Given a positive integer number $n$, determine the maximum number of edges a triangle-free Hamiltonian simple graph on $n$ vertices may have.

Haydn picks two different integers between $1$ and $100$, inclusive, uniformly at random. The probability that their product is divisible by $4$ can be expressed in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

An urn contain a number of yellow and green balls. You extract two balls from the urn (without adding them back) and calculate the probability of both balls being green. Can you choose the number of yellow and green balls such that this probability to be $\frac{1}{4}$?

Eight strangers are preparing to play bridge. How many ways can they be grouped into two bridge games - that is, into unordered pairs of unordered pairs of people?

[b]p1.[/b] The clock is $2$ hours $20$ minutes ahead of the correct time each week. The clock is set to the correct time at midnight Sunday to Monday. What time does this clock show at 6pm correct time on Thursday? [b]p2.[/b] Five cities $A,B,C,D$, and $E$ are located along the straight road in the alphabetical order. The sum of distances from $B$ to $A,C,D$ and $E$ is $20$ miles. The sum of distances from $C$ to the other four cities is $18$ miles. Find the distance between $B$ and $C$. [b]p3.[/b] Does there exist distinct digits $a, b, c$, and $d$ such that $\overline{abc}+\overline{c} = \overline{bda}$? Here $\overline{abc}$ means the three digit number with digits $a, b$, and $c$. [b]p4.[/b] Kuzya, Fyokla, Dunya, and Senya participated in a mathematical competition. Kuzya solved $8$ problems, more than anybody else. Senya solved $5$ problem, less than anybody else. Each problem was solved by exactly $3$ participants. How many problems were there? [b]p5.[/b] Mr Mouse got to the cellar where he noticed three heads of cheese weighing $50$ grams, $80$ grams, and $120$ grams. Mr. Mouse is allowed to cut simultaneously $10$ grams from any two of the heads and eat them. He can repeat this procedure as many times as he wants. Can he make the weights of all three pieces equal? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n\geq 2$ be an integer. Consider $2n$ points around a circle. Each vertex has been tagged with one integer from $1$ to $n$, inclusive, and each one of these integers has been used exactly two times. Isabel divides the points into $n$ pairs, and draws the segments joining them, with the condition that the segments do not intersect. Then, she assigns to each segment the greatest integer between its endpoints. a) Show that, no matter how the points have been tagged, Isabel can always choose the pairs in such a way that she uses exactly $\lceil n/2\rceil$ numbers to tag the segments. b) Can the points be tagged in such a way that, no matter how Isabel divides the points into pairs, she always uses exactly $\lceil n/2\rceil$ numbers to tag the segments? Note. For each real number $x$, $\lceil x\rceil$ denotes the least integer greater than or equal to $x$. For example, $\lceil 3.6\rceil=4$ and $\lceil 2\rceil=2$. [i]Proposed by Victor Domínguez[/i]

[i]Mediovagio[/i] is a computer game that consists in a $3 \times 3$ table in which each of the nine cells has a integer number from $1$ to $n$. When one clicks a cell, the numbers in the clicked cell and in the cells that share an edge with it are increased by $1$ and the sum is evaluated${}\bmod n$. Determine the values of $n$ for which it's possible, with a finite number of clicks, obtain any combination of numbers from an given initial combination. EDIT: I corrected the statement.

A $7 \times 1$ board is completely covered by $m \times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $7 \times 1$ board in which all three colors are used at least once. For example, a $1 \times 1$ red tile followed by a $2 \times 1$ green tile, a $1 \times 1$ green tile, a $2 \times 1$ blue tile, and a $1 \times 1$ green tile is a valid tiling. Note that if the $2 \times 1$ blue tile is replaced by two $1 \times 1$ blue tiles, this results in a different tiling. Find the remainder when $N$ is divided by $1000$.

Given a set $S$ of $n$ variables, a binary operation $\times$ on $S$ is called [i]simple[/i] if it satisfies $(x \times y) \times z = x \times (y \times z)$ for all $x,y,z \in S$ and $x \times y \in \{x,y\}$ for all $x,y \in S$. Given a simple operation $\times$ on $S$, any string of elements in $S$ can be reduced to a single element, such as $xyz \to x \times (y \times z)$. A string of variables in $S$ is called[i] full [/i]if it contains each variable in $S$ at least once, and two strings are [i]equivalent[/i] if they evaluate to the same variable regardless of which simple $\times$ is chosen. For example $xxx$, $xx$, and $x$ are equivalent, but these are only full if $n=1$. Suppose $T$ is a set of strings such that any full string is equivalent to exactly one element of $T$. Determine the number of elements of $T$.

In a group of $k$ people, some are acquainted with each other and some are not. Every evening, one person invites all his acquaintances to a party and introduces them to each other(if they have not already acquainted). Suppose that after each person has arranged at least one party, some two people do not know each other. Prove that they do not meet each other in the next party.

Alice and Bob play the following game on a $100\times 100$ grid, taking turns, with Alice starting first. Initially the grid is empty. At their turn, they choose an integer from $1$ to $100^2$ that is not written yet in any of the cells and choose an empty cell, and place it in the chosen cell. When there is no empty cell left, Alice computes the sum of the numbers in each row, and her score is the maximum of these $100$ numbers. Bob computes the sum of the numbers in each column, and his score is the maximum of these $100$ numbers. Alice wins if her score is greater than Bob's score, Bob wins if his score is greater than Alice's score, otherwise no one wins. Find if one of the players has a winning strategy, and if so which player has a winning strategy. [i]Théo Lenoir, France[/i]

If you draw $4$ points on the unit circle, prove that you can always find two points where their distance between is less than $\sqrt{2}.$

Let $n \ge 3$ be an integer and $A$ be a subset of the real numbers of size n. Denote by $B$ the set of real numbers that are of the form $ x \cdot y$, where $x, y \in A$ and $x\ne y$. At most how many distinct positive primes could $B$ contain (depending on $n$)?

In the country there're $N$ cities and some pairs of cities are connected by two-way airlines (each pair with no more than one). Every airline belongs to one of $k$ companies. It turns out that it's possible to get to any city from any other, but it fails when we delete all airlines belonging to any one of the companies. What is the maximum possible number of airlines in the country ?

Each of two houses $A$ and $B$ is divided into two flats. Several cats and dogs live there. It is known that the fraction of cats in the first flat of $A$ (the ratio between the number of cats and the total number of animals in the flat) is greater than the fraction of cats in the first flat of $B$, and the fraction of cats in the second flat of $A$ is greater than the fraction of cats in the second flat of $B$. Is it true that the fraction of cats in house $A$ is greater than the fraction of cats in house $B$? (AK Kovaldji)

Seventeen football fans were planning to go to Korea to watch the World Cup football match. They selected 17 matches. The conditions of the admission tickets they booked were such that - One person should book at most one admission ticket for one match; - At most one match was same in the tickets booked by every two persons; - There was one person who booked six tickets. How many tickets did those football fans book at most?

Towa has a hand of three different red cards and three different black cards. How many ways can Towa pick a set of three cards from her hand that uses at least one card of each color?

In a school with some students, for any three student, there exists at least one student who are friends with all these three students.(Friendships are mutual) For any friends $A$ and $B$, any two of their common friends are also friends with each other. It's not possible to partition these students into two groups, such that every student in each group are friends with all the students in the other gruop. Prove that any two people who aren't friends with each other, has the same number of common friends.(Each person is a friend of him/herself.)

Let $G$ be a simple connected graph. Each edge has two phases, which is either blue or red. Each vertex are switches that change the colour of every edge that connects the vertex. All edges are initially red. Find all ordered pairs $(n,k)$, $n\ge 3$, such that: a) For all graph $G$ with $n$ vertex and $k$ edges, it is always possible to perform a series of switching process so that all edges are eventually blue. b) There exist a graph $G$ with $n$ vertex and $k$ edges and it is possible to perform a series of switching process so that all edges are eventually blue.

All sets of $49$ distinct positive integers less than or equal to $100$ are considered. Leandro assigned each of these sets a positive integer less than or equal to $100$. Prove that there is a set $L$ of $50$ distinct positive integers less than or equal to $100$, such that for each number $x$ of $L$ the number that Leandro assigned to the set of $49$ numbers $L-\{ x\}$ is different from $x$. Clarification: $L-\{x\}$ denotes the set that results from removing the number $x$ from $L$.

Paint the maximum number of vertices of the cube red so that you cannot select three of the red vertices that form an equilateral triangle.

Some friends formed $6$ football teams, and decided to hold a tournament where each team faces each other exactly once in a match. In each match, whoever wins gets $3$ points, whoever loses gets no points, and if the two teams draw, each gets $1$ point. At the end of the tournament, it was found that the teams' scores were $10$, $9$, $6$, $6$, $4$ and $2$ points. Regarding this tournament, answer the following items, justifying your answer in each one. (a) How many matches ended in a draw in the tournament? (b) Determine, for each of the $6$ teams, the number of wins, draws and losses. (c) If we consider only the matches played between the team that scored $9$ points against the two teams that scored $6$ points, and the one played between the two teams that scored $6$ points, explain why among these three matches, there are at least $2$ draws.

Four points are given inside or on the boundary of a unit square. Prove that at least two of these points are on a mutual distance at most $1.$

Let $ A,B$ denote two distinct fixed points in space. Let $ X, P$ denote variable points (in space), while $ K,N, n$ denote positive integers. Call $ (X,K,N,P)$ admissible if \[ (N \minus{} K) \cdot PA \plus{} K \cdot PB \geq N \cdot PX.\] Call $ (X,K,N)$ admissible if $ (X,K,N,P)$ is admissible for all choices of $ P.$ Call $ (X,N)$ admissible if $ (X,K,N)$ is admissible for some choice of $ K$ in the interval $ 0 < K < N.$ Finally, call $ X$ admissible if $ (X,N)$ is admissible for some choice of $ N, (N > 1).$ Determine: [b](a)[/b] the set of admissible $ X;$ [b](b)[/b] the set of $ X$ for which $ (X, 1989)$ is admissible but not $ (X, n), n < 1989.$

Within a group of $ 2009$ people, every two people has exactly one common friend. Find the least value of the difference between the person with maximum number of friends and the person with minimum number of friends.