Problems

We have a very large chessboard, consisting of white and black squares. We would like to place a stain of a specific shape on this chessboard and we know that the area of this stain is less than the area of one square of the chessboard. Show that it is always possible to place the stain in such a way that it does not cover a vertex of any square.

Does the equation $9^n+9^n+9^n=3^{2025}$ have any integer solutions?

Mark one card with a $1$, two cards with a $2$, ..., fifty cards with a $50$. Put these $1+2+...+50=1275$ cards into a box and shuffle them. How many cards do you need to take from the box to be certain that you will have taken at least $10$ cards with the same mark?

For every pair of integers $a$, $b$, we define an operator $a\otimes b$ with the following three properties.
1. $a\otimes a=a+2$;
2. $a\otimes b=b\otimes a$;
3. $\frac{a\otimes (a+b)}{a\otimes b}=\frac{a+b}{b}.$
Calculate $8\otimes 5$.

During a tournament with six players, each player plays a match against each other player. At each match there is a winner; ties do not occur. A journalist asks five of the six players how many matches each of them has won. The answers given are $4$, $3$, $2$, $2$ and $2$. How many matches have been won by the sixth player?

Let $n$ be an integer (positive or negative). Find all values of $n$, for which $n$ is $4^{\frac{n-1}{n+1}}$ an integer.

Klein tosses $n$ fair coins and Möbius tosses $n+1$ fair coins. What’s the probability that Möbius gets more heads than Klein? (Note that a fair coin is one that comes up heads half the time, and comes up tails the other half of the time).

The letters $A$, $E$ and $T$ each represent different digits from $0$ to $9$ inclusive. We are told that $$ATE\times EAT\times TEA=36239651.$$ What is $A\times E\times T$?

The kingdom of Rabbitland consists of a finite number of cities. No matter how you split the kingdom into two, there is always a train connection from a city in one part of the divide to a city in the other part of the divide. Show that one can in fact travel from any city to any other, possibly changing trains.

A poetry society has 33 members, and each person knows at least 16 people from the society. Show that you can get to know everyone in the society by a series of introductions if you already know someone from the society.