Due to the increasing pandemic situation, the government decides to enforce traffic ban by building $n$ straight lines (infinite at both directions) as "walls", with no two of them are parallel, and no three of them are concurrent. The set of these walls is denoted as $W_n$. For each wall in $W_n$, the government paints one of its sides in blue, and the other side in green. By such, for any two walls, their intersection is divided into four corners: a mono-color corner such that both walls are blue, a mono-color corner such that both walls are green, and two duel-color corners such that the walls are in different colors. To make sure there are paths for emergency, the government opens a door on each intersection of walls so that people can travel from a duel-color corner to the other duel-color corner. These doors are the only ways for people to cross a wall.
Given $W_n$, let $k(W_n)$ be the largest positive integer $k$ such that, no matter how the government paints the walls, we can place $k$ people on the plane so that no two of them can ever meet each other. For each $n$, find all possible values of $k(W_n)$.
Two circles $\Gamma_1$ and $\Gamma_2$ have common external tangents $l_1$ and $l_2$ meeting at $T$. Suppose $l_1$ touches $\Gamma_1$ at $A$ and $l_2$ touches $\Gamma_2$ at $B$. A circle $\Omega$ through $A$ and $B$ intersects $\Gamma_1$ again at $C$ and $\Gamma_2$ again at $D$, such that quadrilateral $ABCD$ is convex.
Suppose lines $AC$ and $BD$ meet at point $X$, while lines $AD$ and $BC$ meet at point $Y$. Show that $T$, $X$, $Y$ are collinear.
Let $ABC$ be a scalene triangle with incenter $I$. The incircle of $ABC$ touches $BC$, $CA$, $AB$ at points $D$, $E$, $F$, respectively. Let $P$ be the foot of the altitude from $D$ to $EF$, and let $M$ be the midpoint of $BC$. The rays $AP$ and $IP$ intersect the circumcircle of triangle $ABC$ again at points $G$ and $Q$, respectively. Show that the incenter of triangle $GQM$ coincides with $D$.
Let $\mathbb{R}_+$ be the set of all positive integers. Find all functions $g: \mathbb{R}_+ \to \mathbb{R}_+$ such that numbers $x, y \in \mathbb{R}_+$ satisfy the equality
$$
xg(x + g(y)) = g(g(xy) + 1).
$$
The set $\{2, 3, 4, \ldots, 2020\}$ is partitioned into triples. In each triple $(a, b, c)$ the numbers were arranged in the ascending order, i.e. $a \lt b \lt c$, and the difference $|b - \frac{a+c}{2}|$ is called the error of this triple. Find the maximal possible sum of errors of all $673$ triples.
For every positive integer $x$, denote by $\kappa(x)$ the number of composite numbers not greater than $x$. Find all positive integers $n$ such that
$$
(\kappa(n))! \cdot \text{lcm}(1,2,\dots,n) \gt (n-1)!
$$
Given a list of integers $2^1 + 1, 2^2 + 1, \dots, 2^{2019} + 1$, Adam chooses two different integers from the list and computes their greatest common divisor. Find the sum of all possible values of this greatest common divisor.
Two circles $\Gamma$ and $\Omega$ intersect at two distinct points $A$ and $B$. Let $P$ be a point on $\Gamma$. The tangent at $P$ to $\Gamma$ meets $\Omega$ at the points $C$ and $D$, where $D$ lies between $P$ and $C$, and $ABCD$ is a convex quadrilateral. The lines $CA$ and $CB$ meet $\Gamma$ again at $E$ and $F$ respectively. The lines $DA$ and $DB$ meet $\Gamma$ again at $S$ and $T$ respectively. Suppose the points $P, E, S, F, B, T, A$ lie on $\Gamma$ in this order. Prove that $PC, ET, SF$ are parallel.
For a positive integer $n$, we consider an $n \times n$-board and tiles with sizes $1 \times 1$, $1 \times 2$, ..., $1 \times n$. In how many ways can exactly $\frac{1}{2}n(n+1)$ squares of the board be coloured red, so that the red squares can be covered by placing the $n$ tiles horizontally on the board, as well as by placing the $n$ tiles vertically on the board? Two colourings which are not identical, but which can be obtained from one another by rotation or reflection, are counted as different colourings.
Determine all pairs $(a, b)$ of positive integers for which
$$
a + b = \varphi(a) + \varphi(b) + \gcd(a, b).
$$
Here, $\varphi(n)$ is the number of integers $k \in \{1, 2, \dots, n\}$ satisfying $\gcd(n, k) = 1$.
Determine all pairs $(a, b)$ of positive integers for which
$$
a + b = \varphi(a) + \varphi(b) + \gcd(a, b).
$$
Here, $\varphi(n)$ is the number of integers $k \in \{1, 2, \dots, n\}$ satisfying $\gcd(n, k) = 1$.
For a positive integer $n$, we consider an $n \times n$-board and tiles with sizes $1 \times 1$, $1 \times 2$, ..., $1 \times n$. In how many ways can exactly $\frac{1}{2}n(n+1)$ squares of the board be coloured red, so that the red squares can be covered by placing the $n$ tiles horizontally on the board, as well as by placing the $n$ tiles vertically on the board? Two colourings which are not identical, but which can be obtained from one another by rotation or reflection, are counted as different colourings.
Let $N \gt 2^{5000}$ be a positive integer. Prove that if $1 \le a_1 \lt \dots \lt a_k \lt 100$ are distinct positive integers then the number
$$
\prod_{i=1}^{k} (N^{a_i} + a_i)
$$
has at least $k$ distinct prime factors.
There are $N$ acute triangles on the plane. Their vertices are all integer points, their areas are all equal to $2^{2020}$, but no two of them are congruent. Find the maximum possible value of $N$.
Note: $(x, y)$ is an integer point if and only if $x$ and $y$ are both integers.
Let $\alpha \ge 1$ be a real number. Hephaestus and Poseidon play a turn-based game on an infinite grid of unit squares. Before the game starts, Poseidon chooses a finite number of cells to be *flooded*. Hephaestus is building a *levee*, which is a subset of unit edges of the grid (called *walls*) forming a connected, non-self-intersecting path or loop.
The game then begins with Hephaestus moving first. On each of Hephaestus's turns, he adds one or more walls to the levee, as long as the total length of the levee is at most $\alpha n$ after his nth turn. On each of Poseidon's turns, every cell which is adjacent to an already flooded cell and with no wall between them becomes flooded as well.
Hephaestus wins if the levee forms a closed loop such that all flooded cells are contained in the interior of the loop — hence stopping the flood and saving the world. For which $\alpha$ can Hephaestus guarantee victory in a finite number of turns no matter how Poseidon chooses the initial cells to flood?
We say a nondegenerate triangle whose angles have measures $\theta_1, \theta_2, \theta_3$ is *quirky* if there exists integers $r_1, r_2, r_3$, not all zero, such that
$$
r_1\theta_1 + r_2\theta_2 + r_3\theta_3 = 0.
$$
Find all integers $n \ge 3$ for which a triangle with side lengths $n-1, n, n+1$ is quirky.
Find the total number of primes $p \lt 100$ such that $\lfloor (2 + \sqrt{5})^p \rfloor - 2^{p+1}$ is divisible by $p$. Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.