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 $y$ .

A $10 \times 10$ square on a grid is split by $80$ unit grid segments into $20$ polygons of equal area (no one of these segments belongs to the boundary of the square). Prove that all polygons are congruent. [i]($6$ points)[/i]

Let $(x_n)$ be a sequence of positive integers defined by $x_1=2$ and $x_{n+1}=2x_n^3+x_n$ for all integers $n\ge1$. Determine the largest power of $5$ that divides $x_{2014}^2+1$.

Let $A$ be any set of $20$ distinct integers chosen from the arithmetic progression $1, 4, 7,\ldots,100$. Prove that there must be two distinct integers in $A$ whose sum is $104$.

There is $207$ boxes on the table which numbered $1,2, \dots , 207$ respectively. Firstly Aslı puts a red ball in each of the $100$ boxes that she chooses and puts a white ball in each of the remaining ones. After that Zehra, writes a pair $(i,j)$ on the blackboard such that $1\leq i \leq j \leq 207$. Finally, Aslı tells Zehra that for every pair; whether the color of the balls which is inside the box which numbered by these numbers are the same or not. Find the least possible value of $N$ such that Zehra can guarantee finding all colors that has been painted to balls in each of the boxes with writing $N$ pairs on the blackboard.

Compute the sum of all integers $x$ for which there exists an integer $y$ such that \[x^3+xy+y^3=503.\] [i]Proposed by Nathan Xiong[/i]

The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\in P$ with integer coefficients is it true that $|4x+3y|\leq 1000$? $\textbf{(A) }38\qquad \textbf{(B) }40\qquad \textbf{(C) }42\qquad \textbf{(D) }44\qquad \textbf{(E) }46\qquad$

Given parallelogram $ABCD$, construct point $F$ so that $CF\perp BC$, as shown. Also $F$ is placed so that $\angle DFC = 120^o$. If $DF = 4$ and $BC =CF = 2$, what is the area of the parallelogram? [img]https://cdn.artofproblemsolving.com/attachments/7/4/0cdb0752760686acb891580da55f55212098fb.jpg[/img]

In a group of $n$ people, every weekend someone organizes a party in which he invites all of his acquaintances. Those who meet at a party become acquainted. After each of the $n$ people has organized a party, there still are two people not knowing each other. Show that these two will never get to know each other at such a party.

[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? $

Suppose $a$ is $150\%$ of $b$. What percent of $a$ is $3b$? $\textbf{(A) } 50 \qquad \textbf{(B) } 66\frac{2}{3} \qquad \textbf{(C) } 150 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 450$

Compute \[ \displaystyle\prod_{n=0}^{\infty} \left( 1 - \left( \frac {1}{2} \right)^{3^n} + \left( \frac {1}{4} \right)^{3^n} \right). \]

Draw a circle that has a given radius $R$ and is tangent to a given line and a given circle. How many solutions does this problem have?

You are playing a game. A $2 \times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3 \times 3$ grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensue that at least one of your guessed squares is covered by the rectangle? $\textbf{(A)}~3\qquad\textbf{(B)}~5\qquad\textbf{(C)}~4\qquad\textbf{(D)}~8\qquad\textbf{(E)}~6$

For $ n\ge 2$, let $ S_1$, $ S_2$, $ \ldots$, $ S_{2^n}$ be $ 2^n$ subsets of $ A \equal{} \{1, 2, 3, \ldots, 2^{n \plus{} 1}\}$ that satisfy the following property: There do not exist indices $ a$ and $ b$ with $ a < b$ and elements $ x$, $ y$, $ z\in A$ with $ x < y < z$ and $ y$, $ z\in S_a$, and $ x$, $ z\in S_b$. Prove that at least one of the sets $ S_1$, $ S_2$, $ \ldots$, $ S_{2^n}$ contains no more than $ 4n$ elements. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

On a plane there is a finite set of $M$ points, no three of which are collinear . Some points are joined to others by line segments, with each point connected to no more than one line segment . If we have a pair of intersecting line segments $AB$ and $CD$ we decide to replace them with $AC$ and $BD$, which are opposite sides of quadrilateral $ABCD$. In the resulting system of segments we decide to perform a similar substitution, if possible, and so on . Is it possible that such substitutions can be carried out indefinitely? (V.E. Kolosov)

Let $ABCD$ be an isosceles trapezoid with $AB = 1$, $BC = DA = 5$, $CD = 7$. Let $P$ be the intersection of diagonals $AC$ and $BD$, and let $Q$ be the foot of the altitude from $D$ to $BC$. Let $PQ$ intersect $AB$ at $R$. Compute $\sin \angle RP D$

[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 $x$.

A sextic funtion $ y \equal{} ax^6 \plus{} bx^5 \plus{} cx^4 \plus{} dx^3 \plus{} ex^2 \plus{} fx \plus{} g\ (a\neq 0)$ touches the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma \ (\alpha < \beta < \gamma ).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\gamma .$ created by kunny

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

Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $A_1,B_1$ and $C_1$ be respectively the midpoints of the arcs $BAC,CBA$ and $ACB$ of $\Gamma$. Show that the inradius of triangle $A_1B_1C_1$ is not less than the inradius of triangle $ABC$.

In a triangle $ABC$, the median from $B$ to $CA$ is perpendicular to the median from $C$ to $AB$. If the median from $A$ to $BC$ is $30$, determine $\frac{BC^2 + CA^2 + AB^2}{100}$.

Let $ABC$ be a non-isosceles triangle with incenter $I$ whose incircle is tangent to $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ at $D$, $E$, $F$, respectively. Denote by $M$ the midpoint of $\overline{BC}$. Let $Q$ be a point on the incircle such that $\angle AQD = 90^{\circ}$. Let $P$ be the point inside the triangle on line $AI$ for which $MD = MP$. Prove that either $\angle PQE = 90^{\circ}$ or $\angle PQF = 90^{\circ}$. [i]Proposed by Evan Chen[/i]

On the sides $(AB)$, $(BC)$, $(CD)$ and $(DA)$ of the regular tetrahedron $ABCD$, one considers the points $M$, $N$, $P$, $Q$, respectively Prove that $$MN \cdot NP \cdot PQ \cdot QM \ge AM \cdot BN \cdot CP \cdot DQ.$$