[b]p1.[/b] Is there an integer such that the product of all whose digits equals $99$ ? [b]p2.[/b] An elevator in a $100$ store building has only two buttons: UP and DOWN. The UP button makes the elevator go $13$ floors up, and the DOWN button makes it go $8$ floors down. Is it possible to go from the $13$th floor to the $8$th floor? [b]p3.[/b] Cut the triangle shown in the picture into three pieces and rearrange them into a rectangle. (Pieces can not overlap.) [img]https://cdn.artofproblemsolving.com/attachments/9/f/359d3b987012de1f3318c3f06710daabe66f28.png[/img] [b]p4.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $5$ rocks in the first pile and $6$ rocks in the second pile. Each of the players in his turn can take either any amount of rocks from one pile or the same amount of rocks from both piles. The winner is the player who takes the last rock. Who does win in this game if Tom starts the game? [b]p5.[/b] In the next long multiplication example each letter encodes its own digit. Find these digits. $\begin{tabular}{ccccc} & & & a & b \\ * & & & c & d \\ \hline & & c & e & f \\ + & & a & b & \\ \hline & c & f & d & f \\ \end{tabular}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider an array of white unit squares arranged in a rectangular grid with $59$ rows of unit squares and $c$ columns of unit squares, for some positive integer $c$. What is the smallest possible value of $c$ such that, if we shade exactly $25$ unit squares in each column black, then there must necessarily be some row with at least $18$ black unit squares?

Each vertex of a finite graph can be coloured either black or white. Initially all vertices are black. We are allowed to pick a vertex $P$ and change the colour of $P$ and all of its neighbours. Is it possible to change the colour of every vertex from black to white by a sequence of operations of this type? Note: A finite graph consists of a finite set of vertices and a finite set of edges between vertices. If there is an edge between vertex $A$ and vertex $B,$ then $A$ and $B$ are neighbours of each other.

Let $G$ be a connected graph and let $X, Y$ be two disjoint subsets of its vertices, such that there are no edges between them. Given that $G/X$ has $m$ connected components and $G/Y$ has $n$ connected components, what is the minimal number of connected components of the graph $G/(X \cup Y)$?

Given several nonnegative integers (repetitions allowed), the allowed operation is to choose a positive integer $a$ and replace each number $b$ greater than or equal to $a$ by $b-a$ (the numbers $a$ , if any, are replaced by $0$). Initially, the integers from $1$ are written on the blackboard until $2013$ inclusive. After a few operations the numbers on the board have a sum equal to $10$. Determine what the numbers that remained on the board could be. Find all the possibilities.

Each element of set $\mathcal{S}$ is colored with multiple colors. A $\textit{rainbow}$ is a subset of $\mathcal{S}$ which has amongst its elements at least $1$ color from each element of $\mathcal{S}$. A $\textit{minimal rainbow}$ is a rainbow where removing any single element gives a non-rainbow. Prove that the union of all minimal rainbows is $\mathcal{S}$. [i]Grisham Paimagam[/i]

There are 5 points on the plane. The following steps are used to construct lines. In step 1, connect all possible pairs of the points; it is found that no two lines are parallel, nor any two lines perpendicular to each other, also no three lines are concurrent. In step 2, perpendicular lines are drawn from each of the five given points to straight lines connecting any two of the other four points. What is the maximum number of points of intersection formed by the lines drawn in step 2, including the 5 given points?

On a chessboard $8\times 8$, $n>6$ Knights are placed so that for any 6 Knights there are two Knights that attack each other. Find the greatest possible value of $n$.

All positive integers from \( 1 \) to \( 2025 \) are written on a board. Mykhailo and Oleksii play the following game. They take turns, starting with Mykhailo, erasing one of the numbers written on the board. The game ends when exactly two numbers remain on the board. If their sum is a perfect square of an integer, Mykhailo wins; otherwise, Oleksii wins. Who wins if both players play optimally? [i]Proposed by Fedir Yudin[/i]

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

Let $A$ be a non-empty subset of the set of all positive integers $N^*$. If any sufficient big positive integer can be expressed as the sum of $2$ elements in $A$(The two integers do not have to be different), then we call that $A$ is a divalent radical. For $x \geq 1$, let $A(x)$ be the set of all elements in $A$ that do not exceed $x$, prove that there exist a divalent radical $A$ and a constant number $C$ so that for every $x \geq 1$, there is always $\left| A(x) \right| \leq C \sqrt{x}$.

Show that no matter how $15$ points are plotted within a circle of radius $2$ (circle border included), there will be a circle with radius $1$ (circle border including) which contains at least three of the $15$ points.

Given a positive integer $n$. An inversion of a permutation is the amount of pairs $(i,j)$ such that $i<j$ and the $i$-th number is smaller than $j$-th number in the permutation. Prove that for every positive integer $k \leq n$ there exist exactly $\frac{n!}{k}$ permutations in which the inversion is divisible by $k$.

Let $S$ be a subset of $\{0,1,2,\ldots,98 \}$ with exactly $m\geq 3$ (distinct) elements, such that for any $x,y\in S$ there exists $z\in S$ satisfying $x+y \equiv 2z \pmod{99}$. Determine all possible values of $m$.

The numbers $1$ to $15$ are each coloured blue or red. Determine all possible colourings that satisfy the following rules: • The number $15$ is red. • If numbers $x$ and $y$ have different colours and $x + y \le 15$, then $x + y$ is blue. • If numbers $x$ and $y$ have different colours and $x \cdot y \le 15$, then $x \cdot y$ is red.

Each of the positive integers $1,2,\dots,n$ is colored in one of the colors red, blue or yellow regarding the following rules: (1) A Number $x$ and the smallest number larger than $x$ colored in the same color as $x$ always have different parities. (2) If all colors are used in a coloring, then there is exactly one color, such that the smallest number in that color is even. Find the number of possible colorings.

In a football tournament there are $8$ teams, each of which plays exacly one match against every other team. If a team $A$ defeats team $B$, then $A$ is awarded $3$ points and $B$ gets $0$ points. If they end up in a tie, they receive $1$ point each. It turned out that in this tournament, whenever a match ended up in a tie, the two teams involved did not finish with the same final score. Find the maximum number of ties that could have happened in such a tournament.

Find the smallest positive integer $n$, such that one can color every cell of a $n \times n$ grid in red, yellow or blue with all the following conditions satisfied: (1) the number of cells colored in each color is the same; (2) if a row contains a red cell, that row must contain a blue cell and cannot contain a yellow cell; (3) if a column contains a blue cell, it must contain a red cell but cannot contain a yellow cell.

Inside a square of sidelength $ 1$ there are finitely many disks that are allowed to overlap. The sum of all circumferences is $ 10$. Show that there is a line intersecting or touching at least $ 4$ disks.

In a simple graph with 300 vertices no two vertices of the same degree are adjacent (boo hoo hoo). What is the maximal possible number of edges in such a graph?

A row of 2021 balls is given. Pasha and Vova play a game, taking turns to perform moves; Pasha begins. On each turn a boy should paint a non-painted ball in one of the three available colors: red, yellow, or green (initially all balls are non-painted). When all the balls are colored, Pasha wins, if there are three consecutive balls of different colors; otherwise Vova wins. Who has a winning strategy?

Let $\Omega$ be a circle disk with radius $1$. Determine the minimum $r$ that has the following property: You can select three points on $\Omega$ so that each circle disk located in $\Omega$ and has a radius greater than $r$ contains at least one of the three points.

Let $E$ be a set of $n$ points in the plane $(n \geq 3)$ whose coordinates are integers such that any three points from $E$ are vertices of a nondegenerate triangle whose centroid doesnt have both coordinates integers. Determine the maximal $n.$

Koshchey opened an account at the bank. Initially, it had 0 rubles. On the first day, Koshchey puts $k>0$ rubles in, and every next day adds one ruble more there than the day before. Each time after Koshchey deposits money into the account, the total amount in the account is divided by two by the bank. Find all such $k{}$ for which the amount on the account will always be an integer number of rubles. [i]Proposed by S. Berlov[/i]

On a piece of paper we have $2019$ statements numbered from $1$ to $2019$. The $n^{th}$ statement is the following: "On this piece of paper there are at most $n$ true statements". How many of the statements are true?