Five pairs of cartoon characters, Donald and Katrien Duck, Asterix and Obelix, Suske and Wiske, Tom and Jerry, Heer Bommel and Tom Poes, sit around a round table with $10$ chairs. The two members of each pair ensure that they sit next to each other. In how many different ways can the ten seats be occupied? Two ways are different if they cannot be transferred to each other by a rotation.

A spherical planet has the equator of length $1$. On this planet, $N$ circular roads of length $1$ each are to be built and used for several trains each. The trains must have the same constant positive speed and never stop or collide. What is the greatest possible sum of lengths of all the trains? The trains are arcs of zero width with endpoints removed (so that if only endpoints of two arcs have coincided then it is not a collision). Solve the problem for : (a) $N=3$ ([i]4 points)[/i] (b) $N=4$ ([i]6 points)[/i] [i]Alexandr Berdnikov [/i]

All cells of the $m\times n$ table are colored either white or black such that all corner cells of any rectangle containing the cells of this table with sides greater than one cell are not the same color. For values $m = 2, 3, 4,$ find all $n$ such that the mentioned coloring is possible.

[b]p1.[/b] Insert "$+$" signs between some of the digits in the following sequence to obtain correct equality: $$1\,\,\,\, 2\,\,\,\, 3\,\,\,\, 4\,\,\,\,5\,\,\,\, 6\,\,\,\, 7 = 100$$ [b]p2.[/b] A square is tiled by smaller squares as shown in the figure. Find the area of the black square in the middle if the perimeter of the big square $ABCD$ is $40$ cm. [img]https://cdn.artofproblemsolving.com/attachments/8/c/d54925cba07f63ec8578048f46e1e730cb8df3.png[/img] [b]p3.[/b] Jack made $3$ quarts of fruit drink from orange and apple juice. $\frac25$ of his drink is orange juice and the rest is apple juice. Nick prefers more orange juice in the drink. How much orange juice should he add to the drink to obtain a drink composed of $\frac35$ of orange juice? [b]p4.[/b] A train moving at $55$ miles per hour meets and is passed by a train moving moving in the opposite direction at $35$ miles per hour. A passenger in the first train sees that the second train takes $8$ seconds to pass him. How long is the second train? [b]p5.[/b] It is easy to arrange $16$ checkers in $10$ rows of $4$ checkers each, but harder to arrange $9$ checkers in $10$ rows of $3$ checkers each. Do both. [b]p6.[/b] Every human that lived on Earth exchanged some number of handshakes with other humans. Show that the number of people that made an odd number of handshakes is even. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Two players, $A$ and $B,$ alternatively take stones from a pile of $n \geq 2$ stones. $A$ plays first and in his first move he must take at least one stone and at most $n-1$ stones. Then each player must take at least one stone and at most as many stones as his opponent took in the previous move. The player who takes the last stone wins. Which player has a winning strategy?

Johnny once saw plums hanging, like eggs so big and numbered according to the first natural numbers. He is the first to pick the plum with number $2$. After that, Jantje picks the plum each time with the smallest number $n$ that satisfies the following two conditions: $\bullet$ $n$ is greater than all numbers on the already picked plums, $\bullet$ $n$ is not the product of two equal or different numbers on already picked plums. We call the numbers on the picked plums plum numbers. Is $100 000$ a plum number? Justify your answer.

Bugs Bunny plays a game in the Euclidean plane. At the $n$-th minute $(n \geq 1)$, Bugs Bunny hops a distance of $F_n$ in the North, South, East, or West direction, where $F_n$ is the $n$-th Fibonacci number (defined by $F_1 = F_2 =1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 3$). If the first two hops were perpendicular, prove that Bugs Bunny can never return to where he started. [i]Proposed by Dylan Toh[/i]

In the diagram shown, how many pathways are there from point $A$ to point $B$ if you are only allowed to travel due East, Southeast, or Southwest? [img]https://cdn.artofproblemsolving.com/attachments/9/1/0a1219fb430c402fef4b7555ddff7c88fec47e.jpg[/img]

There is a list of $n$ football matches. Determine how many possible columns, with an even number of draws, there are.

A finite list of rational numbers is written on a blackboard. In an [i]operation[/i], we choose any two numbers $a$, $b$, erase them, and write down one of the numbers \[ a + b, \; a - b, \; b - a, \; a \times b, \; a/b \text{ (if $b \neq 0$)}, \; b/a \text{ (if $a \neq 0$)}. \] Prove that, for every integer $n > 100$, there are only finitely many integers $k \ge 0$, such that, starting from the list \[ k + 1, \; k + 2, \; \dots, \; k + n, \] it is possible to obtain, after $n - 1$ operations, the value $n!$.

There are $2n$ points on the plane, no three of which lie on the same line. Some segments are drawn between them so that they do not intersect at internal points and any segment with ends among the given points intersects some of the drawn segments at an internal point. Is it true that it is always possible to choose $n$ drawn segments having no common ends?

A rectangle $ D$ is partitioned in several ($ \ge2$) rectangles with sides parallel to those of $ D$. Given that any line parallel to one of the sides of $ D$, and having common points with the interior of $ D$, also has common interior points with the interior of at least one rectangle of the partition; prove that there is at least one rectangle of the partition having no common points with $ D$'s boundary. [i]Author: Kei Irie, Japan[/i]

In the morning, $100$ students study as $50$ groups with two students in each group. In the afternoon, they study again as $50$ groups with two students in each group. No matter how the groups in the morning or groups in the afternoon are established, if it is possible to find $n$ students such that no two of them study together, what is the largest value of $n$? $ \textbf{(A)}\ 42 \qquad\textbf{(B)}\ 38 \qquad\textbf{(C)}\ 34 \qquad\textbf{(D)}\ 25 \qquad\textbf{(E)}\ \text{None of above} $

Evaluate $\sum_{m=0}^{2009}\sum_{n=0}^{m}\binom{2009}{m}\binom{m}{n}$

Consider $16$ points arranged as shown, with horizontal and vertical distances of $1$ between consecutive rows and columns. In how many ways can one choose four of these points such that the distance between every two of those four points is strictly greater than $2$? [asy] for (int x = 0; x < 4; ++x) { for (int y = 0; y < 4; ++y) { dot((x, y)); } } [/asy]

Let $n>2$ be an integer. A deck contains $\frac{n(n-1)}{2}$ cards,numbered \[1,2,3,\cdots , \frac{n(n-1)}{2}\] Two cards form a [i]magic pair[/i] if their numbers are consecutive , or if their numbers are $1$ and $\frac{n(n+1)}{2}$. For which $n$ is it possible to distribute the cards into $n$ stacks in such a manner that, among the cards in any two stacks , there is exactly one [i]magic pair[/i]?

Given a rectangular grid, split into $m\times n$ squares, a colouring of the squares in two colours (black and white) is called valid if it satisfies the following conditions: [list] [*]All squares touching the border of the grid are coloured black. [*]No four squares forming a $2\times 2$ square are coloured in the same colour. [*]No four squares forming a $2\times 2$ square are coloured in such a way that only diagonally touching squares have the same colour.[/list] Which grid sizes $m\times n$ (with $m,n\ge 3$) have a valid colouring?

[b]p1.[/b] Fleming has a list of 8 mutually distinct integers between $90$ to $99$, inclusive. Suppose that the list has median $94$, and that it contains an even number of odd integers. If Fleming reads the numbers in the list from smallest to largest, then determine the sixth number he reads. [b]p2.[/b] Find the number of ordered pairs $(x,y)$ of three digit base-$10$ positive integers such that $x-y$ is a positive integer, and there are no borrows in the subtraction $x-y$. For example, the subtraction on the left has a borrow at the tens digit but not at the units digit, whereas the subtraction on the right has no borrows. $$\begin{tabular}{ccccc} & 4 & 7 & 2 \\ - & 1 & 9 & 1\\ \hline & 2 & 8 & 1 \\ \end{tabular}\,\,\, \,\,\, \begin{tabular}{ccccc} & 3 & 7 & 9 \\ - & 2 & 6 & 3\\ \hline & 1 & 1 & 6 \\ \end{tabular}$$ [b]p3.[/b] Evaluate $$1 \cdot 2 \cdot 3-2 \cdot 3 \cdot 4+3 \cdot 4 \cdot 5- 4 \cdot 5 \cdot 6+ ... +2017 \cdot 2018 \cdot 2019 -2018 \cdot 2019 \cdot 2020+1010 \cdot 2019 \cdot 2021$$ [b]p4.[/b] Find the number of ordered pairs of integers $(a,b)$ such that $$\frac{ab+a+b}{a^2+b^2+1}$$ is an integer. [b]p5.[/b] Lin Lin has a $4\times 4$ chessboard in which every square is initially empty. Every minute, she chooses a random square $C$ on the chessboard, and places a pawn in $C$ if it is empty. Then, regardless of whether $C$ was previously empty or not, she then immediately places pawns in all empty squares a king’s move away from $C$. The expected number of minutes before the entire chessboard is occupied with pawns equals $\frac{m}{n}$ for relatively prime positive integers $m$,$n$. Find $m+n$. A king’s move, in chess, is one square in any direction on the chessboard: horizontally, vertically, or diagonally. [b]p6.[/b] Let $P(x) = x^5-3x^4+2x^3-6x^2+7x+3$ and $a_1,...,a_5$ be the roots of$ P(x)$. Compute $$\sum^5_{k=1}(a^3_k -4a^2_k +a_k +6).$$ [b]p7.[/b] Rectangle $AXCY$ with a longer length of $11$ and square $ABCD$ share the same diagonal $\overline{AC}$. Assume $B$,$X$ lie on the same side of $\overline{AC}$ such that triangle$ BXC$ and square $ABCD$ are non-overlapping. The maximum area of $BXC$ across all such configurations equals $\frac{m}{n}$ for relatively prime positive integers $m$,$n$. Compute $m+n$. [b]p8.[/b] Earl the electron is currently at $(0,0)$ on the Cartesian plane and trying to reach his house at point $(4,4)$. Each second, he can do one of three actions: move one unit to the right, move one unit up, or teleport to the point that is the reflection of its current position across the line $y=x$. Earl cannot teleport in two consecutive seconds, and he stops taking actions once he reaches his house. Earl visits a chronologically ordered sequence of distinct points $(0,0)$, $...$, $(4,4)$ due to his choice of actions. This is called an [i]Earl-path[/i]. How many possible such [i]Earl-paths[/i] are there? [b]p9.[/b] Let $P(x)$ be a degree-$2022$ polynomial with leading coefficient $1$ and roots $\cos \left( \frac{2\pi k}{2023} \right)$ for $k = 1$ , $...$,$2022$ (note $P(x)$ may have repeated roots). If $P(1) =\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, then find the remainder when $m+n$ is divided by $100$. [b]p10.[/b] A randomly shuffled standard deck of cards has $52$ cards, $13$ of each of the four suits. There are $4$ Aces and $4$ Kings, one of each of the four suits. One repeatedly draws cards from the deck until one draws an Ace. Given that the first King appears before the first Ace, the expected number of cards one draws after the first King and before the first Ace is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [b]p11.[/b] The following picture shows a beam of light (dashed line) reflecting off a mirror (solid line). The [i]angle of incidence[/i] is marked by the shaded angle; the[i] angle of reflection[/i] is marked by the unshaded angle. [img]https://cdn.artofproblemsolving.com/attachments/9/d/d58086e5cdef12fbc27d0053532bea76cc50fd.png[/img] The sides of a unit square $ABCD$ are magically distorted mirrors such that whenever a light beam hits any of the mirrors, the measure of the angle of incidence between the light beam and the mirror is a positive real constant $q$ degrees greater than the measure of the angle of reflection between the light beam and the mirror. A light beam emanating from $A$ strikes $\overline{CD}$ at $W_1$ such that $2DW_1 =CW_1$, reflects off of $\overline{CD}$ and then strikes $\overline{BC}$ at $W_2$ such that $2CW_2 = BW_2$, reflects off of $\overline{BC}$, etc. To this end, denote $W_i$ the $i$-th point at which the light beam strikes $ABCD$. As $i$ grows large, the area of $W_iW_{i+1}W_{i+2}W_{i+3}$ approaches $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. [b]p12.[/b] For any positive integer $m$, define $\phi (m)$ the number of positive integers $k \le m$ such that $k$ and $m$ are relatively prime. Find the smallest positive integer $N$ such that $\sqrt{ \phi (n) }\ge 22$ for any integer $n \ge N$. [b]p13.[/b] Let $n$ be a fixed positive integer, and let $\{a_k\}$ and $\{b_k\}$ be sequences defined recursively by $$a_1 = b_1 = n^{-1}$$ $$a_j = j(n- j+1)a_{j-1}\,\,\, , \,\,\, j > 1$$ $$b_j = nj^2b_{j-1}+a_j\,\,\, , \,\,\, j > 1$$ When $n = 2021$, then $a_{2021} +b_{2021} = m \cdot 2017^2$ for some positive integer $m$. Find the remainder when $m$ is divided by $2017$. [b]p14.[/b] Consider the quadratic polynomial $g(x) = x^2 +x+1020100$. A positive odd integer $n$ is called $g$-[i]friendly[/i] if and only if there exists an integer $m$ such that $n$ divides $2 \cdot g(m)+2021$. Find the number of $g$-[i]friendly[/i] positive odd integers less than $100$. [b]p15.[/b] Let $ABC$ be a triangle with $AB < AC$, inscribed in a circle with radius $1$ and center $O$. Let $H$ be the intersection of the altitudes of $ABC$. Let lines $\overline{OH}$, $\overline{BC}$ intersect at $T$. Suppose there is a circle passing through $B$, $H$, $O$, $C$. Given $\cos (\angle ABC-\angle BCA) = \frac{11}{32}$ , then $TO = \frac{m\sqrt{p}}{n}$ for relatively prime positive integers $m$,$n$ and squarefree positive integer $p$. Find $m+n+ p$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given are $n$ different concentric circles on the plane.Inside the disk with the smallest radius (strictly inside it),we consider two distinct points $A,B$.We consider $k$ distinct lines passing through $A$ and $m$ distinct lines passing through $B$.There is no line passing through both $A$ and $B$ and all the lines passing through $k$ intersect with all the lines passing through $B$.The intersections do not lie on some of the circles.Determine the maximum and the minimum number of regions formed by the lines and the circles and are inside the circles.

We call a coloring of an $m\times n$ table ($m,n\ge 5$) in three colors a [i]good coloring[/i] if the following conditions are satisfied: 1) Each cell has the same number of neighboring cells of two other colors; 2) Each corner has no neighboring cells of its color. Find all pairs $(m,n)$ ($m,n\ge 5$) for which there exists a good coloring of $m\times n$ table. [i](I. Manzhulina, B. Rubliov)[/i]

[u]Super Mario 64![/u] Mario is once again on a quest to save Princess Peach. Mario enters Peach's castle and finds himself in a room with $4$ doors. This room is the first in a sequence of $2$ indistinugishable rooms. In each room, $1$ door leads to the next room in the sequence (or, for the second room, into Bowser's level), while the other $3$ doors lead to the first room. [b]p9.[/b] Suppose that in every room, Mario randomly picks a door to walk through. What is the expected number of doors (not including Mario's initial entrance to the first room) through which Mario will pass before he reaches Bowser's level? [b]p10.[/b] Suppose that instead there are $6$ rooms with $4$ doors. In each room, $1$ door leads to the next room in the sequence (or, for the last room, Bowser's level), while the other $3$ doors lead to the first room. Now what is the expected number of doors through which Mario will pass before he reaches Bowser's level? [b]p11.[/b] In general, if there are $d$ doors in every room (but still only $1$ correct door) and $r$ rooms, the last of which leads into Bowser's level, what is the expected number of doors through which Mario will pass before he reaches Bowser's level?

[b]p1.[/b] While computing $7 - 2002 \cdot x$, John accidentally evaluates from left to right $((7 - 2002) \cdot x)$ instead of correctly using order of operations $(7 - (2002 \cdot x))$. If he gets the correct answer anyway, what is $x$? [b]p2.[/b] Given that $$x^2 + y^2 + z^2 = 6$$ $$ \left( \frac{x}{y} + \frac{y}{x} \right)^2 + \left( \frac{y}{z} + \frac{z}{y} \right)^2 + \left( \frac{z}{x} + \frac{x}{z} \right)^2 = 16.5,$$ what is $\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2}$ ? [b]p3.[/b] Evaluate $$\frac{tan \frac{\pi}{4}}{4}+\frac{tan \frac{3\pi}{4}}{8}+\frac{tan \frac{5\pi}{4}}{16}+\frac{tan \frac{7\pi}{4}}{32}+ ...$$ [b]p4.[/b] Note that $2002 = 22 \cdot 91$, and so $2002$ is a multiple of the number obtained by removing its middle $2$ digits. Generalizing this, how many $4$-digit palindromes, $abba$, are divisible by the $2$-digit palindrome, $aa$? [b]p5.[/b] Let $ABCDE$ be a pyramid such that $BCDE$ is a square with side length $2$, and $A$ is $2$ units above the center of $BCDE$. If $F$ is the midpoint of $\overline{DE}$ and $G$ is the midpoint of $\overline{AC}$, what is the length of $\overline{DE}$? [b]p6.[/b] Suppose $a_1, a_2,..., a_{100}$ are real numbers with the property that $$i(a_1 + a_2 +... + a_i) = 1 + (a_{i+1} + a_{i+2} + ... + a_{100})$$ for all $i$. Compute $a_{10}$. [b]p7.[/b] A bug is sitting on one corner of a $3' \times 4' \times 5'$ block of wood. What is the minimum distance nit needs to travel along the block’s surface to reach the opposite corner? [b]p8.[/b] In the number game, a pair of positive integers $(n,m)$ is written on a blackboard. Two players then take turns doing the following: 1. If $n \ge m$, the player chooses a positive integer $c$ such that $n - cm \ge 0$, and replaces $(n,m)$ with $(n - cm,m)$. 2. If $m > n$, the player chooses a positive integer $c$ such that $m - cn \ge 0$, and replaces $(n,m)$ with $(n,m - cn)$. If $m$ or $n$ ever become $0$, the game ends, and the last player to have moved is declared the winner. If $(n,m)$ are originally $(20021000, 2002)$, what choices of $c$ are winning moves for the first player? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There is a grid of height $2$ stretching infinitely in one direction. Between any two edge-adjacent cells of the grid, there is a door that is locked with probability $\frac12$ independent of all other doors. Philip starts in a corner of the grid (in the starred cell). Compute the expected number of cells that Philip can reach, assuming he can only travel between cells if the door between them is unlocked. [img]https://cdn.artofproblemsolving.com/attachments/f/d/fbf9998270e16055f02539bb532b1577a6f92a.png[/img]

There are 101 coins in a circle, each weights 10g or 11g. Prove that there exists a coin such that the total weight of the $k{}$ coins to its left is equal to the total weight of the $k{}$ coins to its right where a) $k = 50$ and b) $k = 49$. [i]Alexandr Gribalko[/i]

Baron Munchausen claims that he got a map of a country that consists of five cities. Each two cities are connected by a direct road. Each road intersects no more than one another road (and no more than once). On the map, the roads are colored in yellow or red, and while circling any city (along its border) one can notice that the colors of crossed roads alternate. Can Baron's claim be true?