Consider the following game for a positive integer $n$. Initially, the numbers $1, 2, \ldots, n$ are written on a board. In each move, two numbers are selected such that their difference is also present on the board. This difference is then erased from the board. (For example, if the numbers $3,6,11$ and $17$ are on the board, then $3$ can be erased as $6 - 3=3$, or $6$ as $17 - 11=6$, or $11$ as $17 - 6=11$.) For which values of $n$ is it possible to end with only one number remaining on the board? [i](Michael Reitmeir)[/i]

Given a permutation $(a_{1}, a_{1},...,a_{n})$ of the numbers $1, 2,...,n$ one may interchange any two consecutive "blocks" - that is, one may transform ($a_{1}, a_{2},...,a_{i}$,$\underbrace {a_{i+1},... a_{i+p},}_{A} $ $ \underbrace{a_{i+p+1},...,a_{i+q},}_{B}...,a_{n}) $ into $ (a_{1}, a_{2},...,a_{i},$ $ \underbrace {a_{i+p+1},...,a_{i+q},}_{B} $ $ \underbrace {a_{i+1},... a_{i+p}}_{A}$$,...,a_{n}) $ by interchanging the "blocks" $A$ and $B$. Find the least number of such changes which are needed to transform $(n, n-1,...,1)$ into $(1,2,...,n)$

In an $6 \times 6$ grid of lattice points, how many ways are there to choose $ 4$ points that are vertices of a nondegenerate quadrilateral with at least one pair of opposite sides parallel to the sides of the grid?

The entries on an $ n$ × $ n$ board are colored black and white like it is usually done in a chessboard, and the upper left hand corner is black. We color the entries on the chess board black according to the following rule: In each step we choose an arbitrary $ 2$×$ 3$ or $ 3$× $ 2$ rectangle that still contains $ 3$ white entries, and we color these three entries black. For which values of $ n$ can the whole board be colored black in a finite number of steps

Let $n$ be a fixed positive integer. There are $n \geq 1$ lamps in a row, some of them are on and some are off. In a single move, we choose a positive integer $i$ ($1 \leq i \leq n$) and switch the state of the first $i$ lamps from the left. Determine the smallest number $k$ with the property that we can make all of the lamps be switched on using at most $k$ moves, no matter what the initial configuration was. [i]Proposed by Viktor Simjanoski and Nikola Velov[/i]

Alice starts with an empty string and randomly appends one of the digits $2$, $0$, $1$, or $8$ until the string ends with $2018$. What is the probability Alice appends less than $9$ digits before stopping?

In the plane, there are $n$ points ($n\ge 4$) where no 3 of them are collinear. Let $A(n)$ be the number of parallelograms whose vertices are those points with area $1$. Prove the following inequality: $A(n)\leq \frac{n^2-3n}{4}$ for all $n\ge 4$

Let $p$ be an odd prime, and put $N=\frac{1}{4} (p^3 -p) -1.$ The numbers $1,2, \dots, N$ are painted arbitrarily in two colors, red and blue. For any positive integer $n \leqslant N,$ denote $r(n)$ the fraction of integers $\{ 1,2, \dots, n \}$ that are red. Prove that there exists a positive integer $a \in \{ 1,2, \dots, p-1\}$ such that $r(n) \neq a/p$ for all $n = 1,2, \dots , N.$ [I]Netherlands[/i]

For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.

a) $r_1, r_2, ... , r_{100}, c_1, c_2, ... , c_{100}$ are distinct reals. The number $r_i + c_j$ is written in position $i, j$ of a $100 \times 100$ array. The product of the numbers in each column is $1$. Show that the product of the numbers in each row is $-1$. b) $r_1, r_2, ... , r_{2n}, c_1, c_2, ... , c_{2n}$ are distinct reals. The number $r_i + c_j$ is written in position $i, j$ of a $2n \times 2n$ array. The product of the numbers in each column is the same. Show that the product of the numbers in each row is also the same.

Express \[ \sum_{k=0}^n (-1)^k (n-k)!(n+k)! \] in closed form.

A natural number $a$ is said [i]to be contained[/i] in the natural number $b$ if it is possible to obtain a by erasing some digits from $b$ (in their decimal representations). For example, $123$ is contained in $901523$, but not contained in $3412$. Does there exist an infinite set of natural numbers such that no number in the set is contained in any other number from the set?

Let us name a move of the chess knight horizontal if it moves two cells horizontally and one vertically, and vertical otherwise. It is required to place the knight on a cell of a ${46} \times {46}$ board and alternate horizontal and vertical moves. Prove that if each cell is visited not more than once then the number of moves does not exceed 2024. Alexandr Gribalko

For any real numbers $a$ and $b>0$, define an [i]extension[/i] of an interval $[a-b,a+b] \subseteq \mathbb{R}$ be $[a-2b, a+2b]$. We say that $P_1, P_2, \ldots, P_k$ covers the set $X$ if $X \subseteq P_1 \cup P_2 \cup \ldots \cup P_k$. Prove that there exists an integer $M$ with the following property: for every finite subset $A \subseteq \mathbb{R}$, there exists a subset $B \subseteq A$ with at most $M$ numbers, so that for every $100$ closed intervals that covers $B$, their extensions covers $A$.

[b]p1.[/b] Evaluate $1+3+5+··· +2019$. [b]p2.[/b] Evaluate $1^2 -2^2 +3^2 -4^2 +...· +99^2 -100^2$. [b]p3. [/b]Find the sum of all solutions to $|2018+|x -2018|| = 2018$. [b]p4.[/b] The angles in a triangle form a geometric series with common ratio $\frac12$ . Find the smallest angle in the triangle. [b]p5.[/b] Compute the number of ordered pairs $(a,b,c,d)$ of positive integers $1 \le a,b,c,d \le 6$ such that $ab +cd$ is a multiple of seven. [b]p6.[/b] How many ways are there to arrange three birch trees, four maple, and five oak trees in a row if trees of the same species are considered indistinguishable. [b]p7.[/b] How many ways are there for Mr. Paul to climb a flight of 9 stairs, taking steps of either two or three at a time? [b]p8.[/b] Find the largest natural number $x$ for which $x^x$ divides $17!$ [b]p9.[/b] How many positive integers less than or equal to $2018$ have an odd number of factors? [b]p10.[/b] Square $MAIL$ and equilateral triangle $LIT$ share side $IL$ and point $T$ is on the interior of the square. What is the measure of angle $LMT$? [b]p11.[/b] The product of all divisors of $2018^3$ can be written in the form $2^a \cdot 2018^b$ for positive integers $a$ and $b$. Find $a +b$. [b]p12.[/b] Find the sum all four digit palindromes. (A number is said to be palindromic if its digits read the same forwards and backwards. [b]p13.[/b] How ways are there for an ant to travel from point $(0,0)$ to $(5,5)$ in the coordinate plane if it may only move one unit in the positive x or y directions each step, and may not pass through the point $(1, 1)$ or $(4, 4)$? [b]p14.[/b] A certain square has area $6$. A triangle is constructed such that each vertex is a point on the perimeter of the square. What is the maximum possible area of the triangle? [b]p15.[/b] Find the value of ab if positive integers $a,b$ satisfy $9a^2 -12ab +2b^2 +36b = 162$. [b]p16.[/b] $\vartriangle ABC$ is an equilateral triangle with side length $3$. Point $D$ lies on the segment $BC$ such that $BD = 1$ and $E$ lies on $AC$ such that $AE = AD$. Compute the area of $\vartriangle ADE$. [b]p17[/b]. Let $A_1, A_2,..., A_{10}$ be $10$ points evenly spaced out on a line, in that order. Points $B_1$ and $B_2$ lie on opposite sides of the perpendicular bisector of $A_1A_{10}$ and are equidistant to $l$. Lines $B_1A_1,...,B_1A_{10}$ and $B_2A_1,...· ,B_2A_{10}$ are drawn. How many triangles of any size are present? [b]p18.[/b] Let $T_n = 1+2+3··· +n$ be the $n$th triangular number. Determine the value of the infinite sum $\sum_{k\ge 1} \frac{T_k}{2^k}$. [b]p19.[/b] An infinitely large bag of coins is such that for every $0.5 < p \le 1$, there is exactly one coin in the bag with probability $p$ of landing on heads and probability $1- p$ of landing on tails. There are no other coins besides these in the bag. A coin is pulled out of the bag at random and when flipped lands on heads. Find the probability that the coin lands on heads when flipped again. [b]p20.[/b] The sequence $\{x_n\}_{n\ge 1}$ satisfies $x1 = 1$ and $(4+ x_1 + x_2 +··· + x_n)(x_1 + x_2 +··· + x_{n+1}) = 1$ for all $n \ge 1$. Compute $\left \lfloor \frac{x_{2018}}{x_{2019}} \right \rfloor$. PS. You had better use hide for answers.

Let $M$ be the set of the integer numbers from the range $[-n, n]$. The subset $P$ of $M$ is called a [i]base subset[/i] if every number from $M$ can be expressed as a sum of some different numbers from $P$. Find the smallest natural number $k$ such that every $k$ numbers that belongs to $M$ form a base subset.

Let $n$ and $k$ be positive integers. Two infinite sequences $\{s_i\}_{i\geq 1}$ and $\{t_i\}_{i\geq 1}$ are [i]equivalent[/i] if, for all positive integers $i$ and $j$, $s_i = s_j$ if and only if $t_i = t_j$. A sequence $\{r_i\}_{i\geq 1}$ has [i]equi-period[/i] $k$ if $r_1, r_2, \ldots $ and $r_{k+1}, r_{k+2}, \ldots$ are equivalent. Suppose $M$ infinite sequences with equi-period $k$ whose terms are in the set $\{1, \ldots, n\}$ can be chosen such that no two chosen sequences are equivalent to each other. Determine the largest possible value of $M$ in terms of $n$ and $k$.

[b]p1.[/b] The final Big $10$ standings for the $1996$ Women's Softball season were 1. Michigan 2. Minnesota З. Iowa 4. Indiana 5. Michigan State 6. Purdue 7. Northwestern 8. Ohio State 9. Penn State 10. Wisconsin (Illinois does not participate in Women's Softball.) When you compare the $1996$ final standings (above) to the final standings for the $1999$ season, you find that the following pairs of teams changed order relative to each other from $1996$ to $1999$ (there are no ties, and no other pairs changed places): (Iowa, Michigan State) (Indiana, Penn State) (Purdue, Wisconsin) (Iowa, Penn State) (Indiana, Wisconsin) (Northwestern, Penn State) (Indiana, Michigan State) (Michigan State, Penn State) (Northwestern, Wisconsin) (Indiana, Purdue) (Purdue, Northwestern) (Ohio State, Penn State) (Indiana, Northwestern) (Purdue, Penn State) (Ohio State, Penn State) (Indiana, Ohio State) Determine as much as you can about the final Big $10$ standings for the $1999$ Women's Softball season. If you cannot determine the standings, explain why you do not have enough information. You must justify your answer. [b]p2.[/b] a) Take as a given that any expression of the form $A \sin t + B \cos t$ ($A>0$) can be put in the form $C \sin (t + D)$, where $C>0$ and $-\pi /2 <D <\pi /2 $. Determine $C$ and $D$ in terms of $A$ and $B$. b) For the values of $C$ and $D$ found in part a), prove that $A \sin t + B \cos t = C \sin (t + D)$. c) Find the maximum value of $3 \sin t +2 \cos t$. [b]pЗ.[/b] А $6$-bу-$6$ checkerboard is completelу filled with $18$ dominoes (blocks of size $1$-bу-$2$). Prove that some horizontal or vertical line cuts the board in two parts but does not cut anу of the dominoes. [b]p4.[/b] a) The midpoints of the sides of a regular hexagon are the vertices of a new hexagon. What is the ratio of the area of the new hexagon to the area of the original hexagon? Justify your answer and simplify as much as possible. b) The midpoints of the sides of a regular $n$-gon ($n >2$) are the vertices of a new $n$-gon. What is the ratio of the area of the new $n$-gon to that of the old? Justify your answer and simplify as much as possible. [b]p5. [/b] You run a boarding house that has $90$ rooms. You have $100$ guests registered, but on any given night only $90$ of these guests actually stay in the boarding house. Each evening a different random set of $90$ guests will show up. You don't know which $90$ it will be, but they all arrive for dinner before you have to assign rooms for the night. You want to give out keys to your guests so that for any set of $90$ guests, you can assign each to a private room without any switching of keys. a) You could give every guest a key to every room. But this requires $9000$ keys. Find a way to hand out fewer than $9000$ keys so that each guest will have a key to a private room. b) What is the smallest number of keys necessary so that each guest will have a key to a private room? Describe how you would distribute these keys and assign the rooms. Prove that this number of keys is as small as possible. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p7[/b] Cindy cuts regular hexagon $ABCDEF$ out of a sheet of paper. She folds $B$ over $AC$, resulting in a pentagon. Then, she folds $A$ over $CF$, resulting in a quadrilateral. The area of $ABCDEF$ is $k$ times the area of the resulting folded shape. Find $k$. [b]p8[/b] Call a sequence $\{a_n\} = a_1, a_2, a_3, . . .$ of positive integers [i]Fib-o’nacci[/i] if it satisfies $a_n = a_{n-1}+a_{n-2}$ for all $n \ge 3$. Suppose that $m$ is the largest even positive integer such that exactly one [i]Fib-o’nacci[/i] sequence satisfies $a_5 = m$, and suppose that $n$ is the largest odd positive integer such that exactly one [i]Fib-o’nacci[/i] sequence satisfies $a_5 = n$. Find $mn$. [b]p9[/b] Compute the number of ways there are to pick three non-empty subsets $A$, $B$, and $C$ of $\{1, 2, 3, 4, 5, 6\}$, such that $|A| = |B| = |C|$ and the following property holds: $$A \cap B \cap C = A \cap B = B \cap C = C \cap A.$$

Find, with proof, the maximum positive integer $k$ for which it is possible to color $6k$ cells of $6 \times 6$ grid such that, for any choice of three distinct rows $R_1$, $R_2$, $R_3$ and three distinct columns $C_1$, $C_2$, $C_3$, there exists an uncolored cell $c$ and integers $1 \le i, j \le 3$ so that $c$ lies in $R_i$ and $C_j$

Let \(n\) be a positive integer and consider an \(n\times n\) square grid. For \(1\le k\le n\), a [i]python[/i] of length \(k\) is a snake that occupies \(k\) consecutive cells in a single row, and no other cells. Similarly, an [i]anaconda[/i] of length \(k\) is a snake that occupies \(k\) consecutive cells in a single column, and no other cells. The grid contains at least one python or anaconda, and it satisfies the following properties: [list] [*]No cell is occupied by multiple snakes. [*]If a cell in the grid is immediately to the left or immediately to the right of a python, then that cell must be occupied by an anaconda. [*]If a cell in the grid is immediately to above or immediately below an anaconda, then that cell must be occupied by a python. [/list] Prove that the sum of the squares of the lengths of the snakes is at least \(n^2\). [i]Proposed by Linus Tang[/i]

Does there exist such a configuration of 22 circles and 22 point, that any circle contains at leats 7 points and any point belongs at least to 7 circles?

Let $P_1, \ldots , P_s$ be arithmetic progressions of integers, the following conditions being satisfied: [b](i)[/b] each integer belongs to at least one of them; [b](ii)[/b] each progression contains a number which does not belong to other progressions. Denote by $n$ the least common multiple of the ratios of these progressions; let $n=p_1^{\alpha_1} \cdots p_k^{\alpha_k}$ its prime factorization. Prove that \[s \geq 1 + \sum^k_{i=1} \alpha_i (p_i - 1).\] [i]Proposed by Dierk Schleicher, Germany[/i]

A two-person game is played with nine boxes arranged in a $3 \times 3$ square and with white and black stones. At each move a player puts three stones, not necessarily of the same colour, in three boxes in either a horizontal or a vertical line. No box can contain stones of different colours: if, for instance, a player puts a white stone in a box containing black stones the white stone and one of the black stones are removed from the box. The game is over when the centrebox and the cornerboxes contain one black stone and the other boxes are empty. At one stage of a game $x$ boxes contained one black stone each and the other boxes were empty. Determine all possible values for $x.$

[b]p1.[/b] Suppose $a_1 \cdot 2 = a_2 \cdot 3 = a_3$ and $a_1 + a_2 + a_3 = 66$. What is $a_3$? [b]p2.[/b] Ankit buys a see-through plastic cylindrical water bottle. However, in coming home, he accidentally hits the bottle against a wall and dents the top portion of the bottle (above the $7$ cm mark). Ankit now wants to determine the volume of the bottle. The area of the base of the bottle is $20$ cm$^2$ . He fills the bottle with water up to the $5$ cm mark. After flipping the bottle upside down, he notices that the height of the empty space is at the $7$ cm mark. Find the total volume (in cm$^3$) of this bottle. [img]https://cdn.artofproblemsolving.com/attachments/1/9/f5735c77b056aaf31b337ea1b777a591807819.png[/img] [b]p3.[/b] If $P$ is a quadratic polynomial with leading coefficient $ 1$ such that $P(1) = 1$, $P(2) = 2$, what is $P(10)$? [b]p4.[/b] Let ABC be a triangle with $AB = 1$, $AC = 3$, and $BC = 3$. Let $D$ be a point on $BC$ such that $BD =\frac13$ . What is the ratio of the area of $BAD$ to the area of $CAD$? [b]p5.[/b] A coin is flipped $ 12$ times. What is the probability that the total number of heads equals the total number of tails? Express your answer as a common fraction in lowest terms. [b]p6.[/b] Moor pours $3$ ounces of ginger ale and $ 1$ ounce of lime juice in cup $A$, $3$ ounces of lime juice and $ 1$ ounce of ginger ale in cup $B$, and mixes each cup well. Then he pours $ 1$ ounce of cup $A$ into cup $B$, mixes it well, and pours $ 1$ ounce of cup $B$ into cup $A$. What proportion of cup $A$ is now ginger ale? Express your answer as a common fraction in lowest terms. [b]p7.[/b] Determine the maximum possible area of a right triangle with hypotenuse $7$. Express your answer as a common fraction in lowest terms. [b]p8.[/b] Debbie has six Pusheens: $2$ pink ones, $2$ gray ones, and $2$ blue ones, where Pusheens of the same color are indistinguishable. She sells two Pusheens each to Alice, Bob, and Eve. How many ways are there for her to do so? [b]p9.[/b] How many nonnegative integer pairs $(a, b)$ are there that satisfy $ab = 90 - a - b$? [b]p10.[/b] What is the smallest positive integer $a_1...a_n$ (where $a_1, ... , a_n$ are its digits) such that $9 \cdot a_1 ... a_n = a_n ... a_1$, where $a_1$, $a_n \ne 0$? [b]p11.[/b] Justin is growing three types of Japanese vegetables: wasabi root, daikon and matsutake mushrooms. Wasabi root needs $2$ square meters of land and $4$ gallons of spring water to grow, matsutake mushrooms need $3$ square meters of land and $3$ gallons of spring water, and daikon need $ 1$ square meter of land and $ 1$ gallon of spring water to grow. Wasabi sell for $60$ per root, matsutake mushrooms sell for $60$ per mushroom, and daikon sell for $2$ per root. If Justin has $500$ gallons of spring water and $400$ square meters of land, what is the maximum amount of money, in dollars, he can make? [b]p12.[/b] A [i]prim [/i] number is a number that is prime if its last digit is removed. A [i]rime [/i] number is a number that is prime if its first digit is removed. Determine how many numbers between $100$ and $999$ inclusive are both prim and rime numbers. [b]p13.[/b] Consider a cube. Each corner is the intersection of three edges; slice off each of these corners through the midpoints of the edges, obtaining the shape below. If we start with a $2\times 2\times 2$ cube, what is the volume of the resulting solid? [img]https://cdn.artofproblemsolving.com/attachments/4/8/856814bf99e6f28844514158344477f6435a3a.png[/img] [b]p14.[/b] If a parallelogram with perimeter $14$ and area $ 12$ is inscribed in a circle, what is the radius of the circle? [b]p15.[/b] Take a square $ABCD$ of side length $1$, and draw $\overline{AC}$. Point $E$ lies on $\overline{BC}$ such that $\overline{AE}$ bisects $\angle BAC$. What is the length of $BE$? [b]p16.[/b] How many integer solutions does $f(x) = (x^2 + 1)(x^2 + 2) + (x^2 + 3)(x + 4) = 2017$ have? [b]p17.[/b] Alice, Bob, Carol, and Dave stand in a circle. Simultaneously, each player selects another player at random and points at that person, who must then sit down. What is the probability that Alice is the only person who remains standing? [b]p18.[/b] Let $x$ be a positive integer with a remainder of $2$ when divided by $3$, $3$ when divided by $4$, $4$ when divided by $5$, and $5$ when divided by $6$. What is the smallest possible such $x$? [b]p19[/b]. A circle is inscribed in an isosceles trapezoid such that all four sides of the trapezoid are tangent to the circle. If the radius of the circle is $ 1$, and the upper base of the trapezoid is $ 1$, what is the area of the trapezoid? [b]p20.[/b] Ray is blindfolded and standing $ 1$ step away from an ice cream stand. Every second, he has a $1/4$ probability of walking $ 1$ step towards the ice cream stand, and a $3/4$ probability of walking $ 1$ step away from the ice cream stand. When he is $0$ steps away from the ice cream stand, he wins. What is the probability that Ray eventually wins? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].