find all functions from the reals to themselves. such that for every real $x,y$. $$f(y-f(x))=f(x)-2x+f(f(y))$$

let $a_1,a_2,...,a_n$,$b_1,b_2,...,b_n$,$c_1,c_2,...,c_n$ be real numbers. prove that $$ \sum_{cyc}{ \sqrt{\sum_{i \in \{1,...,n\} }{ (3a_i-b_i-c_i)^2}}} \ge \sum_{cyc}{\sqrt{\sum_{i \in \{1,2,...,n\}}{a_i^2}}}$$

find all $k$ distinct integers $a_1,a_2,...,a_k$ such that there exists an injective function $f$ from reals to themselves such that for each positive integer $n$ we have $$\{f^n(x)-x| x \in \mathbb{R} \}=\{a_1+n,a_2+n,...,a_k+n\}$$.

We call a polynomial $P(x)$ intresting if there are $1398$ distinct positive integers $n_1,...,n_{1398}$ such that $$P(x)=\sum_{}{x^{n_i}}+1$$Does there exist infinitly many polynomials $P_1(x),P_2(x),...$ such that for each distinct $i,j$ the polynomial $P_i(x)P_j(x)$ is interesting.

Let $ABCD$ be a Rhombus and let $w$ be it's incircle. Let $M$ be the midpoint of $AB$ the point $K$ is on $w$ and inside $ABCD$ such that $MK$ is tangent to $w$. Prove that $CDKM$ is cyclic.

Triangle $ABC$ with it's circumcircle $\Gamma$ is given. Points $D$ and $E$ are chosen on segment $BC$ such that $\angle BAD=\angle CAE$. The circle $\omega$ is tangent to $AD$ at $A$ with it's circumcenter lies on $\Gamma$. Reflection of $A$ through $BC$ is $A'$. If the line $A'E$ meet $\omega$ at $L$ and $K$. Then prove either $BL$ and $CK$ or $BK$ and $CL$ meet on $\Gamma$.

The circle $\Omega$ with center $I_A$, is the $A$-excircle of triangle $ABC$. Which is tangent to $AB,AC$ at $F,E$ respectivly. Point $D$ is the reflection of $A$ through $I_AB$. Lines $DI_A$ and $EF$ meet at $K$. Prove that ,circumcenter of $DKE$ , midpoint of $BC$ and $I_A$ are collinear.

Triangle $ABC$ is given. Let $O$ be it's circumcenter. Let $I$ be the center of it's incircle.The external angle bisector of $A$ meet $BC$ at $D$. And $I_A$ is the $A$-excenter . The point $K$ is chosen on the line $AI$ such that $AK=2AI$ and $A$ is closer to $K$ than $I$. If the segment $DF$ is the diameter of the circumcircle of triangle $DKI_A$, then prove $OF=3OI$.

$1)$. Prove a graph with $2n$ vertices and $n+2$ edges has an independent set of size $n$ (there are $n$ vertices such that no two of them are adjacent ). $2)$.Find the number of graphs with $2n$ vertices and $n+3$ edges , such that among any $n$ vertices there is an edge connecting two of them

For each $n$ find the number of ways one can put the numbers $\{1,2,3,...,n\}$ numbers on the circle, such that if for any $4$ numbers $a,b,c,d$ where $n|a+b-c-d$. The segments joining $a,b$ and $c,d$ do not meet inside the circle. (Two ways are said to be identical , if one can be obtained from rotaiting the other)

Consider a latin square of size $n$. We are allowed to choose a $1 \times 1$ square in the table, and add $1$ to any number on the same row and column as the chosen square (the original square will be counted aswell) , or we can add $-1$ to all of them instead. Can we with doing finitly many operation , reach any latin square of size $n?$

What is the maximum number of subsets of size $5$, taken from the set $A=\{1,2,3,...,20\}$ such that any $2$ of them share exactly $1$ element.

Find all positive integers $n$ such that the following holds. $$\tau(n)|2^{\sigma(n)}-1$$

Find all polynomials $P$ with integer coefficients such that all the roots of $P^n(x)$ are integers. (here $P^n(x)$ means $P(P(...(P(x))...))$ where $P$ is repeated $n$ times)

Find all functions $f$ from positive integers to themselves, such that the followings hold. $1)$.for each positive integer $n$ we have $f(n)<f(n+1)<f(n)+2020$. $2)$.for each positive integer $n$ we have $S(f(n))=f(S(n))$ where $S(n)$ is the sum of digits of $n$ in base $10$ representation.

Prove that for every two positive integers $a,b$ greater than $1$. there exists infinitly many $n$ such that the equation $\phi(a^n-1)=b^m-b^t$ can't hold for any positive integers $m,t$.