Let $r$ and $b$ be positive integers. The game of [i]Monis[/i], a variant of Tetris, consists of a single column of red and blue blocks. If two blocks of the same color ever touch each other, they both vanish immediately. A red block falls onto the top of the column exactly once every $r$ years, while a blue block falls exactly once every $b$ years. (a) Suppose that $r$ and $b$ are odd, and moreover the cycles are offset in such a way that no two blocks ever fall at exactly the same time. Consider a period of $rb$ years in which the column is initially empty. Determine, in terms of $r$ and $b$, the number of blocks in the column at the end. (b) Now suppose $r$ and $b$ are relatively prime and $r+b$ is odd. At time $t=0$, the column is initially empty. Suppose a red block falls at times $t = r, 2r, \dots, (b-1)r$ years, while a blue block falls at times $t = b, 2b, \dots, (r-1)b$ years. Prove that at time $t=rb$, the number of blocks in the column is $\left\lvert 1+2(r-1)(b+r)-8S \right\rvert$, where \[ S = \left\lfloor \frac{2r}{r+b} \right\rfloor + \left\lfloor \frac{4r}{r+b} \right\rfloor + ... + \left\lfloor \frac{(r+b-1)r}{r+b} \right\rfloor . \] [i]Proposed by Sammy Luo[/i]

Isaak and Jeremy play the following game. Isaak says to Jeremy that he thinks a few $2^n$ integers $k_1,..,k_{2^n}$. Jeremy asks questions of the form: ''Is it true that $k_i<k_j$ ?'' in which Isaak answers by always telling the truth. After $n2^{n-1}$ questions, Jeramy must decide whether numbers of Isaak are all distinct each other or not. Prove that Jeremy has bo way to be ''sure'' for his final decision. (UK)

Given a set $X=\{x_1,\ldots,x_n\}$ of natural numbers in which for all $1< i \leq n$ we have $1\leq x_i-x_{i-1}\leq 2$, call a real number $a$ [b]good[/b] if there exists $1\leq j \leq n$ such that $2|x_j-a|\leq 1$. Also a subset of $X$ is called [b]compact[/b] if the average of its elements is a good number. Prove that at least $2^{n-3}$ subsets of $X$ are compact. [i]Proposed by Mahyar Sefidgaran[/i]

Starting at $(0, 0)$, Richard takes $2n+1$ steps, with each step being one unit either East, North, West, or South. For each step, the direction is chosen uniformly at random from the four possibilities. Determine the probability that Richard ends at $(1, 0)$.

Consider a checkered board $2m \times 2n$, $m, n \in \mathbb{Z}_{>0}$. A stone is placed on one of the unit squares on the board, this square is different from the upper right square and from the lower left square. A snail goes from the bottom left square and wants to get to the top right square, walking from one square to other adjacent, one square at a time (two squares are adjacent if they share an edge). Determine all the squares the stone can be in so that the snail can complete its path by visiting each square exactly one time, except the square with the stone, which the snail does not visit.

Sarah buys $3$ gumballs from a gumball machine that contains $10$ orange, $6$ green, and $9$ yellow gumballs. What is the probability that the first gumball is orange, the second is green or yellow, and the third is also orange?

Let $n$ be a positive integer. Find the smallest positive integer $k$ with the property that for any colouring nof the squares of a $2n$ by $k$ chessboard with $n$ colours, there are $2$ columns and $2$ rows such that the $4$ squares in their intersections have the same colour.

For any nonnegative integer $n$, let $S(n)$ be the sum of the digits of $n$. Let $K$ be the number of nonnegative integers $n \le 10^{10}$ that satisfy the equation \[ S(n) = (S(S(n)))^2. \] Find the remainder when $K$ is divided by $1000$.

The number $1$ is written on the blackboard. After that a sequence of numbers is created as follows: at each step each number $a$ on the blackboard is replaced by the numbers $a - 1$ and $a + 1$; if the number $0$ occurs, it is erased immediately; if a number occurs more than once, all its occurrences are left on the blackboard. Thus the blackboard will show $1$ after $0$ steps; $2$ after $1$ step; $1, 3$ after $2$ steps; $2, 2, 4$ after $3$ steps, and so on. How many numbers will there be on the blackboard after $n$ steps?

[b]p1.[/b] Given that $7 \times 22 \times 13 = 2002$, compute $14 \times 11 \times 39$. [b]p2.[/b] Ariel the frog is on the top left square of a $8 \times 10$ grid of squares. Ariel can jump from any square on the grid to any adjacent square, including diagonally adjacent squares. What is the minimum number of jumps required so that Ariel reaches the bottom right corner? [b]p3.[/b] The distance between two floors in a building is the vertical distance from the bottom of one floor to the bottom of the other. In Evans hall, the distance from floor $7$ to floor $5$ is $30$ meters. There are $12$ floors on Evans hall and the distance between any two consecutive floors is the same. What is the distance, in meters, from the first floor of Evans hall to the $12$th floor of Evans hall? [b]p4.[/b] A circle of nonzero radius $ r$ has a circumference numerically equal to $\frac13$ of its area. What is its area? [b]p5.[/b] As an afternoon activity, Emilia will either play exactly two of four games (TwoWeeks, DigBuild, BelowSaga, and FlameSymbol) or work on homework for exactly one of three classes (CS61A, Math 1B, Anthro 3AC). How many choices of afternoon activities does Emilia have? [b]p6.[/b] Matthew wants to buy merchandise of his favorite show, Fortune Concave Decagon. He wants to buy figurines of the characters in the show, but he only has $30$ dollars to spend. If he can buy $2$ figurines for $4$ dollars and $5$ figurines for $8$ dollars, what is the maximum number of figurines that Matthew can buy? [b]p7.[/b] When Dylan is one mile from his house, a robber steals his wallet and starts to ride his motorcycle in the direction opposite from Dylan’s house at $40$ miles per hour. Dylan dashes home at $10$ miles per hour and, upon reaching his house, begins driving his car at $60$ miles per hour in the direction of the robber’s motorcycle. How long, starting from when the robber steals the wallet, does it take for Dylan to catch the robber? Express your answer in minutes. [b]p8.[/b] Deepak the Dog is tied with a leash of $7$ meters to a corner of his $4$ meter by $6$ meter rectangular shed such that Deepak is outside the shed. Deepak cannot go inside the shed, and the leash cannot go through the shed. Compute the area of the region that Deepak can travel to. [img]https://cdn.artofproblemsolving.com/attachments/f/8/1b9563776325e4e200c3a6d31886f4020b63fa.png[/img] [b]p9.[/b] The quadratic equation $a^2x^2 + 2ax -3 = 0$ has two solutions for x that differ by $a$, where $a > 0$. What is the value of $a$? [b]p10.[/b] Find the number of ways to color a $2 \times 2$ grid of squares with $4$ colors such that no two (nondiagonally) adjacent squares have the same color. Each square should be colored entirely with one color. Colorings that are rotations or reflections of each other should be considered different. [b]p11[/b]. Given that $\frac{1}{y^2+5} - \frac{3}{y^4-39} = 0$, and $y \ge 0$, compute $y$. [b]p12.[/b] Right triangle $ABC$ has $AB = 5$, $BC = 12$, and $CA = 13$. Point $D$ lies on the angle bisector of $\angle BAC$ such that $CD$ is parallel to $AB$. Compute the length of $BD$. [img]https://cdn.artofproblemsolving.com/attachments/c/3/d5cddb0e8ac43c35ddfc94b2a74b8d022292f2.png[/img] [b]p13.[/b] Let $x$ and $y$ be real numbers such that $xy = 4$ and $x^2y + xy^2 = 25$. Find the value of $x^3y +x^2y^2 + xy^3$. [b]p14.[/b] Shivani is planning a road trip in a car with special new tires made of solid rubber. Her tires are cylinders that are $6$ inches in width and have diameter $26$ inches, but need to be replaced when the diameter is less than $22$ inches. The tire manufacturer says that $0.12\pi$ cubic inches will wear away with every single rotation. Assuming that the tire manufacturer is correct about the wear rate of their tires, and that the tire maintains its cylindrical shape and width (losing volume by reducing radius), how many revolutions can each tire make before she needs to replace it? [b]p15.[/b] What’s the maximum number of circles of radius $4$ that fit into a $24 \times 15$ rectangle without overlap? [b]p16.[/b] Let $a_i$ for $1 \le i \le 10$ be a finite sequence of $10$ integers such that for all odd $i$, $a_i = 1$ or $-1$, and for all even $i$, $a_i = 1$, $-1$, or $0$. How many sequences a_i exist such that $a_1+a_2+a_3+...+a_{10} = 0$? [b]p17.[/b] Let $\vartriangle ABC$ be a right triangle with $\angle B = 90^o$ such that $AB$ and $BC$ have integer side lengths. Squares $ABDE$ and $BCFG$ lie outside $\vartriangle ABC$. If the area of $\vartriangle ABC$ is $12$, and the area of quadrilateral $DEFG$ is $38$, compute the perimeter of $\vartriangle ABC$. [img]https://cdn.artofproblemsolving.com/attachments/b/6/980d3ba7d0b43507856e581476e8ad91886656.png[/img] [b]p18.[/b] What is the smallest positive integer $x$ such that there exists an integer $y$ with $\sqrt{x} +\sqrt{y} = \sqrt{1025}$ ? [b]p19. [/b]Let $a =\underbrace{19191919...1919}_{19\,\, is\,\,repeated\,\, 3838\,\, times}$. What is the remainder when $a$ is divided by $13$? [b]p20.[/b] James is watching a movie at the cinema. The screen is on a wall and is $5$ meters tall with the bottom edge of the screen $1.5$ meters above the floor. The floor is sloped downwards at $15$ degrees towards the screen. James wants to find a seat which maximizes his vertical viewing angle (depicted below as $\theta$ in a two dimensional cross section), which is the angle subtended by the top and bottom edges of the screen. How far back from the screen in meters (measured along the floor) should he sit in order to maximize his vertical viewing angle? [img]https://cdn.artofproblemsolving.com/attachments/1/5/1555fb2432ee4fe4903accc3b74ea7215bc007.png[/img] PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that there exists a set $ A$ of positive integers with the following property: for any infinite set $ S$ of primes, there exist [i]two[/i] positive integers $ m$ in $ A$ and $ n$ not in $ A$, each of which is a product of $ k$ distinct elements of $ S$ for some $ k \geq 2$.

Zeroes and ones are arranged in all the squares of $n\times n$ table. All the squares of the left column are filled by ones, and the sum of numbers in every figure of the form [asy]size(50); draw((2,1)--(0,1)--(0,2)--(2,2)--(2,0)--(1,0)--(1,2));[/asy] (consisting of a square and its neighbours from left and from below) is even. Prove that no two rows of the table are identical. [i]Proposed by O. Vanyushina[/i]

Let $l_1,l_2,l_3,...,L_n$ be lines in the plane such that no two of them are parallel and no three of them are concurrent. Let $A$ be the intersection point of lines $l_i,l_j$. We call $A$ an "Interior Point" if there are points $C,D$ on $l_i$ and $E,F$ on $l_j$ such that $A$ is between $C,D$ and $E,F$. Prove that there are at least $\frac{(n-2)(n-3)}{2}$ Interior points.($n>2$) note: by point here we mean the points which are intersection point of two of $l_1,l_2,...,l_n$.

Find all pairs $(m, n)$ of positive integers such that the $m \times n$ grid contains exactly $225$ rectangles whose side lengths are odd and whose edges lie on the lines of the grid.

A crossword is a grid of black and white cells such that every white cell belongs to some $2\times 2$ square of white cells. A word in the crossword is a contiguous sequence of two or more white cells in the same row or column, delimited on each side by either a black cell or the boundary of the grid. Show that the total number of words in an $n\times n$ crossword cannot exceed $(n+1)^2/2$. [i]Proposed by Nikolai Beluhov, Bulgaria[/i]

Two people play this game. At the beginning there are numbers 1, 2, 3, 4 in a circle. With each move, the first one adds 1 to two adjacent numbers, and the second swaps any two adjacent numbers. The first one wins if all numbers become equal. Can the second one interfere with him?

A set $S=\{ s_1,s_2,...,s_k\}$ of positive real numbers is "polygonal" if $k\geq 3$ and there is a non-degenerate planar $k-$gon whose side lengths are exactly $s_1,s_2,...,s_k$; the set $S$ is multipolygonal if in every partition of $S$ into two subsets,each of which has at least three elements, exactly one of these two subsets in polygonal. Fix an integer $n\geq 7$. (a) Does there exist an $n-$element multipolygonal set, removal of whose maximal element leaves a multipolygonal set? (b) Is it possible that every $(n-1)-$element subset of an $n-$element set of positive real numbers be multipolygonal?

How many ways can a set of $ n $ items be partitioned into two sets?

For a positive integer $n$, we consider an $n \times n$ board and tiles with dimensions $1 \times 1, 1 \times 2, ..., 1 \times n$. In how many ways exactly can $\frac12 n (n + 1)$ cells of the board are colored red, so that the red squares can all be covered by placing the $n$ tiles all horizontally, but also by placing all $n$ tiles vertically? Two colorings that are not identical, but by rotation or reflection from the board into each other count as different.

Let $G=(V,E)$ be a tree graph with $n$ vertices, and let $P$ be a set of $n$ points in the plane with no three points collinear. Is it true that for any choice of graph $G$ and set $P$, we can embed $G$ in $P$, i.e., we can find a bijection $f:V\to P$ such that when we draw line segment $[f(x),f(y)]$ for all $(x,y)\in E$, no two such segments intersect each other?

Given two integers $h \geq 1$ and $p \geq 2$, determine the minimum number of pairs of opponents an $hp$-member parliament may have, if in every partition of the parliament into $h$ houses of $p$ member each, some house contains at least one pair of opponents.

Let $a_1,a_2,...,a_n$ be pairwise distinct real numbers and $m$ be the number of distinct sums $a_i +a_j$ (where $i \ne j$). Find the least possible value of $m$.

Let $n\geqslant 2$ be a positive integer. Paul has a $1\times n^2$ rectangular strip consisting of $n^2$ unit squares, where the $i^{\text{th}}$ square is labelled with $i$ for all $1\leqslant i\leqslant n^2$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then [i]translate[/i] (without rotating or flipping) the pieces to obtain an $n\times n$ square satisfying the following property: if the unit square in the $i^{\text{th}}$ row and $j^{\text{th}}$ column is labelled with $a_{ij}$, then $a_{ij}-(i+j-1)$ is divisible by $n$. Determine the smallest number of pieces Paul needs to make in order to accomplish this.

Consider a chessboard $6 \times 6$, made up of $36$ single squares. We want to place $6$ chess rooks on this board, one rook on each square, so that there are no two rooks on the same row, nor two rooks on the same column. Note that, once the rooks have been placed in this way, we have that, for every square where a rook has not been placed, there is a rook in the same row as it and a rook in the same column as it. We will say that such rooks are in line with this square. For each of those $30$ houses without rooks, color it green if the two rooks aligned with that same house are the same distance from it, and color it yellow otherwise. For example, when we place the $6$ rooks ($T$) as below, we have: (a) Is it possible to place the rooks so that there are $30$ green squares? (b) Is it possible to place the rooks so that there are $30$ yellow squares? (c) Is it possible to place the rooks so that there are $15$ green and $15$ yellow squares?

A square is cut into 25 smaller squares, exactly 24 of which are unit squares. Find the area of the original square. (V Proizvolov)