# [Solutions] International Zhautykov Mathematical Olympiad 2019

1. Prove that there exist at least $100!$ ways to write $100!$ as sum of elements of set $\{1!,2!,3!...99!\}$. (each number in sum can be two or more times.)
2. Find the biggest real number $C$, such that for every different positive real numbers $a_1,a_2,...,a_{2019}$ that satisfy inequality $$\frac{a_1}{|a_2-a_3|} + \frac{a_2}{|a_3-a_4|} + ... + \frac{a_{2019}}{|a_1-a_2|} > C$$
3. Triangle $ABC$ is given. The median $CM$ intersects the circumference of $ABC$ in $N$. $P$ and $Q$ are chosen on the rays $CA$ and $CB$ respectively, such that $PM$ is parallel to $BN$ and $QM$ is parallel to $AN$. Points $X$ and $Y$ are chosen on the segments $PM$ and $QM$ respectively, such that both $PY$ and $QX$ touch the circumference of $ABC$. Let $Z$ be intersection of $PY$ and $QX$. Prove that, the quadrilateral $MXZY$ is circumscribed.
4. Triangle $ABC$ with $AC=BC$ given and point $D$ is chosen on the side $AC$. $S1$ is a circle that touches $AD$ and extensions of $AB$ and $BD$ with radius $R$ and center $O_1$. $S2$ is a circle that touches $CD$ and extensions of $BC$ and $BD$ with radius $2R$ and center $O_2$. Let $F$ be intersection of the extension of $AB$ and tangent at $O_2$ to circumference of $BO_1O_2$. Prove that $FO_1=O_1O_2$.
5. Natural number $n>1$ is given. Let $I$ be a set of integers that are relatively prime to $n$. Define the function $f:I=>N$. We call a function $k$-periodic if for any $a,b$, $f(a)=f(b)$ whenever $k|a-b$. We know that $f$ is $n$-periodic. Prove that minimal period of $f$ divides all other periods. (Example: if $n=6$ and $f(1)=f(5)$ then minimal period is 1, if $f(1)$ is not equal to $f(5)$ then minimal period is $3$.)
6. We define two types of operation on polynomial of third degree
• switch places of the coefficients of polynomial(including zero coefficients), ex: $$x^3+x^2+3x-2 \Rightarrow -2x^3+3x^2+x+1$$
• replace the polynomial $P(x)$ with $P(x+1)$.
If limitless amount of operations is allowed, is it possible to get $x^3-3x^2+3x-3$ from $x^3-2$?.
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