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]

Let $x, y, z \in R$, find all triples $(x, y, z)$ that satisfy the following system of equations: $2x^2 - 3xy + 2y^2 = 1$ $y^2 - 3yz + 4z^2 = 2$ $z^2 + 3zx - x^2 = 3$

Isabella had a week to read a book for a school assignment. She read an average of $36$ pages per day for the first three days and an average of $44$ pages per day for the next three days. She then finished the book by reading $10$ pages on the last day. How many pages were in the book? $\textbf{(A) }240\qquad\textbf{(B) }250\qquad\textbf{(C) }260\qquad\textbf{(D) }270\qquad \textbf{(E) }280$

The sum of two prime numbers is $85$. What is the product of these two prime numbers? $\textbf{(A) }85\qquad\textbf{(B) }91\qquad\textbf{(C) }115\qquad\textbf{(D) }133\qquad \textbf{(E) }166$

A set of three nonnegative integers $\{x, y, z \}$ with $x<y<z$ is called historic if $\{z-y, y-x\}=\{1776,2001\}$. Show that the set of all nonnegative integers can be written as the union of pairwise disjoint historic sets.

The taxicab distance between points $(x_1, y_1$) and $(x_2, y_2)$ is $|x_2 -x_1|+|y_2 -y_1|$. A regular octagon is positioned in the $xy$ plane so that one of its sides has endpoints $(0, 0)$ and $(1, 0)$. Let $S$ be the set of all points inside the octagon whose taxicab distance from some octagon vertex is at most $2/3$. The area of $S$ can be written as $m/n$ , where $m, n$ are positive integers and $gcd (m, n) = 1$. Find $100m + n$.

In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$

Find all pairs $(m, n)$ of natural numbers for which the numbers $m^2 - 4n$ and $n^2 - 4m$ are both perfect squares.

Each day, the price of the shares of the corporation “Soap Bubble, Limited” either increases or decreases by $n$ percent, where $n$ is an integer such that $0 < n < 100$. The price is calculated with unlimited precision. Does there exist an $n$ for which the price can take the same value twice?

Let $ ABCD$ be a convex quadrilateral with $ AB\equal{}AD$ and $ BC\equal{}CD$. On the sides $ AB,BC,CD,DA$ we consider points $ K,L,L_1,K_1$ such that quadrilateral $ KLL_1K_1$ is rectangle. Then consider rectangles $ MNPQ$ inscribed in the triangle $ BLK$, where $ M\in KB,N\in BL,P,Q\in LK$ and $ M_1N_1P_1Q_1$ inscribed in triangle $ DK_1L_1$ where $ P_1$ and $ Q_1$ are situated on the $ L_1K_1$, $ M$ on the $ DK_1$ and $ N_1$ on the $ DL_1$. Let $ S,S_1,S_2,S_3$ be the areas of the $ ABCD,KLL_1K_1,MNPQ,M_1N_1P_1Q_1$ respectively. Find the maximum possible value of the expression: $ \frac{S_1\plus{}S_2\plus{}S_3}{S}$

Let $ A,B,C,$ and $ D$ denote four points in space and $ AB$ the distance between $ A$ and $ B$, and so on. Show that \[ AC^2\plus{}BD^2\plus{}AD^2\plus{}BC^2 \ge AB^2\plus{}CD^2.\]

Prove that two points in a compact metric space can be joined with a rectifiable arc if and only if there exists a positive number $ K$ such that, for any $ \varepsilon>0$, these points can be connected with an $ \varepsilon$-chain not longer that $ K$. [i]M. Bognar[/i]