Several distinct real numbers are written on a blackboard. Peter wants to create an algebraic expression such that among its values there would be these and only these numbers. He may use any real numbers, brackets, signs $+, -, \times$ and a special sign $\pm$. Usage of $\pm$ is equivalent to usage of $+$ and $-$ in all possible combinations. For instance, the expression $5 \pm 1$ results in $\{4, 6\}$, while $(2 \pm 0.5) \pm 0.5$ results in $\{1, 2, 3\}$. Can Peter construct an expression if the numbers on the blackboard are : (a) $1, 2, 4$ ? [i]($2$ points)[/i] (b) any $100$ distinct real numbers ? [i]($6$ points)[/i]

Find all pairs of positive integers $(m,n)$ such that one can partition a $m\times n$ board with $1\times 2$ or $2\times 1$ dominoes and draw one of the diagonals on each of the dominos so that none of the diagonals share endpoints.

Given any integer $n\geq 3$. A finite series is called $n$-series if it satisfies the following two conditions $1)$ It has at least $3$ terms and each term of it belongs to $\{ 1,2,...,n\}$ $2)$ If series has $m$ terms $a_1,a_2,...,a_m$ then $(a_{k+1}-a_k)(a_{k+2}-a_k)<0$ for all $k=1,2,...,m-2$ How many $n$-series are there $?$

In the rectangular coordinate system every point with integer coordinates is called laticeal point. Let $P_n(n, n + 5)$ be a laticeal point and denote by $f(n)$ the number of laticeal points on the open segment $(OP_n)$, where the point $0(0,0)$ is the coordinates system origine. Calculate the number $f(1) +f(2) + f(3) + ...+ f(2002) + f(2003)$.

Let $A$ be a finite set, and $A_1,A_2,\cdots, A_n$ are subsets of $A$ with the following conditions: (1) $|A_1|=|A_2|=\cdots=|A_n|=k$, and $k>\frac{|A|}{2}$; (2) for any $a,b\in A$, there exist $A_r,A_s,A_t\,(1\leq r<s<t\leq n)$ such that $a,b\in A_r\cap A_s\cap A_t$; (3) for any integer $i,j\, (1\leq i<j\leq n)$, $|A_i\cap A_j|\leq 3$. Find all possible value(s) of $n$ when $k$ attains maximum among all possible systems $(A_1,A_2,\cdots, A_n,A)$.

* A bus route has $14$ stops (counting the first and the last). A bus cannot carry more than $25$ passengers. We assume that a passenger takes a bus from $A$ to $B$ if (s)he enters the bus at $A$ and gets off at $B$. Prove that for any bus route: a) there are $8$ distinct stops $A_1, B_1, A_2, B_2, A_3, B_3, A_4, B_4$ such that no passenger rides from $A_k$ to $B_k$ for all $k = 1, 2, 3, 4$ (#) b) there might not exist $10$ distinct stops $A_1, B_1, . . . , A_5, B_5$ with property (#).

For $n$ people, if it is known that (a) there exist two people knowing each other among any three people, and (b) there exist two people not knowing each other among any four people. Find the maximum of $n$. Here, we assume that if $A$ knows $B$, then $B$ knows $A$.

Given positive integers $n,k$, $n \ge 2$. Find the minimum constant $c$ satisfies the following assertion: For any positive integer $m$ and a $kn$-regular graph $G$ with $m$ vertices, one could color the vertices of $G$ with $n$ different colors, such that the number of monochrome edges is at most $cm$.

The $ 2^N$ vertices of the $ N$-dimensional hypercube $ \{0,1\}^N$ are labelled with integers from $ 0$ to $ 2^N \minus{} 1$, by, for $ x \equal{} (x_1,x_2,\ldots ,x_N)\in \{0,1\}^N$, \[v(x) \equal{} \sum_{k \equal{} 1}^{N}x_k2^{k \minus{} 1}.\] For which values $ n$, $ 2\leq n \leq 2^n$ can the vertices with labels in the set $ \{v|0\leq v \leq n \minus{} 1\}$ be connected through a Hamiltonian circuit, using edges of the hypercube only? [i]E. Bazavan & C. Talau[/i]

Let $\Omega$ be a circle in a plane. $2022$ pink points and $2565$ blue points are placed inside $\Omega$ such that no point has two colors and no two points are collinear with the center of $\Omega$. Prove that there exists a sector of $\Omega$ such that the angle at the center is acute and the number of blue points inside the sector is greater than the number of pink points by exactly $100$. (Note: such sector may contain no pink points.)

Prove that the set of positive integers cannot be partitioned into three nonempty subsets such that, for any two integers $x,y$ taken from two different subsets, the number $x^2-xy+y^2$ belongs to the third subset.

Let $ p\ge 2 $ be a natural number that divides $ \binom{p}{k} , $ for any natural number $ k $ smaller than $ p. $ Prove that: [b]a)[/b] $ p $ is prime. [b]b)[/b] $ p^2 $ divides $ -2+\binom{2p}{p} . $

There are $15$ people attending math team: $12$ students and $3$ captains. One of the captains brings $33$ identical snacks. A nonnegative number of names (students and/or captains) are written on the NO SNACK LIST. At the end of math team, all students each get n snacks, and all captains get $n +1$ snacks, unless the person’s name is written on the board. After everyone’s snacks are distributed, there are none left. Find the number of possible integer values of $n$.

Let $f$ be a function on non-negative integers defined as follows $$f(2n)=f(f(n))~~~\text{and}~~~f(2n+1)=f(2n)+1$$ [b](a)[/b] If $f(0)=0$ , find $f(n)$ for every $n$. [b](b)[/b] Show that $f(0)$ cannot equal $1$. [b](c)[/b] For what non-negative integers $k$ (if any) can $f(0)$ equal $2^k$ ?

Show that there exists a set $ A$ of positive integers with the following property: for any infinite set $ S$ of primes, there exist [i]two[/i] positive integers $ m$ in $ A$ and $ n$ not in $ A$, each of which is a product of $ k$ distinct elements of $ S$ for some $ k \geq 2$.

On a line are given $n$ blue and $n$ red points. Prove that the sum of distances between pairs of points of the same color does not exceed the sum of distances between pairs of points of different colors. (O. Musin)

Let $n\in\mathbb{Z}^+$. Arrange $n$ students $A_1,A_2,...,A_n$ on a circle such that the distances between them are.equal. They each receives a number of candies such that the total amount of candies is $m\geq n$. A configuration is called [i]balance[/i] if for an arbitrary student $A_i$, there will always be a regular polygon taking $A_i$ as one of its vertices, and every student standing at the vertices of this polygon has an equal number of candies. a) Given $n$, find the least $m$ such that we can create a balance configuration. b) In a [i]move[/i], a student can give a candy to the student standing next to him (no matter left or right) on one condition that the receiver has less candies than the giver. Prove that if $n$ is the product of at most $2$ prime numbers and $m$ satisfies the condition in a), then no matter how we distribute the candies at the beginning, one can always create a balance configuration after a finite number of moves.

A triangular pie has the same shape as its box, except that they are mirror images of each other. We wish to cut the pie in two pieces which can t together in the box without turning either piece over. How can this be done if [list][b](a)[/b] one angle of the triangle is three times as big as another; [b](b)[/b] one angle of the triangle is obtuse and is twice as big as one of the acute angles?[/list]

A Magician has a deck of $52$ cards. Spectators want to know the order of cards in the deck(without specifying face-up or face-down). They are allowed to ask the questions “How many cards are there between such-and-such card and such-and-such card?” One of the spectators knows the card order. Find the minimal number of questions he needs to ask to be sure that the other spectators can learn the card order. (9)

A table with $m$ rows and $n$ columns is given. At any move one chooses some empty cells such that any two of them lie in different rows and columns, puts a white piece in any of those cells and then puts a black piece in the cells whose rows and columns contain white pieces. The game is over if it is not possible to make a move. Find the maximum possible number of white pieces that can be put on the table.

2025 IMO leaders are discussing $100$ problems in a meeting. For each $i = 1, 2,\ldots , 100$, each leader will talk about the $i$-th problem for $i$-th minutes. The chair can assign one leader to talk about a problem of his choice, but he would have to wait for the leader to complete the entire talk of that problem before assigning the next leader and problem. The next leader can be the same leader. The next problem can be a different problem. Each leader’s longest idle time is the longest consecutive time that he is not talking. Find the minimum positive integer $T$ so that the chair can ensure that the longest idle time for any leader does not exceed $T$. [i]Proposed by usjl[/i]

There are $20$ pupils in the Backwoods school. Any two of them have a common grandfather. Prove that there exist $14$ pupils all of whom have a common grandfather. (AV Shapovalov)

You are given a square $n \times n$. The centers of some of some $m$ of its $1\times 1$ cells are marked. It turned out that there is no convex quadrilateral with vertices at these marked points. For each positive integer $n \geq 3$, find the largest value of $m$ for which it is possible. [i]Proposed by Oleksiy Masalitin, Fedir Yudin[/i]

$N$ straight lines are drawn on a plane. The $N$ lines can be partitioned into set of lines such that if a line $l$ belongs to a partition set then all lines parallel to $l$ make up the rest of that set. For each $n>=1$,let $a_n$ denote the number of partition sets of size $n$. Now that $N$ lines intersect at certain points on the plane. For each $n>=2$ let $b_n$ denote the number of points that are intersection of exactly $n$ lines. Show that $\sum_{n>= 2}(a_n+b_n)$$\binom{n}{2}$ $=$ $\binom{N}{2}$

[u]Round 9[/u] [b]p25.[/b] Find the largest prime factor of $1031301$. [b]p26.[/b] Let $ABCD$ be a trapezoid such that $AB \parallel CD$, $\angle ABC = 90^o$ , $AB = 5$, $BC = 20$, $CD = 15$. Let $X$, $Y$ be the intersection of the circle with diameter $BC$ and segment $AD$. Find the length of $XY$. [b]p27.[/b] A string consisting of $1$’s, $2$’s, and $3$’s is said to be a superpermutation of the string $123$ if it contains every permutation of $123$ as a contiguous substring. Find the smallest possible length of such a superpermutation. [u]Round 10[/u] [b]p28.[/b] Suppose that we have a function $f (x) = x^3 -3x^2 +3x$, and for all $n \ge 1$, $f^n(x)$ is defined by the function $f$ applied $n$ times to $x$. Find the remainder when $f^5(2019)$ is divided by $100$. [b]p29.[/b] A function $f : {1,2, . . . ,10} \to {1,2, . . . ,10}$ is said to be happy if it is a bijection and for all $n \in {1,2, . . . ,10}$, $|n - f (n)| \le 1$. Compute the number of happy functions. [b]p30.[/b] Let $\vartriangle LMN$ have side lengths $LM = 15$, $MN = 14$, and $NL = 13$. Let the angle bisector of $\angle MLN$ meet the circumcircle of $\vartriangle LMN$ at a point $T \ne L$. Determine the area of $\vartriangle LMT$ . [u]Round 11[/u] [b]p31.[/b] Find the value of $$\sum_{d|2200} \tau (d),$$ where $\tau (n)$ denotes the number of divisors of $n$, and where $a|b$ means that $\frac{b}{a}$ is a positive integer. [b]p32.[/b] Let complex numbers $\omega_1,\omega_2, ...,\omega_{2019}$ be the solutions to the equation $x^{2019}-1 = 0$. Evaluate $$\sum^{2019}_{i=1} \frac{1}{1+ \omega_i}.$$ [b]p33.[/b] Let $M$ be a nonnegative real number such that $x^{x^{x^{...}}}$ diverges for all $x >M$, and $x^{x^{x^{...}}}$ converges for all $0 < x \le M$. Find $M$. [u]Round 12[/u] [b]p34.[/b] Estimate the number of digits in ${2019 \choose 1009}$. If your estimate is $E$ and the actual value is $A$, your score for this problem will be $$\max \, \left( 0, \left \lfloor 15-10 \cdot \left|\log_{10} \left( \frac{A}{E} \right) \right| \right \rfloor \right).$$ [b]p35.[/b] You may submit any integer $E$ from $1$ to $30$. Out of the teams that submit this problem, your score will be $$\frac{E}{2 \, (the\,\, number\,\, of\,\, teams\,\, who\,\, chose\,\, E)}$$ [b]p36.[/b] We call a $m \times n$ domino-tiling a configuration of $2\times 1$ dominoes on an $m\times n$ cell grid such that each domino occupies exactly $2$ cells of the grid and all cells of the grid are covered. How many $8 \times 8$ domino-tilings are there? If your estimate is $E$ and the actual value is $A$, your score for this problem will be $$\max \, \left( 0, \left \lfloor 15-10 \cdot \left|\log_{10} \left( \frac{A}{E} \right) \right| \right \rfloor \right).$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166016p28809598]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3166019p28809679]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].