Let $S$ be the set of points inside and on the boarder of a regular haxagon with side length 1. Find the least constant $r$, such that there exists one way to colour all the points in $S$ with three colous so that the distance between any two points with same colour is less than $r$.

Sir Alex is coaching a soccer team of $n$ players of distinct heights. He wants to line them up so that for each player $P$, the total number of players that are either to the left of $P$ and taller than $P$ or to the right of $P$ and shorter than $P$ is even. In terms of $n$, how many possible orders are there? [i]Michael Ren[/i]

Two points are chosen uniformly at random on the sides of a square with side length 1. If $p$ is the probability that the distance between them is greater than $1$, what is $\lfloor 100p \rfloor$? (Note: $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)

Find all positive integers $n$ for which it is possible to color some cells of an infinite grid of unit squares red, such that each rectangle consisting of exactly $n$ cells (and whose edges lie along the lines of the grid) contains an odd number of red cells. [i]Proposed by Merlijn Staps[/i]

Compute the number of nonempty subsets $S$ of $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ such that $\frac{\max \,\, S + \min \,\,S}{2}$ is an element of $S$.

$(1)$ Start with $a$ white balls and $b$ black balls. $(2)$ Draw one ball at random. $(3)$ If the ball is white, then stop. Otherwise, add two black balls and go to step $2$. Let $S$ be the number of draws before the process terminates. For the cases $a = b = 1$ and $a = b = 2$ only, find $a_n = P(S = n), b_n = P(S \le n), \lim_{n\to\infty} b_n$, and the expectation value of the number of balls drawn: $E(S) =\displaystyle\sum_{n\ge1} na_n.$

In the following image is a beehive lattice of hexagons. Each cell is colored in one of three colors Red, Blue, or Green (denoted by the letters $R, B, G$). The frame is colored according to the instructions in the image, and the rest of the hexagons are colored however one wants. Is there necessarily a point where three hexagons of different colors meet?

[b](a)[/b] One black and one white pawn are placed on a chessboard. You may move the pawns in turn to the neighbouring empty squares of the chessboard using vertical and horizontal moves. Can you arrange the moves so that every possible position of the two pawns will appear on the chessboard exactly once? [b](b)[/b] Same question, but you don’t have to move the pawns in turn.

Let $n$ be a positive integer. Let $G (n)$ be the number of $x_1,..., x_n, y_1,...,y_n \in \{0,1\}$, for which the number $x_1y_1 + x_2y_2 +...+ x_ny_n$ is even, and similarly let $U (n)$ be the number for which this sum is odd. Prove that $$\frac{G(n)}{U(n)}= \frac{2^n + 1}{2^n - 1}.$$

Find the smallest $n \in \mathbb{N}$ such that if any 5 vertices of a regular $n$-gon are colored red, there exists a line of symmetry $l$ of the $n$-gon such that every red point is reflected across $l$ to a non-red point.

On a blackboard, there are $17$ integers not divisible by $17$. Alice and Bob play a game. Alice starts and they alternately play the following moves: $\bullet$ Alice chooses a number $a$ on the blackboard and replaces it with $a^2$ $\bullet$ Bob chooses a number $b$ on the blackboard and replaces it with $b^3$. Alice wins if the sum of the numbers on the blackboard is a multiple of $17$ after a finite number of steps. Prove that Alice has a winning strategy. (Daniel Holmes)

There are $15$ people, including Petruk, Gareng, and Bagong, which will be partitioned into $6$ groups, randomly, that consists of $3, 3, 3, 2, 2$, and $2$ people (orders are ignored). Determine the probability that Petruk, Gareng, and Bagong are in a group.

A finite set $S{}$ of $n{}$ points is given in the plane. No three points lie on the same line. The number of non-self-intersecting closed $n{}$-link polylines with vertices at these points will be denoted by $f(S).$ Prove that [list=a] [*]$f(S)>0$ for all sets $S{};$ [*]$f(S)=1$ if and only if all the points of $S{}$ lie on the convex hull of $S{};$ [*]if $f(S)>1$ then $f(S)\geqslant n-1$, with equality if and only if one point of $S$ lies inside the convex hull; [*]if exactly two points of $S{}$ lie inside the convex hull, then\[f(S)\geqslant\frac{(n-2)(n-3)}{2}.\] [/list]Let $n\geqslant 3.$ Denote by $F(n)$ the largest possible value of the function $f(S)$ over all admissible sets $S{}$ of $n{}$ points. Prove that \[F(n)\geqslant3\cdot 2^{(n-8)/3}.\][i]Proposed by E. Bakaev and D. Magzhanov[/i]

New license plates consist of two letters, three digits, and two letters (from the English alphabet of$ 26$ letters). What is the largest possible number of such license plates if it is required that every two of them differ at no less than two positions?

[u]Round 6[/u] [b]p16.[/b] Let $a_1, a_2, ... , a_{2011}$ be a sequence of numbers such that $a_1 = 2011$ and $a_1+a_2+...+a_n = n^2 \cdot a_n$ for $n = 1, 2, ... 2011$. (That is, $a_1 = 1^2\cdot a_1$, $a_1 + a_2 = 2^2 \cdot a_2$, $...$) Compute $a_{2011}$. [b]p17.[/b] Three rectangles, with dimensions $3 \times 5$, $4 \times 2$, and $6 \times 4$, are each divided into unit squares which are alternately colored black and white like a checkerboard. Each rectangle is cut along one of its diagonals into two triangles. For each triangle, let m be the total black area and n the total white area. Find the maximum value of $|m - n|$ for the $6$ triangles. [b]p18.[/b] In triangle $ABC$, $\angle BAC = 90^o$, and the length of segment $AB$ is $2011$. Let $M$ be the midpoint of $BC$ and $D$ the midpoint of $AM$. Let $E$ be the point on segment $AB$ such that $EM \parallel CD$. What is the length of segment $BE$? [u]Round 7[/u] [b]p19.[/b] How many integers from $1$ to $100$, inclusive, can be expressed as the difference of two perfect squares? (For example, $3 = 2^2 - 1^2$). [b]p20.[/b] In triangle $ABC$, $\angle ABC = 45$ and $\angle ACB = 60^o$. Let $P$ and $Q$ be points on segment $BC$, $F$ a point on segment $AB$, and $E$ a point on segment $AC$ such that $F Q \parallel AC$ and $EP \parallel AB$. Let $D$ be the foot of the altitude from $A$ to $BC$. The lines $AD$, $F Q$, and $P E$ form a triangle. Find the positive difference, in degrees, between the largest and smallest angles of this triangle. [b]p21.[/b] For real number $x$, $\lceil x \rceil$ is equal to the smallest integer larger than or equal to $x$. For example, $\lceil 3 \rceil = 3$ and $\lceil 2.5 \rceil = 3$. Let $f(n)$ be a function such that $f(n) = \left\lceil \frac{n}{2}\right\rceil + f\left( \left\lceil \frac{n}{2}\right\rceil\right)$ for every integer $n$ greater than $1$. If $f(1) = 1$, find the maximum value of $f(k) - k$, where $k$ is a positive integer less than or equal to $2011$. [u]Round 8[/u] The answer to each of the three questions in this round depends on the answer to one of the other questions. There is only one set of correct answers to these problems; however, each question will be scored independently, regardless of whether the answers to the other questions are correct. [b]p22.[/b] Let $W$ be the answer to problem 24 in this guts round. Let $f(a) = \frac{1}{1 -\frac{1}{1- \frac{1}{a}}}$. Determine$|f(2) + ... + f(W)|$. [b]p23.[/b] Let $X$ be the answer to problem $22$ in this guts round. How many odd perfect squares are less than $8X$? [b]p24.[/b] Let $Y$ be the answer to problem $23$ in this guts round. What is the maximum number of points of intersections of two regular $(Y - 5)$-sided polygons, if no side of the first polygon is parallel to any side of the second polygon? [u]Round 9[/u] [b]p25.[/b] Cross country skiers $s_1, s_2, s_3, ..., s_7$ start a race one by one in that order. While each skier skis at a constant pace, the skiers do not all ski at the same rate. In the course of the race, each skier either overtakes another skier or is overtaken by another skier exactly two times. Find all the possible orders in which they can finish. Write each possible finish as an ordered septuplet $(a, b, c, d, e, f, g)$ where $a, b, c, d, e, f, g$ are the numbers $1-7$ in some order. (So a finishes first, b finishes second, etc.) [b]p26.[/b] Archie the Alchemist is making a list of all the elements in the world, and the proportion of earth, air, fire, and water needed to produce each. He writes the proportions in the form E:A:F:W. If each of the letters represents a whole number from $0$ to $4$, inclusive, how many different elements can Archie list? Note that if Archie lists wood as $2:0:1:2$, then $4:0:2:4$ would also produce wood. In addition, $0:0:0:0$ does not produce an element. [b]p27.[/b] Let $ABCD$ be a rectangle with $AB = 10$ and $BC = 12$. Let $M$ be the midpoint of $CD$, and $P$ be the point on $BM$ such that $DP = DA$. Find the area of quadrilateral $ABPD$. [u]Round 10[/u] [b]p28.[/b] David the farmer has an infinitely large grass-covered field which contains a straight wall. He ties his cow to the wall with a rope of integer length. The point where David ties his rope to the wall divides the wall into two parts of length $a$ and $b$, where $a > b$ and both are integers. The rope is shorter than the wall but is longer than $a$. Suppose that the cow can reach grass covering an area of $\frac{165\pi}{2}$. Find the ratio $\frac{a}{b}$ . You may assume that the wall has $0$ width. [b]p29.[/b] Let $S$ be the number of ordered quintuples $(a, b, x, y, n)$ of positive integers such that $$\frac{a}{x}+\frac{b}{y}=\frac{1}{n}$$ $$abn = 2011^{2011}$$ Compute the remainder when $S$ is divided by $2012$. [b]p30.[/b] Let $n$ be a positive integer. An $n \times n$ square grid is formed by $n^2$ unit squares. Each unit square is then colored either red or blue such that each row or column has exactly $10$ blue squares. A move consists of choosing a row or a column, and recolor each unit square in the chosen row or column – if it is red, we recolor it blue, and if it is blue, we recolor it red. Suppose that it is possible to obtain fewer than $10n$ blue squares after a sequence of finite number of moves. Find the maximum possible value of $n$. PS. You should use hide for answers. First rounds have been posted [url=https://artofproblemsolving.com/community/c4h2786905p24497746]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On an $n\times n$ chart, where $n \geq 4$, stand "$+$" signs in the cells of the main diagonal and "$-$" signs in all the other cells. You can change all the signs in one row or in one column, from $-$ to $+$ or from $+$ to $-$. Prove that you will always have $n$ or more $+$ signs after finitely many operations.

Find the number of $10$-tuples $(a_1,a_2,\dots,a_9,a_{10})$ of integers such that $|a_1|\leq 1$ and \[a_1^2+a_2^2+a_3^2+\cdots+a_{10}^2-a_1a_2-a_2a_3-a_3a_4-\cdots-a_9a_{10}-a_{10}a_1=2.\]

In a certain kingdom, the king has decided to build $25$ new towns on $13$ uninhabited islands so that on each island there will be at least one town. Direct ferry connections will be established between any pair of new towns which are on different islands. Determine the least possible number of these connections.

$2021$ points are marked on a circle. $2021$ segments with marked endpoints are drawn. After that one counts the number of different points where some $2$ drawn segments intersect(endpoints of segments do [b]not[/b] count as intersections) Find the maximum number one can get.

A given triangle is divided into $n$ triangles in such a way that any line segment which is a side of a tiling triangle is either a side of another tiling triangle or a side of the given triangle. Let $s$ be the total number of sides and $v$ be the total number of vertices of the tiling triangles (counted without multiplicity). (a) Show that if $n$ is odd then such divisions are possible, but each of them has the same number $v$ of vertices and the same number $s$ of sides. Express $v$ and $s$ as functions of $n$. (b) Show that, for $n$ even, no such tiling is possible

Find the number of sequences of $2005$ terms with the following properties: (i) No three consecutive terms of the sequence are equal, (ii) Every term equals either $1$ or $-1$, (iii) The sum of all terms of the sequence is at least $666$.

Let $m$ be a positive integer. Two players, Axel and Elina play the game HAUKKU ($m$) proceeds as follows: Axel starts and the players choose integers alternately. Initially, the set of integers is the set of positive divisors of a positive integer $m$ .The player in turn chooses one of the remaining numbers, and removes that number and all of its multiples from the list of selectable numbers. A player who has to choose number $1$, loses. Show that the beginner player, Axel, has a winning strategy in the HAUKKU ($m$) game for all $m \in Z_{+}$. PS. As member Loppukilpailija noted, it should be written $m>1$, as the statement does not hold for $m = 1$.

The closed interval $A = [0, 50]$ is the union of a finite number of closed intervals, each of length $1$. Prove that some of the intervals can be removed so that those remaining are mutually disjoint and have total length greater than $25$. Note: For reals $a\le b$, the closed interval $[a, b] := \{x\in \mathbb{R}:a\le x\le b\}$ has length $b-a$; disjoint intervals have [i]empty [/i]intersection.

In a simple graph, there are two disjoint set of vertices $A$ and $B$ where $A$ has $k$ and $B$ has $2016$ vertices. Four numbers are written to each vertex using the colors red, green, blue and black. There is no any edge at the beginning. For each vertex in $A$, we first choose a color and then draw all edges from this vertex to the vertices in $B$ having a larger number with the chosen color. It is known that for each vertex in $B$, the set of vertices in $A$ connected to this vertex are different. Find the minimal possible value of $k$.

Boeotia is comprised of $3$ islands which are home to $2019$ towns in total. Each flight route connects three towns, each on a different island, providing connections between any two of them in both directions. Any two towns in the country are connected by at most one flight route. Find the maximal number of flight routes in the country