In Wonderland there are at least $5$ towns. Some towns are connected directly by roads or railways. Every town is connected to at least one other town and for any four towns there exists some direct connection between at least three pairs of towns among those four. When entering the public transportation network of this land, the traveller must insert one gold coin into a machine, which lets him use a direct connection to go to the next town. But if the traveller continues travelling from some town with the same method of transportation that took him there, and he has paid a gold coin to get to this town, then going to the next town does not cost anything, but instead the traveller gains the coin he last used back. In other cases he must pay just like when starting travelling. Prove that it is possible to get from any town to any other town by using at most $2$ gold coins.

What are the positive integer numbers that we are able to obtain in $2007$ distinct ways, when the sum is at least out of two positive consecutive integers? What is the smallest of all of them? Example: the number 9 is written in exactly two such distinct ways: $9 = 4 + 5$ $9 = 2 + 3 + 4.$

Given a set $S \subset R^+$, $S \ne \emptyset$ such that for all $a, b, c \in S$ (not necessarily distinct) then $a^3 + b^3 + c^3 - 3abc$ is rational number. Prove that for all $a, b \in S$ then $\frac{a - b}{a + b}$ is also rational.

Let $ABC$ be a triangle with $AB \ne AC$. The incirle of triangle $ABC$ is tangent to $BC, CA, AB$ at $D, E, F$, respectively. The perpendicular line from $D$ to $EF$ intersects $AB$ at $X$. The second intersection point of circumcircles of triangles $AEF$ and $ABC$ is $T$. Prove that $TX \perp T F$

For any $a,b,c>0$ satisfying $a+b+c+ab+ac+bc= 3$, prove that \[2\leq a+b+c+abc\leq 3\] [i]Proposed by Mohammad Ahmadi[/i]

Let $ABC$ be an isosceles triangle ($AB=AC$) with incenter $I$. Circle $\omega$ passes through $C$ and $I$ and is tangent to $AI$. $\omega$ intersects $AC$ and circumcircle of $ABC$ at $Q$ and $D$, respectively. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $CQ$. Prove that $AD$, $MN$ and $BC$ are concurrent. [i]Proposed by Alireza Dadgarnia[/i]

[b]p1.[/b] You are given three buckets with a capacity to hold $8$, $5$, and $3$ quarts of water, respectively. Initially, the first bucket is filled with $8$ quarts of water, while the remaining two buckets are empty. There are no markings on the buckets, so you are only allowed to empty a bucket into another one or to fill a bucket to its capacity using the water from one of the other buckets. (a) Describe a procedure by which we can obtain exactly $6$ quarts of water in the first bucket. (b) Describe a procedure by which we can obtain exactly $4$ quarts of water in the first bucket. [b]p2.[/b] A point in the plane is called a lattice point if its coordinates are both integers. A triangle whose vertices are all lattice points is called a lattice triangle. In each case below, give explicitly the coordinates of the vertices of a lattice triangle $T$ that satisfies the stated properties. (a) The area of $T$ is $1/2$ and two sides of $T$ have length greater than $2011$. (b) The area of $T$ is $1/2$ and the three sides of $T$ each have length greater than $2011$. [b]p3.[/b] Alice and Bob play several rounds of a game. In the $n$-th round, where $n = 1, 2, 3, ...$, the loser pays the winner $2^{n-1}$ dollars (there are no ties). After $40$ rounds, Alice has a profit of $\$2011$ (and Bob has lost $\$2011$). How many rounds of the game did Alice win, and which rounds were they? Justify your answer. [b]p4.[/b] Each student in a school is assigned a $15$-digit ID number consisting of a string of $3$’s and $7$’s. Whenever $x$ and $y$ are two distinct ID numbers, then $x$ and $y$ differ in at least three entries. Show that the number of students in the school is less than or equal to $2048$. [b]p5.[/b] A triangle $ABC$ has the following property: there is a point $P$ in the plane of $ABC$ such that the triangles $PAB$, $PBC$ and $PCA$ all have the same perimeter and the same area. Prove that: (a) If $P$ is not inside the triangle $ABC$, then $ABC$ is a right-angled triangle. (b) If $P$ is inside the triangle $ABC$, then $ABC$ is an equilateral triangle. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine the numbers $x,y$, with $x$ integer and $y$ rational, for which equality holds: $$5(x^2+xy+y^2) = 7(x+2y)$$

Prove that the equation $(3x+4y)(4x+5y)=7^z$ doesn't have solution in natural numbers.

Two integers are called [i]relatively prime[/i] if they share no common factors other than $1.$ Determine the sum of all positive integers less than $162$ that are relatively prime to $162.$

The area of a circle is doubled when its radius $ r$ is increased by $ n$. Then $ r$ equals: $ \textbf{(A)}\ n(\sqrt{2} \plus{} 1)\qquad \textbf{(B)}\ n(\sqrt{2} \minus{} 1)\qquad \textbf{(C)}\ n\qquad \textbf{(D)}\ n(2 \minus{} \sqrt{2})\qquad \textbf{(E)}\ \frac{n\pi}{\sqrt{2} \plus{} 1}$

Find all triples $(a,b,c)$ of real numbers such that the following system holds: $$\begin{cases} a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\a^2+b^2+c^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\end{cases}$$ [i]Proposed by Dorlir Ahmeti, Albania[/i]

For each positive integer $k$ let $a_k$ be the largest divisor of $k$ which is not divisible by $3$. Let $s_n=a_1+a_2+\dots+a_n$. Show that: (a) The number $s_n$ is divisible by $3$ iff the number of ones in the ternary expansion of $n$ is divisible by $3$. (b) There are infinitely many $n$ for which $s_n$ is divisible by $3^3$.

If $a @ b = \frac{a\times b}{a+b}$, for $a,b$ positive integers, then what is $5 @10$? $\textbf{(A)}\ \frac{3}{10} \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ \frac{10}{3} \qquad\textbf{(E)}\ 50$

Let $n$ be positive integer. Let $S_1,S_2,\cdots,S_k$ be a collection of $2n$-element subsets of $\{1,2,3,4,...,4n-1,4n\}$ so that $S_{i}\cap S_{j}$ contains at most $n$ elements for all $1\leq i<j\leq k$. Show that $$k\leq 6^{(n+1)/2}$$

Let $ABC$ be a triangle inscribed in the circle $(O)$. The bisector of $\angle BAC$ cuts the circle $(O)$ again at $D$. Let $DE$ be the diameter of $(O)$. Let $G$ be a point on arc $AB$ which does not contain $C$. The lines $GD$ and $BC$ intersect at $F$. Let $H$ be a point on the line $AG$ such that $FH \parallel AE$. Prove that the circumcircle of triangle $HAB$ passes through the orthocenter of triangle $HAC$.

In triangle $ABC, \angle A= 45^o, BH$ is the altitude, the point $K$ lies on the $AC$ side, and $BC = CK$. Prove that the center of the circumscribed circle of triangle $ABK$ coincides with the center of an excircle of triangle $BCH$.

[b]a)[/b] show that every curve $f:I \longrightarrow S^2$ is homotop with a path with another curve in $S^2$ like $g$ such that Image of $g$, doesn't contain all of $S^2$. [b]b)[/b] conclude that $S^2$ is simple connected. [b]c)[/b] construct a topological space such that it's fundamental group is $\mathbb Z_2$.

Let $T$ be the set of all lattice points (i.e., all points with integer coordinates) in three-dimensional space. Two such points $(x, y, z)$ and $(u, v,w)$ are called [i]neighbors[/i] if $|x - u| + |y - v| + |z - w| = 1$. Show that there exists a subset $S$ of $T$ such that for each $p \in T$ , there is exactly one point of $S$ among $p$ and its [i]neighbors[/i].

The positive integer $m$ is a multiple of $101$, and the positive integer $n$ is a multiple of $63$. Their sum is $2018$. Find $m - n$.

For all reals $a, b, c,d $ prove the following inequality: $$\frac{a + b + c + d}{(1 + a^2)(1 + b^2)(1 + c^2)(1 + d^2)}< 1$$

The motion of the particles of a fluid in the plane is specified by the following components of velocity $$\frac{dx}{dt}=y+2x(1-x^2 -y^2),\;\; \frac{dy}{dt}=-x.$$ Sketch the shape of the trajectories near the origin. Discuss what happens to an individual particle as $t\to \infty$, and justify your conclusion.

Determine for which positive integers $n$ there exists a positive integer $A$ such that • $A$ is divisible by $2022$, • the decimal expression of $A$ contains only digits $0$ and $7$, • the decimal expression of $A$ contains [i]exactly[/i] $n$ times the digit $7$.

A [i]cactus[/i] is a finite simple connected graph where no two cycles share an edge. Show that in a nonempty cactus, there must exist a vertex which is part of at most one cycle. [i]Kevin Yang[/i]

We call a pair $(a,b)$ of positive integers, $a<391$, [i]pupusa[/i] if $$\textup{lcm}(a,b)>\textup{lcm}(a,391)$$ Find the minimum value of $b$ across all [i]pupusa[/i] pairs. Fun Fact: OMCC 2017 was held in El Salvador. [i]Pupusa[/i] is their national dish. It is a corn tortilla filled with cheese, meat, etc.