Alice and Bob play the following game: They start with non-empty piles of coins. Taking turns, with Alice playing first, each player choose a pile with an even number of coins and moves half of the coins of this pile to the other pile. The game ends if a player cannot move, in which case the other player wins. Determine all pairs $(a,b)$ of positive integers such that if initially the two piles have $a$ and $b$ coins respectively, then Bob has a winning strategy. Proposed by Dimitris Christophides, Cyprus

Write a positive integer in each box so that: All six numbers are different. The sum of the six numbers is $100$. If each number is multiplied by its neighbor (in a clockwise direction) and the six results of those six multiplications are added, the smallest possible value is obtained. Explain why a lower value cannot be obtained. [img]https://cdn.artofproblemsolving.com/attachments/7/1/6fdadd6618f91aa03cdd6720cc2d6ee296f82b.gif[/img]

There are $N\ge3$ letters arranged in a circle, and each letter is one of $L$, $T$ and $F$. For a letter, we can do the following operation: if its neighbors are the same, then change it to the same letter too; otherwise, change it so that it is different from both its neighbors. Show that for any initial state, one can perform finitely many operations to achieve a stable state. Here, a stable state means that any operation does not change any of the $N$ letters. (ltf0501)

Trilandia is a very unusual city. The city has the shape of an equilateral triangle of side lenght 2012. The streets divide the city into several blocks that are shaped like equilateral triangles of side lenght 1. There are streets at the border of Trilandia too. There are 6036 streets in total. The mayor wants to put sentinel sites at some intersections of the city to monitor the streets. A sentinel site can monitor every street on which it is located. What is the smallest number of sentinel sites that are required to monitor every street of Trilandia?

[b]p1.[/b] Three camps are located in the vertices of an equilateral triangle. The roads connecting camps are along the sides of the triangle. Captain America is inside the triangle and he needs to know the distances between camps. Being able to see the roads he has found that the sum of the shortest distances from his location to the roads is 50 miles. Can you help Captain America to evaluate the distances between the camps? [b]p2.[/b] $N$ regions are located in the plane, every pair of them have a non-empty overlap. Each region is a connected set, that means every two points inside the region can be connected by a curve all points of which belong to the region. Iron Man has one charge remaining to make a laser shot. Is it possible for him to make the shot that goes through all $N$ regions? [b]p3.[/b] Money in Wonderland comes in $\$5$ and $\$7$ bills. (a) What is the smallest amount of money you need to buy a slice of pizza that costs $\$1$ and get back your change in full? (The pizza man has plenty of $\$5$ and $\$7$ bills.) For example, having $\$7$ won't do since the pizza man can only give you $\$5$ back. (b) Vending machines in Wonderland accept only exact payment (do not give back change). List all positive integer numbers which CANNOT be used as prices in such vending machines. (That is, find the sums of money that cannot be paid by exact change.) [b]p4.[/b] (a) Put $5$ points on the plane so that each $3$ of them are vertices of an isosceles triangle (i.e., a triangle with two equal sides), and no three points lie on the same line. (b) Do the same with $6$ points. [b]p5.[/b] Numbers $1,2,3,…,100$ are randomly divided in two groups $50$ numbers in each. In the first group the numbers are written in increasing order and denoted $a_1,a_2, ..., a_{50}$. In the second group the numberss are written in decreasing order and denoted $b_1,b_2, ..., b_{50}$. Thus $a_1<a_2<...<a_{50}$ and $ b_1>b_2>...>b_{50}$. Evaluate $|a_1-b_1|+|a_2-b_2|+...+|a_{50}-b_{50}|$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The harmonic table is a triangular array: $1$ $\frac 12 \qquad \frac 12$ $\frac 13 \qquad \frac 16 \qquad \frac 13$ $\frac 14 \qquad \frac 1{12} \qquad \frac 1{12} \qquad \frac 14$ Where $a_{n,1} = \frac 1n$ and $a_{n,k+1} = a_{n-1,k} - a_{n,k}$ for $1 \leq k \leq n-1.$ Find the harmonic mean of the $1985^{th}$ row.

Nine skiers left the start line in turn and covered the distance, each at their own constant speed. Could it turn out that each skier participated in exactly four overtakes? (In each overtaking, exactly two skiers participate - the one who is overtaking, and the one who is being overtaken.)

Let $n\ge 2$ be a positive integer. Find whether there exist $n$ pairwise nonintersecting nonempty subsets of $\{1, 2, 3, \ldots \}$ such that each positive integer can be expressed in a unique way as a sum of at most $n$ integers, all from different subsets.

Among all possible representations of the positive integer $ n$ as $ n\equal{}\sum_{i\equal{}1}^k a_i$ with positive integers $ k, a_1 < a_2 < ...<a_k$, when will the product $ \prod_{i\equal{}1}^k a_i$ be maximum?

There are $19$ lines in the plane dividing the plane into exactly $97$ pieces. (a) Prove that among these pieces there is at least one triangle. (b) Show that it is indeed possible to place $19$ lines in the above way.

A company has a secret safe box that is locked by six locks. Several copies of the keys are distributed among the directors of the company. Each key can unlock exactly one lock. Each director has three keys for three different locks. No two directors can unlock the same three locks. No two directors together can unlock the safe. What is the maximum possible number of directors in the company?

Positive integers $n$ and $k$ satisfying $n\geq 2k+1$ are known to Alice. There are $n$ cards with numbers from $1$ to $n$, randomly shuffled as a deck, face down. On her turn, she does the following in order: (i) She first flips over the top card of the deck, and puts it face up on the table. (ii) Then, if Alice has not signed any card, she can sign the newest card now. The game ends after $2k+1$ turns, and Alice must have signed on some card. Let $A$ be the number on the signed cards, and $M$ be the $(k+1)^{\textup{st}}$ largest number among all $2k+1$ face-up cards. Alice's score is $|M-A|$, and she wants the score to be as close to zero as possible. For each $(n,k)$, find the smallest integer $d=d(n,k)$ such that Alice has a strategy to guarantee her score no greater than $d$. [i]Proposed by usjl[/i]

How many coins can be placed on a $10 \times 10$ board (each at the center of its square, at most one per square) so that no four coins form a rectangle with sides parallel to the sides of the board?

Each edge of a complete graph with $101$ vertices is marked with $1$ or $-1$. It is known that absolute value of the sum of numbers on all the edges is less than $150$. Prove that the graph contains a path visiting each vertex exactly once such that the sum of numbers on all edges of this path is zero. [i](Y. Caro, A. Hansberg, J. Lauri, C. Zarb)[/i]

Determine the greatest positive integer $k$ that satisfies the following property: The set of positive integers can be partitioned into $k$ subsets $A_1, A_2, \ldots, A_k$ such that for all integers $n \geq 15$ and all $i \in \{1, 2, \ldots, k\}$ there exist two distinct elements of $A_i$ whose sum is $n.$ [i]Proposed by Igor Voronovich, Belarus[/i]

Sheldon was really annoying Leonard. So to keep him quiet, Leonard decided to do something. He gave Sheldon the following grid $\begin{tabular}{|c|c|c|c|c|c|} \hline 1 & 1 & 1 & 1 & 1 & 0\\ \hline 1 & 1 & 1 & 1 & 0 & 0\\ \hline 1 & 1 & 1 & 0 & 0 & 0\\ \hline 1 & 1 & 0 & 0 & 0 & 1\\ \hline 1 & 0 & 0 & 0 & 1 & 0\\ \hline 0 & 0 & 0 & 1 & 0 & 0\\ \hline \end{tabular}$ and asked him to transform it to the new grid below $\begin{tabular}{|c|c|c|c|c|c|} \hline 1 & 2 & 18 &24 &28 &30\\ \hline 21 & 3 & 4 &16 &22 &26\\ \hline 23 &19 & 5 & 6 &14 &20\\ \hline 32 &25 &17 & 7 & 8 &12\\ \hline 33 &34 &27 &15 & 9 &10\\ \hline 35 &31 &36 &29 &13 &11\\ \hline \end{tabular}$ by only applying the following algorithm: $\bullet$ At each step, Sheldon must choose either two rows or two columns. $\bullet$ For two columns $c_1, c_2$, if $a,b$ are entries in $c_1, c_2$ respectively, then we say that $a$ and $b$ are corresponding if they belong to the same row. Similarly we define corresponding entries of two rows. So for Sheldon's choice, if two corresponding entries have the same parity, he should do nothing to them, but if they have different parities, he should add 1 to both of them. Leonard hoped this would keep Sheldon occupied for some time, but Sheldon immediately said, "But this is impossible!". Was Sheldon right? Justify.

Colour a $20000\times 20000$ square grid using 2000 different colours with 1 colour in each square. Two squares are neighbours if they share a vertex. A path is a sequence of squares so that 2 successive squares are neighbours. Mark $k$ of the squares. For each unmarked square $x$, there is exactly 1 marked square $y$ of the same colour so that $x$ and $y$ are connected by a path of squares of the same colour. For any 2 marked squares of the same colour, any path connecting them must pass through squares of all the colours. Find the maximum value of $k$.

$A$ and $B$ play a game, given an integer $N$, $A$ writes down $1$ first, then every player sees the last number written and if it is $n$ then in his turn he writes $n+1$ or $2n$, but his number cannot be bigger than $N$. The player who writes $N$ wins. For which values of $N$ does $B$ win? [i]Proposed by A. Slinko & S. Marshall, New Zealand[/i]

If you have $65$ points in a plane, we will make the lines that passes by any two points in this plane and we obtain exactly $2015$ distinct lines, prove that least $4$ points are collinears!!

Consider all sets of $n$ distinct positive integers, no three of which form an arithmetic progression. Prove that among all such sets there is one which has the largest sum of the reciprocals of its elements.

On the infinite chessboard several rectangular pieces are placed whose sides run along the grid lines. Each two have no squares in common, and each consists of an odd number of squares. Prove that these pieces can be painted in four colours such that two pieces painted in the same colour do not share any boundary points.

Let $X$ be a set of $11$ integers. Prove that one can find a nonempty subset $\{a_1, a_2, \cdots , a_k \}$ of $X$ such that $3$ divides $k$ and $9$ divides the sum $\sum_{i=1}^{k} 4^i a_i$.

There are $13$ marked points on the circumference of a circle with radius $13$. Prove that we can choose three of the marked points which form a triangle with area less than $13$.

Markers in the colors violet, cyan, octarine and gamma were placed on all fields of a $41\times 5$ chessboard. Show that there are four squares of the same color that form the vertices of a rectangle whose edges are parallel to those of the board.

Let $n\ge 2$ be a positive integer, and consider an $n\times n$ grid of $n^2$ equilateral triangles. Two triangles are adjacent if they share at least one vertex. Each triangle is colored red or blue, splitting the grid into regions. Find, with proof, the minimum number of triangles in the largest region. [i]Rohan Bodke[/i]