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]

Let $N$ be a positive integer, and consider an $N \times N$ grid. A [i]right-down path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell below the previous cell in the sequence. A [i]right-up path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell above the previous cell in the sequence. Prove that the cells of the $N \times N$ grid cannot be partitioned into less than $N$ right-down or right-up paths. For example, the following partition of the $5 \times 5$ grid uses $5$ paths. [asy] size(4cm); draw((5,-1)--(0,-1)--(0,-2)--(5,-2)--(5,-3)--(0,-3)--(0,-4)--(5,-4),gray+linewidth(0.5)+miterjoin); draw((1,-5)--(1,0)--(2,0)--(2,-5)--(3,-5)--(3,0)--(4,0)--(4,-5),gray+linewidth(0.5)+miterjoin); draw((0,0)--(5,0)--(5,-5)--(0,-5)--cycle,black+linewidth(2.5)+miterjoin); draw((0,-1)--(3,-1)--(3,-2)--(1,-2)--(1,-4)--(4,-4)--(4,-3)--(2,-3)--(2,-2),black+linewidth(2.5)+miterjoin); draw((3,0)--(3,-1),black+linewidth(2.5)+miterjoin); draw((1,-4)--(1,-5),black+linewidth(2.5)+miterjoin); draw((4,-3)--(4,-1)--(5,-1),black+linewidth(2.5)+miterjoin); [/asy] [i]Proposed by Zixiang Zhou, Canada[/i]

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

A square grid on the Euclidean plane consists of all points $(m,n)$, where $m$ and $n$ are integers. Is it possible to cover all grid points by an infinite family of discs with non-overlapping interiors if each disc in the family has radius at least $5$?

[b]p1.[/b] Consider a triangular grid: nodes of the grid are painted black and white. At a single step you are allowed to change colors of all nodes situated on any straight line (with the slope $0^o$ ,$60^o$, or $120^o$ ) going through the nodes of the grid. Can you transform the combination in the left picture into the one in the right picture in a finite number of steps? [img]https://cdn.artofproblemsolving.com/attachments/3/a/957b199149269ce1d0f66b91a1ac0737cf3f89.png[/img] [b]p2.[/b] Find $x$ satisfying $\sqrt{x\sqrt{x \sqrt{x ...}}} = \sqrt{2022}$ where it is an infinite expression on the left side. [b]p3.[/b] $179$ glasses are placed upside down on a table. You are allowed to do the following moves. An integer number $k$ is fixed. In one move you are allowed to turn any $k$ glasses . (a) Is it possible in a finite number of moves to turn all $179$ glasses into “bottom-down” positions if $k=3$? (b) Is it possible to do it if $k=4$? [b]p4.[/b] An interval of length $1$ is drawn on a paper. Using a compass and a simple ruler construct an interval of length $\sqrt{93}$. [b]p5.[/b] Show that $5^{2n+1} + 3^{n+2} 2^{n-1} $ is divisible by $19$ for any positive integer $n$. [b]p6.[/b] Solve the system $$\begin{cases} \dfrac{xy}{x+y}=1-z \\ \dfrac{yz}{y+z}=2-x \\ \dfrac{xz}{x+z}=2-y \end{cases}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n>1$ be a positive integer. Each cell of an $n\times n$ table contains an integer. Suppose that the following conditions are satisfied: [list=1] [*] Each number in the table is congruent to $1$ modulo $n$. [*] The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^2$. [/list] Let $R_i$ be the product of the numbers in the $i^{\text{th}}$ row, and $C_j$ be the product of the number in the $j^{\text{th}}$ column. Prove that the sums $R_1+\hdots R_n$ and $C_1+\hdots C_n$ are congruent modulo $n^4$.

A wizard kidnaps $31$ members from party $A$, $28$ members from party $B$, $23$ members from party $C$, and $19$ members from party $D$, keeping them isolated in individual rooms in his castle, where he forces them to work. Every day, after work, the kidnapped people can walk in the park and talk with each other. However, when three members of three different parties start talking with each other, the wizard reconverts them to the fourth party (there are no conversations with $4$ or more people involved). a) Find out whether it is possible that, after some time, all of the kidnapped people belong to the same party. If the answer is yes, determine to which party they will belong. b) Find all quartets of positive integers that add up to $101$ that if they were to be considered the number of members from the four parties, it is possible that, after some time, all of the kidnapped people belong to the same party, under the same rules imposed by the wizard.

Players $A$ and $B$ play a game with $N \geq 2012$ coins and $2012$ boxes arranged around a circle. Initially $A$ distributes the coins among the boxes so that there is at least $1$ coin in each box. Then the two of them make moves in the order $B,A,B,A,\ldots $ by the following rules: [b](a)[/b] On every move of his $B$ passes $1$ coin from every box to an adjacent box. [b](b)[/b] On every move of hers $A$ chooses several coins that were [i]not[/i] involved in $B$'s previous move and are in different boxes. She passes every coin to an adjacent box. Player $A$'s goal is to ensure at least $1$ coin in each box after every move of hers, regardless of how $B$ plays and how many moves are made. Find the least $N$ that enables her to succeed.

Let $ABCD$ be a cyclic quadrilateral in which internal angle bisectors $\angle ABC$ and $\angle ADC$ intersect on diagonal $AC$. Let $M$ be the midpoint of $AC$. Line parallel to $BC$ which passes through $D$ cuts $BM$ at $E$ and circle $ABCD$ in $F$ ($F \neq D$ ). Prove that $BCEF$ is parallelogram [i]Proposed by Steve Dinh[/i]

Let $n\ge 1$ be an odd integer. Alice and Bob play the following game, taking alternating turns, with Alice playing first. The playing area consists of $n$ spaces, arranged in a line. Initially all spaces are empty. At each turn, a player either • places a stone in an empty space, or • removes a stone from a nonempty space $s,$ places a stone in the nearest empty space to the left of $s$ (if such a space exists), and places a stone in the nearest empty space to the right of $s$ (if such a space exists). Furthermore, a move is permitted only if the resulting position has not occurred previously in the game. A player loses if he or she is unable to move. Assuming that both players play optimally throughout the game, what moves may Alice make on her first turn?

Let $ABC$ be a triangle and let $X$,$Y$,$Z$ be interior points on the sides $BC$, $CA$, $AB$, respectively. Show that the magnified image of the triangle $XYZ$ under a homothety of factor $4$ from its centroid covers at least one of the vertices $A$, $B$, $C$.

[b]p1.[/b] Consider a triangular grid: nodes of the grid are painted black and white. At a single step you are allowed to change colors of all nodes situated on any straight line (with the slope $0^o$ ,$60^o$, or $120^o$ ) going through the nodes of the grid. Can you transform the combination in the left picture into the one in the right picture in a finite number of steps? [img]https://cdn.artofproblemsolving.com/attachments/3/a/957b199149269ce1d0f66b91a1ac0737cf3f89.png[/img] [b]p2.[/b] Find $x$ satisfying $\sqrt{x\sqrt{x \sqrt{x ...}}} = \sqrt{2022}$ where it is an infinite expression on the left side. [b]p3.[/b] $179$ glasses are placed upside down on a table. You are allowed to do the following moves. An integer number $k$ is fixed. In one move you are allowed to turn any $k$ glasses . (a) Is it possible in a finite number of moves to turn all $179$ glasses into “bottom-down” positions if $k=3$? (b) Is it possible to do it if $k=4$? [b]p4.[/b] An interval of length $1$ is drawn on a paper. Using a compass and a simple ruler construct an interval of length $\sqrt{93}$. [b]p5.[/b] Show that $5^{2n+1} + 3^{n+2} 2^{n-1} $ is divisible by $19$ for any positive integer $n$. [b]p6.[/b] Solve the system $$\begin{cases} \dfrac{xy}{x+y}=1-z \\ \dfrac{yz}{y+z}=2-x \\ \dfrac{xz}{x+z}=2-y \end{cases}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are $100$ points on the circumference of a circle, arbitrarily labelled by $1,2,\ldots,100$. For each three points, call their triangle [b]clockwise[/b] if the increasing order of them is in clockwise order. Prove that it is impossible to have exactly $2017$ [b]clockwise[/b] triangles.

A frog begins at $P_0 = (0,0)$ and makes a sequence of jumps according to the following rule: from $P_n=(x_n,y_n)$, the frog jumps to $P_{n+1}$, which may be any of the points $(x_n+7, y_n+2)$, $(x_n+2,y_n+7)$, $(x_n-5, y_n-10)$, or $(x_n-10,y_n-5)$. There are $M$ points $(x,y)$ with $|x|+|y| \le 100$ that can be reached by a sequence of such jumps. Find the remainder when $M$ is divided by $1000$.