[b]p1.[/b] An animal farm has geese and pigs with a total of $30$ heads and $84$ legs. Find the number of pigs and geese on this farm. [b]p2.[/b] What is the maximum number of $1 \times 1$ squares of a $7 \times 7$ board that can be colored black in such a way that the black squares don’t touch each other even at their corners? Show your answer on the figure below and explain why it is not possible to get more black squares satisfying the given conditions. [img]https://cdn.artofproblemsolving.com/attachments/d/5/2a0528428f4a5811565b94061486699df0577c.png[/img] [b]p3.[/b] Decide whether it is possible to divide a regular hexagon into three equal not necessarily regular hexagons? A regular hexagon is a hexagon with equal sides and equal angles. [img]https://cdn.artofproblemsolving.com/attachments/3/7/5d941b599a90e13a2e8ada635e1f1f3f234703.png[/img] [b]p4.[/b] A rectangle is subdivided into a number of smaller rectangles. One observes that perimeters of all smaller rectangles are whole numbers. Is it possible that the perimeter of the original rectangle is not a whole number? [b]p5.[/b] Place parentheses on the left hand side of the following equality to make it correct. $$ 4 \times 12 + 18 : 6 + 3 = 50$$ [b]p6.[/b] Is it possible to cut a $16\times 9$ rectangle into two equal parts which can be assembled into a square? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $f(x)=x^2-6x+5$. On the plane $(x, y)$ draw a set of points $M(x, y)$ whose coordinates satisfy the inequalities $$\begin{cases} f(x)+f(y)\le 0 \\ f(x)-f(y)\ge 0 \end{cases}$$

[u]Round 1 [/u] [b]p1.[/b] In the Hundred Acre Wood, all the animals are either knights or liars. Knights always tell the truth and liars always lie. One day in the Wood, Winnie-the-Pooh, a knight, decides to visit his friend Rabbit, also a noble knight. Upon arrival, Pooh finds his friend sitting at a round table with $5$ other guests. One-by-one, Pooh asks each person at the table how many of his two neighbors are knights. Surprisingly, he gets the same answer from everybody! "Oh bother!" proclaims Pooh. "I still don't have enough information to figure out how many knights are at this table." "But it's my birthday," adds one of the guests. "Yes, it's his birthday!" agrees his neighbor. Now Pooh can tell how many knights are at the table. Can you? [b]p2.[/b] Harry has an $8 \times 8$ board filled with the numbers $1$ and $-1$, and the sum of all $64$ numbers is $0$. A magical cut of this board is a way of cutting it into two pieces so that the sum of the numbers in each piece is also $0$. The pieces should not have any holes. Prove that Harry will always be able to find a magical cut of his board. (The picture shows an example of a proper cut.) [img]https://cdn.artofproblemsolving.com/attachments/4/b/98dec239cfc757e6f2996eef7876cbfd79d202.png[/img] [b]p3.[/b] Several girls participate in a tennis tournament in which each player plays each other player exactly once. At the end of the tournament, it turns out that each player has lost at least one of her games. Prove that it is possible to find three players $A$, $B$, and $C$ such that $A$ defeated $B$, $B$ defeated $C$, and $C$ defeated $A$. [b]p4.[/b] $120$ bands are participating in this year's Northwest Grunge Rock Festival, and they have $119$ fans in total. Each fan belongs to exactly one fan club. A fan club is called crowded if it has at least $15$ members. Every morning, all the members of one of the crowded fan clubs start arguing over who loves their favorite band the most. As a result of the fighting, each of them leaves the club to join another club, but no two of them join the same one. Is it true that, no matter how the clubs are originally arranged, all these arguments will eventually stop? [b]p5.[/b] In Infinite City, the streets form a grid of squares extending infinitely in all directions. Bonnie and Clyde have just robbed the Infinite City Bank, located at the busiest intersection downtown. Bonnie sets off heading north on her bike, and, $30$ seconds later, Clyde bikes after her in the same direction. They each bike at a constant speed of $1$ block per minute. In order to throw off any authorities, each of them must turn either left or right at every intersection. If they continue biking in this manner, will they ever be able to meet? [u]Round 2 [/u] [b]p6.[/b] In a certain herd of $33$ cows, each cow weighs a whole number of pounds. Farmer Dan notices that if he removes any one of the cows from the herd, it is possible to split the remaining $32$ cows into two groups of equal total weight, $16$ cows in each group. Show that all $33$ cows must have the same weight. [b]p7.[/b] Katniss is thinking of a positive integer less than $100$: call it $x$. Peeta is allowed to pick any two positive integers $N$ and $M$, both less than $100$, and Katniss will give him the greatest common divisor of $x+M$ and $N$ . Peeta can do this up to seven times, after which he must name Katniss' number $x$, or he will die. Can Peeta ensure his survival? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The equation $x^3-3x^2+1=0$ has three real solutions $x_1<x_2<x_3$. Show that for any positive integer $n$, the number $\left\lceil x_3^n\right\rceil$ is a multiple of $3$.

Let $n \ge 2$ be an integer. Find all real numbers $a$ such that there exist real numbers $x_1,x_2,\dots,x_n$ satisfying \[x_1(1-x_2)=x_2(1-x_3)=\dots=x_n(1-x_1)=a.\] [i]Proposed by Walther Janous and Gerhard Kirchner, Innsbruck.[/i]

Find all real numbers $x$ such that $\lfloor x^2-2x \rfloor+2\lfloor x \rfloor=\lfloor x \rfloor^2$. (For a real number $x$, $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$.)

Let $P(x) = x^4 + x^3 + x^2 + x + 1$. Show that there exist two non-constant polynomials $Q(y)$ and $R(y)$ with integer coefficients such that for all $Q(y) \cdot R(y)= P(5y^2)$ for all $y$ .

[b]a)[/b] Find the real numbers $ x,y $ such that the set $ \{ x,y \}\cup\left\{ 31/20,29/30,27/40,11/120 \right\} $ contains six elements that can represent an arithmetic progression. [b]b)[/b] Let be four real numbers in arithmetic progression $ b_1<b_2<b_3<b_4. $ Are there sets $ S $ of $ 6 $ elements that represent an arithmetic progression such that $ \left\{ b_1,b_2,b_3,b_4 \right\}\subset S? $

[u]Round 1[/u] [b]p1.[/b] Three snails – Alice, Bobby, and Cindy – were racing down a road. Whenever one snail passed another, it waved at the snail it passed. During the race, Alice waved $3$ times and was waved at twice. Bobby waved $4$ times and was waved at $3$ times. Cindy waved $5$ times. How many times was she waved at? [b]p2.[/b] Sherlock and Mycroft are playing Battleship on a $4\times 4$ grid. Mycroft hides a single $3\times 1$ cruiser somewhere on the board. Sherlock can pick squares on the grid and fire upon them. What is the smallest number of shots Sherlock has to fire to guarantee at least one hit on the cruiser? [b]p3.[/b] Thirty girls – $13$ of them in red dresses and $17$ in blue dresses – were dancing in a circle, hand-in-hand. Afterwards, each girl was asked if the girl to her right was in a blue dress. Only the girls who had both neighbors in red dresses or both in blue dresses told the truth. How many girls could have answered “Yes”? [b]p4.[/b] Herman and Alex play a game on a $5\times 5$ board. On his turn, a player can claim any open square as his territory. Once all the squares are claimed, the winner is the player whose territory has the longer border. Herman goes first. If both play their best, who will win, or will the game end in a draw? [img]https://cdn.artofproblemsolving.com/attachments/5/7/113d54f2217a39bac622899d3d3eb51ec34f1f.png[/img] [b]p5.[/b] Is it possible to find $2014$ distinct positive integers whose sum is divisible by each of them? [u]Round 2[/u] [b]p6.[/b] Hermione and Ron play a game that starts with 129 hats arranged in a circle. They take turns magically transforming the hats into animals. On each turn, a player picks a hat and chooses whether to change it into a badger or into a raven. A player loses if after his or her turn there are two animals of the same species right next to each other. Hermione goes first. Who loses? [b]p7.[/b] Three warring states control the corner provinces of the island whose map is shown below. [img]https://cdn.artofproblemsolving.com/attachments/e/a/4e2f436be1dcd3f899aa34145356f8c66cda82.png[/img] As a result of war, each of the remaining $18$ provinces was occupied by one of the states. None of the states was able to occupy any province on the coast opposite their corner. The states would like to sign a peace treaty. To do this, they each must send ambassadors to a place where three provinces, one controlled by each state, come together. Prove that they can always find such a place to meet. For example, if the provinces are occupied as shown here, the squares mark possible meeting spots. [img]https://cdn.artofproblemsolving.com/attachments/e/b/81de9187951822120fc26024c1c1fbe2138737.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine whether or not two polynomials $P, Q$ with degree no less than 2018 and with integer coefficients exist such that $$P(Q(x))=3Q(P(x))+1$$ for all real numbers $x$.

The following applies: $$a, b, c, d > 0 , a + b < c + d$$ Prove that $$ac + bd > ab.$$

A loader has a waggon and a little cart. The waggon can carry up to 1000 kg, and the cart can carry only up to 1 kg. A finite number of sacks with sand lie in a storehouse. It is known that their total weight is more than 1001 kg, while each sack weighs not more than 1 kg. What maximum weight of sand can the loader carry in the waggon and the cart, regardless of particular weights of sacks? [i]Author: M.Ivanov, D.Rostovsky, V.Frank[/i]

Find all functions $ f: \mathbb{N} \to \mathbb{Z} $ satisfying $$ n \mid f\left(m\right) \Longleftrightarrow m \mid \sum\limits_{d \mid n}{f\left(d\right)} $$ holds for all positive integers $ m,n $

Let $a_0,$ $a_1,$ $\ldots,$ $a_n$ be complex numbers such that [center]$|a_nz^n+a_{n-1}z^{n-1}+\ldots+a_1z+a_0| \le 1,$ for any $z \in \mathbb{C}$ with $|z|=1.$[/center] Prove that $|a_k| \le 1$ and $|a_0+a_1+\ldots+a_n-(n+1)a_k| \le n,$ for any $k=\overline{0,n}.$

Tickets for the football game are $\$10$ for students and $\$15$ for non-students. If $3000$ fans attend and pay $\$36250$, how many students went?

Let $n$ be a natural number, $n \ge 5$, and $a_1, a_2, . . . , a_n$ real numbers such that all possible sums $a_i + a_j$, where $1 \le i < j \le n$, form $\frac{n(n-1)}{2}$ consecutive members of an arithmetic progression when taken in some order. Prove that $a_1 = a_2 = . . . = a_n$.

Let $a$, $b$ be two distinct real numbers and let $c$ be a positive real numbers such that $a^4 - 2019a = b^4 - 2019b = c$. Prove that $- \sqrt{c} < ab < 0$.

Which number is greater, $\frac{1996^{1995}+1}{1996^{1996}+1}$ or $ \frac{1996^{1996}+1}{1996^{1997}+1}$ ?

Define a sequence $a_n$ such that $a_n = a_{n-1} - a_{n-2}$. Let $a_1 = 6$ and $a_2 = 5$. Find $\Sigma_{n=1}^{1000}a_n$.

Prove that if $ n $ is an integer greater than $ 4 $, then $ 2^n $ is greater than $ n^2 $.

Prove that the numbers $1,2,...,1998$ cannot be separated into three classes whose sums of elements are divisible by $2000,3999$, and $5998$, respectively.

Prove that for every acute angle $\alpha$, $\sin (\cos \alpha) < \cos(\sin \alpha)$.

Blondes, brunettes, redheads and brown-haired women participate in the Olympiad. There are twice as many redheads as brown-haired. Blondes and redheads make up a quarter of the total number of participants, and Brown-haired and Blondes one fifth part. Prove that the number of Brunettes is divisible by $7$.

Find all polynomials $P(x, y)$ with real coefficients which for all real numbers $x$ and $y$ satisfy $P(x + y, x - y) = 2P(x, y)$.

Prove that $1280000401$ is composite.