We define a beehive of order $n$ as follows: a beehive of order 1 is one hexagon To construct a beehive of order $n$, take a beehive of order $n-1$ and draw a layer of hexagons in the exterior of these hexagons. See diagram for examples of $n=2,3$ Initially some hexagons are infected by a virus. If a hexagon has been infected, it will always be infected. Otherwise, it will be infected if at least 5 out of the 6 neighbours are infected. Let $f(n)$ be the minimum number of infected hexagons in the beginning so that after a finite time, all hexagons become infected. Find $f(n)$.

Sugar Station sells $44$ different kinds of candies, packaged one to a box. Each box is priced at a positive integer number of cents, and it costs $\$1.51$ to buy one of every kind. (There is no discount based on the number of candies in a purchase.) Unfortunately, Anna only has $\$0.75$. [list=a] [*] Show that Anna can buy at least $22$ boxes, each containing a different candy. [*] Show that Anna can do even better, buying at least $25$ boxes, each containing a different candy. [/list]

A grid measuring $10 \times 11$ is given. How many "crosses" covering five unit squares can be placed on the grid? (pictured right) so that no two of them cover the same square? [img]https://cdn.artofproblemsolving.com/attachments/a/7/8a5944233785d960f6670e34ca7c90080f0bd6.png[/img]

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 have a complete graph with $n$ vertices. We have to color the vertices and the edges in a way such that: no two edges pointing to the same vertice are of the same color; a vertice and an edge pointing him are coloured in a different way. What is the minimum number of colors we need?

There are $24$ apples in $4$ boxes. An optimistic worm is convinced that he can eat no more than half of the apples such that there will be $3$ boxes with equal number of apples. Is it possible that he is wrong?

Let $n,m$ be positive integers. Let $A_1,A_2,A_3, ... ,A_m$ be sets such that $A_i \subseteq \{1, 2, 3, . . . , n\}$ and $|A_i| = 3$ for all $i$ (i.e. $A_i$ consists of three different positive integers each at most $n$). Suppose for all $i < j$ we have $|A_i \cap A_j | \le 1$ (i.e. $A_i$ and $A_j$ have at most one element in common). (a) Prove that $m \le \frac{n(n-1)}{ 6}$ . (b) Show that for all $n \ge3$ it is possible to have $m \ge \frac{(n-1)(n-2)}{ 6}$ .

For each positive integer $n$ determine the maximum number of points in space creating the set $A$ which has the following properties: $1)$ the coordinates of every point from the set $A$ are integers from the range $[0, n]$ $2)$ for each pair of different points $(x_1,x_2,x_3), (y_1,y_2,y_3)$ belonging to the set $A$ it is satisfied at least one of the following inequalities $x_1< y_1, x_2<y_2, x_3<y_3$ and at least one of the following inequalities $x_1>y_1, x_2>y_2,x_3>y_3$.

$240$ students are participating a big performance show. They stand in a row and face to their coach. The coach askes them to count numbers from left to right, starting from $1$. (Of course their counts be like $1,2,3,...$)The coach askes them to remember their number and do the following action: First, if your number is divisible by $3$ then turn around. Then, if your number is divisible by $5$ then turn around. Finally, if your number is divisible by $7$ then turn around. (a) How many students are face to coach now? (b) What is the number of the $66^{\text{th}}$ student counting from left who is face to coach?

Exactly $4n$ numbers in set $A= \{ 1,2,3,...,6n \} $ of natural numbers painted in red, all other in blue. Proved that exist $3n$ consecutive natural numbers from $A$, exactly $2n$ of which numbers is red.

The vertices of the regular $n$-gon and its center are marked. Two players play the following game: they, in turn, select a vertex and connect it by a segment to either the adjacent vertex or the center. The winner I a player if after his maveit is possible to get any marked point from any other moving along the segments. For each $n>2$ determine who has a winning strategy.

(a) Two people perform a card trick. The first performer takes $5$ cards from a $52$-card deck (previously shuffled by a member of the audience) , looks at them, and arranges them in a row from left to right: one face down (not necessarily the first one) , the others face up . The second performer guesses correctly the card which is face down. Prove that the performers can agree on a system which always makes this possible. (b) For their second trick, the first performer arranges four cards in a row, face up, the fifth card is kept hidden. Can they still agree on a system which enables the second performer to correctly guess the hidden card? (G Galperin)

Let $ n$ be an integer, $ n\geq 2$, and the integers $ a_1,a_2,\ldots,a_n$, such that $ 0 < a_k\leq k$, for all $ k \equal{} 1,2,\ldots,n$. Knowing that the number $ a_1 \plus{} a_2 \plus{} \cdots \plus{} a_n$ is even, prove that there exists a choosing of the signs $ \plus{}$, respectively $ \minus{}$, such that \[ a_1 \pm a_2 \pm \cdots \pm a_n\equal{} 0. \]

Given a positive integer $n\geq 2$. Let $A=\{ (a,b)\mid a,b\in \{ 1,2,…,n\} \}$ be the set of points in Cartesian coordinate plane. How many ways to colour points in $A$, each by one of three fixed colour, such that, for any $a,b\in \{ 1,2,…,n-1\}$, if $(a,b)$ and $(a+1,b)$ have same colour, then $(a,b+1)$ and $(a+1,b+1)$ also have same colour.

Leonard has a hand clock with only hour and minute hands. Determine the number of minutes in a day where the angle between the clock hands is not more than 1 degree. Both clock hands move continuously and at a constant speed. [i]Proposed by Giorgi Arabidze, Georgia [/i]

Prove that $$\sum \limits_{0\leq i<j\leq n}(i+j) {{n}\choose{i}}{{n}\choose{j}}=n\left (2^{2n-1}-{{2n-1}\choose{n}} \right )$$

A circular disc is divided into $12$ equal sectors and one of $6$ different colours is used to colour each sector. No two adjacent sectors can have the same colour. Find the number of such distinct colorings possible.

An $m \times n$ chessboard where $m \le n$ has several black squares such that no two rows have the same pattern. Determine the largest integer $k$ such that we can always color $k$ columns red while still no two rows have the same pattern.

We colored the $n^2$ unit squares of an $n\times n$ square lattice such that in each $2\times 2$ square, at least two of the four unit squares have the same color. What is the largest number of colors we could have used? [i]Based on a problem of the Dürer Competition[/i]

[b]p1.[/b] Prove that the equation $x^6 - 143x^5 - 917x^4 + 51x^3 + 77x^2 + 291x + 1575 = 0$ has no integer solutions. [b]p2.[/b] There are $81$ wheels in a storage marked by their two types, say first and second type. Wheels of the same type weigh equally. Any wheel of the second type is much lighter than a wheel of the first type. It is known that exactly one wheel is marked incorrectly. Show that it can be detected with certainty after four measurements on a balance scale. [b]p3.[/b] Rob and Ann multiplied the numbers from $1$ to $100$ and calculated the sum of digits of this product. For this sum, Rob calculated the sum of its digits as well. Then Ann kept repeating this operation until he got a one-digit number. What was this number? [b]p4.[/b] Rui and Jui take turns placing bishops on the squares of the $ 8\times 8$ chessboard in such a way that bishops cannot attack one another. (In this game, the color of the rooks is irrelevant.) The player who cannot place a rook loses the game. Rui takes the first turn. Who has a winning strategy, and what is it? [b]p5.[/b] The following figure can be cut along sides of small squares into several (more than one) identical shapes. What is the smallest number of such identical shapes you can get? [img]https://cdn.artofproblemsolving.com/attachments/8/e/9cd09a04209774dab34bc7f989b79573453f35.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n\geq 2$ be an integer. Alice and Bob play a game concerning a country made of $n$ islands. Exactly two of those $n$ islands have a factory. Initially there is no bridge in the country. Alice and Bob take turns in the following way. In each turn, the player must build a bridge between two different islands $I_1$ and $I_2$ such that: $\bullet$ $I_1$ and $I_2$ are not already connected by a bridge. $\bullet$ at least one of the two islands $I_1$ and $I_2$ is connected by a series of bridges to an island with a factory (or has a factory itself). (Indeed, access to a factory is needed for the construction.) As soon as a player builds a bridge that makes it possible to go from one factory to the other, this player loses the game. (Indeed, it triggers an industrial battle between both factories.) If Alice starts, then determine (for each $n\geq 2$) who has a winning strategy. ([i]Note:[/i] It is allowed to construct a bridge passing above another bridge.)

Captain Jack and his pirate men plundered six chests of treasure $(A_1,A_2,A_3,A_4,A_5,A_6)$. Every chest $A_i$ contains $a_i$ coins of gold, and all $a_i$s are pairwise different $(i=1,2,\cdots ,6)$. They place all chests according to a layout (see the attachment) and start to alternately take out one chest a time between the captain and a pirate who serves as the delegate of the captain’s men. A rule must be complied with during the game: only those chests that are not adjacent to other two or more chests are allowed to be taken out. The captain will win the game if the coins of gold he obtains are not less than those of his men in the end. Let the captain be granted to take chest firstly, is there a certain strategy for him to secure his victory?

There are $n$ teams in a football league. During a championship, every two teams play exactly one match, but no team can play more than one match in a week. At least, how many weeks are necessary for the championship to be held? Give an schedule for such a championship.

Let $A$ be a finite set of positive numbers , $B=\{\frac{a+b}{c+d} |a,b,c,d \in A \}$. Show that: $\left | B \right | \ge 2\left | A \right |^2-1 $, where $|X| $ be the number of elements of the finite set $X$. (High School Affiliated to Nanjing Normal University )

Let $S$ be a set of $2020$ distinct points in the plane. Let \[M=\{P:P\text{ is the midpoint of }XY\text{ for some distinct points }X,Y\text{ in }S\}.\] Find the least possible value of the number of points in $M$.