Let $P_1P_2\ldots P_{2021}$ be a convex polygon with 2021 vertices such that, at every vertex $P_i$, the 2018 diagonals from $P_i$ divide the angle $\widehat P_i$ into 2019 equal angles.
Prove that $P_1P_2\ldots P_{2021}$ is a regular polygon, that is, all its angles have the same measure and all its sides have the same length.
Let $f:\mathbb R\to\mathbb R$ be a nondecreasing function such that $f(y)-f(x)\lt y-x$ for all real numbers $x,y$ with $y\gt x$. Let $(u_n)_{n\geq0}$ be a sequence satisfying
$$
u_{n+2}=f(u_{n+1})-f(u_n)
$$
for every integer $n\geq0$.
Prove that $(u_n)_{n\geq0}$ converges to $0$, that is, for every real $\varepsilon\gt0$, there is an integer $N$ such that $|u_n|\leq\varepsilon$ for every integer $n\geq N$.
Let $ABCD$ be a convex quadrilateral whose angles at $B$ and $D$ are obtuse and whose angles at $A$ and $C$ are equal. Let $E$ and $F$ be the reflections of $A$ across the lines $BC$ and $CD$, respectively. Let $K$ and $L$ be the intersections of the line $BD$ with the lines $AE$ and $AF$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other.
Pierre and Clara play the following game. First Clara chooses an integer $c$. Pierre then chooses a prime $p\geq c$ and writes two integers $a$ and $b$ on the board. Clara may perform the following operation: she chooses one of the two integers on the board, say $n$, erases it, and replaces it with an integer $m$ such that $p$ divides
$$(n^2-m+1)(m^2-n+1).$$
Clara wins if, after finitely many such operations, she makes the two integers on the board equal. Otherwise she loses and Pierre wins.
The sequences $(a_n)_{n\geq0}$ and $(b_n)_{n\geq0}$ are defined by
$$\begin{aligned}
a_0&=0,&a_n&=2a_{\lfloor n/2\rfloor}+n&&\text{for }n\geq1,\\
b_0&=0,&b_n&=3b_{\lfloor n/3\rfloor}+n&&\text{for }n\geq1.
\end{aligned}$$
Prove that the sequence with general term $2^{a_n}-3^{b_n}$ changes sign infinitely often.
Let $ABC$ be an isosceles triangle with $AB=AC$, and let $D$ be a point of the segment $BC$ such that $BD\neq CD$. Let $P$ and $Q$ be the perpendicular projections of $D$ onto the lines $AB$ and $AC$, respectively. Finally, let $E$ be the intersection other than $A$ of the circumcircles of $ABC$ and $APQ$.
Prove that if the lines $EP$, $AC$, and the perpendicular bisector of $PQ$ are concurrent, then $ABC$ is right-angled at $A$.
For every integer $k\geq0$, let $F_k$ be the $k$th Fibonacci number, defined by $F_0=0$, $F_1=1$, and $F_k=F_{k-2}+F_{k-1}$ for $k\geq2$. Let $n\geq2$ be an integer, and let $S$ be a set of integers with the following property:
For every integer $k$ with $2\leq k\leq n$, the set $S$ contains two integers $x,y$ such that $x-y=F_k$.
What is the smallest possible number of elements of such a set $S$?
Find the largest integer $n\geq 3$ for which there exists a set $\mathcal S$ of $n$ points in the plane with the following property: every triangle (including a degenerate triangle) whose vertices belong to $\mathcal S$ is isosceles but not equilateral.
Let $n\geq 6$ be an integer. In the plane there are $n$ pairwise disjoint disks $D_1,D_2,\ldots,D_n$, whose radii satisfy $r_1\geq r_2\geq\cdots\geq r_n$. For every integer $i$ with $1\leq i\leq n$, let $P_i$ be a point in the interior of $D_i$. Let $A$ be any point in the plane. Prove that
$$AP_1+AP_2+\cdots+AP_n\geq r_6+r_7+\cdots+r_n.$$
Let $\mathbb R_{\gt0}$ denote the set of strictly positive real numbers. Find all functions $f:\mathbb R_{\gt0}\to\mathbb R_{\gt0}$ such that
$$f(x+f(xy))+y=f(x)f(y)+1$$
for all strictly positive real numbers $x,y$.
Find all functions $f:\mathbb N_{\geq1}\to\mathbb N_{\geq0}$ satisfying both conditions:
1. $f(xy)=f(x)+f(y)$ for all integers $x,y\geq1$;
2. there are infinitely many integers $n\geq1$ such that $f(k)=f(n-k)$ for every integer $k$ with $1\leq k\leq n-1$.
Here $\mathbb N_{\geq0}$ and $\mathbb N_{\geq1}$ denote the nonnegative and positive integers, respectively.
Morgane writes the integers $3$, $4$, and $12$ on a board. She then repeatedly makes the following change: she chooses two numbers $a,b$ on the board, erases them, and replaces them by $(3a+4b)/5$ and $(4a-3b)/5$.
Let $ABC$ be a triangle. Let $H_A$ be the foot of the altitude from $A$, and let $A'$ be the midpoint of $BC$. Let $Q_A$ be the reflection of $H_A$ about $A'$. Define $Q_B,Q_C$ analogously. Finally, let $R$ be the intersection other than $Q_A$ of the circumcircles of $Q_AQ_BC$ and $Q_ABQ_C$.
Let $ABC$ be a triangle with $\angle A\gt90^\circ$. Let $BE$ bisect $\angle ABC$, with $E\in AC$, and let $CF$ bisect $\angle ACB$, with $F\in AB$. Set $BE\cap CF=\{I\}$. Given that $IE=IF$, prove that $ABC$ is isosceles.
Is it possible to find a nonconstant polynomial $P(x,y)$ such that $P([\alpha],[3\alpha])=0$ for every real number $\alpha$? Here $[\mu]$ denotes the greatest integer less than or equal to $\mu$.
Amelia wrote down a sequence of consecutive positive integers, erased one integer, and scrambled the rest, leaving the sequence below. What integer did she erase?
$$
6,12,1,3,11,10,8,15,13,9,7,4,14,5,2
$$
The graphs of the equations
$$
\begin{aligned}
y & = -x + 8 \\
173y & = -289x + 2021
\end{aligned}
$$
on the Cartesian plane intersect at $(a, b)$. Find $a + b$.
A domino has a left end and a right end, each of a certain color. Alice has four dominos, colored red-red, red-blue, blue-red, and blue-blue. Find the number of ways to arrange the dominos in a row end-to-end such that adjacent ends have the same color. The dominos cannot be rotated.