IZhO 2019

Prove that there are at least 100! ways to partition the number 100! into summands from the set $\{1!, 2!, 3!, \ldots, 99$ !}. (Partitions differing in the order of summands are considered the same; any summand can be taken multiple times. We remind that $n!=1 \cdot 2 \cdot \ldots \cdot n$.)

Find the biggest real number $C$ , such that for every different positive real numbers $a_1,a_2...a_{2019}$ that satisfy inequality : $\frac{a_1}{|a_2-a_3|} + \frac{a_2}{|a_3-a_4|} + ... + \frac{a_{2019}}{|a_1-a_2|} > C$

The extension of median $C M$ of the triangle $A B C$ intersects its circumcircle $\omega$ at $N$. Let $P$ and $Q$ be the points on the rays $C A$ and $C B$ respectively such that $P M \| B N$ and $Q M \| A N$. Let $X$ and $Y$ be the points on the segments $P M$ and $Q M$ respectively such that $P Y$ and $Q X$ are tangent to $\omega$. The segments $P Y$ and $Q X$ intersect at $Z$. Prove that the quadrilateral $M X Z Y$ is circumscribed.

An isosceles triangle $A B C$ with $A C=B C$ is given. Point $D$ is chosen on the side $A C$. The circle $S_{1}$ of radius $R$ with the center $O_{1}$ touches the segment $A D$ and the extensions of $B A$ and $B D$ over the points $A$ and $D$, respectively. The circle $S_{2}$ of radius $2 R$ with the center $O_{2}$ touches the segment $D C$ and the extensions of $B D$ and $B C$ over the points $D$ and $C$, respectively. Let the tangent to the circumcircle of the triangle $B O_{1} O_{2}$ at the point $O_{2}$ intersect the line $B A$ at point $F$. Prove that $O_{1} F=O_{1} O_{2}$.

Let $n>1$ be a positive integer. A function $f: I \rightarrow \mathbb{Z}$ is given, where $I$ is the set of all integers coprime with $n$. ( $\mathbb{Z}$ is the set of integers). A positive integer $k$ is called a period of the function $f$ if $f(a)=f(b)$ for all $a, b \in I$ such that $a \equiv b(\bmod k)$. It is known that $n$ is a period of $f$. Prove that the minimal period of the function $f$ divides all its periods.

On a polynomial of degree three it is allowed to perform the following two operations arbitrarily many times:

(i) reverse the order of its coefficients including zeroes (for instance, from the polynomial $x^{3}-2 x^{2}-3$ we can obtain $-3 x^{3}-2 x+1$;

(ii) change polynomial $P(x)$ to the polynomial $P(x+1)$.

Is it possible to obtain the polynomial $x^{3}-3 x^{2}+3 x-3$ from the polynomial $x^{3}-2$ ?

The answer is no.