Let $S = \{1,..., 999\}$. Determine the smallest integer $m$. for which there exist $m$ two-sided cards $C_1$,..., $C_m$ with the following properties: $\bullet$ Every card $C_i$ has an integer from $S$ on one side and another integer from $S$ on the other side. $\bullet$ For all $x,y \in S$ with $x\ne y$, it is possible to select a card $C_i$ that shows $x$ on one of its sides and another card $C_j$ (with $i \ne j$) that shows $y$ on one of its sides.

An economist and a statistician play a game on a calculator which does only one operation. The calculator displays only positive integers and it is used in the following way: Denote by $n$ an integer that is shown on the calculator. A person types an integer, $m$, chosen from the set $\{ 1, 2, . . . , 99 \}$ of the first $99$ positive integers, and if $m\%$ of the number $n$ is again a positive integer, then the calculator displays $m\%$ of $n$. Otherwise, the calculator shows an error message and this operation is not allowed. The game consists of doing alternatively these operations and the player that cannot do the operation looses. How many numbers from $\{1, 2, . . . , 2019\}$ guarantee the winning strategy for the statistician, who plays second? For example, if the calculator displays $1200$, the economist can type $50$, giving the number $600$ on the calculator, then the statistician can type $25$ giving the number $150$. Now, for instance, the economist cannot type $75$ as $75\%$ of $150$ is not a positive integer, but can choose $40$ and the game continues until one of them cannot type an allowed number [i]Proposed by Serbia [/i]

Let $n \geq 3$ be an integer. John and Mary play the following game: First John labels the sides of a regular $n$-gon with the numbers $1, 2,\ldots, n$ in whatever order he wants, using each number exactly once. Then Mary divides this $n$-gon into triangles by drawing $n-3$ diagonals which do not intersect each other inside the $n$-gon. All these diagonals are labeled with number $1$. Into each of the triangles the product of the numbers on its sides is written. Let S be the sum of those $n - 2$ products. Determine the value of $S$ if Mary wants the number $S$ to be as small as possible and John wants $S$ to be as large as possible and if they both make the best possible choices.

We have $2015$ marbles in a box, where each marble has one color from red, green or blue. At each step, we are allowed to take $2$ different colored marbles, then replace it with $2$ marbles with the third color. For example, we take one blue marble and one green marble, and we fill with $2$ red marbles. Prove that we can always do a series of steps so that all marbles in the box have the same color.

[u]Round 1[/u] [b]p1.[/b] Nineteen witches, all of different heights, stand in a circle around a campfire. Each witch says whether she is taller than both of her neighbors, shorter than both, or in-between. Exactly three said “I am taller.” How many said “I am in-between”? [b]p2.[/b] Alex is writing a sequence of $A$’s and $B$’s on a chalkboard. Any $20$ consecutive letters must have an equal number of $A$’s and $B$’s, but any 22 consecutive letters must have a different number of $A$’s and $B$’s. What is the length of the longest sequence Alex can write?. [b]p3.[/b] A police officer patrols a town whose map is shown. The officer must walk down every street segment at least once and return to the starting point, only changing direction at intersections and corners. It takes the officer one minute to walk each segment. What is the fastest the officer can complete a patrol? [img]https://cdn.artofproblemsolving.com/attachments/a/3/78814b37318adb116466ede7066b0d99d6c64d.png[/img] [b]p4.[/b] A zebra is a new chess piece that jumps in the shape of an “L” to a location three squares away in one direction and two squares away in a perpendicular direction. The picture shows all the moves a zebra can make from its given position. Is it possible for a zebra to make a sequence of $64$ moves on an $8\times 8$ chessboard so that it visits each square exactly once and returns to its starting position? [img]https://cdn.artofproblemsolving.com/attachments/2/d/01a8af0214a2400b279816fc5f6c039320e816.png[/img] [b]p5.[/b] Ann places the integers $1, 2,..., 100$ in a $10 \times 10$ grid, however she wants. In each round, Bob picks a row or column, and Ann sorts it from lowest to highest (left-to-right for rows; top-to-bottom for columns). However, Bob never sees the grid and gets no information from Ann. After eleven rounds, Bob must name a single cell that is guaranteed to contain a number that is at least $30$ and no more than $71$. Can he find a strategy to do this, no matter how Ann originally arranged the numbers? [u]Round 2[/u] [b]p6.[/b] Evelyn and Odette are playing a game with a deck of $101$ cards numbered $1$ through $101$. At the start of the game the deck is split, with Evelyn taking all the even cards and Odette taking all the odd cards. Each shuffles her cards. On every move, each player takes the top card from her deck and places it on a table. The player whose number is higher takes both cards from the table and adds them to the bottom of her deck, first the opponent’s card, then her own. The first player to run out of cards loses. Card $101$ was played against card $2$ on the $10$th move. Prove that this game will never end. [img]https://cdn.artofproblemsolving.com/attachments/8/1/aa16fe1fb4a30d5b9e89ac53bdae0d1bdf20b0.png[/img] [b]p7.[/b] The Vogon spaceship Tempest is descending on planet Earth. It will land on five adjacent buildings within a $10 \times 10$ grid, crushing any teacups on roofs of buildings within a $5 \times 1$ length of blocks (vertically or horizontally). As Commander of the Space Force, you can place any number of teacups on rooftops in advance. When the ship lands, you will hear how many teacups the spaceship breaks, but not where they were. (In the figure, you would hear $4$ cups break.) What is the smallest number of teacups you need to place to ensure you can identify at least one building the spaceship landed on? [img]https://cdn.artofproblemsolving.com/attachments/8/7/2a48592b371bba282303e60b4ff38f42de3551.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that the circle with radius $2$ can be completely covered with $7$ unit circles

For positive integers $a$ and $b$, an $(a,b)$-shuffle of a deck of $a+b$ cards is any shuffle that preserves the relative order of the top $a$ cards and the relative order of the bottom $b$ cards. Let $n$, $k$, $a_1$, $a_2$, $\dots$, $a_k$, $b_1$, $b_2$, $\dots$, $b_k$ be fixed positive integers such that $a_i+b_i=n$ for all $1\leq i\leq k$. Big Bird has a deck of $n$ cards and will perform an $(a_i,b_i)$-shuffle for each $1\leq i\leq k$, in ascending order of $i$. Suppose that Big Bird can reverse the order of the deck. Prove that Big Bird can also achieve any of the $n!$ permutations of the cards. [i]Linus Tang[/i]

There are two airlines A and B and finitely many airports. For each pair of airports, there is exactly one airline among A and B whose flights operates in both directions. Each airline plans to develop world travel packages which pass each airport exactly once using only its flights. Let $a$ and $b$ be the number of possible packages which belongs to A and B respectively. Prove that $a-b$ is a multiple of $4$. The official statement of the problem has been changed. The above is the form which appeared during the contest. Now the condition 'the number of airports is no less than 4'is attached. Cite the following link. [url]https://artofproblemsolving.com/community/c6h2923697p26140823[/url]

Compute the number of triples $(f,g,h)$ of permutations on $\{1,2,3,4,5\}$ such that \begin{align*} & f(g(h(x))) = h(g(f(x))) = g(x) \\ & g(h(f(x))) = f(h(g(x))) = h(x), \text{ and } \\ & h(f(g(x))) = g(f(h(x))) = f(x), \\ \end{align*} for all $x\in \{1,2,3,4,5\}$.

Every positive integer can be represented as a sum of one or more consecutive positive integers. For each $n$ , find the number of such represententation of $n$.

A circle is divided into $n$ sectors ($n \geq 3$). Each sector can be filled in with either $1$ or $0$. Choose any sector $\mathcal{C}$ occupied by $0$, change it into a $1$ and simultaneously change the symbols $x, y$ in the two sectors adjacent to $\mathcal{C}$ to their complements $1-x$, $1-y$. We repeat this process as long as there exists a zero in some sector. In the initial configuration there is a $0$ in one sector and $1$s elsewhere. For which values of $n$ can we end this process?

Two digits one are written at the ends of a segment. In the middle, their sum is written, the number 2. Then, in the middle between each two neighboring numbers written, their sum is written again, and so on, 1973 times. How many times will the number 1973 be written? [i]Proposed by G. Halperin[/i]

$n\geq2$ and $E=\left \{ 1,2,...,n \right \}. A_1,A_2,...,A_k$ are subsets of $E$, such that for all $1\leq{i}<{j}\leq{k}$ Exactly one of $A_i\cap{A_j},A_i'\cap{A_j},A_i\cap{A_j'},A_i'\cap{A_j'}$ is empty set. What is the maximum possible $k$?

[b]p1.[/b] James has $8$ Instagram accounts, $3$ Facebook accounts, $4$ QQ accounts, and $3$ YouTube accounts. If each Instagram account has $19$ pictures, each Facebook account has $5$ pictures and $9$ videos, each QQ account has a total of $17$ pictures, and each YouTube account has $13$ videos and no pictures, how many pictures in total does James have in all these accounts? [b]p2.[/b] If Poonam can trade $7$ shanks for $4$ shinks, and she can trade $10$ shinks for $17$ shenks. How many shenks can Poonam get if she traded all of her $105$ shanks? [b]p3.[/b] Jerry has a bag with $3$ red marbles, $5$ blue marbles and $2$ white marbles. If Jerry randomly picks two marbles from the bag without replacement, the probability that he gets two different colors can be expressed as a fraction $\frac{m}{n}$ in lowest terms. What is $m + n$? [b]p4.[/b] Bob's favorite number is between $1200$ and $4000$, divisible by $5$, has the same units and hundreds digits, and the same tens and thousands digits. If his favorite number is even and not divisible by $3$, what is his favorite number? [b]p5.[/b] Consider a unit cube $ABCDEFGH$. Let $O$ be the center of the face $EFGH$. The length of $BO$ can be expressed in the form $\frac{\sqrt{a}}{b}$, where $a$ and $b$ are simplified to lowest terms. What is $a + b$? [b]p6.[/b] Mr. Eddie Wang is a crazy rich boss who owns a giant company in Singapore. Even though Mr. Wang appears friendly, he finds great joy in firing his employees. His immediately fires them when they say "hello" and/or "goodbye" to him. It is well known that $1/2$ of the total people say "hello" and/or "goodbye" to him everyday. If Mr. Wang had $2050$ employees at the end of yesterday, and he hires $2$ new employees at the beginning of each day, in how many days will Mr. Wang first only have $6$ employees left? [b]p7.[/b] In $\vartriangle ABC$, $AB = 5$, $AC = 6$. Let $D,E,F$ be the midpoints of $\overline{BC}$, $\overline{AC}$, $\overline{AB}$, respectively. Let $X$ be the foot of the altitude from $D$ to $\overline{EF}$. Let $\overline{AX}$ intersect $\overline{BC}$ at $Y$ . Given $DY = 1$, the length of $BC$ is $\frac{p}{q}$ for relatively prime positive integers $p, q$: Find $p + q$. [b]p8.[/b] Given $\frac{1}{2006} = \frac{1}{a} + \frac{1}{b}$ where $a$ is a $4$ digit positive integer and $b$ is a $6$ digit positive integer, find the smallest possible value of $b$. [b]p9.[/b] Pocky the postman has unlimited stamps worth $5$, $6$ and $7$ cents. However, his post office has two very odd requirements: On each envelope, an odd number of $7$ cent stamps must be used, and the total number of stamps used must also be odd. What is the largest amount of postage money Pocky cannot make with his stamps, in cents? [b]p10.[/b] Let $ABCDEF$ be a regular hexagon with side length $2$. Let $G$ be the midpoint of side $DE$. Now let $O$ be the intersection of $BG$ and $CF$. The radius of the circle inscribed in triangle $BOC$ can be expressed in the form $\frac{a\sqrt{b}-\sqrt{c}}{d} $ where $a$, $b$, $c$, $d$ are simplified to lowest terms. What is $a + b + c + d$? [b]p11.[/b] Estimation (Tiebreaker): What is the total number of characters in all of the participants' email addresses in the Accuracy Round? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a convex polygon $P$ some diagonals have been drawn, without intersections inside $P$. Show that there exist at least two vertices of $P$, neither one of them being an endpoint of any one of those diagonals.

Let $n$ be a fixed positive integer. The points $A_1$, $A_2$, $\ldots$, $A_{2n}$ are on a straight line. Color each point blue or red according to the following procedure: draw $n$ pairwise disjoint circumferences, each with diameter $A_iA_j$ for some $i \neq j$ and such that every point $A_k$ belongs to exactly one circumference. Points in the same circumference must be of the same color. Determine the number of ways of coloring these $2n$ points when we vary the $n$ circumferences and the distribution of the colors.

Six straight lines are given in space. Among any three of them, two are perpendicular. Show that the given lines can be labeled $\ell_1,...,\ell_6$ in such a way that $\ell_1, \ell_2, \ell_3$ are pairwise perpendicular, and so are $\ell_4, \ell_5, \ell_6$.

Seyed has 998 white coins, a red coin, and an unusual coin with one red side and one white side. He can not see the color of the coins instead he has a scanner which checks if all of the coin sides touching the scanner glass are white. Is there any algorithm to find the red coin by using the scanner at most 17 times? [i]Proposed by Seyed Reza Hosseini[/i]

An island has $10$ cities, where some of the possible pairs of cities are connected by roads. A [i]tour route[/i] is a route starting from a city, passing exactly eight out of the other nine cities exactly once each, and returning to the starting city. (In other words, it is a loop that passes only nine cities instead of all ten cities.) For each city, there exists a tour route that doesn't pass the given city. Find the minimum number of roads on the island.

Let $S$ be a finite set of points on a plane, where no three points are collinear, and the convex hull of $S$, $\Omega$, is a $2016-$gon $A_1A_2\ldots A_{2016}$. Every point on $S$ is labelled one of the four numbers $\pm 1,\pm 2$, such that for $i=1,2,\ldots , 1008,$ the numbers labelled on points $A_i$ and $A_{i+1008}$ are the negative of each other. Draw triangles whose vertices are in $S$, such that any two triangles do not have any common interior points, and the union of these triangles is $\Omega$. Prove that there must exist a triangle, where the numbers labelled on some two of its vertices are the negative of each other.

Let $m$ boxes be given, with some balls in each box. Let $n < m$ be a given integer. The following operation is performed: choose $n$ of the boxes and put $1$ ball in each of them. Prove: [i](a) [/i]If $m$ and $n$ are relatively prime, then it is possible, by performing the operation a finite number of times, to arrive at the situation that all the boxes contain an equal number of balls. [i](b)[/i] If $m$ and $n$ are not relatively prime, there exist initial distributions of balls in the boxes such that an equal distribution is not possible to achieve.

In an $m\times n$ rectangular chessboard,there is a stone in the lower leftmost square. Two persons A,B move the stone alternately. In each step one can move the stone upward or rightward any number of squares. The one who moves it into the upper rightmost square wins. Find all $(m,n)$ such that the first person has a winning strategy.

Let $n$ be a positive integer. In a coordinate grid, a path from $(0,0)$ to $(2n,2n)$ consists of $4n$ consecutive unit steps $(1,0)$ or $(0,1)$. Prove that the number of paths that divide the square with vertices $(0,0),(2n,0),(2n,2n),(0,2n)$ into 2 regions with even areas is $$\frac{{4n \choose 2n} + {2n \choose n}}{2}$$

Let $a_1<a_2<\cdots<a_n$ be positive integers. Some colouring of $\mathbb{Z}$ is periodic with period $t$ such that for each $x\in \mathbb{Z}$ exactly one of $x+a_1,x+a_2,\dots,x+a_n$ is coloured. Prove that $n\mid t$. [i]Andrei Radulescu-Banu[/i]

If there are four numbers $(a,b,c,d)$ in four registers of the calculating machine, they turn into $(a-b,b-c,c-d,d-a)$ numbers whenever you press the button. Prove that if not all the initial numbers are equal, machine will obtain at least one number more than $1985$ after some number of the operations.