A number is called [i]trilegal[/i] if its digits belong to the set \(\{1, 2, 3\}\) and if it is divisible by \(99\). How many trilegal numbers with \(10\) digits are there?

Let $(a,b)$ be a pair of natural numbers. Henning and Paul play the following game. At the beginning there are two piles of $a$ and $b$ coins respectively. We say that $(a,b)$ is the [i]starting position [/i]of the game. Henning and Paul play with the following rules: $\bullet$ They take turns alternatively where Henning begins. $\bullet$ In every step each player either takes a positive integer number of coins from one of the two piles or takes same natural number of coins from both piles. $\bullet$ The player how take the last coin wins. Let $A$ be the set of all positive integers like $a$ for which there exists a positive integer $b<a$ such that Paul has a wining strategy for the starting position $(a,b)$. Order the elements of $A$ to construct a sequence $a_1<a_2<a_3<\dots$ $(a)$ Prove that $A$ has infinity many elements. $(b)$ Prove that the sequence defined by $m_k:=a_{k+1}-a_{k}$ will never become periodic. (This means the sequence $m_{k_0+k}$ will not be periodic for any choice of $k_0$)

A student firstly wrote $x=3$ on the board. For each procces, the stutent deletes the number x and replaces it with either $(2x+4)$ or $(3x+8)$ or $(x^2+5x)$. Is this possible to make the number $(20^{17}+2016)$ on the board? \\ (Explain your answer) \\ [hide=Note]This type of the question is well known but I am going to make a collection so, :blush: [/hide]

Determine the maximal length $L$ of a sequence $a_1,\dots,a_L$ of positive integers satisfying both the following properties: [list=disc] [*]every term in the sequence is less than or equal to $2^{2023}$, and [*]there does not exist a consecutive subsequence $a_i,a_{i+1},\dots,a_j$ (where $1\le i\le j\le L$) with a choice of signs $s_i,s_{i+1},\dots,s_j\in\{1,-1\}$ for which \[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\] [/list]

A finite family of finite sets $F$ is given, satisfying two conditions: (i) if $A, B \in F$, then $A \cup B \in F$; (ii) if $A \in F$, then the number of elements $| A |$ is not a multiple of $3$. Prove that you can specify at most two elements so that every set of the family $F$ contains at least one of them.

Let $ M \equal{} \lbrace 2, 3, 4, \ldots\, 1000 \rbrace$. Find the smallest $ n \in \mathbb{N}$ such that any $ n$-element subset of $ M$ contains 3 pairwise disjoint 4-element subsets $ S, T, U$ such that [b]I.[/b] For any 2 elements in $ S$, the larger number is a multiple of the smaller number. The same applies for $ T$ and $ U$. [b]II.[/b] For any $ s \in S$ and $ t \in T$, $ (s,t) \equal{} 1$. [b]III.[/b] For any $ s \in S$ and $ u \in U$, $ (s,u) > 1$.

$A$ and $B$ are any two subsets of $\{1, 2,...,n - 1\}$ such that $|A| +|B|> n - 1$. Prove that one can find $a$ in $A$ and $b$ in $B$ such that $a + b = n$.

Let $n$ $\geq$ $2$ and $k$ $\geq$ $2$ be positive integers. A cat and a mouse are playing [i]Wim[/i], which is a stone removal game. The game starts with $n$ stones and they take turns removing stones, with the cat going first. On each turn they are allowed to remove $1$, $2$, $\dotsb$, or $k$ stones, and the player who cannot remove any stones on their turn loses. \\\\ A raccoon finds Wim very boring and creates [i]Wim 2[/i], which is Wim but with the following additional rule: [i]You cannot remove the same number of stones that your opponent removed on the previous turn[/i]. \\\\Find all values of $k$ such that for every $n$, the cat has a winning strategy in Wim if and only if it has a winning strategy in Wim 2.

In a country, there are some cities linked together by roads. The roads just meet each other inside the cities. In each city, there is a board which showing the shortest length of the road originating in that city and going through all other cities (the way can go through some cities more than one times and is not necessary to turn back to the originated city). Prove that 2 random numbers in the boards can't be greater or lesser than 1.5 times than each other.

[b]p1.[/b] Solve the equation $x^2 - x - cos y+1.25 =0$. [b]p2.[/b] Solve the inequality: $\left| \frac{x - 2}{x - 3}\right| \le x$ [b]p3.[/b] Bilbo and Dwalin are seated at a round table of radius $R$. Bilbo places a coin of radius $r$ at the center of the table, then Dwalin places a second coin as near to the table’s center as possible without overlapping the first coin. The process continues with additional coins being placed as near as possible to the center of the table and in contact with as many coins as possible without overlap. The person who places the last coin entirely on the table (no overhang) wins the game. Assume that $R/r$ is an integer. (a) Who wins, Bilbo or Dawalin? Please justify your answer. (b) How many coins are on the table when the game ends? [b]p4.[/b] In the center of a square field is an orc. Four elf guards are on the vertices of that square. The orc can run in the field, the elves only along the sides of the square. Elves run $\$1.5$ times faster than the orc. The orc can kill one elf but cannot fight two of them at the same time. Prove that elves can keep the orc from escaping from the field. [b]p5.[/b] Nine straight roads cross the Mirkwood which is shaped like a square, with an area of $120$ square miles. Each road intersects two opposite sides of the square and divides the Mirkwood into two quadrilaterals of areas $40$ and $80$ square miles. Prove that there exists a point in the Mirkwood which is an intersection of at least three roads. PS. You should use hide for answers.

The sequence $n_1<n_2<\ldots < n_k$ consists of all positive integers $n$ for which in a square $n \times n$ one can mark $10$ cells such that in any square $3 \times 3$ an odd amount of cells are marked. Find $n_{k-2}$.

Prove that, if every three consecutive vertices of a convex $n{}$-gon, $n\geqslant 4$, span a triangle of area at least 1, then the area of the $n{}$-gon is (strictly) greater than $(n\log_2 n)/4-1/2.$ [i]Radu Bumbăcea & Călin Popescu[/i]

Treasure was buried in a single cell of an $M\times N$ ($2\le M$, $N$) grid. Detectors were brought to find the cell with the treasure. For each detector, you can set it up to scan a specific subgrid $[a,b]\times[c,d]$ with $1\le a\le b\le M$ and $1\le c\le d\le N$. Running the detector will tell you whether the treasure is in the region or not, though it cannot say where in the region the treasure was detected. You plan on setting up $Q$ detectors, which may only be run simultaneously after all $Q$ detectors are ready. In terms of $M$ and $N$, what is the minimum $Q$ required to gaurantee to determine the location of the treasure?

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?

Let $S$ be a set of $n$ points in the plane such that for any two points $(a, b), (c, d) \in S$, we have that $| a - c | \cdot | b - d | \ge 1$. Show that [list] [*] (a) If $S = \{ Q_1, Q_2, Q_3\}$ such that for any point $Q_i$ in $S$, this point doesn't lie in the axis-aligned rectangle with corners as the other two points, show that the area of $\triangle Q_1Q_2Q_3$ is at least $\frac{\sqrt{5}}{2}$. [*] (b) If all points in $S$ lie in an axis-aligned square with sidelength $a$, then $|S| \le \frac{a^2}{\sqrt{5}} + 2a + 1$. [/list]

Knights, who always tell truth, and liars, who always lie, live on an island. They have been distributed into two teams $A$ and $B$ for a game of tennis, and team $A$ had more members than team $B$. Two players from different teams started the game, whenever a player loses the game, he leaves it forever and he is replaces by a member of his team (that has never played before). The team, all of whose members left the game, loses. After the tournament, every member of team $A$ was asked: "Is it true that you have lost to a liar in some game?", and every member of team $B$ was asked: "Is it true that you have won at least two games, in which your opponent was a knight?". It turns out that every single answer was positive. Which team won?

Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

In a school there are $2007$ male and $2007$ female students. Each student joins not more than $100$ clubs in the school. It is known that any two students of opposite genders have joined at least one common club. Show that there is a club with at least $11$ male and $11$ female members.

Let $M$ be a set of $n \ge 4$ points in the plane, no three of which are collinear. Initially these points are connected with $n$ segments so that each point in $M$ is the endpoint of exactly two segments. Then, at each step, one may choose two segments $AB$ and $CD$ sharing a common interior point and replace them by the segments $AC$ and $BD$ if none of them is present at this moment. Prove that it is impossible to perform $n^3 /4$ or more such moves. [i]Proposed by Vladislav Volkov, Russia[/i]

The cells of some rectangular $M \times n$ array are colored, each by one of two colors, so that for any two columns the number of pairs of cells situated on a same row and bearing the same color is less than the number of pairs of cells situated on a same row and bearing different colors. i) Prove that if $M=2011$ then $n \leq 2012$ (a model for the extremal case $n=2012$ does indeed exist, but you are not asked to exhibit one). ii) Prove that if $M=2011=n$, each of the colors appears at most $1006\cdot 2011$ times, and at least $1005\cdot 2011$ times. iii) Prove that if however $M=2012$ then $n \leq 1007$. ([i]Dan Schwarz[/i])

On an infinite sheet of tiles, an infinite number of $1 \times 2$ tile rectangles are placed, their edges follow the lines of the tiles, and they do not touch each other, not even the corners. Is it true that the remaining checkered sheet can be completely covered with $1 \times 2$ checkered rectangles? [hide=original wording]Uz bezgalīgas rūtiņu lapas ir novietoti bezgaglīgi daudzi 1 x 2 rūtiņu taisnstūri, to malas iet pa rūtiņu līnijām, un tie nesaskaras cits ar citu pat ne ar stūriem. Vai tiesa, ka atlikušo rūtiņu lapu var pilnībā noklāt ar 1 x 2 rūtiņu tainstūriem? [/hide]

Albert writes $2025$ numbers $a_1, \ldots, a_{2025}$ in a circle on a blackboard. Initially, each of the numbers is uniformly and independently sampled at random from the interval $[0,1].$ Then, each second, he [i]simultaneously[/i] replaces $a_i$ with $\max(a_{i-1},a_i,a_{i+1})$ for all $i = 1, 2, \ldots, 2025$ (where $a_0 = a_{2025}$ and $a_{2026} = a_1$). Compute the expected value of the number of distinct values remaining after $100$ seconds.

Let $n$ be a fixed positive integer. Ana and Banana are playing a game. First, Ana picks a subset $S$ of $\{1,2,\ldots,n\}$. Then for each $k=1,2,\ldots,n$, she tells Banana how many numbers from $k-1$ to $k+1$ she has picked (i.e. $\lvert S \cap \{k-1,k,k+1\}\rvert$). Then Banana guesses $S$; she wins if her guess is correct and she loses otherwise. (a) Determine all $n$ for which Banana will win regardless of what Ana chooses. (b) For the values of $n$ for which Ana can win, determine the number of sets $S$ she can choose so as to do so.

Let $(v_1, ..., v_{2^n})$ be the vertices of an $n$-dimensional hypercube. Label each vertex $v_i$ with a real number $x_i$. Label each edge of the hypercube with the product of labels of the two vertices it connects. Let $S$ be the sum of the labels of all the edges. Over all possible labelings, find the minimum possible value of $\frac{S}{x^2_1+ x^2_2+ ...+ x^2_n}$ in terms of $ n$. Note: an $n$ dimensional hypercube is a graph on $2^n$ vertices labeled labeled with the binary strings of length $n$, where two vertices have an edge between them if and only if their labels differ in exactly one place. For instance, the vertices $100$ and $101$ on the $3$ dimensional hypercube are connected, but the vertices $100$ and $111$ are not.

Let's call it a [i] staircase of height [/i]$n$, a figure consisting from all square cells $n\times n$ lying no higher diagonals (the figure shows a [i]staircase of height [/i] $4$ ). In how many different ways can a [i]staircase of height[/i] $n$ can be divided into several rectangles whose sides go along the grid lines, but the areas are different in pairs? [img]https://cdn.artofproblemsolving.com/attachments/f/0/f66d7e9ada0978e8403fbbd8989dc1b201f2cd.png[/img]