Find two roots of the equation $$5x^6 - 16x^4 - 33x^3 - 40x^2 +8 = 0,$$ whose product is equal to $1$.

Two integers $0 < k < n$ and distinct real numbers $a_1, a_2, \dots ,a_n$ are given. Define the sets as the following, where all indices are modulo $n$. \begin{align*} A &= \{ 1 \le i \le n : a_i > a_{i-k}, a_{i-1}, a_{i+1}, a_{i+k} \text{ or } a_i < a_{i-k}, a_{i-1}, a_{i+1}, a_{i+k} \} \\ B &= \{ 1 \le i \le n : a_i > a_{i-k}, a_{i+k} \text{ and } a_i < a_{i-1}, a_{i+1} \} \\ C &= \{ 1 \le i \le n ; a_i > a_{i-1}, a_{i+1} \text{ and } a_i < a_{i-k}, a_{i+k} \} \end{align*} Prove that $\lvert A \rvert \ge \lvert B \rvert + \lvert C \rvert$.

On a circular billiard table a ball rebounds from the rails as if the rail was the tangent to the circle at the point of impact. A regular hexagon with its vertices on the circle is drawn on a circular billiard table. A (point-shaped) ball is placed somewhere on the circumference of the hexagon, but not on one of its edges. Describe a periodical track of this ball with exactly four points at the rails. With how many different directions of impact can the ball be brought onto such a track?

Let $A_1,A_2,\ldots,A_n$ be subsets of a finite set $S$ such that $|A_j|=8$ for each $j$. For a subset $B$ of $S$ let $F(B)=\{j \mid 1\le j\le n \ \ \text{and} \ A_j \subset B\}$. Suppose for each subset $B$ of $S$ at least one of the following conditions holds [list][b](a)[/b] $|B| > 25$, [b](b)[/b] $F(B)={\O}$, [b](c)[/b] $\bigcap_{j\in F(B)} A_j \neq {\O}$.[/list] Prove that $A_1\cap A_2 \cap \cdots \cap A_n \neq {\O}$.

A positive integer $n$ is good if $n$ can be written as the sum of $2004$ positive integers $a_1$, $a_2$, ..., $a_{2004}$ such that $1 \le a_1 < a_2<...<a_{2004}$ and $a_i$ divides $a_{i+1}$ for $i=1$, $2$, ..., $2003$. Show that there are only finitely many positive integers that are not good.

The tangents to the circumcircle of the acute triangle $ABC$, passing through $B$ and $C$, meet at point $F$. The points $M$, $L$, and $N$ are the feet of perpendiculars from the vertex $A$ to the lines $FB$, $FC$, and $BC$, respectively. Prove the inequality $AM+AL\ge 2AN$. [i](V. Karamzin)[/i]

$A, B, C$, and $D$ are all on a circle, and $ABCD$ is a convex quadrilateral. If $AB = 13$, $BC = 13$, $CD = 37$, and $AD = 47$, what is the area of $ABCD$?

Prove that if the distance from a point inside a convex polyhedra with $n$ faces to the vertices of the polyhedra is at most 1, then the sum of the distances from this point to the faces of the polyhedra is smaller than $n-2$. [i]Calin Popescu[/i]

Natural $p$ and $q$ are relatively prime. The $[0,1]$ is divided onto $(p+q)$ equal segments. Prove that every segment except two marginal contain exactly one from the $(p+q-2)$ numbers $$\{1/p, 2/p, ... , (p-1)/p, 1/q, 2/q, ... , (q-1)/q\}$$

The king assembled 300 wizards and gave them the following challenge. For this challenge, 25 colors can be used, and they are known to the wizards. Each of the wizards receives a hat of one of those 25 colors. If for each color the number of used hats would be written down then all these number would be different, and the wizards know this. Each wizard sees what hat was given to each other wizard but does not see his own hat. Simultaneously each wizard reports the color of his own hat. Is it possible for the wizards to coordinate their actions beforehand so that at least 150 of them would report correctly?

Complex numbers $|z_1|=2,|z_2|=3$, and the intersection angle between the vectors corresponding to $z_1,z_2$ is $60^{\circ}$, then $\frac{|z_1+z_2|}{|z_1-z_2|}=$________.

Towns $A_1, A_2, . . . , A_{30}$ lie on line $MN$. The distances between the consecutive towns are equal. Each of the towns is the point of origin of a straight highway. The highways are on the same side of $MN$ and form the following angles with it: [img]https://cdn.artofproblemsolving.com/attachments/a/f/6cfcac497bdd729b966705f1060bd4b1caba25.png[/img] Thirty cars start simultaneously from these towns along the highway at the same constant speed. Each intersection has a gate. As soon as the first (in time, not in number) car passes the intersection the gate closes and blocks the way for all other cars approaching this intersection. Which cars will pass all intersections and which will be stopped? Note: This refers to angles measured counterclockwise from straight MN to the corresponding road.

Find all polynomials $ p\in\mathbb Z[x]$ such that $ (m,n)\equal{}1\Rightarrow (p(m),p(n))\equal{}1$

Let $ f$ be a polynomial with integer coefficients. Assume that there exists integers $ a$ and $ b$ such that $ f(a)\equal{}41$ and $ f(b)\equal{}49$. Prove that there exists an integer $ c$ such that $ 2009$ divides $ f(c)$. [i]Nanang Susyanto, Jogjakarta[/i]

Terry the Tiger lives on a cube-shaped world with edge length $2$. Thus he walks on the outer surface. He is tied, with a leash of length $2$, to a post located at the center of one of the faces of the cube. The surface area of the region that Terry can roam on the cube can be represented as $\frac{p \pi}{q} + a\sqrt{b}+c$ for integers $a, b, c, p, q$ where no integer square greater than $1$ divides $b, p$ and $q$ are coprime, and $q > 0$. What is $p + q + a + b + c$? (Terry can be at a location if the shortest distance along the surface of the cube between that point and the post is less than or equal to $2$.)

How many ways are there to arrange the numbers $\{ 1,2,3,4,5,6,7,8 \}$ in a circle so that every two adjacent elements are relatively prime? Consider rotations and reflections of the same arrangement to be indistinguishable.

Let $a_1,a_2,...,a_n$ be pairwise distinct real numbers and $m$ be the number of distinct sums $a_i +a_j$ (where $i \ne j$). Find the least possible value of $m$.

There are $270$ students at Colfax Middle School, where the ratio of boys to girls is $5 : 4$. There are $180$ students at Winthrop Middle School, where the ratio of boys to girls is $4 : 5$. The two schools hold a dance and all students from both schools attend. What fraction of the students at the dance are girls? $ \textbf{(A)} \dfrac7{18} \qquad\textbf{(B)} \dfrac7{15} \qquad\textbf{(C)} \dfrac{22}{45} \qquad\textbf{(D)} \dfrac12 \qquad\textbf{(E)} \dfrac{23}{45} $

Solve the following system of linear equations with unknown $x_1,x_2 \ldots, x_n \ (n \geq 2)$ and parameters $c_1,c_2, \ldots , c_n:$ \[2x_1 -x_2 = c_1;\]\[-x_1 +2x_2 -x_3 = c_2;\]\[-x_2 +2x_3 -x_4 = c_3;\]\[\cdots \qquad \cdots \qquad \cdots \qquad\]\[-x_{n-2} +2x_{n-1} -x_n = c_{n-1};\]\[-x_{n-1} +2x_n = c_n.\]

It is desired to construct a right triangle in the coordinate plane so that its legs are parallel to the $x$ and $y$ axes and so that the medians to the midpoints of the legs lie on the lines $y = 3x + 1$ and $y = mx + 2$. The number of different constants $m$ for which such a triangle exists is $ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \text{more than 3} $

Determine the largest natural number whose all decimal digits are different and which is divisible by each of its digits.

Let $m$ and $n$ be integers greater than 1. Prove that $\left\lfloor \dfrac{mn}{6} \right\rfloor$ non-overlapping 2-by-3 rectangles can be placed in an $m$-by-$n$ rectangle. Note: $\lfloor x \rfloor$ means the greatest integer that is less than or equal to $x$.

Let $ABC$ be an acute triangle with $BC = 4$ and $AC = 5$. Let $D$ be the midpoint of $BC$, $E$ be the foot of the altitude from $B$ to $AC$, and $F$ be the intersection of the angle bisector of $\angle BCA$ with segment $AB$. Given that $AD$, $BE$, and $CF$ meet at a single point $P$, compute the area of triangle $ABC$. Express your answer as a common fraction in simplest radical form.

In a football championship with $2021$ teams, each team play with another exactly once. The score of the match(es) is three points to the winner, one point to both players if the match end in draw(tie) and zero point to the loser. The final of the tournament will be played by the two highest score teams. Brazil Football Club won the first match, and it has the advantage if in the final score it draws with any other team. Determine the least score such that Brazil Football Club has a [b]chance[/b] to play the final match.

Show that $$ \left\{ X\in\mathcal{M}_2\left( \mathbb{Z}_3 \right)\left| \begin{pmatrix} 1&1\\2&2 \end{pmatrix} X\begin{pmatrix} 1&2\\2&1 \end{pmatrix} =0 \right. \right\} $$ is a multiplicative ring. [i]Cătălin Țigăeru[/i]