There are 2002 towns in a kingdom. Some of the towns are connected by roads in such a manner that, if all roads from one city closed, one can still travel between any two cities. Every year, the kingdom chooses a non-self-intersecting cycle of roads, founds a new town, connects it by roads with each city from the chosen cycle, and closes all the roads from the original cycle. After several years, no non-self-intersecting cycles remained. Prove that at that moment there are at least 2002 towns, exactly one road going out from each of them.

Find all $f:\mathbb{R}\to\mathbb{R}$ functions such that $$f(x+y)^3=(x+2y)f(x^2)+f(f(y))(x^2+3xy+y^2)$$ for all real numbers $x,y$

There are 2012 chameleons sitting at a round table. At the beginning each has the color red or green. After every full minute, each chamaleon, which has two neighbors of the same color, changes its color from red to green or from green to red. All others keep their color. Show that after $2012$ minutes there are at least $2$ chameleons that have the same often changed color. [hide=original wording]Es sitzen 2012 Chamaleons an einem runden Tisch. Am Anfang besitzt jedes die Farbe rot oder grun. Nach jeder vollen Minute wechselt jedes Cham aleon, welches zwei gleichfarbige Nachbarn hat, seine Farbe von rot zu grun respektive von gr un zu rot. Alle anderen behalten ihre Farbe. Zeige, dass es nach 2012 Minuten mindestens 2 Chamaleons gibt, welche gleich oft die Farbe gewechselt haben.[/hide]

31. Define a number to be an anti-palindrome if, when written in base $3$ as $a_na_{n-1}\ldots a_0$, then $a_i+a_{n-i} = 2$ for any $0 \le i \le n$. Find the number of anti-palindromes less than $3^{12}$ such that no two consecutive digits in base 3 are equal. 32. Let $C_{k,n}$ denote the number of paths on the Cartesian plane along which you can travel from $(0, 0)$ to $(k, n)$, given the following rules: 1) You can only travel directly upward or directly rightward 2) You can only change direction at lattice points 3) Each horizontal segment in the path must be at most $99$ units long. Find $$\sum_{j=0}^\infty C_{100j+19,17}$$ 33. Camille the snail lives on the surface of a regular dodecahedron. Right now he is on vertex $P_1$ of the face with vertices $P_1, P_2, P_3, P_4, P_5$. This face has a perimeter of $5$. Camille wants to get to the point on the dodecahedron farthest away from $P_1$. To do so, he must travel along the surface a distance at least $L$. What is $L^2$?

Suppose that all collisions are instantaneous and perfectly elastic. After a long time, which of the following is true? (A) The center block is moving to the left. (B) The center block is moving to the right. (C) The center block is at rest somewhere to the left of its initial position. (D) The center block is at rest at its initial position. (E) The center block is at rest somewhere to the right of its initial position.

A set $M$ of points in the plane will be called obtuse, if any 3 points from $M$ are the vertices of an obtuse triangle. a.) Prove: For each finite obtuse set $M$ there is a point in the plane with the following property: $P$ is no element from $M$ and $M \cup \{P\}$ is also obtuse. b.) Determine whether the statement from a.) will remain valid, if it is replaced by infinite.

Find all polynomials $P(x)$ with integer coefficients such that for all positive number $n$ and prime $p$ satisfying $p\nmid nP(n)$, we have $ord_p(n)\ge ord_p(P(n))$.

In a square $ABCD$ of side $2$ units, $E$ is the midpoint of $AD$ and $F$ on $BE$ such that $CF\perp BE$, then the quadrilateral $CDEF$ has an area of (A)$2$ (B)$2.2$ (C)$\sqrt{5}$ (D)None of these

Given a convex quadrilateral $ABCD$ with $\angle BCD=90^\circ$. Let $E$ be the midpoint of $AB$. Prove that $2EC \leqslant AD+BD$.

On the party every boy gave $1$ candy to every girl and every girl gave $1$ candy to every boy. Then every boy ate $2$ candies and every girl ate $3$ candies. It is known that $\frac 14$ of all candies was eaten. Find the greatest possible number of children on the party.

A finite list of rational numbers is written on a blackboard. In an [i]operation[/i], we choose any two numbers $a$, $b$, erase them, and write down one of the numbers \[ a + b, \; a - b, \; b - a, \; a \times b, \; a/b \text{ (if $b \neq 0$)}, \; b/a \text{ (if $a \neq 0$)}. \] Prove that, for every integer $n > 100$, there are only finitely many integers $k \ge 0$, such that, starting from the list \[ k + 1, \; k + 2, \; \dots, \; k + n, \] it is possible to obtain, after $n - 1$ operations, the value $n!$.

Calvin and Hobbes play a game. First, Hobbes picks a family $F$ of subsets of $\{1, 2, . . . , 2020\}$, known to both players. Then, Calvin and Hobbes take turns choosing a number from $\{1, 2, . . . , 2020\}$ which is not already chosen, with Calvin going first, until all numbers are taken (i.e., each player has $1010$ numbers). Calvin wins if he has chosen all the elements of some member of $F$, otherwise Hobbes wins. What is the largest possible size of a family $F$ that Hobbes could pick while still having a winning strategy?

An altitude of a pentagon is the perpendicular drop from a vertex to the opposite side. A median of a pentagon is the line joining a vertex to the midpoint of the opposite side. If the five altitudes and the five medians all have the same length, prove that the pentagon is regular.

Given points $A,B,M,N$ on the circumference. Two chords $[MA_1]$ and $[MA_2]$ are orthogonal to lines $(NA)$ and $(NB)$ respectively. Prove that $(AA_1)$ and $(BB_1)$ lines are parallel.

Given a positive integer $n$, all of its positive integer divisors are written on a board. Two players $A$ and $B$ play the following game: Each turn, each player colors one of these divisors either red or blue. They may choose whichever color they wish, but they may only color numbers that have not been colored before. The game ends once every number has been colored. $A$ wins if the product of all of the red numbers is a perfect square, or if no number has been colored red, $B$ wins otherwise. If $A$ goes first, determine who has a winning strategy for each $n$.

The axis of a parabola is its axis of symmetry and its vertex is its point of intersection with its axis. Find: the equation of the parabola which touches $y = 0$ at $(1,0)$ and $x = 0$ at $(0,2);$ the equation of its axis; and its vertex.

a) Given $100$ numbers $a_1, ..., a_{100}$ such that $\begin{cases} a_1 - 3a_2 + 2a_3 \ge 0, \\ a_2 - 3a_3 + 2a_4 \ge 0, \\ a_3 - 3a_4 + 2a_5 \ge 0, \\ ... \\ a_{99} - 3a_{100} + 2a_1 \ge 0, \\ a_{100} - 3a_1 + 2a_2 \ge 0 \end{cases}$ prove that the numbers are equal. b) Given numbers $a_1=1, ..., a_{100}$ such that $\begin{cases} a_1 - 4a_2 + 3a_3 \ge 0, \\ a_2 - 4a_3 + 3a_4 \ge 0, \\ a_3 - 4a_4 + 3a_5 \ge 0, \\ ... \\ a_{99} - 4a_{100} + 3a_1 \ge 0, \\ a_{100} - 4a_1 + 3a_2 \ge 0 \end{cases}$ Find $a_2, a_3, ... , a_{100}.$

Show: when a prime number is divided by $30,$ the remainder is either $1$ or a prime number. Is a similar claim true, when the divisor is $60$ or $90$?

Suppose that $f:\{1, 2,\ldots ,1600\}\rightarrow\{1, 2,\ldots ,1600\}$ satisfies $f(1)=1$ and \[f^{2005}(x)=x\quad\text{for}\ x=1,2,\ldots ,1600. \] $(a)$ Prove that $f$ has a fixed point different from $1$. $(b)$ Find all $n>1600$ such that any $f:\{1,\ldots ,n\}\rightarrow\{1,\ldots ,n\}$ satisfying the above condition has at least two fixed points.

Let be given $n\ge 3$ distinct points in the plane. Is it always possible to find a circle which passes through three of the points and contains none of the remaining points (a) inside the circle. (b) inside the circle or on its boundary?

How many ways are there to arrange three indistinguishable rooks on a $ 6 \times 6$ board such that no two rooks are attacking each other? (Two rooks are attacking each other if and only if they are in the same row or the same column.)

A permutation $(a_1, a_2, a_3, \dots, a_{2012})$ of $(1, 2, 3, \dots, 2012)$ is selected at random. If $S$ is the expected value of \[ \sum_{i = 1}^{2012} | a_i - i |, \] then compute the sum of the prime factors of $S$. [i]Proposed by Aaron Lin[/i]

A circle $\omega$ touches the sides $A$B and $BC$ of a triangle $ABC$ and intersects its side $AC$ at $K$. It is known that the tangent to $\omega$ at $K$ is symmetrical to the line $AC$ with respect to the line $BK$. What can be the difference $AK -CK$ if $AB = 9$ and $BC = 11$?

Consider $\vartriangle ABC$ and a point $M$ inside it. We move $M$ parallel to $BC$ until $M$ meets $CA$, then parallel to $AB$ until it meets $BC$, then parallel to $CA$, and so on. Prove that $M$ traverses a self-intersecting closed broken line and find the number of its straight segments.

Let $ABC$ be an isosceles triangle with $AC=BC$, whose incentre is $I$. Let $P$ be a point on the circumcircle of the triangle $AIB$ lying inside the triangle $ABC$. The lines through $P$ parallel to $CA$ and $CB$ meet $AB$ at $D$ and $E$, respectively. The line through $P$ parallel to $AB$ meets $CA$ and $CB$ at $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ intersect on the circumcircle of the triangle $ABC$. [i]Proposed by Hojoo Lee, Korea[/i]