The real numbers $a$, $b$ and $c$ verify that the polynomial $p(x)=x^4+ax^3+bx^2+ax+c$ has exactly three distinct real roots; these roots are equal to $\tan y$, $\tan 2y$ and $\tan 3y$, for some real number $y$. Find all possible values of $y$, $0\leq y < \pi$.

Let $ABC$ be an acute triangle with incenter $I$ and circumcenter $O$. Assume that $\angle OIA = 90^{\circ}$. Given that $AI = 97$ and $BC = 144$, compute the area of $\triangle ABC$.

In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.

There are $14$ players participating at a chess tournament, each playing one game with every other player. After the end of the tournament, the players were ranked in descending order based on their points. The sum of the points of the first three players is equal with the sum of the points of the last nine players. What is the highest possible number of draws in the tournament.(For a victory the player gets $1$ point, for a loss $0$ points, in a draw both players get $0,5$ points.)

A square-shaped pizza with side length $30$ cm is cut into pieces (not necessarily rectangular). All cuts are parallel to the sides, and the total length of the cuts is $240$ cm. Show that there is a piece whose area is at least $36$ cm$^2$

As part of his effort to take over the world, Edward starts producing his own currency. As part of an effort to stop Edward, Alex works in the mint and produces $1$ counterfeit coin for every $99$ real ones. Alex isn't very good at this, so none of the counterfeit coins are the right weight. Since the mint is not perfect, each coin is weighed before leaving. If the coin is not the right weight, then it is sent to a lab for testing. The scale is accurate $95\%$ of the time, $5\%$ of all the coins minted are sent to the lab, and the lab's test is accurate $90\%$ of the time. If the lab says a coin is counterfeit, what is the probability that it really is?

If you roll four standard, fair six-sided dice, the top faces of the dice can show just one value (for example, $3333$), two values (for example, $2666$), three values (for example, $5215$), or four values (for example, $4236$). The mean number of values that show is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Compute the number of positive four-digit multiples of $11$ whose sum of digits (in base ten) is divisible by $11$.

Given a polygon with $n$ sides, we assign the numbers $0,1,...,n-1$ to the vertices, and to each side is assigned the sum of the numbers assigned to its ends. The figure shows an example for $n = 5$. Notice that the numbers assigned to the sides are still in arithmetic progression. [img]https://cdn.artofproblemsolving.com/attachments/c/0/975969e29a7953dcb3e440884461169557f9a7.png[/img] $\bullet$ Make the respective assignment for a $9$-sided polygon, and generalize for odd $n$. $\bullet$ Prove that this is not possible if $n$ is even.

Given a triangle $ABC$ with symmedians $AD,BE,CF$ concurrent at [i]Lemoine[/i] point $L(D,E,F$ are on the sides $BC,CA,AB,$ respectively$).$ Prove that: $LA+LB+LC\ge 2(LD+LE+LF).$

Let $x_0, x_1, \ldots$ be a sequence of real numbers such that $x_n = \frac{1 + x_{n -1}}{x_{n - 2}}$ for $n \geq 2$. Find the number of ordered pairs of positive integers $(x_0, x_1)$ such that the sequence gives $x_{2018} = \frac{1}{1000}$.

Given an endless supply of white, blue and red cubes. In a circle arrange any $N$ of them. The robot, standing in any place of the circle, goes clockwise and, until one cube remains, constantly repeats this operation: destroys the two closest cubes in front of him and puts a new one behind him a cube of the same color if the destroyed ones are the same, and the third color if the destroyed two are different colors. We will call the arrangement of the cubes [i]good [/i] if the color of the cube remaining at the very end does not depends on where the robot started. We call $N$ [i]successful [/i] if for any choice of $N$ cubes all their arrangements are good. Find all successful $N$. I. Bogdanov

We consider the nonzero matrices $A_0, A_1, \ldots, A_n \in \mathcal{M}_2(\mathbb{R}),$ $n \ge 2,$ with the properties: $A_0 \neq aI_2$ for any $a \in \mathbb{R}$ and $A_0A_k=A_kA_0$ for $k= \overline{1,n}.$ Prove that a) $\det \left(\sum\limits_{k=1}^n A_k^2 \right) \ge 0$; b) If $\det \left(\sum\limits_{k=1}^n A_k^2 \right) = 0$ and $A_2 \ne aA_1$ for any $a \in \mathbb{R},$ then $\sum\limits_{k=1}^n A_k^2=O_2.$

Cai and Lai are eating cookies. Their cookies are in the shape of $2$ regular hexagons glued to each other, and the cookies have area $18$ units. They each make a cut along the $2$ long diagonals of a cookie; this now makes four pieces for them to eat and enjoy. What is the minimum area among the four pieces? [i]Proposed by Richard Chen[/i]

Call the [i]improvement [/i] of a positive number its replacement by a power of two. (i.e. one of the numbers $1, 2, 4, 8, ...$), for which it increases, but not more than than $3$ times. Given $2^{100}$ positive numbers with a sum of $2^{100}$. Prove that you can erase some of them, and [i]improve [/i] each of the other numbers so that the sum the resulting numbers were again $2^{100}$.

An isosceles triangle $T$ has the following property: it is possible to draw a line through one of the three vertices of $T$ that splits it into two smaller isosceles triangles $R$ and $S$, neither of which are similar to $T$. Find all possible values of the vertex (apex) angle of $T$.

A grasshopper can jump along a checkered strip for $8, 9$ or $10$ cells in any direction. A natural number $n$ is called jumpable if the grasshopper can start from some cell of a strip of length $n$ and visit every cell exactly once. Find at least one non-jumpable number $n > 50$. [i](Egor Bakaev)[/i]

Circles $S_1$ and $S_2$ meet at points $A$ and $B$. A line through $A$ is parallel to the line through the centers of $S_1$ and $S_2$ and meets $S_1$ and $S_2$ again $C$ and $D$ respectively. Circle $S_3$ having $CD$ as its diameter meets $S_1$ and $S_2$ again at $P$ and $Q$ respectively. Prove that lines $CP$, $DQ$, and $AB$ are concurent.

Let $S$ be the set of colorings of a $100 \times 100$ grid where each square is colored black or white and no $2\times2$ subgrid is colored like a chessboard. A random such coloring is chosen: what is the probability there is a path of black squares going from the top row to the bottom row where any two consecutive squares in the path are adjacent? [i]Proposed by Evan Chang (squareman), USA [/i]

A triangular pyramid $ O.ABC$ with base $ ABC$ has the property that the lengths of the altitudes from $ A$, $ B$ and $ C$ are not less than $ \frac{OB \plus{}OC}{2}$, $ \frac{OC \plus{} OA}{2}$ and $ \frac{OA \plus{} OB}{2}$, respectively. Given that the area of $ ABC$ is $ S$, calculate the volume of the pyramid.

A natural number is written in each cell of an $8 \times 8$ board. It turned out that for any tiling of the board with dominoes, the sum of numbers in the cells of each domino is different. Can it happen that the largest number on the board is no greater than $32$? [i](N. Chernyatyev)[/i] (Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])

It is given $n$ positive integers. Product of any one of them with sum of remaining numbers increased by $1$ is divisible with sum of all $n$ numbers. Prove that sum of squares of all $n$ numbers is divisible with sum of all $n$ numbers

Find all real $x$ that satisfy $\sqrt[3]{20x+\sqrt[3]{20x+13}}=13$.

There are $n$ people, and given that any $2$ of them have contacted with each other at most once. In any group of $n-2$ of them, any one person of the group has contacted with other people in this group for $3^k$ times, where $k$ is a non-negative integer. Determine all the possible value of $n.$

Let $(b_n)_{n \ge 0}$ be a sequence of positive integers satisfying $b_n=d\left(\sum_{i=0}^{n-1} b_k\right)$ for all $n \ge 1$. (By $d(m)$ we denote the number of positive divisors of $m$.) a) Prove that $(b_n)_{n \ge 0}$ is unbounded. b) Prove that there are infinitely many $n$ such that $b_n>b_{n+1}$.