Is it true that any two rectangles of equal area can be placed in the plane such that any horizontal line intersecting at least one of them will also intersect the other, and the segments of intersection will be equal?

Each integer in $\{1, 2, 3, . . . , 2020\}$ is coloured in such a way that, for all positive integers $a$ and $b$ such that $a + b \leq 2020$, the numbers $a$, $b$ and $a + b$ are not coloured with three different colours. Determine the maximum number of colours that can be used. [i]Massimiliano Foschi, Italy[/i]

Ten gamblers start playing with the same amount of money. In turn they cast five dice. If the dice show a total of $n$, the player must pay each other player $\frac{1}{n}$ times the sum which that player owns at the moment. They throw and pay one after the other. At the $10^{\text{th}}$ round (i.e. after each player has cast the five die once), the dice shows a total of $12$ and after the payment it turns out that every player has exactly the same sum as he had in the beginning. Is it possible to determine the totals shown by the dice at the nine former rounds?

Every cell of table $4 \times 4$ is colored into white. It is permitted to place the cross (pictured below) on the table such that its center lies on the table (the whole figure does not need to lie on the table) and change colors of every cell which is covered into opposite (white and black). Find all $n$ such that after $n$ steps it is possible to get the table with every cell colored black.

[hide=Intro]The answers to each of the ten questions in this section are integers containing only the digits $ 1$ through $ 8$, inclusive. Each answer can be written into the grid on the answer sheet, starting from the cell with the problem number, and continuing across or down until the entire answer has been written. Answers may cross dark lines. If the answers are correctly filled in, it will be uniquely possible to write an integer from $ 1$ to $ 8$ in every cell of the grid, so that each number will appear exactly once in every row, every column, and every marked $2$ by $4$ box. You will get $7$ points for every correctly filled answer, and a $15$ point bonus for filling in every gridcell. It will help to work back and forth between the grid and the problems, although every problem is uniquely solvable on its own. Please write clearly within the boxes. No points will be given for a cell without a number, with multiple numbers, or with illegible handwriting.[/hide] [img]https://cdn.artofproblemsolving.com/attachments/9/b/f4db097a9e3c2602b8608be64f06498bd9d58c.png[/img] [b]1 ACROSS:[/b] Jack puts $ 10$ red marbles, $ 8$ green marbles and 4 blue marbles in a bag. If he takes out $11$ marbles, what is the expected number of green marbles taken out? [b]2 DOWN:[/b] What is the closest integer to $6\sqrt{35}$ ? [b]3 ACROSS: [/b]Alan writes the numbers $ 1$ to $64$ in binary on a piece of paper without leading zeroes. How many more times will he have written the digit $ 1$ than the digit $0$? [b]4 ACROSS:[/b] Integers a and b are chosen such that $-50 < a, b \le 50$. How many ordered pairs $(a, b)$ satisfy the below equation? $$(a + b + 2)(a + 2b + 1) = b$$ [b]5 DOWN: [/b]Zach writes the numbers $ 1$ through $64$ in binary on a piece of paper without leading zeroes. How many times will he have written the two-digit sequence “$10$”? [b]6 ACROSS:[/b] If you are in a car that travels at $60$ miles per hour, $\$1$ is worth $121$ yen, there are $8$ pints in a gallon, your car gets $10$ miles per gallon, a cup of coffee is worth $\$2$, there are 2 cups in a pint, a gallon of gas costs $\$1.50$, 1 mile is about $1.6$ kilometers, and you are going to a coffee shop 32 kilometers away for a gallon of coffee, how much, in yen, will it cost? [b]7 DOWN:[/b] Clive randomly orders the letters of “MIXING THE LETTERS, MAN”. If $\frac{p}{m^nq}$ is the probability that he gets “LMT IS AN EXTREME THING” where p and q are odd integers, and $m$ is a prime number, then what is $m + n$? [b]8 ACROSS:[/b] Joe is playing darts. A dartboard has scores of $10, 7$, and $4$ on it. If Joe can throw $12$ darts, how many possible scores can he end up with? [b]9 ACROSS:[/b] What is the maximum number of bounded regions that $6$ overlapping ellipses can cut the plane into? [b]10 DOWN:[/b] Let $ABC$ be an equilateral triangle, such that $A$ and $B$ both lie on a unit circle with center $O$. What is the maximum distance between $O$ and $C$? Write your answer be in the form $\frac{a\sqrt{b}}{c}$ where $b$ is not divisible by the square of any prime, and $a$ and $c$ share no common factor. What is $abc$ ? PS. You had better use hide for answers.

$n \ge 4$ real numbers are arranged in a circle. It turned out that for any four consecutive numbers $a, b, c, d$, that lie on the circle in this order, holds $a+d = b+c$. For which $n$ does it follow that all numbers on the circle are equal? [i]Proposed by Oleksiy Masalitin[/i]

Vasya has $100$ cards of $3$ colors, and there are not more than $50$ cards of same color. Prove that he can create $10\times 10$ square, such that every cards of same color have not common side.

We are given a row of $n\geq7$ tiles. In the leftmost 3 tiles, there is a white piece each, and in the rightmost 3 tiles, there is a black piece each. The white and black players play in turns (the white starts). In each move, a player may take a piece of their color, and move it to an adjacent tile, so long as it's not occupied by a piece of the [u]same color[/u]. If the new tile is empty, nothing happens. If the tile is occupied by a piece of the [u]opposite color[/u], both pieces are destroyed (both white and black). The player who destroys the last two pieces wins the game. Which player has a winning strategy, and what is it? (The answer may depend on $n$)

Consider increasing integer sequences with elements from $1,\ldots,10^6$. Such a sequence is [i]Adriatic[/i] if its first element equals 1 and if every element is at least twice the preceding element. A sequence is [i]Tyrrhenian[/i] if its final element equals $10^6$ and if every element is strictly greater than the sum of all preceding elements. Decide whether the number of Adriatic sequences is smaller than, equal to, or greater than the number of Tyrrhenian sequences. (Proposed by Gerhard Woeginger, Austria)

Find the smallest natural number $n$ that the following statement holds : Let $A$ be a finite subset of $\mathbb R^{2}$. For each $n$ points in $A$ there are two lines including these $n$ points. All of the points lie on two lines.

The city of Bridge Village has some highways. Highways are closed curves that have intersections with each other or themselves in $4$-way crossroads. Mr.Bridge Lover, mayor of the city, wants to build a bridge on each crossroad in order to decrease the number of accidents. He wants to build the bridges in such a way that in each highway, cars pass above a bridge and under a bridge alternately. By knowing the number of highways determine that this action is possible or not. [i]Proposed by Erfan Salavati[/i]

Let $S$ be a set of $N \ge 3$ points in the plane. Assume that no $3$ points in $S$ are collinear. The segments with both endpoints in $S$ are colored in two colors. Prove that there is a set of $N - 1$ segments of the same color which don't intersect except in their endpoints such that no subset of them forms a polygon with positive area.

[u]Round 1 [/u] [b]p1.[/b] Ben the bear has an algorithm he runs on positive integers- each second, if the integer is even, he divides it by $2$, and if the integer is odd, he adds $1$. The algorithm terminates after he reaches $1$. What is the least positive integer n such that Ben's algorithm performed on n will terminate after seven seconds? (For example, if Ben performed his algorithm on $3$, the algorithm would terminate after $3$ seconds: $3 \to 4 \to 2 \to 1$.) [b]p2.[/b] Suppose that a rectangle $R$ has length $p$ and width $q$, for prime integers $p$ and $q$. Rectangle $S$ has length $p + 1$ and width $q + 1$. The absolute difference in area between $S$ and $R$ is $21$. Find the sum of all possible values of $p$. [b]p3.[/b] Owen the origamian takes a rectangular $12 \times 16$ sheet of paper and folds it in half, along the diagonal, to form a shape. Find the area of this shape. [u]Round 2[/u] [b]p4.[/b] How many subsets of the set $\{G, O, Y, A, L, E\}$ contain the same number of consonants as vowels? (Assume that $Y$ is a consonant and not a vowel.) [b]p5.[/b] Suppose that trapezoid $ABCD$ satisfies $AB = BC = 5$, $CD = 12$, and $\angle ABC = \angle BCD = 90^o$. Let $AC$ and $BD$ intersect at $E$. The area of triangle $BEC$ can be expressed as $\frac{a}{b}$, for positive integers $a$ and $b$ with $gcd(a, b) = 1$. Find $a + b$. [b]p6.[/b] Find the largest integer $n$ for which $\frac{101^n + 103^n}{101^{n-1} + 103^{n-1}}$ is an integer. [u]Round 3[/u] [b]p7.[/b] For each positive integer n between $1$ and $1000$ (inclusive), Ben writes down a list of $n$'s factors, and then computes the median of that list. He notices that for some $n$, that median is actually a factor of $n$. Find the largest $n$ for which this is true. [b]p8.[/b] ([color=#f00]voided[/color]) Suppose triangle $ABC$ has $AB = 9$, $BC = 10$, and $CA = 17$. Let $x$ be the maximal possible area of a rectangle inscribed in $ABC$, such that two of its vertices lie on one side and the other two vertices lie on the other two sides, respectively. There exist three rectangles $R_1$, $R_2$, and $R_3$ such that each has an area of $x$. Find the area of the smallest region containing the set of points that lie in at least two of the rectangles $R_1$, $R_2$, and $R_3$. [b]p9.[/b] Let $a, b,$ and $c$ be the three smallest distinct positive values of $\theta$ satisfying $$\cos \theta + \cos 3\theta + ... + \cos 2021\theta = \sin \theta+ \sin 3 \theta+ ... + \sin 2021\theta. $$ What is $\frac{4044}{\pi}(a + b + c)$? [color=#f00]Problem 8 is voided. [/color] PS. You should use hide for answers.Rounds 4-5 have been posted [url=https://artofproblemsolving.com/community/c4h3131422p28368457]here [/url] and 6-7 [url=https://artofproblemsolving.com/community/c4h3131434p28368604]here [/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On a circle are given $n\ge 3$ points. At most, how many parts can the segments with the endpoints at these $n$ points divide the interior of the circle into?

For the team, power, and tournament rounds, BMT divided up the teams into $14$ rooms. You sign up to proctor all $3$ rounds, but you cannot proctor in the same room more than once. How many ways can you be assigned for rooms for the $3$ rounds?

Each cell in a $13 \times 13$ grid table is painted in black or white. Each move consists of choosing a subsquare of size either $2 \times 2$ or $9 \times 9$, and painting all white cells of the choosen subsquare black, and painting all its black cells white. It is always possible to get all cells of the original square black, after a finite number of such moves ?

Daniel painted a rectangular painting measuring $2$ meters by $4$ meters with four colors. Knowing that he used more than two colors to paint the four corners of the painting, prove that he painted of the same color two points that are at least $\sqrt5$ meters

Rectangle on a checked paper with length of a unit square side being $1$ Is divided into domino figures( two unit square sharing a common edge). Prove that you colour all corners of squares on the edge of rectangle and inside rectangle with $3$ colours such that for any two corners with distance $1$ the following conditions hold: they are coloured in different colour if the line connecting the two corners is on the border of two domino figures and coloured in same colour if the line connecting the two corners is inside a domino figure.

A bag has $3$ white and $7$ black marbles. Arjun picks out one marble without replacement and then a second. What is the probability that Arjun chooses exactly $1$ white and $1$ black marble?

Prove that we can color every subset with $n$ element of a set with $3n$ elements with $8$ colors . In such a way that there are no $3$ subsets $A,B,C$ with the same color where : $$|A \cap B| \le 1,|A \cap C| \le 1,|B \cap C| \le 1$$ Proposed by [i]Morteza Saghafian[/i] and [i]Amir Jafari[/i]

A torpedo set consists of $2$ pieces of $1 \times 4$, $4$ pieces of $1 \times 3$, $6$ pieces of $1 \times 2$ and $ 8$ pieces of $1 \times 1$ ships. a) Can one put the whole set to a $10 \times 10$ table so that the ships do not even touch with corners? (The ships can be placed both horizontally and vertically.) b) Can we solve this problem if we change $4$ pieces of $1 \times 1$ ships to $3$ pieces of $1 \times 2$ ships? c) Can we solve the problem if we change the remaining $4$ pieces of $1 \times 1$ ships to one piece of $1 \times 3$ ship and one piece of $1 \times 2$ ship? (So the number of pieces are $2, 5, 10, 0$.)

A finite set $\{a_1, a_2, ... a_k\}$ of positive integers with $a_1 < a_2 < a_3 < ... < a_k$ is named [i]alternating [/i] if $i+a$ for $i = 1, 2, 3, ..., k$ is even. The empty set is also considered to be alternating. The number of alternating subsets of $\{1, 2, 3,..., n\}$ is denoted by $A(n)$. Develop a method to determine $A(n)$ for every $n \in N$ and calculate hence $A(33)$.

In the plane we are given $3 \cdot n$ points ($n>$1) no three collinear, and the distance between any two of them is $\leq 1$. Prove that we can construct $n$ pairwise disjoint triangles such that: The vertex set of these triangles are exactly the given 3n points and the sum of the area of these triangles $< 1/2$.

There are $24$ different pencils, $4$ different colors, and $6$ pencils of each color. They were given to $6$ children in such a way that each got $4$ pencils. What is the least number of children that you can randomly choose so that you can guarantee that you have pencils of all colors. P.S. for 10 grade gives same problem with $40$ pencils, $10$ of each color and $10$ children.

Each of $1000$ elves has a hat, red on the inside and blue on the outside or vise versa. An elf with a hat that is red outside can only lie, and an elf with a hat that is blue outside can only tell the truth. One day every elf tells every other elf, “Your hat is red on the outside.” During that day, some of the elves turn their hats inside out at any time during the day. (An elf can do that more than once per day.) Find the smallest possible number of times any hat is turned inside out.