Lazim rolls two $24$-sided dice. From the two rolls, Lazim selects the die with the highest number. $N$ is an integer not greater than $24$. What is the largest possible value for $N$ such that there is a more than $50$% chance that the die Lazim selects is larger than or equal to $N$?

A pedestrian walked for $3.5$ hours. In every period of one hour’s duration he walked $5$ kilometres. Is it true that his average speed was $5$ kilometres per hour? (NN Konstantinov, Moscow)

A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$

In a square of side length $60$, $121$ distinct points are given. Show that among them there exists three points which are vertices of a triangle with an area not exceeding $30$.

The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection. Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$. [i]Proposed by Warut Suksompong, Thailand[/i]

Let $S$ be a set consisting of $n$ elements, $F$ a set of subsets of $S$ consisting of $2^{n-1}$ subsets such that every three such subsets have a non-empty intersection. a) Show that the intersection of all subsets of $F$ is not empty. b) If you replace the number of sets from $2^{n-1}$ with $2^{n-1}-1$, will the previous answer change?

A student read the book with $480$ pages two times. If he in the second time for every day read $16$ pages more than in the first time and he finished it $5$ days earlier than in the first time. For how many days did he read the book in the first time?

In each field of 2009*2009 table you can write either 1 or -1. Denote Ak multiple of all numbers in k-th row and Bj the multiple of all numbers in j-th column. Is it possible to write the numbers in such a way that $ \sum_{i\equal{}1}^{2009}{Ai}\plus{} \sum_{i\equal{}1}^{2009}{Bi}\equal{}0$?

Let $a_1,a_2,...,a_9$ be nine real numbers, not necessarily distinct, with average $m$. Let $A$ denote the number of triples $1 \le i < j < k \le 9$ for which $a_i + a_j + a_k \ge 3m$. What is the minimum possible value of $A$? [i]Proposed by Ray Li[/i]

Santa Claus plays a guessing game with Marvin before giving him his present. He hides the present behind one of $100$ doors, numbered from $1$ to $100$. Marvin can point at a door, and then Santa Claus will reply with one of the following words: $\bullet$ "hot" if the present lies behind the guessed door, $\bullet$ "warm" if the guess is not exact but the number of the guessed door differs from that of the present’s door by at most $5$, $\bullet$ "cold" if the numbers of the two doors differ by more than $5$. At least how many such guesses does Marvin need, so that he can be certain about where his present is? Marvin does not necessarily need to make a "hot" guess, just to know the correct door with $100\%$ certainty.

There are $2021$ points on a circle. Kostya marks a point, then marks the adjacent point to the right, then he marks the point two to its right, then three to the next point's right, and so on. Which move will be the first time a point is marked twice? [i]K. Kokhas[/i]

Each of the $50$ students in a class sent greeting cards to $25$ of the others. Prove that there exist two students who greeted each other.

Let $ S \equal{} \{x_1, x_2, \ldots, x_{k \plus{} l}\}$ be a $ (k \plus{} l)$-element set of real numbers contained in the interval $ [0, 1]$; $ k$ and $ l$ are positive integers. A $ k$-element subset $ A\subset S$ is called [i]nice[/i] if \[ \left |\frac {1}{k}\sum_{x_i\in A} x_i \minus{} \frac {1}{l}\sum_{x_j\in S\setminus A} x_j\right |\le \frac {k \plus{} l}{2kl}\] Prove that the number of nice subsets is at least $ \dfrac{2}{k \plus{} l}\dbinom{k \plus{} l}{k}$. [i]Proposed by Andrey Badzyan, Russia[/i]

There are 20 people at a party. Each person holds some number of coins. Every minute, each person who has at least 19 coins simultaneously gives one coin to every other person at the party. (So, it is possible that $A$ gives $B$ a coin and $B$ gives $A$ a coin at the same time.) Suppose that this process continues indefinitely. That is, for any positive integer $n$, there exists a person who will give away coins during the $n$th minute. What is the smallest number of coins that could be at the party? [i]Proposed by Ray Li[/i]

Nine pawns forming a $3$ by $3$ square are placed in the lower left hand corner of an $8$ by $8$ chessboard. Any pawn may jump over another one standing next to it into a free square, i .e. may be reflected symmetrically with respect to a neighb our's centre (jumps may be horizontal , vertical or diagonal) . It is required to rearrange the nine pawns in another corner of the chessboard (in another $3$ by $3$ square) by means of such jumps. Can the pawns be thus re-arranged in the (a) upper left hand corner? (b) upper right hand corner? (J . E . Briskin)

On the circumference of a circle, there are $3n$ colored points that divide the circle on $3n$ arches, $n$ of which have lenght $1$, $n$ of which have length $2$ and the rest of them have length $3$ . Prove that there are two colored points on the same diameter of the circle.

In a pond there are $n \geq 3$ stones arranged in a circle. A princess wants to label the stones with the numbers $1, 2, \dots, n$ in some order and then place some toads on the stones. Once all the toads are located, they start jumping clockwise, according to the following rule: when a toad reaches the stone labeled with the number $k$, it waits for $k$ minutes and then jumps to the adjacent stone. What is the greatest number of toads for which the princess can label the stones and place the toads in such a way that at no time are two toads occupying a stone at the same time? [b]Note:[/b] A stone is considered occupied by two toads at the same time only if there are two toads that are on the stone for at least one minute.

Show that one can color all the points of a plane using only two colors such that no line segment has all points of the same color.

In a school tennis tournament with $ m \ge 2$ participants, each match consists of 4 sets. A player who wins more than half of all sets during a match gets 2 points for this match. A player who wins exactly half of all sets during the match gets 1 point, and a player who wins less than half of all sets gets 0 points. During the tournament, each participant plays exactly one match against each remaining player. Find the least number of participants m for which it is possible that some participant wins more sets than any other participant but obtains less points than any other participant.

Find the coefficient of $a^5b^5c^5d^6$ in the expansion of the following expression $(bcd +acd +abd +abc)^7$

The numbers $1, 2, 3, ..., 1000$ are written on the board. Two people take turns erasing one number at a time. The game ends when two numbers remain on the board. If their sum is divisible by three, then the one who made the first move wins. if not, then his partner. Which one will win if played correctly?

A positive integers has exactly $81$ divisors, which are located in a $9 \times 9$ table such that for any two numbers in the same row or column one of them is divisible by the other one. Find the maximum possible number of distinct prime divisors of $n$

Snow White and seven dwarfs live in their house in the forest. During several days some dwarfs worked in the diamond mine, while others were collecting mushrooms. Each dwarf each day was doing only one type of job. It is known that in any two consecutive days there are exactly three dwarfs which did both types of job. Also, for any two days at least one dwarf did both types of job. What is maximum amount of days which this situation could last? [i]M. Karpuk[/i]

(a) Several identical napkins, each in the shape of a regular hexagon, are put on a table (the napkins may overlap). Each napkin has one side which is parallel to a fixed line. Is it always possible to hammer a few nails into the table so that each napkin is nailed with exactly one nail? (b) The same question for regular pentagons. (A Kanel)

All sides and diagonals of a convex $n$-gon, $n\ge 3$, are coloured one of two colours. Show that there exist $\left[\frac{n+1}{3}\right]$ pairwise disjoint monochromatic segments. [i](Two segments are disjoint if they do not share an endpoint or an interior point).[/i]