Let $n$ be a positive integer. A square $ABCD$ is partitioned into $n^2$ unit squares. Each of them is divided into two triangles by the diagonal parallel to $BD$. Some of the vertices of the unit squares are colored red in such a way that each of these $2n^2$ triangles contains at least one red vertex. Find the least number of red vertices. [i](4th Middle European Mathematical Olympiad, Team Competition, Problem 4)[/i]

Prove that every polygon that has a center of symmetry can be dissected into a square such that it is divided into finitely many polygonal pieces, and all the pieces can only be translated. (In other words, the original polygon can be divided into polygons $A_1,A_2,\dotsc ,A_n$, a square can be divided into polygons a $B_1,B_2,\dotsc ,B_n$ such that for $1\leqslant i\leqslant n$ polygon $B_i$ is a translated copy of polygon $A_i$.)

[b]N[/b]ew this year at HMNT: the exciting game of RNG baseball! In RNG baseball, a team of infinitely many people play on a square field, with a base at each vertex; in particular, one of the bases is called the home base. Every turn, a new player stands at home base and chooses a number n uniformly at random from $\{0, 1, 2, 3, 4\}$. Then, the following occurs: • If $n>0$, then the player and everyone else currently on the field moves (counterclockwise) around the square by n bases. However, if in doing so a player returns to or moves past the home base, he/she leaves the field immediately and the team scores one point. • If $n=0$ (a strikeout), then the game ends immediately; the team does not score any more points. What is the expected number of points that a given team will score in this game?

For which $n\in\{3900, 3901,\cdots, 3909\}$ can the set $\{1, 2, . . . , n\}$ be partitioned into (disjoint) triples in such a way that in each triple one of the numbers equals the sum of the other two?

Find all elements $n \in A = \{2,3,...,2016\} \subset \mathbb{N}$ such that: every number $m \in A$ smaller than $n$, and coprime with $n$, must be a prime number

[u]Round 5 [/u] [b]p13.[/b] Suppose $\vartriangle ABC$ is an isosceles triangle with $\overline{AB} = \overline{BC}$, and $X$ is a point in the interior of $\vartriangle ABC$. If $m \angle ABC = 94^o$, $m\angle ABX = 17^o$, and $m\angle BAX = 13^o$, then what is $m\angle BXC$ (in degrees)? [b]p14.[/b] Find the remainder when $\sum^{2019}_{n=1} 1 + 2n + 4n^2 + 8n^3$ is divided by $2019$. [b]p15.[/b] How many ways can you assign the integers $1$ through $10$ to the variables $a, b, c, d, e, f, g, h, i$, and $j$ in some order such that $a < b < c < d < e, f < g < h < i$, $a < g, b < h, c < i$, $f < b, g < c$, and $h < d$? [u]Round 6 [/u] [b]p16.[/b] Call an integer $n$ equi-powerful if $n$ and $n^2$ leave the same remainder when divided by 1320. How many integers between $1$ and $1320$ (inclusive) are equi-powerful? [b]p17.[/b] There exists a unique positive integer $j \le 10$ and unique positive integers $n_j$ , $n_{j+1}$, $...$, $n_{10}$ such that $$j \le n_j < n_{j+1} < ... < n_{10}$$ and $${n_{10} \choose 10}+ {n_9 \choose 9}+ ... + {n_j \choose j}= 2019.$$ Find $n_j + n_{j+1} + ... + n_{10}$. [b]p18.[/b] If $n$ is a randomly chosen integer between $1$ and $390$ (inclusive), what is the probability that $26n$ has more positive factors than $6n$? [u]Round 7[/u] [b]p19.[/b] Suppose $S$ is an $n$-element subset of $\{1, 2, 3, ..., 2019\}$. What is the largest possible value of $n$ such that for every $2 \le k \le n$, $k$ divides exactly $n - 1$ of the elements of $S$? [b]p20.[/b] For each positive integer $n$, let $f(n)$ be the fewest number of terms needed to write $n$ as a sum of factorials. For example, $f(28) = 3$ because $4! + 2! + 2! = 28$ and 28 cannot be written as the sum of fewer than $3$ factorials. Evaluate $f(1) + f(2) + ... + f(720)$. [b]p21.[/b] Evaluate $\sum_{n=1}^{\infty}\frac{\phi (n)}{101^n-1}$ , where $\phi (n)$ is the number of positive integers less than or equal to n that are relatively prime to $n$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2788993p24519281]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A wizard kidnaps $31$ members from party $A$, $28$ members from party $B$, $23$ members from party $C$, and $19$ members from party $D$, keeping them isolated in individual rooms in his castle, where he forces them to work. Every day, after work, the kidnapped people can walk in the park and talk with each other. However, when three members of three different parties start talking with each other, the wizard reconverts them to the fourth party (there are no conversations with $4$ or more people involved). a) Find out whether it is possible that, after some time, all of the kidnapped people belong to the same party. If the answer is yes, determine to which party they will belong. b) Find all quartets of positive integers that add up to $101$ that if they were to be considered the number of members from the four parties, it is possible that, after some time, all of the kidnapped people belong to the same party, under the same rules imposed by the wizard.

Every cell of a $4\times 4$ square grid net, has $1\times 1$ size. Is it possible to represent this net as a union of the following sets: a) Eight broken lines of length five each? b) Five broken lines of length eight each?

A pack of $2003$ circus flees are deployed by their circus trainer, named Gogu, on a sufficiently large table, in such a way that they are not all lying on the same line. He now wants to get his Ph.D. in fleas training, so Gogu has thought the fleas the following trick: we chooses two fleas and tells one of them to jump over the other one, such that any flea jumps as far as twice the initial distance between him and the flea over which he is jumping. The Ph.D. circus committee has but only one task to Gogu: he has to make the his flees gather around on the table such that every flea represents a vertex of a convex polygon. Can Gogu get his Ph.D., no matter of how the fleas were deployed?

Two people play a game on a graph with $2022$ points. Initially, there are no edges in the graph. They take turns and connect two non-neighbouring vertices with an edge. Whoever makes the graph connected loses. Which player has a winning strategy? [i]ST, danny2915[/i]

Suppose that each point on the plane is colored either white or black. Show that there exists an equilateral triangle with the side length equal to $1$ or $\sqrt{3}$ whose three vertices are in the same color.

Signals used for communication are binary sequences of length $10$. Unfortunately, the receiving device got broken so that it cannot distinguish between two signals unless those differ in more than five places. What is the largest possible number of signals that can still be used to prevent ambiguities?

In a checkered square, there is a closed door between any two cells adjacent by side. A beetle starts from some cell and travels through cells, passing through doors; she opens a closed door in the direction she is moving and leaves that door open. Through an open door, the beetle can only pass in the direction the door is opened. Prove that if at any moment the beetle wants to return to the starting cell, it is possible for her to do that.

$16$ progamers are playing in a single elimination tournament. Each player has a different skill level and when two play against each other the one with the higher skill level will always win. Each round, each progamer plays a match against another and the loser is eliminated. This continues until only one remains. How many different progamers can reach the round that has $2$ players remaining?

[b]a)[/b] Prove that for all $m,n\in \mathbb N$ there exists a natural number $a$ such that if we color every $3$-element subset of the set $\mathcal A=\{1,2,3,...,a\}$ using $2$ colors red and green, there exists an $m$-element subset of $\mathcal A$ such that all $3$-element subsets of it are red or there exists an $n$-element subset of $\mathcal A$ such that all $3$-element subsets of it are green. [b]b)[/b] Prove that for all $m,n\in \mathbb N$ there exists a natural number $a$ such that if we color every $k$-element subset ($k>3$) of the set $\mathcal A=\{1,2,3,...,a\}$ using $2$ colors red and green, there exists an $m$-element subset of $\mathcal A$ such that all $k$-element subsets of it are red or there exists an $n$-element subset of $\mathcal A$ such that all $k$-element subsets of it are green.

Let $A_1$, $A_2$, $\cdots$, $A_m$ be $m$ subsets of a set of size $n$. Prove that $$ \sum_{i=1}^{m} \sum_{j=1}^{m}|A_i|\cdot |A_i \cap A_j|\geq \frac{1}{mn}\left(\sum_{i=1}^{m}|A_i|\right)^3.$$

You are given an odd number $n\ge 3$. For every pair of integers $(i, j)$ with $1\le i \le j \le n$ there is a domino, with $i$ written on one its end and with $j$ written on another (there are $\frac{n(n+1)}{2}$ domino overall). Amin took this dominos and started to put them in a row so that numbers on the adjacent sides of the dominos are equal. He has put $k$ dominos in this way, got bored and went away. After this Anton came to see this $k$ dominos, and he realized that he can't put all the remaining dominos in this row by the rules. For which smallest value of $k$ is this possible? [i]Oleksii Masalitin[/i]

Nelson is having his friend drop his unique bouncy ball from a $12$ foot building, and Nelson will only catch the ball at the peak of its trajectory between bounces. On any given bounce, there is an $80\%$ chance that the next peak occurs at $\frac13$ the height of the previous peak and a $20\%$ chance that the next peak occurs at $3$ times the height of the previous peak (where the first peak is at $12$ feet). If Nelson can only reach $4$ feet into the air and will catch the ball as soon as possible, the probability that Nelson catches the ball after exactly $13$ bounces is $2^a \times 3^b \times 5^c \times 7^d \times 11^e$ for integers $a, b, c, d$, and $e$. Find $|a| + |b| + |c| + |d| + |e|$.

A $5 \times 5$ table is called regular f each of its cells contains one of four pairwise distinct real numbers,such that each of them occurs exactly one in every $2 \times 2$ subtable.The sum of all numbers of a regular table is called the total sum of the table.With any four numbers,one constructs all possible regular tables,computes their total sums and counts the distinct outcomes.Determine the maximum possible count.

Whether there are integers $a_1$, $a_2$, $\cdots$, that are different from each other, satisfying: (1) For $\forall k\in\mathbb N_+$, $a_{k^2}>0$ and $a_{k^2+k}<0$; (2) For $\forall n\in\mathbb N_+$, $\left| a_{n+1}-a_n\right|\leqslant 2023\sqrt n$?

There are $200$ boxes on the table. In the beginning, each of the boxes contains a positive integer (the integers are not necessarily distinct). Every minute, Alice makes one move. A move consists of the following. First, she picks a box $X$ which contains a number $c$ such that $c = a + b$ for some numbers $a$ and $b$ which are contained in some other boxes. Then she picks a positive integer $k > 1$. Finally, she removes $c$ from $X$ and replaces it with $kc$. If she cannot make any mobes, she stops. Prove that no matter how Alice makes her moves, she won't be able to make infinitely many moves.

On an infinite grid, a square with four vertices lie at $(m, n)$, $(m-1, n)$, $(m,n-1)$, $(m-1, n-1)$ is denoted as cell $(m,n)$ $(m, n \in Z)$. Some marbles are dropped on some cell. Each cell may have more than one marble or have no marble at all. Consider a "move" can be conducted in one of two following ways: i) Remove one marble from cell $(m,n)$ (if there is marble at that cell), then add one marble to each of cell $(m - 1, n- 2)$ and cell $(m -2, n - 1)$. ii) Remove two marbles from cell $(m,n)$ (if there is marble at that cell), then add one marble to each of cell $(m +1, n - 2)$ and cell $(m - 2, n +1)$. Assume that initially, there are $n$ marbles at the cell $(1,n), (2,n - 1),..., (n, 1)$ (each cell contains one marble). Can we conduct an finite amount of moves such that both cells $(n + 1, n)$ and $(n, n + 1)$ have marbles?

66 players take part in the chess tournament, each player plays one game against each other, and the games take place in four cities. Prove that three players play all their games in the same city.

Let $T$ be the answer to question $17$, and let $N =\frac{24}{T}$. Leanne flips a fair coin $N$ times. Let $X$ be the number of times that within a series of three consecutive flips, there were exactly two heads or two tails. What is the expected value of $X$?

Let $m,n>2$ be integers. A regular ${n}$-sided polygon region $\mathcal T$ on a plane contains a regular ${m}$-sided polygon region with a side length of ${}{}{}1$. Prove that any regular ${m}$-sided polygon region $\mathcal S$ on the plane with side length $\cos{\pi}/[m,n]$ can be translated inside $\mathcal T.$ In other words, there exists a vector $\vec\alpha,$ such that for each point in $\mathcal S,$ after translating the vector $\vec\alpha$ at that point, it fall into $\mathcal T.$ Note: The polygonal area includes both the interior and boundaries. [i]Created by Bin Wang[/i]