Neel and Roshan are going to the Newark Liberty International Airport to catch separate flights. Neel plans to arrive at some random time between 5:30 am and 6:30 am, while Roshan plans to arrive at some random time between 5:40 am and 6:40 am. The two want to meet, however briefly, before going through airport security. As such, they agree that each will wait for $n$ minutes once he arrives at the airport before going through security. What is the smallest $n$ they can select such that they meet with at least 50% probability? The answer will be of the form $a + b\sqrt{c}$ for integers $a$, $b$, and $c$, where $c$ has no perfect square factor other than $1$. Report $a + b + c.$

A mink is standing in the center of a field shaped like a regular polygon. The field is surrounded by a fence, and the mink can only exit through the vertices of the polygon. A dog is standing on one of the vertices, and can move along the fence. The mink wants to escape the field, while the dog tries to prevent it. Each of them moves with constant velocity. For what ratio of velocities could the mink escape if: a. The field is a regular triangle? b. The field is a square?

Let $(X, <)$ be an arbitrary ordered set. Show that the elements of $X$ can be coloured by two colours in such a way that between any two points of the same colour there is a point of the opposite colour. (translated by L. Erdős)

On an infinite (in both directions) strip of squares, indexed by the integers, are placed several stones (more than one may be placed on a single square). We perform a sequence of moves of one of the following types: (a) Remove one stone from each of the squares $n - 1$ and $n$ and place one stone on square $n + 1$. (b) Remove two stones from square $n$ and place one stone on each of the squares $n + 1$, $n - 2$. Prove that any sequence of such moves will lead to a position in which no further moves can be made, and moreover that this position is independent of the sequence of moves. [i]D. Fon-der-Flaas[/i]

Let $T$ be a non-empty finite subset of positive integers $\ge 1$. A subset $S$ of $T$ is called [b]good [/b] if for every integer $t\in T$ there exists an $s$ in $S$ such that $gcd(t,s) >1$. Let \[A={(X,Y)\mid X\subseteq T,Y\subseteq T,gcd(x,y)=1 \text{for all} x\in X, y\in Y}\] Prove that : $a)$ If $X_0$ is not [b]good[/b] then the number of pairs $(X_0,Y)$ in $A$ is [b]even[/b]. $b)$ the number of good subsets of $T$ is [b]odd[/b].

Determine all positive integers $ n$ for which there is a coloring of all points in space so that each of the following conditions is satisfied: (i) Each point is painted in exactly one color. (ii) Exactly $ n$ colors are used. (iii) Each line is painted in at most two different colors.

[u]Discrete Math Round[/u] [b]p1.[/b] A class of six students has to split into two indistinguishable teams of three people. Compute the number of distinct team arrangements that can result. [b]p2.[/b] What is the probability that a randomly chosen factor of $2016$ is a perfect square? [b]p3.[/b] Compute the remainder when $$5\underbrace{666...6666}_{2016 \,\, sixes}5$$ is divided by $17$. [b]p4.[/b] At an M&M factory, two types of M&Ms are produced, red and blue. The M&Ms are transported individually on a conveyor belt. Anna is watching the conveyor belt, and has determined that four out of every five red M&Ms are followed by a blue one, while one out of every six blue M&Ms is followed by a red one. What proportion of the M&Ms are red? [b]p5.[/b] Three cards are chosen from a standard deck of $52$ without replacing them. Given that the ace of spades was chosen, what is the expected number of aces chosen? [b]p6.[/b] Moor decides that he needs a new email address, and forms the address by taking some permutation of the $12$ letters $MMMOOOOOORRR$. How many permutations of the letters will contain $MOOR$ in this exact order at least once? [b]p7.[/b] Suppose that the $8$ corners of a cube can be colored either red, green, or blue. We call a coloring of the cube rotationally symmetric if the cube can be rotated along a single axis parallel to an edge of a cube either $90^o$, $180^o$, or $270^o$, and reach the original coloring. How many rotationally symmetric colorings exist using the $3$ colors? Assume that any colorings which are identical after rotation are equivalent. [b]p8.[/b] Let $x = \frac{1}{9} + \frac{1}{99} + \frac{1}{999} + ...+ \frac{1}{999999999}$ . Compute the number of digits in the first $3000$ decimal places of the base $10$ representation of $x$ which are greater than or equal to $8$. [b]p9.[/b] Two $20$-sided dice are rolled. Their outcomes are independent and take uniformly distributed integer values from $1$ to $20$, inclusive. For each roll, let $x$ be (the sum of the dice) $\times $ (the positive difference of the dice). What is the expected value of $x$? [b]p10.[/b] Compute $$\sum^{1000}_{a=1} \sum^{1000}_{b=1} \sum^{1000}_{c=1} \left\lfloor \frac{1000}{lcm (a, b, c)} \right \rfloor \phi (a) \phi (b) \phi(c)$$ where $\phi (n) = | \{k : 1 \le k \le n, gcd (k, n) = 1\} |$ counts the integers coprime to $n$ that are less than or equal to $n$. [u]Discrete Math Tiebreaker[/u] [b]Tie 1.[/b] A certain elementary school has $48$ students in the third grade that must be organized into three classes of $16$ students each. There are three troublemakers in the grade. If the students are assigned independently and randomly to classes, what is the probability that all three trou blemakers are assigned to the same $16$ student class? [b]Tie 2.[/b] A $4$-digit number $x$ has the property that the expected value of the integer obtained from switching any two digits in $x$ is $4625$. Given that the sum of the digits of $x$ is $20$, compute $x$. [b]Tie 3.[/b] Let $S$ be the set of factors of $10^5$. The number of subsets of $S$ with a least common multiple of $10^5$ can be written as $2^n * m$, where $n$ and $m$ are positive integers and $m$ is not divisible by $2$. Compute $m + n$. PS. You should use hide for answers.

Let $f(k)$ be the number of integers $n$ satisfying the following conditions: (i) $0\leq n < 10^k$ so $n$ has exactly $k$ digits (in decimal notation), with leading zeroes allowed; (ii) the digits of $n$ can be permuted in such a way that they yield an integer divisible by $11$. Prove that $f(2m) = 10f(2m-1)$ for every positive integer $m$. [i]Proposed by Dirk Laurie, South Africa[/i]

[b]p1.[/b] Let $f(x) = \frac{3x^3+7x^2-12x+2}{x^2+2x-3}$ . Find all integers $n$ such that $f(n)$ is an integer. [b]p2.[/b] How many ways are there to arrange $10$ trees in a line where every tree is either a yew or an oak and no two oak trees are adjacent? [b]p3.[/b] $20$ students sit in a circle in a math class. The teacher randomly selects three students to give a presentation. What is the probability that none of these three students sit next to each other? [b]p4.[/b] Let $f_0(x) = x + |x - 10| - |x + 10|$, and for $n \ge 1$, let $f_n(x) = |f_{n-1}(x)| - 1$. For how many values of $x$ is $f_{10}(x) = 0$? [b]p5.[/b] $2$ red balls, $2$ blue balls, and $6$ yellow balls are in a jar. Zion picks $4$ balls from the jar at random. What is the probability that Zion picks at least $1$ red ball and$ 1$ blue ball? [b]p6.[/b] Let $\vartriangle ABC$ be a right-angled triangle with $\angle ABC = 90^o$ and $AB = 4$. Let $D$ on $AB$ such that $AD = 3DB$ and $\sin \angle ACD = \frac35$ . What is the length of $BC$? [b]p7.[/b] Find the value of of $$\dfrac{1}{1 +\dfrac{1}{2+ \dfrac{1}{1+ \dfrac{1}{2+ \dfrac{1}{1+ ...}}}}}$$ [b]p8.[/b] Consider all possible quadrilaterals $ABCD$ that have the following properties; $ABCD$ has integer side lengths with $AB\parallel CD$, the distance between $\overline{AB}$ and $\overline{CD}$ is $20$, and $AB = 18$. What is the maximum area among all these quadrilaterals, minus the minimum area? [b]p9.[/b] How many perfect cubes exist in the set $\{1^{2018},2^{2017}, 3^{2016},.., 2017^2, 2018^1\}$? [b]p10.[/b] Let $n$ be the number of ways you can fill a $2018\times 2018$ array with the digits $1$ through $9$ such that for every $11\times 3$ rectangle (not necessarily for every $3 \times 11$ rectangle), the sum of the $33$ integers in the rectangle is divisible by $9$. Compute $\log_3 n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In the $100 \times 100$ table in every cell there is natural number. All numbers in same row or column are different. Can be that for every square sum of numbers, that are in angle cells, is square number ?

There are two boxes containing balls. One of them contains $m$ balls, and the other contains $n$ balls, where $m, n > 0$. Two actions are permitted: (i) Remove an equal number of balls from both boxes. (ii) Increase the number of balls in one of the boxes by a factor $k$. Is it possible to remove all of the balls from both boxes with just these two actions, 1. if $k = 2$? 2. if $k = 3$?

A directed graph has each vertex with outdegree 2. Prove that it is possible to split the vertices into 3 sets so that for each vertex $v$, $v$ is not simultaneously in the same set with both of the vertices that it points to. [i]David Yang.[/i] [hide="Stronger Version"]See [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=42&t=492100]here[/url].[/hide]

Let be given $32$ positive integers with the sum $120$, none of which is greater than $60.$ Prove that these integers can be divided into two disjoint subsets with the same sum of elements.

Let $ n$ be a positive integer. We consider the set $ P$ of all partitions of $ n$ into a sum of positive integers (the order is irrelevant). For every partition $ \alpha$, let $ a_{k}(\alpha)$ be the number of summands in $ \alpha$ that are equal to $ k, k = 1,2,...,n.$ Prove that $ \sum_{\alpha\in P}\frac{1}{1^{a_{1}(\alpha)}a_{1}(\alpha)!\cdot 2^{a_{2}(\alpha)}a_{2}(\alpha)!...n^{a_{n}(\alpha)}a_{n}(\alpha)!}=1.$

A graph $G$ with $n$ vertex is called [i]good [/i] if every vertex could be labelled with distinct positive integers which are less than or equal $\lfloor \frac{n^2}{4} \rfloor$ such that there exists a set of nonnegative integers $D$ with the following property: there exists an edge between $2$ vertices if and only if the difference of their labels is in $D$. Show that there exists a positive integer $N$ such that for every $n \ge N$, there exist a not-good graph with $n$ vertices.

Two players play a game on an infinite board that consists of unit squares. Player $I$ chooses a square and marks it with $O$. Then player $II$ chooses another square and marks it with $X$. They play until one of the players marks a whole row or a whole column of five consecutive squares, and this player wins the game. If no player can achieve this, the result of the game is a tie. Show that player $II$ can prevent player $I$ from winning.

Aws plays a solitaire game on a fifty-two card deck: whenever two cards of the same color are adjacent, he can remove them. Aws wins the game if he removes all the cards. If Aws starts with the cards in a random order, what is the probability for him to win?

Initially, the point-like particles $A, B$ and $C{}$ are located respectively at the points $(0,0), (1,0)$ and $(0,1)$ in the coordinate plane. Every minute some two particles repel each other along the straight line connecting their current positions, moving the same (positive) distance. [list=a] [*]Can the particle $A{}$ be at the point $(3,3)$? What about the point $(2,3)$? [*]Can the particles $B{}$ and $C{}$ be at the same time at the points $(0,100)$ and $(100,0)$ respectively? [/list] [i]Proposed by K. Krivosheev[/i]

The intramural squash league has 5 players, namely Albert, Bassim, Clara, Daniel, and Eugene. Albert has played one game, Bassim has played two games, Clara has played 3 games, and Daniel has played 4 games. Assuming no two players in the league play each other more than one time, how many games has Eugene played?

Let [n] be the set {1, 2,. . . , n}. For any $a, b \in N$, denote $G (a, b)$ by a graph (not directed) defined by the following rule: the vertices have the form (i, f), where $i \in [a]$, and $f: [a] \to [b]$. A vertex (i, f) and a vertex (j, g) are connected if $i \neq j$, and $f (k) \neq g (k)$ holds exactly for k strictly between i and j. Prove that for any $c \in N$ there is $a, b \in N$ such that the vertices of G (a, b) cannot be well-colored with $c$ colors.

In a football tournament of one round (each team plays each other once, $2$ points for win , $1$ point for draw and $0$ points for loss), $28$ teams compete. During the tournament more than $75\%$ of the matches finished in a draw . Prove that there were two teams who finished with the same number of points. (M . Vora, gymnasium student , Hungary)

A particle moves from $(0, 0)$ to $(n, n)$ directed by a fair coin. For each head it moves one step east and for each tail it moves one step north. At $(n, y), y < n$, it stays there if a head comes up and at $(x, n), x < n$, it stays there if a tail comes up. Let$ k$ be a fixed positive integer. Find the probability that the particle needs exactly $2n+k$ tosses to reach $(n, n).$

A polygon is decomposed into triangles by drawing some non-intersecting interior diagonals in such a way that for every pair of triangles of the triangulation sharing a common side, the sum of the angles opposite to this common side is greater than $180^o$. a) Prove that this polygon is convex. b) Prove that the circumcircle of every triangle used in the decomposition contains the entire polygon. [i]Proposed by Morteza Saghafian - Iran[/i]

Given a $8$x$8$ square board a) Prove that: for any ways to color the board, we are always be able to find a rectangle consists of $8$ squares such that these squares are not colored. b) Prove that: we can color $7$ squares on the board such that for any rectangles formed by $\geq 9$ squares, there are at least $1$ colored square.

Let $a,b$ be integers with $0<a<b$. A set $\{x,y,z\}$ of non-negative integers is [i]olympic[/i] if $x<y<z$ and if $\{z-y,y-x\}=\{a,b\}$. Show that the set of all non-negative integers is the union of pairwise disjoint olympic sets.