King Albrecht founded a family. In the family everyone has exactly $ 8$ children. The only, but really important rule is that among the grandchildren of any person at most $x$ can be named Bela. (None of Albrecht’s children is called Bela.) For which $x$ is it possible that after a certain time each newborn in the family has at least one direct ancestor in the Royal family called Bela. No two of Albrecht’s descendants (including himself) have a common child.

[b]p17.[/b] Let the roots of the polynomial $f(x) = 3x^3 + 2x^2 + x + 8 = 0$ be $p, q$, and $r$. What is the sum $\frac{1}{p} +\frac{1}{q} +\frac{1}{r}$ ? [b]p18.[/b] Two students are playing a game. They take a deck of five cards numbered $1$ through $5$, shuffle them, and then place them in a stack facedown, turning over the top card next to the stack. They then take turns either drawing the card at the top of the stack into their hand, showing the drawn card to the other player, or drawing the card that is faceup, replacing it with the card on the top of the pile. This is repeated until all cards are drawn, and the player with the largest sum for their cards wins. What is the probability that the player who goes second wins, assuming optimal play? [b]p19.[/b] Compute the sum of all primes $p$ such that $2^p + p^2$ is also prime. [b]p20.[/b] In how many ways can one color the $8$ vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A school has three classes. Some pairs of children from different classes are enemies (there are no enemies in a class). It is known that every child from the first class has as many enemies in the second class as in the third; the same is true for other classes. Prove that the number of pairs of children from classes having a common enemy is not less than the number of pairs of children being enemies.

In each cell of an \(n \times n\) grid, one of the numbers \(0\), \(1,\) or \(2\) must be written. Determine all positive integers \(n\) for which there exists a way to fill the \(n \times n\) grid such that, when calculating the sum of the numbers in each row and each column, the numbers \(1, 2, \ldots, 2n\) are obtained in some order.

Prove that $\sum \frac{1}{i_1i_2 \ldots i_k} = n$ is taken over all non-empty subsets $\left\{i_1,i_2, \ldots, i_k\right\}$ of $\left\{1,2,\ldots,n\right\}$. (The $k$ is not fixed, so we are summing over all the $2^n-1$ possible nonempty subsets.)

A group of $ 100$ students numbered $ 1$ through $ 100$ are playing the following game. The judge writes the numbers $ 1$, $ 2$, $ \ldots$, $ 100$ on $ 100$ cards, places them on the table in an arbitrary order and turns them over. The students $ 1$ to $ 100$ enter the room one by one, and each of them flips $ 50$ of the cards. If among the cards flipped by student $ j$ there is card $ j$, he gains one point. The flipped cards are then turned over again. The students cannot communicate during the game nor can they see the cards flipped by other students. The group wins the game if each student gains a point. Is there a strategy giving the group more than $ 1$ percent of chance to win?

Pedro and Juan are playing the following game: $-$ There are $2$ piles of rocks, with $X$ rocks in one pile and $Y$ rocks in the other pile ($X < 12, Y < 11$). $-$ Each player can draw: -- $1$ rock from one of the piles, or -- $2$ rocks from one of the piles, or -- $1$ rock from each pile, or -- $2$ rock from one pile and $1$ from the other pile. Each player must perform one of these four operations in their turns. The looser is the one who takes the last rock. Pedro plays first and has a winning strategy. What are the three maximum possible values of ($X+Y$)?

In how many ways can the king in the chessboard reach the eighth rank in $7$ moves from its original square on the first row?

[u]Round 1[/u] [b]p1.[/b] Is it possible to draw some number of diagonals in a convex hexagon so that every diagonal crosses EXACTLY three others in the interior of the hexagon? (Diagonals that touch at one of the corners of the hexagon DO NOT count as crossing.) [b]p2.[/b] A $ 3\times 3$ square grid is filled with positive numbers so that (a) the product of the numbers in every row is $1$, (b) the product of the numbers in every column is $1$, (c) the product of the numbers in any of the four $2\times 2$ squares is $2$. What is the middle number in the grid? Find all possible answers and show that there are no others. [b]p3.[/b] Each letter in $HAGRID$'s name represents a distinct digit between $0$ and $9$. Show that $$HAGRID \times H \times A\times G\times R\times I\times D$$ is divisible by $3$. (For example, if $H=1$, $A=2$, $G=3$, $R = 4$, $I = 5$, $D = 64$, then $HAGRID \times H \times A\times G\times R\times I\times D= 123456\times 1\times2\times3\times4\times5\times 6$). [b]p4.[/b] You walk into a room and find five boxes sitting on a table. Each box contains some number of coins, and you can see how many coins are in each box. In the corner of the room, there is a large pile of coins. You can take two coins at a time from the pile and place them in different boxes. If you can add coins to boxes in this way as many times as you like, can you guarantee that each box on the table will eventually contain the same number of coins? [b]p5.[/b] Alex, Bob and Chad are playing a table tennis tournament. During each game, two boys are playing each other and one is resting. In the next game the boy who lost a game goes to rest, and the boy who was resting plays the winner. By the end of tournament, Alex played a total of $10$ games, Bob played $15$ games, and Chad played $17$ games. Who lost the second game? [u]Round 2[/u] [b]p6.[/b] After going for a swim in his vault of gold coins, Scrooge McDuck decides he wants to try to arrange some of his gold coins on a table so that every coin he places on the table touches exactly three others. Can he possibly do this? You need to justify your answer. (Assume the gold coins are circular, and that they all have the same size. Coins must be laid at on the table, and no two of them can overlap.) [b]p7.[/b] You have a deck of $50$ cards, each of which is labeled with a number between $1$ and $25$. In the deck, there are exactly two cards with each label. The cards are shuffled and dealt to $25$ students who are sitting at a round table, and each student receives two cards. The students will now play a game. On every move of the game, each student takes the card with the smaller number out of his or her hand and passes it to the person on his/her right. Each student makes this move at the same time so that everyone always has exactly two cards. The game continues until some student has a pair of cards with the same number. Show that this game will eventually end. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

How many colorings of an $n$-gon in $p \ge 2$ colors are there such that no two neighboring vertices have the same color?

1.1 Let $f(A)$ denote the difference between the maximum value and the minimum value of a set $A$. Find the sum of $f(A)$ as $A$ ranges over the subsets of $\{1, 2, \dots, n\}$. 1.2 All cells of an $8 × 8$ board are initially white. A move consists of flipping the color (white to black or vice versa) of cells in a $1\times 3$ or $3\times 1$ rectangle. Determine whether there is a finite sequence of moves resulting in the state where all $64$ cells are black. 1.3 Prove that for all positive integers $m$, there exists a positive integer $n$ such that the set $\{n, n + 1, n + 2, \dots , 3n\}$ contains exactly $m$ perfect squares.

$a_1,a_2,\ldots,a_n$ is a sequence of positive integers that has at least $\frac {2n}{3}+1$ distinct numbers and each positive integer has occurred at most three times in it. Prove that there exists a permutation  $b_1,b_2,\ldots,b_n$ of $a_i $'s such that all the $n$ sums $b_i+b_{i+1}$ are distinct ($1\le i\le n $ , $b_{n+1}\equiv b_1 $) [i]Proposed by Mohsen Jamali[/i]

Find all triples $(a, b, c)$ of positive integers with $a+b+c = 10$ such that there are $a$ red, $b$ blue and $c$ green points (all different) in the plane satisfying the following properties: $\bullet$ for each red point and each blue point we consider the distance between these two points, the sum of these distances is $37$, $\bullet$ for each green point and each red point we consider the distance between these two points, the sum of these distances is $30$, $\bullet$ for each blue point and each green point we consider the distance between these two points, the sum of these distances is $1$.

Each face of a cube is divided into $n^2$ equal squares. The vertices of the squares are called [i]nodes[/i], so each face has $(n+1)^2$ nodes. $(a)$ If $n=2$,does there exist a closed polygonal line whose links are sids of the squares and which passes through each node exactly once? $(b)$ Prove that, for each $n$, such a polygonal line divides the surface area of the cube into two equal parts

Let $a_1, a_2, \cdots, a_n$ be a sequence of nonnegative integers. For $k=1,2,\cdots,n$ denote \[ m_k = \max_{1 \le l \le k} \frac{a_{k-l+1} + a_{k-l+2} + \cdots + a_k}{l}. \] Prove that for every $\alpha > 0$ the number of values of $k$ for which $m_k > \alpha$ is less than $\frac{a_1+a_2+ \cdots +a_n}{\alpha}.$

Gourmet Jan compared $n$ restaurants ($n$ is a positive integer). Each pair of restaurants was compared in two categories: tastiness of food and quality of service. For some pairs Jan couldn't tell which restaurant was better in one category, but never in two categories. Moreover, if Jan thought restaurant $A$ was better than restaurant $B$ in one category and restaurant $B$ was better than restaurant $C$ in the same category, then $A$ is also better than $C$ in that category. Prove there exists a restaurant $R$ such that every other restaurant is worse than $R$ in at least one category.

A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations? $ \textbf{(A)}\ 15\qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 70 $

In Miskolc there are two tram lines: line $1$ runs between Tiszai railway station and UpperMajláth, while line $2$ runs between Tiszai railway station and the Ironworks. The timetable for trams leaving Tiszai railway station is as follows: tram $ 1$ leaves at every minute ending in a $0$ or $6$, and tram $2$ leaves at every minute ending in a $3$. There are three types of passengers waiting for the trams: those who will take tram $ 1$ only, those who will take tram $2$ only and those who will take any tram. Every minute there is a constant number of passengers of each type arriving at the station. (This number is not necessarily the same for the different types.) Also, every tram departs with an equal number of passengers from Tiszai railway station. How many passengers are there on a departing tram, if we know that every minute there are $3$ passengers arriving at the station who will take tram $2$ only?

Let $ \mathcal{A}_n$ denote the set of all mappings $ f: \{1,2,\ldots ,n \} \rightarrow \{1,2,\ldots, n \}$ such that $ f^{-1}(i) :=\{ k \colon f(k)=i\ \} \neq \varnothing$ implies $ f^{-1}(j) \neq \varnothing, j \in \{1,2,\ldots, i \} .$ Prove \[ |\mathcal{A}_n| = \sum_{k=0}^{\infty} \frac{k^n}{2^{k+1}}.\] [i]L. Lovasz[/i]

Initially, on the lower left and right corner of a $2018\times 2018$ board, there're two horses, red and blue, respectively. $A$ and $B$ alternatively play their turn, $A$ start first. Each turn consist of moving their horse ($A$-red, and $B$-blue) by, simultaneously, $20$ cells respect to one coordinate, and $17$ cells respect to the other; while preserving the rule that the horse can't occupied the cell that ever occupied by any horses in the game. The player who can't make the move loss, who has the winning strategy?

Consider a $7\times7$ chessboard that starts out with all the squares white. We start painting squares black, one at a time, according to the rule that after painting the first square, each newly painted square must be adjacent along a side to only the square just previously painted. The final figure painted will be a connected “snake” of squares. (a) Show that it is possible to paint $31$ squares. (b) Show that it is possible to paint $32$ squares. (c) Show that it is possible to paint $33$ squares.

Let $S$ be a set of $n$ points in the plane such that any two points of $S$ are at least $1$ unit apart. Prove there is a subset $T$ of $S$ with at least $\frac{n}{7}$ points such that any two points of $T$ are at least $\sqrt{3}$ units apart.

A fair coin is repeatedly flipped until $2019$ consecutive coin flips are the same. Compute the probability that the first and last flips of the coin come up differently.

Let $n$ be a positive integer. In how many ways can we mark cells on an $n\times n$ board such that no two rows and no two columns have the same number of marked cells? [i]Selim Bahadir, Turkey[/i]

Two people throw coins. One throws his coin $10$ times, the other throws his $11$ times . What is the probability that the second coin fell showing "heads" a greater number of times than the first? (For those not familiar with Probability Theory this question could have been reformulated thus : Consider various arrangements of a $21$ digit number in which each digit must be a " $1$ " or a "$2$" . Among all such numbers what fraction of them will have more occurrences of the digit "$2$" among the last $11$ digits than among the first $10$?) (S. Fomin , Leningrad)