In the fictional country of Mahishmati, there are $50$ cities, including a capital city. Some pairs of cities are connected by two-way flights. Given a city $A$, an ordered list of cities $C_1,\ldots, C_{50}$ is called an [i]antitour[/i] from $A$ if [list] [*] every city (including $A$) appears in the list exactly once, and [*] for each $k\in \{1,2,\ldots, 50\}$, it is impossible to go from $A$ to $C_k$ by a sequence of exactly $k$ (not necessarily distinct) flights. [/list] Baahubali notices that there is an antitour from $A$ for any city $A$. Further, he can take a sequence of flights, starting from the capital and passing through each city exactly once. Find the least possible total number of antitours from the capital city. [i]Proposed by Sutanay Bhattacharya[/i]

A graph is called a [i]self-intersecting [/i]graph if it is isomorphic to a graph whose every edge is a segment and every two edges intersect. Notice that no edge contains a vertex except its two endings. a) Find all $ n$'s for which the cycle of length $ n$ is self-intersecting. b) Prove that in a self-intersecting graph $ |E(G)|\leq|V(G)|$. c) Find all self-intersecting graphs. [img]http://i35.tinypic.com/x43s5u.png[/img]

There are $n \geq 1$ notebooks, numbered from $1$ to $n$, stacked in a pile. Zahar repeats the following operation: he randomly chooses a notebook whose number $k$ does not correspond to its location in this stack, counting from top to bottom, and returns it to the $k$th position, counting from the top, without changing the location of the other notebooks. If there is no such notebook, he stops. Is it guaranteed that Zahar will arrange all the notebooks in ascending order of numbers in a finite number of operations? [i]Proposed by Zahar Naumets[/i]

Determine the largest integer $n\geq 3$ for which the edges of the complete graph on $n$ vertices can be assigned pairwise distinct non-negative integers such that the edges of every triangle have numbers which form an arithmetic progression.

In every row of a grid $100 \times n$ is written a permutation of the numbers $1,2 \ldots, 100$. In one move you can choose a row and swap two non-adjacent numbers with difference $1$. Find the largest possible $n$, such that at any moment, no matter the operations made, no two rows may have the same permutations.

On a board there is a regular polygon $A_1A_2\ldots A_{99}.$ Ana and Barbu alternatively occupy empty vertices of the polygon and write down triangles on a list: Ana only writes obtuse triangles, while Barbu only writes acute ones. At the first turn, Ana chooses three vertices $X,Y$ and $Z$ and writes down $\triangle XYZ.$ Then, Barbu chooses two of $X,Y$ and $Z,$ for example $X$ and $Y$, and an unchosen vertex $T$, and writes down $\triangle XYT.$ The game goes on and at each turn, the player must choose a new vertex $R$ and write down $\triangle PQR$, where $P$ is the last vertex chosen by the other player, and $Q$ is one of the other vertices of the last triangle written down by the other player. If one player cannot perform a move, then the other one wins. If both people play optimally, determine who has a winning strategy.

Let $r\geq 2$ be a positive integer, and let each positive integer be painted in one of $r$ different colors. For every positive integer $n$ and every pair of colors $a$ and $b$, if the difference between the number of divisors of $n$ that are painted in color $a$ and the number of divisors of $n$ that are painted in color $b$ is at most $1$, find all possible values of $r$.

Three numbered tiles are arranged in a tray as shown: [img]https://cdn.artofproblemsolving.com/attachments/d/0/d449364f92b7fae971fd348a82bafd25aa8ea1.jpg[/img] Show that we cannot interchange the $1$ and the $3$ by a sequence of moves where we slide a tile to the adjacent vacant space.

All positive integers are distributed among two disjoint sets $N_{1}$ and $N_{2}$ such that no difference of two numbers belonging to the same set is a prime greater than 100. Find all such distributions. [i]Proposed by N. Sedrakyan[/i]

Two squirrels, Bushy and Jumpy, have collected $2021$ walnuts for the winter. Jumpy numbers the walnuts from 1 through $2021$, and digs $2021$ little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts, and Bushy decides to interfere with Jumpy. The two take turns to reorder the walnuts. Each time, Bushy chooses $1232$ walnuts and reorders them and then Jumpy chooses $n$ walnuts to reorder. Find the least positive integer $n$ such that whatever Bushy does, Jumpy can ensure that the $i$th hole from the left has the $i$th walnut [i]Ift0501[/i]

Twenty cubes have been coloured in the following way: There are two red faces opposite each other, two blue faces opposite each other and two green faces opposite each other. The cubes have been glued together as shown in the figure. Two faces that are glued together always have the same colour. The figure shows the colours of some of the faces. Which colours are possible for the face marked with the symbol $\times$? [img]https://cdn.artofproblemsolving.com/attachments/8/2/6127db5bfdce7a749d730fe3626499582f62ba.png[/img]

There are $n$ matches lying on a table, forming $n$ one-match piles. Adam wants to combine them into a single pile of $n$ matches. He will do this using $n-1$ operations, each of which consists of combining two piles into one. Adam has made a deal with Bartek that every time he combines a pile of $a$ matches with a pile of $b$ matches, he will receive $a\cdot b$ candies from Bartek. What is the greatest number of candies that Adam can receive after performing $n-1$ operations? Justify your answer.

[u]Set 1[/u] [b]p1.[/b] Assume the speed of sound is $343$ m/s. Anastasia and Bananastasia are standing in a field in front of you. When they both yell at the same time, you hear Anastasia’s yell $5$ seconds before Bananastasia’s yell. If Bananastasia yells first, and then Anastasia yells when she hears Bananastasia yell, you hear Anastasia’s yell $5$ seconds after Bananastasia’s yell. What is the distance between Anastasia and Bananastasia in meters? [b]p2.[/b] Michelle picks a five digit number with distinct digits. She then reverses the digits of her number and adds that to her original number. What is the largest possible sum she can get? [b]p3.[/b] Twain is trying to crack a $4$-digit number combination lock. They know that the second digit must be even, the third must be odd, and the fourth must be different from the previous three. If it takes Twain $10$ seconds to enter a combination, how many hours would it take them to try every possible combination that satisfies these rules? PS. You should use hide for answers.

Twenty cubes of the same size and appearance are made of either aluminum or of heavier duralumin. How can one find the number of duralumin cubes using not more than $11$ weighings on a balance without weights? (We assume that all cubes can be made of aluminum, but not all of duralumin.)

Two friends $A$ and $B$ must solve the following puzzle. Each of them receives a number from the set $\{1,2,…,250\}$, but they don’t see the number that the other received. The objective of each friend is to discover the other friend’s number. The procedure is as follows: each friend, by turns, announces various not necessarily distinct positive integers: first $A$ says a number, then $B$ says one, $A$ says a number again, etc., in such a way that the sum of all the numbers said is $20$. Demonstrate that there exists a strategy that $A$ and $B$ have previously agreed on such that they can reach the objective, no matter which number each one received at the beginning of the puzzle.

Let $S$ be the set of $10$-tuples of non-negative integers that have sum $2019$. For any tuple in $S$, if one of the numbers in the tuple is $\geq 9$, then we can subtract $9$ from it, and add $1$ to the remaining numbers in the tuple. Call thus one operation. If for $A,B\in S$ we can get from $A$ to $B$ in finitely many operations, then denote $A\rightarrow B$. (1) Find the smallest integer $k$, such that if the minimum number in $A,B\in S$ respectively are both $\geq k$, then $A\rightarrow B$ implies $B\rightarrow A$. (2) For the $k$ obtained in (1), how many tuples can we pick from $S$, such that any two of these tuples $A,B$ that are distinct, $A\not\rightarrow B$.

A sequence $(x_n)_{n= 1}^{\infty}$ satisfies $x_1 = 1$ and for each $n > 1, x_n = \pm (n-1)x_{n-1} \pm (n-2)x_{n-2} \pm ... \pm 2x_2 \pm x_1$. Prove that the signs ” $\pm$” can be chosen so that $x_n \ne 12$ holds only for finitely many $n$.

Sabine has a very large collection of shells. She decides to give part of her collection to her sister. On the first day, she lines up all her shells. She takes the shells that are in a position that is a perfect square (the first, fourth, ninth, sixteenth, etc. shell), and gives them to her sister. On the second day, she lines up her remaining shells. Again, she takes the shells that are in a position that is a perfect square, and gives them to her sister. She repeats this process every day. The $27$th day is the first day that she ends up with fewer than $1000$ shells. The $28$th day she ends up with a number of shells that is a perfect square for the tenth time. What are the possible numbers of shells that Sabine could have had in the very beginning?

Let $d$ be an even positive integer. John writes the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2 $ on the blackboard and then chooses three of them, let them be ${a_1}, {a_2}, {a_3}$, erases them and writes the number $1+ \displaystyle\sum_{1\le i<j\leq 3} |{a_i} -{a_j}|$ He continues until two numbers remain written on on the blackboard. Prove that the sum of squares of those two numbers is different than the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2$. (Albania)

How many 3-digit numbers have the property that the sum of their digits is even?

Let $n \ge 2$ be an integer. There are $n$ houses in a town. All distances between pairs of houses are different. Every house sends a visitor to the house closest to it. Find all possible values of $n$ (with full justification) for which we can design a town with $n$ houses where every house is visited.

Determine the largest natural number $ N $ having the following property: every $ 5\times 5 $ array consisting of pairwise distinct natural numbers from $ 1 $ to $ 25 $ contains a $ 2\times 2 $ subarray of numbers whose sum is, at least, $ N. $ [i]Demetres Christofides[/i] and [i]Silouan Brazitikos[/i]

Peter plays a solitaire game with a deck of cards, some of which are face-up while the others are face-down. Peter loses if all the cards are face-down. As long as at least one card is face up, Peter must choose a stack of consecutive cards from the deck, so that the top and the bottom cards of the stack are face-up. They may be the same card. Then Peter turns the whole stack over and puts it back into the deck in exactly the same place as before. Prove that Peter always loses. (A Shapovalov)

Problem: A person has seven friends and invites a different subset of three friends to dinner every night for one week (seven days). In how many ways can this be done so that all friends are invited at least once?

[u]Round 1[/u] [b]p1.[/b] At a fortune-telling exam, $13$ witches are sitting in a circle. To pass the exam, a witch must correctly predict, for everybody except herself and her two neighbors, whether they will pass or fail. Each witch predicts that each of the $10$ witches she is asked about will fail. How many witches could pass? [b]p2.[/b] Out of $152$ coins, $7$ are counterfeit. All counterfeit coins have the same weight, and all real coins have the same weight, but counterfeit coins are lighter than real coins. How can you find $19$ real coins if you are allowed to use a balance scale three times? [b]p3.[/b] The digits of a number $N$ increase from left to right. What could the sum of the digits of $9 \times N$ be? [b]p4.[/b] The sides and diagonals of a pentagon are colored either blue or red. You can choose three vertices and flip the colors of all three lines that join them. Can every possible coloring be turned all blue by a sequence of such moves? [img]https://cdn.artofproblemsolving.com/attachments/5/a/644aa7dd995681fc1c813b41269f904283997b.png[/img] [b]p5.[/b] You have $100$ pancakes, one with a single blueberry, one with two blueberries, one with three blueberries, and so on. The pancakes are stacked in a random order. Count the number of blueberries in the top pancake and call that number $N$. Pick up the stack of the top $N$ pancakes and flip it upside down. Prove that if you repeat this counting-and-flipping process, the pancake with one blueberry will eventually end up at the top of the stack. [u]Round 2[/u] [b]p6.[/b] A circus owner will arrange $100$ fleas on a long string of beads, each flea on her own bead. Once arranged, the fleas start jumping using the following rules. Every second, each flea chooses the closest bead occupied by one or more of the other fleas, and then all fleas jump simultaneously to their chosen beads. If there are two places where a flea could jump, she jumps to the right. At the start, the circus owner arranged the fleas so that, after some time, they all gather on just two beads. What is the shortest amount of time it could take for this to happen? [b]p7.[/b] The faraway land of Noetheria has $2016$ cities. There is a nonstop flight between every pair of cities. The price of a nonstop ticket is the same in both directions, but flights between different pairs of cities have different prices. Prove that you can plan a route of $2015$ consecutive flights so that each flight is cheaper than the previous one. It is permissible to visit the same city several times along the way. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].