Let regular hexagon is divided into $6n^2$ regular triangles. Let $2n$ coins are put in different triangles such, that no any two coins lie on the same layer (layer is area between two consecutive parallel lines). Let also triangles are painted like on the chess board. Prove that exactly $n$ coins lie on black triangles. [img]https://cdn.artofproblemsolving.com/attachments/0/4/96503a10351b0dc38b611c6ee6eb945b5ed1d9.png[/img]

Do there exist $5$ points in the space, such that for all $n\in\{1,2,\ldots,10\}$ there exist two of them at distance between them $n$?

Saskia and her sisters have been given a large number of pearls. The pearls are white, black and red, not necessarily the same number of each color. Each white pearl is worth $5$ Ducates, each black one is worth $7$, and each red one is worth $12$. The total worth of the pearls is $2107$ Ducates. Saskia and her sisters split the pearls so that each of them gets the same number of pearls and the same total worth, but the color distribution may vary among the sisters. Interestingly enough, the total worth in Ducates that each of the sisters holds equals the total number of pearls split between the sisters. Saskia is particularly fond of the red pearls, and therefore makes sure that she has as many of those as possible. How many pearls of each color has Saskia?

Prove that for arbitary integer $ n > 16$, there exists the set $ S$ that contains $ n$ positive integers and has the following property:if the subset $ A$ of $ S$ satisfies for arbitary $ a,a'\in A, a\neq a', a \plus{} a'\notin S$ holds, then $ |A|\leq4\sqrt n.$

Compute the number of five-digit positive integers whose digits have exactly $30$ distinct permutations (the permutations do not necessarily have to be valid five-digit integers). [i]Proposed by David Sun[/i]

Some of the towns in a country are connected with bidirectional paths, where each town can be reached by any other by going through these paths. From each town there are at least $n \geq 3$ paths. In the country there is no such route that includes all towns exactly once. Find the least possible number of towns in this country (Answer depends from $n$).

Prove that any group of people can be divided into two disjoint groups $A$ and $B$ such that any member from $A$ has at least half of his acquaintances in $B$ and any member from $B$ has at least half of his acquaintances in $A$ (acquaintance is reciprocal).

Let $A=\{1,2,\ldots,n\}$. Find the number of unordered triples $(X,Y,Z)$ that satisfy $X\bigcup Y \bigcup Z=A$

Consider a sequence of words , consisting of the letters $A$ and $B$ . The first word in the sequence is "$A$" . The k-th word i s obtained from the $(k-1)$-th by means of the following transformation : each $A$ is substituted by $AAB$ , and each $B$ is substituted by $A$. It is easily seen that every word is an initial part of the next word. The initial parts of these words coincide to give a sequence of letters $AABAABAAA BAABAAB...$ (a) In which place of this sequence is the $1000$-th letter $A$? (b ) Prove that this sequence is not periodic. (V . Galperin , Moscows)

An infinite grid with two rows is divided into unit squares. One of the cells in the second row is colored red and all other cells in the grid are white. Initially, we are in the red cell. In one move, we can move from one cell to an adjacent cell (sharing a side). Find the number of sequences of \( n \) moves such that no cell is visited more than once. (In particular, it is not allowed to return to the red cell after several moves.)

In a mathematical olympiad students received marks for any of the four areas: algebra, geometry, number theory and combinatorics. Any two of the students have distinct marks for all four areas. A group of students is called [i]nice [/i] if all students in the group can be ordered in increasing order simultaneously of at least two of the four areas. Find the least positive integer N, such that among any N students there exist a [i]nice [/i] group of ten students.

Let $n$ and $t$ be positive integers. What is the number of ways to place $t$ dominoes $(1\times 2$ or $2\times 1$ rectangles) in a $2\times n$ table so that there is no $2\times 2$ square formed by $2$ dominoes and each $2\times 3$ rectangle either does not have a horizontal domino in the middle and last cell in the first row or does not have a horizontal domino in the first and middle cell in the second row (or both)?

Alan is bored one day and decides to write down all the divisors of $1260^2$ on a wall. After writing down all of them, he realizes he wrote them on the wrong wall and needs to erase all his work. Every second, he picks a random divisor which is still on the wall and instantly erases it and every number that divides it. What is the expected time it takes for Alan to erase everything on the wall?

An equilateral triangle $\mathcal{T}{}$ with side 111 is divided by straight lines parallel to its sides into equilateral triangles with side 1. The vertices of these small triangles, except the centre of $\mathcal{T}{}$ are marked. Call a set of several marked points [i]linear[/i] if[list=i][*]the marked points lie on a line $\ell$ parallel to one of the sides of the triangle $\mathcal{T}$ and; [*]if two marked points on $\ell$ are in this set, every other marked point inbetween them is in the set. [/list]How many ways are there to split all the marked points into 111 linear sets?

Consider a sequence of positive integers with total sum $20$ such that no number and no sum of a set of consecutive numbers is equal to $3$. Is it possible for such a sequence to contain more than $10$ numbers? (Alexandr Shapovalov)

There is a board with 32 rows and 10 columns. Pablo writes 1 or -1 in each box. Matías, with Pablo's board in view, chooses one or more columns, and in each of the chosen columns, changes all of Pablo's numbers to their opposites (where there is 1 he puts -1 and where there is -1 he puts 1) . In the other columns, leave Pablo's numbers. Matías wins if he manages to make his board have each of the rows different from all the rows on Pablo's board. Otherwise, that is, if any row on Matías's board is equal to any row on Pablo's board, Pablo wins. If both play perfectly, determine which of the two is assured of victory.

Suppose you have ten pairs of red socks, ten pairs of blue socks, and ten pairs of green socks in your drawer. You need to go to a party soon, but the power is currently off in your room. It is completely dark, so you cannot see any colors and unfortunately the socks are identically shaped. What is the minimum number of socks you need to take from the drawer in order to guarantee that you have at least one pair of socks whose colors match?

Determine the greatest positive integer $ n$ such that in three-dimensional space, there exist n points $ P_{1},P_{2},\cdots,P_{n},$ among $ n$ points no three points are collinear, and for arbitary $ 1\leq i < j < k\leq n$, $ P_{i}P_{j}P_{k}$ isn't obtuse triangle.

p1. The parabola $y = ax^2 + bx$, $a < 0$, has a vertex $C$ and intersects the $x$-axis at different points $A$ and $B$. The line $y = ax$ intersects the parabola at different points $A$ and $D$. If the area of triangle $ABC$ is equal to $|ab|$ times the area of ​​triangle $ABD$, find the value of $ b$ in terms of $a$ without use the absolute value sign. p2. It is known that $a$ is a prime number and $k$ is a positive integer. If $\sqrt{k^2-ak}$ is a positive integer, find the value of $k$ in terms of $a$. p3. There are five distinct points, $T_1$, $T_2$, $T_3$, $T_4$, and $T$ on a circle $\Omega$. Let $t_{ij}$ be the distance from the point $T$ to the line $T_iT_j$ or its extension. Prove that $\frac{t_{ij}}{t_{jk}}=\frac{TT_i}{TT_k}$ and $\frac{t_{12}}{t_{24}}=\frac{t_{13}}{t_{34}}$ [img]https://cdn.artofproblemsolving.com/attachments/2/8/07fff0a36a80708d6f6ec6708f609d080b44a2.png[/img] p4. Given a $7$-digit positive integer sequence $a_1, a_2, a_3, ..., a_{2017}$ with $a_1 < a_2 < a_3 < ...<a_{2017}$. Each of these terms has constituent numbers in non-increasing order. Is known that $a_1 = 1000000$ and $a_{n+1}$ is the smallest possible number that is greater than $a_n$. As For example, we get $a_2 = 1100000$ and $a_3 = 1110000$. Determine $a_{2017}$. p5. At the oil refinery in the Duri area, pump-1 and pump-2 are available. Both pumps are used to fill the holding tank with volume $V$. The tank can be fully filled using pump-1 alone within four hours, or using pump-2 only in six hours. Initially both pumps are used simultaneously for $a$ hours. Then, charging continues using only pump-1 for $ b$ hours and continues again using only pump-2 for $c$ hours. If the operating cost of pump-1 is $15(a + b)$ thousand per hour and pump-2 operating cost is $4(a + c)$ thousand per hour, determine $ b$ and $c$ so that the operating costs of all pumps are minimum (express $b$ and $c$ in terms of $a$). Also determine the possible values ​​of $a$.

Sofia and Viktor are playing the following game on a $2022 \times 2022$ board: - Firstly, Sofia covers the table completely by dominoes, no two are overlapping and all are inside the table; - Then Viktor without seeing the table, chooses a positive integer $n$; - After that Viktor looks at the table covered with dominoes, chooses and fixes $n$ of them; - Finally, Sofia removes the remaining dominoes that aren't fixed and tries to recover the table with dominoes differently from before. If she achieves that, she wins, otherwise Viktor wins. What is the minimum number $n$ for which Viktor can always win, no matter the starting covering of dominoes. [i]Proposed by Viktor Simjanoski[/i]

Several strips and a circle of radius $1$ are drawn on the plane. The sum of the widths of the strips is $100$. Prove that one can translate each strip parallel to itself so that together they cover the circle. (M Smurov )

We are given $101$ rectangles with sides of integer lengths not exceeding $100$ . Prove that among these $101$ rectangles there are $3$ rectangles, say $A , B$ and $C$ such that $A$ will fit inside $B$ and $B$ inside $C$. ( N . Sedrakyan, Yerevan)

There are $12$ members in a club. The members created some small groups, which satisfy the following: - The small group consists of $3$ or $4$ people. - Also, for two arbitrary members, there exists exactly one small group that has both members. Prove that all members are in the same number of small groups.

Let $G$ be a simple graph with $n$ vertices and $m$ edges. Two vertices are called [i]neighbours[/i] if there is an edge between them. It turns out the $G$ does not contain any cycles of length from 3 to $2k$ (inclusive), where $k\geq2$ is a given positive integer. a) Prove that it is possible to pick a non-empty set $S$ of vertices of $G$ such that every vertex in $S$ has at least $\left\lceil \frac mn \right\rceil$ neighbours that are in $S$. ($\lceil x\rceil$ denotes the smallest integer larger than or equal to $x$.) b) Suppose a set $S$ as described in (a) is chosen. Let $H$ be the graph consisting of the vertices in $S$ and the edges between those vertices only. Let $v$ be a vertex of $H$. Prove that at least $\left\lceil \left(\frac mn -1\right)^k \right\rceil$ vertices of $H$ can be reached by starting at $v$ and travelling across the edges of $H$ for at most $k$ steps. (Note that $v$ itself satisfies this condition, since it can be reached by starting at $v$ and travelling along the edges of $H$ for 0 steps.)

An alphabet consists of $n$ letters. What is the maximal length of a word if we know that any two consecutive letters $a,b$ of the word are different and that the word cannot be reduced to a word of the kind $abab$ with $a\neq b$ by removing letters.