The checkered plane is painted black and white, after a chessboard fashion. A polygon $\Pi$ of area $S$ and perimeter $P$ consists of some of these unit squares (i.e., its sides go along the borders of the squares). Prove the polygon $\Pi$ contains not more than $\dfrac {S} {2} + \dfrac {P} {8}$, and not less than $\dfrac {S} {2} - \dfrac {P} {8}$ squares of a same color. (Alexander Magazinov)

A class consists of $7$ boys and $13$ girls. During the first three months of the school year, each boy has communicated with each girl at least once. Prove that there exist two boys and two girls such that both boys communicated with both girls first time in the same month.

[u]Round 1[/u] [b]p1.[/b] Alex is on a week-long mining quest. Each morning, she mines at least $1$ and at most $10$ diamonds and adds them to her treasure chest (which already contains some diamonds). Every night she counts the total number of diamonds in her collection and finds that it is divisible by either $22$ or $25$. Show that she miscounted. [b]p2.[/b] Hermione set out a row of $11$ Bertie Bott’s Every Flavor Beans for Ron to try. There are $5$ chocolateflavored beans that Ron likes and $6$ beans flavored like earwax, which he finds disgusting. All beans look the same, and Hermione tells Ron that a chocolate bean always has another chocolate bean next to it. What is the smallest number of beans that Ron must taste to guarantee he finds a chocolate one? [b]p3.[/b] There are $101$ pirates on a pirate ship: the captain and $100$ crew. Each pirate, including the captain, starts with $1$ gold coin. The captain makes proposals for redistributing the coins, and the crew vote on these proposals. The captain does not vote. For every proposal, each crew member greedily votes “yes” if he gains coins as a result of the proposal, “no” if he loses coins, and passes otherwise. If strictly more crew members vote “yes” than “no,” the proposal takes effect. The captain can make any number of proposals, one after the other. What is the largest number of coins the captain can accumulate? [b]p4.[/b] There are $100$ food trucks in a circle and $10$ gnomes who sample their menus. For the first course, all the gnomes eat at different trucks. For each course after the first, gnome #$1$ moves $1$ truck left or right and eats there; gnome #$2$ moves $2$ trucks left or right and eats there; ... gnome #$10$ moves $10$ trucks left or right and eats there. All gnomes move at the same time. After some number of courses, each food truck had served at least one gnome. Show that at least one gnome ate at some food truck twice. [b]p5.[/b] The town of Lumenville has $100$ houses and is preparing for the math festival. The Tesla wiring company lays lengths of power wire in straight lines between the houses so that power flows between any two houses, possibly by passing through other houses.The Edison lighting company hangs strings of lights in straight lines between pairs of houses so that each house is connected by a string to exactly one other. Show that however the houses are arranged, the Edison company can always hang their strings of lights so that the total length of the strings is no more than the total length of the power wires the Tesla company used. [img]https://cdn.artofproblemsolving.com/attachments/9/2/763de9f4138b4dc552247e9316175036c649b6.png[/img] [u]Round 2[/u] [b]p6.[/b] What is the largest number of zeros that could appear at the end of $1^n + 2^n + 3^n + 4^n$, where n can be any positive integer? [b]p7.[/b] A tennis academy has $2023$ members. For every group of 1011 people, there is a person outside of the group who played a match against everyone in it. Show there is someone who has played against all $2022$ other members. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] We define $a \oplus b = \frac{ab}{a+b}$. Compute $(3 \oplus 5) \oplus (5 \oplus 4)$. [b]p2.[/b] Let $ABCD$ be a quadrilateral with $\angle A = 45^o$ and $\angle B = 45^o$. If $BC = 5\sqrt2$, $AD = 6\sqrt2$, and $AB = 18$, find the length of side $CD$. [b]p3.[/b] A positive real number $x$ satisfies the equation $x^2 + x + 1 + \frac{1}{x }+\frac{1}{x^2} = 10$. Find the sum of all possible values of $x + 1 + \frac{1}{x}$. [b]p4.[/b] David writes $6$ positive integers on the board (not necessarily distinct) from least to greatest. The mean of the first three numbers is $3$, the median of the first four numbers is $4$, the unique mode of the first five numbers is $5$, and the range of all 6 numbers is $6$. Find the maximum possible value of the product of David’s $6$ integers. [b]p5.[/b] Let $ABCD$ be a convex quadrilateral such that $\angle A = \angle B = 120^o$ and $\angle C = \angle D = 60^o$. There exists a circle with center $I$ which is tangent to all four sides of $ABCD$. If $IA \cdot IB \cdot IC \cdot ID = 240$, find the area of quadrilateral $ABCD$. [b]p6.[/b] The letters $EXETERMATH$ are placed into cells on an annulus as shown below. How many ways are there to color each cell of the annulus with red, blue, green, or yellow such that each letter is always colored the same color and adjacent cells are always colored differently? [img]https://cdn.artofproblemsolving.com/attachments/3/5/b470a771a5279a7746c06996f2bb5487c33ecc.png[/img] [b]p7.[/b] Let $ABCD$ be a square, and let $\omega$ be a quarter circle centered at $A$ passing through points $B$ and $D$. Points $E$ and $F$ lie on sides $BC$ and $CD$ respectively. Line $EF$ intersects $\omega$ at two points, $G$ and $H$. Given that $EG = 2$, $GH = 16$ and $HF = 9$, find the length of side $AB$. [b]p8.[/b] Let x be equal to $\frac{2022! + 2021!}{2020! + 2019! + 2018!}$ . Find the closest integer to $2\sqrt{x}$. [b]p9.[/b] For how many ordered pairs of positive integers $(m, n)$ is the absolute difference between $lcm(m, n)$ and $gcd(m, n)$ equal to $2023$? [b]p10.[/b] There are $2023$ distinguishable frogs sitting on a number line with one frog sitting on $i$ for all integers $i$ between $-1011$ and $1011$, inclusive. Each minute, every frog randomly jumps either one unit left or one unit right with equal probability. After $1011$ minutes, over all possible arrangements of the frogs, what is the average number of frogs sitting on the number $0$? [b]p11.[/b] Albert has a calculator initially displaying $0$ with two buttons: the first button increases the number on the display by one, and the second button returns the square root of the number on the display. Each second, he presses one of the two buttons at random with equal probability. What is the probability that Albert’s calculator will display the number $6$ at some point? [b]p12.[/b] For a positive integer $k \ge 2$, let $f(k)$ be the number of positive integers $n$ such that n divides $(n-1)!+k$. Find $$f(2) + f(3) + f(4) + f(5) + ... + f(100).$$ [b]p13.[/b] Mr. Atf has nine towers shaped like rectangular prisms. Each tower has a $1$ by $1$ base. The first tower as height $1$, the next has height $2$, up until the ninth tower, which has height $9$. Mr. Atf randomly arranges these $9$ towers on his table so that their square bases form a $3$ by $3$ square on the surface of his table. Over all possible solids Mr. Atf could make, what is the average surface area of the solid? [b]p14.[/b] Let $ABCD$ be a cyclic quadrilateral whose diagonals are perpendicular. Let $E$ be the intersection of $AC$ and $BD$, and let the feet of the altitudes from $E$ to the sides $AB$, $BC$, $CD$, $DA$ be $W, X, Y , Z$ respectively. Given that $EW = 2EY$ and $EW \cdot EX \cdot EY \cdot EZ = 36$, find the minimum possible value of $\frac{1}{[EAB]} +\frac{1}{[EBC]}+\frac{1}{[ECD]} +\frac{1}{[EDA]}$. The notation $[XY Z]$ denotes the area of triangle $XY Z$. [b]p15.[/b] Given that $x^2 - xy + y^2 = (x + y)^3$, $y^2 - yz + z^2 = (y + z)^3$, and $z^2 - zx + x^2 = (z + x)^3$ for complex numbers $x, y, z$, find the product of all distinct possible nonzero values of $x + y + z$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Noah has 8 species of animals to fit into 4 cages of the ark. He plans to put species in each cage. It turns out that, for each species, there are at most 3 other species with which it cannot share the accomodation. Prove that there is a way to assign the animals to their cages so that each species shares with compatible species.

For a positive integer $n$ let $t_n$ be the number of unordered triples of non-empty and pairwise disjoint subsets of a given set with $n$ elements. For example, $t_3 = 1$. Find a closed form formula for $t_n$ and determine the last digit of $t_{2022}$. (I also give here that $t_4 = 10$, for a reader to check his/her understanding of the problem statement.)

Ivan has a $n \times n$ board. He colors some of the squares black such that every black square has exactly two neighbouring square that are also black. Let $d_n$ be the maximum number of black squares possible, prove that there exist some real constants $a$, $b$, $c\ge 0$ such that; $$an^2-bn\le d_n\le an^2+cn.$$ [i]Proposed by Ivan Chan Kai Chin[/i]

For each positive integer $ n$, let $ f(n)$ denote the number of ways of representing $ n$ as a sum of powers of 2 with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance, $ f(4) \equal{} 4$, because the number 4 can be represented in the following four ways: 4; 2+2; 2+1+1; 1+1+1+1. Prove that, for any integer $ n \geq 3$ we have $ 2^{\frac {n^2}{4}} < f(2^n) < 2^{\frac {n^2}2}$.

We consider on a circle a finite number of real numbers with the sum strictly greater than $0$. Of all the sums that have as terms numbers on consecutive positions on the circle, let $S$ be the largest sum and $s$ the smallest sum. Show that $S + s> 0$.

Consider the set $A = \{1, 2, 3, ..., 2n - 1\}$, where $n \ge 2$ is a positive integer. We remove from the set $A$ at least $n - 1$ elements such that: • if $a \in A$ has been removed, and $2a \in A$, then $2a$ has also been removed, • if $a, b \in A (a \ne b)$ have been removed and $a + b \in A$, then $a + b$ has also been removed. Which numbers have to be removed such that the sum of the remaining numbers is maximum?

Every square on a $n\times n$ chessboard is colored with red, blue, or green. Each red square has at least one green square adjacent to it, each green square has at least one blue square adjacent to it, and each blue square has at least one red square adjacent to it. Let $R$ be the number of red squares. Prove that $\displaystyle \frac{n^2}{11} \le R \le \frac{2n^2}{3}$.

What is the maximum number of acute interior angles a non-overlapping planar $2017$-gon can have?

Let $a < b < c < d$ be real numbers and $(x,y, z,t)$ be any permutation of $a$,$b$, $c$ and $d$. What are the maximum and minimum values of the expression $$(x - y)^2 + (y- z)^2 + (z - t)^2 + (t - x)^2?$$

[b]p1.[/b] a) There are barrels weighing $1, 2, 3, 4, ..., 19, 20$ pounds. Is it possible to distribute them equally (by weight) into three trucks? b) The same question for barrels weighing $1, 2, 3, 4, ..., 9, 10$ pounds. [b]p2.[/b] There are apples and pears in the basket. If you add the same number of apples there as there are now pears (in pieces), then the percentage of apples will be twice as large as what you get if you add as many pears to the basket as there are now apples. What percentage of apples are in the basket now? [b]p3.[/b] What is the smallest number of integers from $1000$ to $1500$ that must be marked so that any number $x$ from $1000$ to $1500$ differs from one of the marked numbers by no more than $10\% $of the value of $x$? [b]p4.[/b] Draw a perpendicular from a given point to a given straight line, having a compass and a short ruler (the length of the ruler is significantly less than the distance from the point to the straight line; the compass reaches from the point to the straight line “with a margin”). [b]p5.[/b] There is a triangle on the chessboard (left figure). It is allowed to roll it around the sides (in this case, the triangle is symmetrically reflected relative to the side around which it is rolled). Can he, after a few steps, take the position shown in right figure? [img]https://cdn.artofproblemsolving.com/attachments/f/5/eeb96c92f30b837e7ed2cdf7cf77b0fbb8ceda.png[/img] [b]p6.[/b] The natural number $a$ is less than the natural number $b$. In this case, the sum of the digits of number $a$ is $100$ less than the sum of the digits of number $b$. Prove that between the numbers $ a$ and $b$ there is a number whose sum of digits is $43$ more than the sum of the digits of $a$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

A particle can travel at speeds up to $ \frac{2m}{s}$ along the $ x$-axis, and up to $ \frac{1m}{s}$ elsewhere in the plane. Provide a labelled sketch of the region which can be reached within one second by the particle starting at the origin.

On a $ 9\times 9$ board, divided into $1\times 1$ squares, pieces of the form Each piece covers exactly $3$ squares. (a) Starting from the empty board, what is the maximum number of pieces that can be placed? (b) Starting from the board with $3$ pieces already placed as shown in the diagram below, what is the maximum number of pieces that can be placed? [img]https://cdn.artofproblemsolving.com/attachments/d/4/3bd010828accb2d1811d49eb17fa69662ff60d.gif[/img]

Ayala and Barvaz play a game: Ayala initially gives Barvaz two $100\times100$ tables of positive integers, such that the product of numbers in each table is the same. In one move, Barvaz may choose a row or column in one of the tables, and change the numbers in it (to some positive integers), as long as the total product remains the same. Barvaz wins if after $N$ such moves, he manages to make the two tables equal to each other, and otherwise Ayala wins. a. For which values of $N$ does Barvaz have a winning strategy? b. For which values of $N$ does Barvaz have a winning strategy, if all numbers in Ayalah’s tables must be powers of $2$?

Submit a positive integer $n$ less than $10^5$. Let the sum of the valid submissions from all teams to this question be $S$. If you submit an invalid answer, you will receive $0$ points. Otherwise, your score will be $ \max \left(0,\lfloor 25 - \frac{|S'-n|}{10} \rfloor \right)$ , where $S'$ is the sum of the squares of the digits of $S$.

Each of the $48$ faces of eight $1\times 1\times 1$ cubes is randomly painted either blue or green. The probability that these eight cubes can then be assembled into a $2\times 2\times 2$ cube in a way so that its surface is solid green can be written $\frac{p^m}{q^n}$ , where $p$ and $q$ are prime numbers and $m$ and $n$ are positive integers. Find $p + q + m + n$.

There are $n$ points on a circle, such that each line segment connecting two points is either red or blue. $P_iP_j$ is red if and only if $P_{i+1} P_{j+1}$ is blue, for all distinct $i, j$ in $\left\{1, 2, ..., n\right\}$. (a) For which values of $n$ is this possible? (b) Show that one can get from any point on the circle to any other point, by doing a maximum of 3 steps, where one step is moving from a point to another point through a red segment connecting these points.

Let be given $n$ positive integer. Lets write with $a_n$ the number of positive integer pairs $(x,y)$ such that $x+y$ is even and $1\leq x\leq y\leq n$. Lets write with $b_n$ the number of positive integer pairs $(x,y)$ such that $x+y\leq n+1$ and $1\leq x\leq y\leq n$.

In chess, a knight placed on a chess board can move by jumping to an adjacent square in one direction (up, down, left, or right) then jumping to the next two squares in a perpendicular direction. We then say that a square in a chess board [i]can be attacked[/i] by a knight if the knight can end up on that square after a move. Thus, depending on where a knight is placed, it can attack as many as eight squares, or maybe even less. In a $10 \times 10$ chess board, what is the maximum number of knights that can be placed such that each square on the board can be attacked by at most one knight?

Each of the squares in a $50 \times 50$ square board is filled with a number from $1$ to $50$ so that each of the numbers $1,2, ..., 50$ appears exactly $50$ times. Prove that there is a row or a column containing at least $8$ distinct numbers.

A $7 \times 1$ board is completely covered by $m \times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $7 \times 1$ board in which all three colors are used at least once. For example, a $1 \times 1$ red tile followed by a $2 \times 1$ green tile, a $1 \times 1$ green tile, a $2 \times 1$ blue tile, and a $1 \times 1$ green tile is a valid tiling. Note that if the $2 \times 1$ blue tile is replaced by two $1 \times 1$ blue tiles, this results in a different tiling. Find the remainder when $N$ is divided by $1000$.

Let $n\geqslant 2$ be an integer. A [i]Welsh darts board[/i] is a disc divided into $2n$ equal sectors, half of them being red and the other half being white. Two Welsh darts boards are [i]matched[/i] if they have the same radius and they are superimposed so that each sector of the first board comes exactly over a sector of the second board. Suppose that two given Welsh darts boards can be matched so that more than half of the paurs of superimposed sectors have different colours. Prove that these Welsh darts boards can be matched so that at least $2\lfloor n/2\rfloor +2$ pairs of superimposed sectors have the same colour.