A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five. Using the four words “Hi”, “hey”, “hello”, and “haiku”, How many haikus Can somebody make? (Repetition is allowed, Order does matter.) [i]Proposed by Jeff Lin[/i]

We are given two point sets $A$ and $B$ which are both composed of finite disjoint arcs on the unit circle. Moreover, the length of each arc in $B$ is equal to $\dfrac{\pi}{m}$ ($m \in \mathbb{N}$). We denote by $A^j$ the set obtained by a counterclockwise rotation of $A$ about the center of the unit circle for $\dfrac{j\pi}{m}$ ($j=1,2,3,\dots$). Show that there exists a natural number $k$ such that $l(A^k\cap B)\ge \dfrac{1}{2\pi}l(A)l(B)$.(Here $l(X)$ denotes the sum of lengths of all disjoint arcs in the point set $X$)

$1994$ points from the plane are given so that any $100$ of them can be selected $98$ that can be rounded (some points may be at the border of the circle) with a diameter of $1$. Determine the smallest number of circles with radius $1$, sufficient to cover all $1994$

All points in the plane that have both integer coordinates are painted, using the colors red, green, and yellow. If the points are painted so that there is at least one point of each color. Prove that there are always three points $X$, $Y$ and $Z$ of different colors, such that $\angle XYZ = 45^{\circ} $

Suppose $a_1, a_2, ..., a_r$ are integers with $a_i \geq 2$ for all $i$ such that $a_1 + a_2 + ... + a_r = 2010$. Prove that the set $\{1,2,3,...,2010\}$ can be partitioned in $r$ subsets $A_1, A_2, ..., A_r$ each with $a_1, a_2, ..., a_r$ elements respectively, such that the sum of the numbers on each subset is divisible by $2011$. Decide whether this property still holds if we replace $2010$ by $2011$ and $2011$ by $2012$ (that is, if the set to be partitioned is $\{1,2,3,...,2011\}$).

Label sides of a regular $n$-gon in clockwise direction in order 1,2,..,n. Determine all integers n ($n\geq 4$) satisfying the following conditions: (1) $n-3$ non-intersecting diagonals in the $n$-gon are selected, which subdivide the $n$-gon into $n-2$ non-overlapping triangles; (2) each of the chosen $n-3$ diagonals are labeled with an integer, such that the sum of labeled numbers on three sides of each triangles in (1) is equal to the others;

The first quadrant of the Cartesian $0-x-y$ plane can be considered to be divided into an infinite set of squares of unit side length, arranged in rows and columns , formed by the axes and lines $x = i$ and $y = j$ , where $i$ and $j$ are non-negative integers. Is it possible to write a natural number $(1,2, 3,...)$ in each square , so that each row and column contains each natural number exactly once? (V . S . Shevelev)

Two players take turns placing the numbers $1, 2, 3,. . . , 24$, in each of the $24$ squares on the surface of a $2 \times 2 \times 2$ cube (each number can be placed once). The second player wants the sum of the numbers in each cell the rings of $8$ cells encircling the cube were identical. Will he be able to the first player to stop him?

There are $n$ hoops on a circle. Rik numbers all hoops with a natural number so that all numbers from $1$ to $n$ occur exactly once. Then he makes one walk from hoop to hoop. He starts in hoop $1$ and then follows the following rule: if he gets to hoop $k$, then he walks to the hoop that places $k$ clockwise without getting into the intermediate hoops. The walk ends when Rik has to walk to a hoop he has already been to. The length of the walk is the number of hoops he passed on the way. For example, for $n = 6$ Rik can take a walk of length $5$ as the hoops are numbered as shown in the figure. [img]https://cdn.artofproblemsolving.com/attachments/2/a/3d4b7edbba4d145c7e00368f9b794f39572dc5.png[/img] (a) Determine for every even $n$ how Rik can number the hoops so that he has one walk of length $n$. (b) Determine for every odd $n$ how Rik can number the hoops so that he has one walk of length $n - 1$. (c) Show that for an odd $n$ there is no such numbering of the hoops that Rik can make a walk of length $n$.

The sweeties shop called "Olympiad" sells boxes of $6,9$ or $20$ chocolates. Groups of students from a school that is near the shop collect money to buy a chocolate for each student; to make this they buy a box and than give to everybody a chocolate. Like this students can create groups of $15=6+9$ students, $38=2*9+20$ students, etc. The seller has promised to the students that he can satisfy any group of students, and if he will need to open a new box of chocolate for any group (like groups of $4,7$ or $10$ students) than he will give all the chocolates for free to this group. Can there be constructed the biggest group that profits free chocolates, and if so, how many students are there in this group?

A square is divided into $25$ small squares. We draw diagonals of some of the small squares so that no two diagonals share a common point (not even a common endpoint). What is the largest possible number of diagonals that we can draw? (I Rubanov)

There are several coins in a row, and the [i]allowed move[/i] is to remove exactly one coin from the row, which can either be the first or the last. In the initial distribution, there are $n$ coins with not necessarily equal values. Ana and María alternate turns. Ana starts, making two moves, then María makes one move, then Ana makes two moves, and so on until no coins remain: Ana makes two moves and María makes one. (Only in the last move can Ana take one coin if only one coin is left.) Ana's goal is to ensure she takes at least $\dfrac{2}{3}$ of the total value of the coins. Determine if Ana can achieve her goal with certainty if [list=a] [*]$n=2013$ [*]$n=2014$ [/list] If the answer is yes, provide a strategy to achieve it; if the answer is no, give a specific sequence of coins and explain how María prevents Ana from achieving her goal.

Given a $15\times 15$ chessboard. We draw a closed broken line without self-intersections such that every edge of the broken line is a segment joining the centers of two adjacent cells of the chessboard. If this broken line is symmetric with respect to a diagonal of the chessboard, then show that the length of the broken line is $\leq 200$.

We are given $n$ distinct points in the plane. Consider the number $\tau(n)$ of segments of length 1 joining pairs of these points. Show that $\tau(n)\leq \frac{n^{2}}3$.

Paraflea makes jumps on the plane, starting from the origin $(0, 0)$. From point $(x, y)$ it may jump to another point of the form $(x + p, y + p^2)$, where $p$ is any positive real number. (The value of $p$ may differ for each jump.) a) Is there any point in quadrant $I$ which cannot be reached by the flea? (Quadrant $I$ contains points $(x, y)$ for which $x$ and $y$ are positive real numbers.) b) What is the minimum number of jumps that the flea must make from the origin so that it gets to the point $(100, 1)$?

On the circle, 99 points are marked, dividing this circle into 99 equal arcs. Petya and Vasya play the game, taking turns. Petya goes first; on his first move, he paints in red or blue any marked point. Then each player can paint on his own turn, in red or blue, any uncolored marked point adjacent to the already painted one. Vasya wins, if after painting all points there is an equilateral triangle, all three vertices of which are colored in the same color. Could Petya prevent him?

Find all pairs $(m, n)$ of positive integers such that the $m \times n$ grid contains exactly $225$ rectangles whose side lengths are odd and whose edges lie on the lines of the grid.

In a club with $30$ members, every member initially had a hat. One day each member sent his hat to a different member (a member could have received more than one hat). Prove that there exists a group of $10$ members such that no one in the group has received a hat from another one in the group.

Ana and Bora are each given a sufficiently long paper strip, one with letter $A$ written , and the other with letter $B$. Every minute, one of them (not necessarily one after another) writes either on the left or on the right to the word on his/her strip the word written on the other strip. Prove that the day after, one will be able to cut word on Ana's strip into two words and exchange their places, obtaining a palindromic word.

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

Let $S$ be a set of finite number of points in the plane any 3 of which are not linear and any 4 of which are not concyclic. A coloring of all the points in $S$ to red and white is called [i]discrete coloring[/i] if there exists a circle which encloses all red points and excludes all white points. Determine the number of [i]discrete colorings[/i] for each set $S$.

Given positive integer $ n \ge 5 $ and a convex polygon $P$, namely $ A_1A_2...A_n $. No diagonals of $P$ are concurrent. Proof that it is possible to choose a point inside every quadrilateral $ A_iA_jA_kA_l (1\le i<j<k<l\le n) $ not on diagonals of $P$, such that the $ \tbinom{n}{4} $ points chosen are distinct, and any segment connecting these points intersect with some diagonal of P.

$3n-1$ points are given in the plane, no three are collinear. Prove that one can select $2n$ of them whose convex hull is not a triangle.

A $10$ digit positive integer is called a $\emph{cute}$ number if its digits are from the set $\{1,2,3\}$ and every two consecutive digits differ by $1$. [list=a] [*]Prove that exactly $5$ digits of a cute number are equal to $2$. [*]Find the total number of cute numbers. [*]Prove that the sum of all cute numbers is divisible by $1408$.[/list]

A given rectangle $ R$ is divided into $mn$ small rectangles by straight lines parallel to its sides. (The distances between the parallel lines may not be equal.) What is the minimum number of appropriately selected rectangles’ areas that should be known in order to determine the area of $ R$?