[b]p25.[/b] You are given that $1000!$ has $2568$ decimal digits. Call a permutation $\pi$ of length $1000$ good if $\pi(2i) > \pi (2i - 1)$ for all $1 \le i \le 500$ and $\pi (2i) > \pi (2i + 1)$ for all $1 \le i \le 499$. Let $N$ be the number of good permutations. Estimate $D$, the number of decimal digits in $N$. You will get $\max \left( 0, 25 - \left\lceil \frac{|D-X|}{10} \right\rceil \right)$ points, where $X$ is the true answer. [b]p26.[/b] A year is said to be [i]interesting [/i] if it is the product of $3$, not necessarily distinct, primes (for example $2^2 \cdot 5$ is interesting, but $2^2 \cdot 3 \cdot 5$ is not). How many interesting years are there between $ 5000$ and $10000$, inclusive? For an estimate of $E$, you will get $\max \left( 0, 25 - \left\lceil \frac{|E-X|}{10} \right\rceil \right)$ points, where $X$ is the true answer. [b]p27.[/b] Sam chooses $1000$ random lattice points $(x, y)$ with $1 \le x, y \le 1000$ such that all pairs $(x, y)$ are distinct. Let $N$ be the expected size of the maximum collinear set among them. Estimate $\lfloor 100N \rfloor$. Let $S$ be the answer you provide and $X$ be the true value of $\lfloor 100N \rfloor$. You will get $\max \left( 0, 25 - \left\lceil \frac{|S-X|}{10} \right\rceil \right)$ points for your estimate. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that $\sum \frac{1}{i_1i_2 \ldots i_k} = n$ is taken over all non-empty subsets $\left\{i_1,i_2, \ldots, i_k\right\}$ of $\left\{1,2,\ldots,n\right\}$. (The $k$ is not fixed, so we are summing over all the $2^n-1$ possible nonempty subsets.)

Let $P_1, P_2, \ldots, P_n$ be distinct $2$-element subsets of $\{1, 2, \ldots, n\}$. Suppose that for every $1 \le i < j \le n$, if $P_i \cap P_j \neq \emptyset$, then there is some $k$ such that $P_k = \{i, j\}$. Prove that if $a \in P_i$ for some $i$, then $a \in P_j$ for exactly one value of $j$ not equal to $i$.

A dot-trapezium consists of several rows of dots such that each row contains one more dot than the row immediately above (apart from the top row). For example here is a dot-trapezium consisting of $15$ dots, having $3$ rows and $4$ dots in the top row. [asy] //wonderfully scuffed asymptote code , please don't laugh at me. constructed from the diagram at https://www.mathsolympiad.org.nz/competitions/nzmo/problems/nzmo1_2024.pdf //top row dot((.05,.1)); dot((-.05,.1)); dot((-.15,.1)); dot((.15,.1)); //middle row dot((0,0)); dot((.1,0)); dot((-.1,0)); dot((.2,0)); dot((-.2,0)); //bottom row dot((.05,-.1)); dot((-.05,-.1)); dot((-.15,-.1)); dot((.15,-.1)); dot((.25,-.1)); dot((-.25,-.1)); [/asy] A positive integer $n$ is called a trapezium-number if there exists a dot-trapezium consisting of exactly $n$ dots, with at least two rows and at least two dots in the top row. How many trapezium-numbers are there less than $100$?

Ella lays out $16$ coins heads up in a $4\times 4$ grid as shown. [img]https://cdn.artofproblemsolving.com/attachments/3/3/a728be9c51b27f442109cc8613ef50d61182a0.png[/img] On a move, Ella can flip all the coins in any row, column, or diagonal (including small diagonals such as $H_1$ & $H_4$). If rotations are considered distinct, how many distinct grids of coins can she create in a finite number of moves?

We have a chessboard and we call a $1\times1$ square a room. A robot is standing on one arbitrary vertex of the rooms. The robot starts to move and in every one movement, he moves one side of a room. This robot has $2$ memories $A,B$. At first, the values of $A,B$ are $0$. In each movement, if he goes up, $1$ unit is added to $A$, and if he goes down, $1$ unit is waned from $A$, and if he goes right, the value of $A$ is added to $B$, and if he goes left, the value of $A$ is waned from $B$. Suppose that the robot has traversed a traverse (!) which hasn’t intersected itself and finally, he has come back to its initial vertex. If $v(B)$ is the value of $B$ in the last of the traverse, prove that in this traverse, the interior surface of the shape that the robot has moved on its circumference is equal to $|v(B)|$.

A configuration of $4027$ points in the plane is called Colombian if it consists of $2013$ red points and $2014$ blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is good for a Colombian configuration if the following two conditions are satisfied: i) No line passes through any point of the configuration. ii) No region contains points of both colors. Find the least value of $k$ such that for any Colombian configuration of $4027$ points, there is a good arrangement of $k$ lines. Proposed by [i]Ivan Guo[/i] from [i]Australia.[/i]

Let $ m\geq n\geq 4$ be two integers. We call a $ m\times n$ board filled with 0's or 1's [i]good[/i] if 1) not all the numbers on the board are 0 or 1; 2) the sum of all the numbers in $ 3\times 3$ sub-boards is the same; 3) the sum of all the numbers in $ 4\times 4$ sub-boards is the same. Find all $ m,n$ such that there exists a good $ m\times n$ board.

We are given an infinite deck of cards, each with a real number on it. For every real number $x$, there is exactly one card in the deck that has $x$ written on it. Now two players draw disjoint sets $A$ and $B$ of $100$ cards each from this deck. We would like to define a rule that declares one of them a winner. This rule should satisfy the following conditions: 1. The winner only depends on the relative order of the $200$ cards: if the cards are laid down in increasing order face down and we are told which card belongs to which player, but not what numbers are written on them, we can still decide the winner. 2. If we write the elements of both sets in increasing order as $A =\{ a_1 , a_2 , \ldots, a_{100} \}$ and $B= \{ b_1 , b_2 , \ldots , b_{100} \}$, and $a_i > b_i$ for all $i$, then $A$ beats $B$. 3. If three players draw three disjoint sets $A, B, C$ from the deck, $A$ beats $B$ and $B$ beats $C$ then $A$ also beats $C$. How many ways are there to define such a rule? Here, we consider two rules as different if there exist two sets $A$ and $B$ such that $A$ beats $B$ according to one rule, but $B$ beats $A$ according to the other. [i]Proposed by Ilya Bogdanov, Russia[/i]

The country has $n \ge 3$ airports, some pairs of which are connected by bidirectional flights. Every day, the government closes the airport with the strictly highest number of flights going out of it. What is the maximum number of days this can continue? [i]Proposed by Fedir Yudin[/i]

Find all finite sequences $(x_0, x_1, \ldots,x_n)$ such that for every $j$, $0 \leq j \leq n$, $x_j$ equals the number of times $j$ appears in the sequence.

Consider a complete graph on $4046$ nodes, whose edges are colored in some colors. Let's call this graph $k$-good if we can split all its nodes into $2023$ pairs so that there are exactly $k$ distinct colors among the colors of $2023$ edges that connect the nodes from the same pairs. Is it possible that the graph is $999$-good and $1001$-good but not $1000$-good? [i]Proposed by Anton Trygub[/i]

Compute the number of ways to color $3$ cells in a $3\times 3$ grid so that no two colored cells share an edge.

Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.

To connect to the OFM site, Alice must choose a password. The latter must be consisting of $n$ characters among the following $27$ characters: $$A, B, C, . . ., Y , Z, \#$$ We say that a password $m$ is [i]redundant [/i] if we can color in red and blue a block of consecutive letters of $m$ in such a way that the word formed from the red letters is identical to the word formed from blue letters. For example, the password $H\#ZBZJBJZ$ is redundant, because it contains the [color=#00f]ZB[/color][color=#f00]Z[/color][color=#00f]J[/color][color=#f00]BJ[/color] block, where the word $ZBJ$ appears in both blue and red. At otherwise, the $ABCACB$ password is not redundant. Show that, for any integer $n \ge 1$, there exist at least $18^n$ passwords of length $n$, that is to say formed of $n$ characters each, which are not redundant.

$k,n$ are two arbitrary positive integers. Prove that there exists at least $(k-1)(n-k+1)$ positive integers that can be produced by $n$ number of $k$'s and using only $+,-,\times, \div$ operations and adding parentheses between them, but cannot be produced using $n-1$ number of $k$'s. [i]Proposed by Aryan Tajmir[/i]

A $9\times 7$ rectangle is tiled with tiles of the two types: L-shaped tiles composed by three unit squares (can be rotated repeatedly with $90^\circ$) and square tiles composed by four unit squares. Let $n\ge 0$ be the number of the $2 \times 2 $ tiles which can be used in such a tiling. Find all the values of $n$.

In a card game, each card is associated with a numerical value from 1 to 100, with each card beating less, with one exception: 1 beats 100. The player knows that 100 cards with different values lie in front of him. The dealer who knows the order of these cards can tell the player which card beats the other for any pair of cards he draws. Prove that the dealer can make one hundred such messages, so that after that the player can accurately determine the value of each card.

Given that $a_1, a_2, \ldots,a_{2020}$ are integers, find the maximal number of subsequences $a_i,a_{i+1}, ..., a_j$ ($0<i\leq j<2021$) with with sum $2021$

A and B plays the following game: they choose randomly $k$ integers from $\{1,2,\dots,100\}$; if their sum is even, A wins, else B wins. For what values of $k$ does A and B have the same chance of winning?

Dylan has a $100\times 100$ square, and wants to cut it into pieces of area at least $1$. Each cut must be a straight line (not a line segment) and must intersect the interior of the square. What is the largest number of cuts he can make?

Let $A$ be the set of all binary sequences of length $n$ and denote $o =(0, 0, \ldots , 0) \in A$. Define the addition on $A$ as $(a_1, \ldots , a_n)+(b_1, \ldots , b_n) =(c_1, \ldots , c_n)$, where $c_i = 0$ when $a_i = b_i$ and $c_i = 1$ otherwise. Suppose that $f\colon A \to A$ is a function such that $f(0) = 0$, and for each $a, b \in A$, the sequences $f(a)$ and $f(b)$ differ in exactly as many places as $a$ and $b$ do. Prove that if $a$ , $b$, $c \in A$ satisfy $a+ b + c = 0$, then $f(a)+ f(b) + f(c) = 0$.

There are $2022$ signs arranged in a straight line. Mark tasks Auto to color each sign with either red or blue with the following condition: for any given sequence of length $1011$ whose each term is either red or blue, Auto can always remove $1011$ signs from the line so that the remaining $1011$ signs match the given color sequence without changing the order. Determine the number of ways Auto can color the signs to satisfy Mark's condition.

$ 1994$ girls are seated at a round table. Initially one girl holds $ n$ tokens. Each turn a girl who is holding more than one token passes one token to each of her neighbours. a.) Show that if $ n < 1994$, the game must terminate. b.) Show that if $ n \equal{} 1994$ it cannot terminate.

How many positive integers smaller than $1000$ and not divisible by $5$ and by $7$ are there?