Ana and Beto play the following game with a stick of length $15$. Ana starts, and on her first turn, she cuts the stick into two pieces with integer lengths. Then, on each player's turn, they must cut one of the pieces, of their choice, into two new pieces with integer lengths. The player who, on their turn, leaves at least one piece with length equal to $1$ loses. Determine which of the two players has a winning strategy.

Each of the positive integers from $1$ to $2023,$ inclusive, are randomly colored either blue or red. For each nonempty subset of $S=\{1,2,\cdots,2023\},$ we define the score of that subset to be the positive difference between the number of blue integers and the number of red integers in that subset. Let $X$ be the expected value of the sum of the scores of all the nonempty subsets of $S$. What is the maximum integer $k$ such that $2^k$ divides $2^{2023}\cdot X$? [i]Proposed by Kyle Lee[/i]

A grasshopper is traveling on the coordinate plane, starting at the origin $(0, 0)$. Each hop, the grasshopper chooses to move $1$ unit up, down, left, or right with equal probability. The grasshopper hops $4$ times and stops at point $P$. Compute the probability that it is possible to return to the origin from $P$ in at most $3$ hops.

Find the number of all $\text{permutations}$ of $\left \{ 1,2,\cdots ,n \right \}$ like $p$ such that there exists a unique $i \in \left \{ 1,2,\cdots ,n \right \}$ that : $$p(p(i)) \geq i$$

A triangle is composed of circular cells arranged in $5784$ rows: the first row has one cell, the second has two cells, and so on (see the picture). The cells are divided into pairs of adjacent cells (circles touching each other), so that each cell belongs to exactly one pair. A pair of adjacent cells is called [b]diagonal[/b] if the two cells in it [i]aren't[/i] in the same row. What is the minimum possible amount of diagonal pairs in the division? An example division into pairs is depicted in the image.

There are $n$ triangles inscribed in a circle and all $3n$ of their vertices are different. Prove that it is possible to put a boy in one of the vertices in each triangle, and a girl in the other, so that boys and girls alternate on a circle.

A square with horizontal and vertical sides is drawn on the plane. It held several segments parallel to the sides, and there are no two segments which lie on one line or intersect at an interior point for both segments. It turned out that the segments cuts square into rectangles, and any vertical line intersecting the square and not containing segments of the partition intersects exactly $ k $ rectangles of the partition, and any horizontal line intersecting the square and not containing segments of the partition intersects exactly $\ell$ rectangles. How much the number of rectangles can be? [i]I. Bogdanov, D. Fon-Der-Flaass[/i]

Bengt wants to put out crosses and rings in the squares of an $n \times n$-square, so that it is exactly one ring and exactly one cross in each row and in each column, and no more than one symbol in each box. Mona wants to stop him by setting a number in advance ban on crosses and a number of bans on rings, maximum one ban in each square. She want to use as few bans as possible of each variety. To succeed in preventing Bengt, how many prohibitions she needs to use the least of the kind of prohibitions she uses the most of?

Through an informant, the police know the meeting place of a group of criminals. The identity of the different elements of the group is, however, unknown. Inspector Loureiro's task is to arrest the leader of the group. The inspector knows that the leader of the group is the shortest of the five members of the group, all of them of different heights, who will be present at the meeting. After the meeting, the bandits - as a precautionary measure - leave the building separately with an interval of $15$ minutes. As the inspector doesn't know which of them is the shortest, he decides to let the first two criminals out, and arrest the first of the following who is shorter than those who left until that moment. What is the probability that Inspector Loureiro will arrest the right person?

An $(n^2+n+1) \times (n^2+n+1)$ matrix of zeros and ones is given. If no four ones are vertices of a rectangle, prove that the number of ones does not exceed $(n + 1)(n^2 + n + 1).$

Prove that from $x + y = 1 \ (x, y \in \mathbb R)$ it follows that \[x^{m+1} \sum_{j=0}^n \binom{m+j}{j} y^j + y^{n+1} \sum_{i=0}^m \binom{n+i}{i} x^i = 1 \qquad (m, n = 0, 1, 2, \ldots ).\]

Robert has five beads in his hand, with the letters $C, M, I, M,$ and $C,$ and he wants to make a circular bracelet spelling "$CMIMC.$" However, the power went out, so Robert can no longer see the beads in his hand. Thus, he puts the five beads on the bracelet randomly, hoping that the bracelet, when possibly rotated or flipped, spells out "$CMIMC.$" What is the probability that this happens? (Robert doesn’t care whether some letters appear upside down or backwards.)

Given a set $A$ which contains $n$ elements. For any two distinct subsets $A_{1}$, $A_{2}$ of the given set $A$, we fix the number of elements of $A_1 \cap A_2$. Find the sum of all the numbers obtained in the described way.

Let $n \geq 2$ be an integer. Switzerland and Liechtenstein are performing their annual festive show. There is a field divided into $n \times n$ squares, in which the bottom-left square contains a red house with $k$ Swiss gymnasts, and the top-right square contains a blue house with $k$ Liechtensteiner gymnasts. Every other square only has enough space for a single gymnast at a time. Each second either a Swiss gymnast or a Liechtensteiner gymnast moves. The Swiss gymnasts move to either the square immediately above or to the right and the Liechtensteiner gymnasts move either to the square immediately below or to the left. The goal is to move all the Swiss gymnasts to the blue house and all the Liechtensteiner gymnasts to the red house, with the caveat that a gymnast cannot enter a house until all the gymnasts of the other nationality have left. Determine the largest $k$ in terms of $n$ for which this is possible.

Consider $n>9$ lines on the plane such that no two lines are parallel. Show that there exist at least $n/9$ lines such that the angle between any two of the lines is at most $20^\circ$.

$3n-1$ points are given in the plane, no three are collinear. Prove that one can select $2n$ of them whose convex hull is not a triangle.

In a party, there are $2n + 1$ people. It's well known that for every group of $n$ people, there exist a person(out of the group) who knows all them(the $n$ people of the group). Show that there exist a person who knows all the people in the party.

Let $G$ be a $2n-$vertices simple graph such that in any partition of the set of vertices of $G$ into two $n-$vertices sets $V_1$ and $V_2$, the number of edges from a vertex in $V_1$ to another vertex in $V_1$ is equal to the number of edges from a vertex in $V_2$ to another vertex in $V_2$. Prove that all the vertices have equal degrees.

Determine the number of distinct sequences of letters of length 1997 which use each of the letters $A$, $B$, $C$ (and no others) an odd number of times.

On planet Zork there are some cities. For every city there is a city at the diametrically opposite point. Certain roads join the cities on Zork. If there is a road between cities $P$ and $Q$, then there is also a road between the cities $P'$ and $Q'$ diametrically opposite to $P$ and $Q$. In plus, the roads do not cross each other and for any two cities $P$ and $Q$ it is possible to travel from $P$ to $Q$. The prices of Kriptonita in Urghs (the planetary currency) in two towns connected by a road differ by at most 100. Prove that there exist two diametrically opposite cities in which the prices of Kriptonita differ by at most 100 Urghs.

Let $n \in N$. The land of Draconis has more than $2^n$ dungeons. Between two different Dungeons have exactly one path, but each path is a one-way street. Total $n$ Dragon cults share the territory; each path is controlled by exactly one cult. It is said that a dragon cult $K$ has established itself in a dungeon $D$ if there is both one a path beginning in $D$ and one a path ending in $D$, both of which are controlled by $K$ . Prove that there is a cult $K$, which has at least one dungeon controlled.

Let $n \geq 2$ be an integer. A lead soldier is moving across the unit squares of a $n \times n$ grid, starting from the corner square. Before each move to the neighboring square, the lead soldier can (but doesn't need to) turn left or right. Determine the smallest number of turns, which it needs to do, to visit every square of the grid at least once. At the beginning the soldier's back is faced at the edge of the grid.

Let $S$ be a square with sides length $100$. Let $L$ be a path within $S$ which does not meet itself and which is composed of line segments $A_0A_1,A_1A_2,A_2A_3,\ldots,A_{n-1}A_n$ with $A_0=A_n$. Suppose that for every point $P$ on the boundary of $S$ there is a point of $L$ at a distance from $P$ no greater than $\frac {1} {2}$. Prove that there are two points $X$ and $Y$ of $L$ such that the distance between $X$ and $Y$ is not greater than $1$ and the length of the part of $L$ which lies between $X$ and $Y$ is not smaller than $198$.

A sequence of seven digits is randomly chosen in a weekly lottery. Every digit can be any of the digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9.$ What is the probability of having at most five different digits in the sequence?

Karakade has three flash drives of each of the six capacities $1, 2, 4, 8, 16, 32$ gigabytes. She gives each of her $6$ servants three flash drives of different capacities. Prove that either there are two capacities where each servant has at most one of the two capacities, or all servants have flash drives with different sums of capacities.