots a_k$ be all numbers less than $n$ and coprime to $n$ in increasing order. Find the set of values the function $f(n)=gcd(a_1^3-1, a_2^3-1, \cdots, a_k^3-1)$.

Given is a graph $G$ of $n+1$ vertices, which is constructed as follows: initially there is only one vertex $v$, and one a move we can add a vertex and connect it to exactly one among the previous vertices. The vertices have non-negative real weights such that $v$ has weight $0$ and each other vertex has a weight not exceeding the avarage weight of its neighbors, increased by $1$. Prove that no weight can exceed $n^2$.

The positive integers $a$ and $b$ are such that $a+k$ is divisible by $b+k$ for all positive integers numbers $k<b$. Prove that $a-k$ is divisible by $b-k$ for all positive integers $k<b$.

$12$ schoolchildren are engaged in a circle of patriotic songs, each of them knows a few songs (maybe none). We will say that a group of schoolchildren can sing a song if at least one member of the group knows it. Supervisor the circle noticed that any group of $10$ circle members can sing exactly $20$ songs, and any group of $8$ circle members - exactly $16$ songs. Prove that the group of all $12$ circle members can sing exactly $24$ songs.

Given is a trapezoid $ABCD$, $AD \parallel BC$. The angle bisectors of the two pairs of opposite angles meet at $X, Y$. Prove that $AXYD$ and $BXYC$ are cyclic.

We will say that a point of the plane $(u, v)$ lies between the parabolas $y = f(x)$ and $y = g(x)$ if $f(u) \leq v \leq g(u)$. Find the smallest real $p$ for which the following statement is true: for any segment, the ends and the midpoint of which lie between the parabolas $y = x^2$ and $y=x^2+1$, then they lie entirely between the parabolas $y=x^2$ and $y=x^2+p$.

Given are $n$ distinct natural numbers. For any two of them, the one is obtained from the other by permuting its digits (zero cannot be put in the first place). Find the largest $n$ such that it is possible all these numbers to be divisible by the smallest of them?