A rectangle is tiled with rectangles of size $6\times 1$. Prove that one of its side lengths is divisible by $6$.

There are 2000 cities in the country. Every city is connected by non-stop two-way airlines with some other cities, and for each city, the number of airlines originating from it is a factor of two. (i.e. $1$, $2$, $4$, $8$, $...$). For each city $A$, the statistician calculated the number routes with no more than one transfer connecting $A$ with other cities, and then summed up the results for all $2000$ cities. He got $100,000$. Prove that the statistician was wrong.

Mom brought Andriy and Olesya $4$ balls with the numbers $1, 2, 3$ and $4$ written on them (one on each ball). She held $2$ balls in each hand and did not know which numbers were written on the balls in each hand. The mother asked Andriy to take a ball with a higher number from each hand, and then to keep the ball with the lower number from the two balls he took. After that, she asked Olesya to take two other balls, and out of these two, keep the ball with the higher number. Does the mother know with certainty, which child has the ball with the higher number? [i]Proposed by Bogdan Rublov[/i]

A rectangle is divided into several similar isosceles triangles. Determine the possible values of the angles of the triangles.

[i](7 pts)[/i] Fix a positive integer $n$. Pick $4n$ equally spaced points on a circle and color them alternately blue and red. You use $n$ blue chords to pair the $2n$ blue points, and you use $n$ red chords to pair the $2n$ red points. If some blue chord intersects some other red chord, then such a pair of chords is called a "good pair." (a) [i](1 pts)[/i] For the case $n = 3$, explicitly show that there are at least $3$ distinct ways to pair the $2n$ blue points and the $2n$ red points such that there are a total of $3$ good pairs ($2$ configurations of chord pairings are [i]not[/i] considered distinct if one of them can be "rotated" to the other). (b) [i](6 pts)[/i] Now suppose that $n$ is arbitrary. Find, with proof, the minimum number of good pairs under all possible configurations of chord pairings.

[b]p1.[/b] Solve: $INK + INK + INK + INK + INK + INK = PEN$ ($INK$ and $PEN$ are $3$-digit numbers, and different letters stand for different digits). [b]p2. [/b]Two people play a game. They put $3$ piles of matches on the table: the first one contains $1$ match, the second one $3$ matches, and the third one $4$ matches. Then they take turns making moves. In a move, a player may take any nonzero number of matches FROM ONE PILE. The player who takes the last match from the table loses the game. a) The player who makes the first move can win the game. What is the winning first move? b) How can he win? (Describe his strategy.) [b]p3.[/b] The planet Naboo is under attack by the imperial forces. Three rebellion camps are located at the vertices of a triangle. The roads connecting the camps are along the sides of the triangle. The length of the first road is less than or equal to $20$ miles, the length of the second road is less than or equal to $30$ miles, and the length of the third road is less than or equal to $45$ miles. The Rebels have to cover the area of this triangle with a defensive field. What is the maximal area that they may need to cover? [b]p4.[/b] Money in Wonderland comes in $\$5$ and $\$7$ bills. What is the smallest amount of money you need to buy a slice of pizza that costs $\$ 1$ and get back your change in full? (The pizza man has plenty of $\$5$ and $\$7$ bills.) For example, having $\$7$ won't do, since the pizza man can only give you $\$5$ back. [b]p5.[/b] (a) Put $5$ points on the plane so that each $3$ of them are vertices of an isosceles triangle (i.e., a triangle with two equal sides), and no three points lie on the same line. (b) Do the same with $6$ points. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Oriol has a finite collection of cards, each one with a positive integer written on it. We say the collection is $n$-[i]complete[/i] if for any integer $k$ from $1$ to $n$ (inclusive), he can choose some cards such that the sum of the numbers on them is exactly $k$. Suppose that Oriol's collection is $n$-complete, but it stops being $n$-complete if any card is removed from it. What is the maximum possible sum of the numbers on all the cards? [i]Proposed by Ariel García[/i]

Prove that for any natural number $ m $ there exists a natural number $ n $ such that any $ n $ distinct points on the plane can be partitioned into $ m $ non-empty sets whose [i]convex hulls[/i] have a common point. The [i] convex hull [/i] of a finite set $ X $ of points on the plane is the set of points lying inside or on the boundary of at least one convex polygon with vertices in $ X $, including degenerate ones, that is, the segment and the point are considered convex polygons. No three vertices of a convex polygon are collinear. The polygon contains its border.

The number 2021 is fantabulous. For any positive integer $m$, if any element of the set $\{m, 2m+1, 3m\}$ is fantabulous, then all the elements are fantabulous. Does it follow that the number $2021^{2021}$ is fantabulous?

There are some real numbers on the board (at least two). In every step we choose two of them, for example $a$ and $b$, and then we replace them with $\frac{ab}{a+b}$. We continue until there is one number. Prove that the last number does not depend on which order we choose the numbers to erase.

Let $n\geq 2$ be a given integer. Initially, we write $n$ sets on the blackboard and do a sequence of moves as follows: choose two sets $A$ and $B$ on the blackboard such that none of them is a subset of the other, and replace $A$ and $B$ by $A\cap B$ and $A\cup B$. This is called a $\textit{move}$. Find the maximum number of moves in a sequence for all possible initial sets.

There are nine points in the data square, of which no three are collinear. Prove that three of them are vertices of a triangle with an area not exceeding $ \frac{1}{8} $ the area of a square.

In a club, some pairs of members are friends. Given an integer $k \geqslant 3$, we say a club is $k$-[i]friendly[/i] if, in any group of $k$ members, they can be seated at a round table such that each pair of neighbors are friends. [list=a] [*]Prove that if a club is $6$-friendly, then it is also $7$-friendly. [*]Is it true that if a club is $9$-friendly, then it is also $10$-friendly? [/list]

Let $n$ be an even positive integer. A sequence of $n$ real numbers is called complete if for every integer $m$ with $1 \leq m \leq n$ either the sum of the first $m$ terms of the sum or the sum of the last $m$ terms is integral. Determine the minimum number of integers in a complete sequence of $n$ numbers.

When counting from $3$ to $201$, $53$ is the $51^{\text{st}}$ number counted. When counting backwards from $201$ to $3$, $53$ is the $n^{\text{th}}$ number counted. What is $n$? $\textbf{(A) }146\qquad \textbf{(B) } 147\qquad\textbf{(C) } 148\qquad\textbf{(D) }149\qquad\textbf{(E) }150$

[b]p1.[/b] Give a fake proof that $0 = 1$ on the back of this page. The most convincing answer to this question at this test site will receive a point. [b]p2.[/b] It is often said that once you assume something false, anything can be derived from it. You may assume for this question that $0 = 1$, but you can only use other statements if they are generally accepted as true or if your prove them from this assumption and other generally acceptable mathematical statements. With this in mind, on the back of this page prove that every number is the same number. [b]p3.[/b] Suppose you write out all integers between $1$ and $1000$ inclusive. (The list would look something like $1$, $2$, $3$, $...$ , $10$, $11$, $...$ , $999$, $1000$.) Which digit occurs least frequently? [b]p4.[/b] Pick a real number between $0$ and $1$ inclusive. If your response is $r$ and the standard deviation of all responses at this site to this question is $\sigma$, you will receive $r(1 - (r - \sigma)^2)$ points. [b]p5.[/b] Find the sum of all possible values of $x$ that satisfy $243^{x+1} = 81^{x^2+2x}$. [b]p6.[/b] How many times during the day are the hour and minute hands of a clock aligned? [b]p7.[/b] A group of $N + 1$ students are at a math competition. All of them are wearing a single hat on their head. $N$ of the hats are red; one is blue. Anyone wearing a red hat can steal the blue hat, but in the process that person’s red hat disappears. In fact, someone can only steal the blue hat if they are wearing a red hat. After stealing it, they would wear the blue hat. Everyone prefers the blue hat over a red hat, but they would rather have a red hat than no hat at all. Assuming that everyone is perfectly rational, find the largest prime $N$ such that nobody will ever steal the blue hat. [b]p8.[/b] On the back of this page, prove there is no function f$(x)$ for which there exists a (finite degree) polynomial $p(x)$ such that $f(x) = p(x)(x + 3) + 8$ and $f(3x) = 2f(x)$. [b]p9.[/b] Given a cyclic quadrilateral $YALE$ with $Y A = 2$, $AL = 10$, $LE = 11$, $EY = 5$, what is the area of $YALE$? [b]p10.[/b] About how many pencils are made in the U.S. every year? If your answer to this question is $p$, and our (good) estimate is $\rho$, then you will receive $\max(0, 1 -\frac 12 | \log_{10}(p) - \log_{10}(\rho)|)$ points. [b]p11.[/b] The largest prime factor of $520, 302, 325$ has $5$ digits. What is this prime factor? [b]p12.[/b] The previous question was on the individual round from last year. It was one of the least frequently correctly answered questions. The first step to solving the problem and spotting the pattern is to divide $520, 302, 325$ by an appropriate integer. Unfortunately, when solving the problem many people divide it by $n$ instead, and then they fail to see the pattern. What is $n$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]Problem 3.[/b] Let $n\geq 3$ is given natural number, and $M$ is the set of the first $n$ primes. For any nonempty subset $X$ of $M$ with $P(X)$ denote the product of its elements. Let $N$ be a set of the kind $\ds\frac{P(A)}{P(B)}$, $A\subset M, B\subset M, A\cap B=\emptyset$ such that the product of any 7 elements of $N$ is integer. What is the maximal number of elements of $N$? [i]Alexandar Ivanov[/i]

A school has three senior classes of $M$ students each. Every student knows at least $\frac{3}{4}M$ people in each of the other two classes. Prove that the school can send $M$ non-intersecting teams to the olympiad so that each team consists of $3$ students from different classes who know each other. [i]Proposed by C. Magyar, R. Martin[/i]

Consider a regular $2020$-gon circumscribed into a circle of radius $ 1$. Given three vertices of this polygon such that they form an isosceles triangle, let $X$ be the expected area of the isosceles triangle they create. $X$ can be written as $\frac{1}{m \tan((2\pi)/n)}$ where $m$ and $n$ are integers. Compute $m + n$.

A convex $2004$-sided polygon $P$ is given such that no four vertices are cyclic. We call a triangle whose vertices are vertices of $P$ thick if all other $2001$ vertices of $P$ lie inside the circumcircle of the triangle, and thin if they all lie outside its circumcircle. Prove that the number of thick triangles is equal to the number of thin triangles.

Andriyko has rectangle desk and a lot of stripes that lie parallel to sides of the desk. For every pair of stripes we can say that first of them is under second one. In desired configuration for every four stripes such that two of them are parallel to one side of the desk and two others are parallel to other side, one of them is under two other stripes that lie perpendicular to it. Prove that Andriyko can put stripes one by one such way that every next stripe lie upper than previous and get desired configuration. [i]Proposed by Denys Smirnov[/i]

Tomek invited to a remote birthday part $11$ of his friends who will join the meeting one by one. Tomek chose the guests in such a way that, regardless of the order in which they will join, always the newcomer knew at least half of the people already present, including Tomek. Prove that among of invited guests, there is one who knows all of Tom's other $10$ friends. Caution: We assume that if person A knows person $B$, then $B$ also knows $A$. [hide=original wording]Tomek zaprosił na zdalne przyjęcie urodzinowe 11 swoich znajomych, którzy kolejno będą dołączać do spotkania. Tomek dobrał gości w taki sposób, aby niezależnie od kolejności w jakiej będą dołączać, zawsze nowo przybyła osoba znała co najmniej połowę już obecnych osób, wliczając Tomka. Wykaż, że wśród zaproszonych gości istnieje taki, który zna wszystkich pozostałych 10 znajomych Tomka. Uwaga: Przyjmujemy, że jeśli osoba A zna osobę B, to również B zna A.[/hide]

Given a circle and $2006$ points lying on this circle. Albatross colors these $2006$ points in $17$ colors. After that, Frankinfueter joins some of the points by chords such that the endpoints of each chord have the same color and two different chords have no common points (not even a common endpoint). Hereby, Frankinfueter intends to draw as many chords as possible, while Albatross is trying to hinder him as much as he can. What is the maximal number of chords Frankinfueter will always be able to draw?

In a $m\times n$ chessboard ($m,n\ge 2$), some dominoes are placed (without overlap) with each domino covering exactly two adjacent cells. Show that if no more dominoes can be added to the grid, then at least $2/3$ of the chessboard is covered by dominoes. [i]Proposed by DVDthe1st, mzy and jjax[/i]

Some marbles are distributed over $2n + 1$ bags. Suppose that, whichever bag is removed, it is possible to divide the remaining bags into two groups of $n$ bags such that the number of marbles in each group is the same. Prove that all the bags contain the same number of marbles.