Let $k\geq 2$ be a natural number. Suppose that $a_1, a_2, \dots a_{2021}$ is a monotone decreasing sequence of non-negative numbers such that \[\sum_{i=n}^{2021}a_i\leq ka_n\]for all $n=1,2,\dots 2021$. Prove that $a_{2021}\leq 4(1-\frac{1}{k})^{2021}a_1$.

Let $ABC$ be an acute triangle such that $AB<AC$. Denote by $A'$ the reflection of $A$ with respect to $BC$. The circumcircle of $A'BC$ meets the rays $AB$ and $AC$ at $D$ and $E$ respectively, such that $B$ is between $A$ and $D$, and $E$ is between $A$ and $C$. Denote by $P$ and $Q$ the midpoints of the segments $CD$ and $BE$, and let $S$ be the midpoint of $BC$. Show that the lines $BC$ and $AA'$ meet on the circumcircle of $PQS$. Proposed by Nikola Velov

A group of people is said to be good if every member has an even number (zero included) of acquaintances in it. Prove that any group of people can be partitioned into two (possibly empty) parts such that each part is good.

Let $S=\{1, 2, 3, \dots 2021\}$ and $f:S \to S$ be a function such that $f^{(n)}(n)=n$ for each $n \in S$. Find all possible values for $f(2021)$. (Here, $f^{(n)}(n) = \underbrace{f(f(f\dots f(}_{n \text{ times} }n)))\dots))$.) Proposed by Viktor Simjanoski

Determine all functions $f:\mathbb{N}\to \mathbb{N}$ such that for all $a, b \in \mathbb{N}$ the following conditions hold: $(i)$ $f(f(a)+b) \mid b^a-1$; $(ii)$ $f(f(a))\geq f(a)-1$.

Let $ABC$ be an acute triangle such that $AB<AC$ with orthocenter $H$. The altitudes $BH$ and $CH$ intersect $AC$ and $AB$ at $B_{1}$ and $C_{1}$. Denote by $M$ the midpoint of $BC$. Let $l$ be the line parallel to $BC$ passing through $A$. The circle around $ CMC_{1}$ meets the line $l$ at points $X$ and $Y$, such that $X$ is on the same side of the line $AH$ as $B$ and $Y$ is on the same side of $AH$ as $C$. The lines $MX$ and $MY$ intersect $CC_{1}$ at $U$ and $V$ respectively. Show that the circumcircles of $ MUV$ and $ B_{1}C_{1}H$ are tangent. Proposed by Nikola Velov