On a magical island there are lions, wolves and goats. Wolves can eat goats while lions can eat both wolves and goats. But if a lion eats a wolf, the lion becomes a goat. Likewise if a wolf eats a goat, the wolf becomes a lion. And if a lion eats a goat, the lion becomes a wolf. Initially on the island there are $17$ goats, $55$ wolves and $6$ lions. If they start eating until they no longer possible to eat more, what is the maximum number of animals that they can stay alive?

In a certain country there are 9 cities and two airline companies: AeroSol and AeroLuna. Between each pair of cities there are flights from one and only one of them. the two companies. Furthermore, for any triple of cities $X$, $Y$,$ Z$ σt least one of the flights between them is served by AeroLuna. It is possible to find $4$ cities such that all flights between them be served by AeroLuna?

A tourist traveling on a bus from the VIAZUL company arrived at the station in the city of Cienfuegos and immediately set out to take a walking tour of the city. After visiting the Terry theater in this city, decided to return to the station but walking only for blocks traveled an odd number of times in his previous walk. Can he do this on his return regardless. of the initial route?

Let $A$ and $B$ be two subsets of $\{1, 2, 3, 4, ..., 100\}$, such that $|A| = |B|$ and $A\cap B =\emptyset$. If $n \in A$ implies that $2n + 2 \in B$, determine the largest possible value of $ |A \cup B|$.

In a certain forest there are at least three crossroads, and for any three crossroads of roads A, B and C there is a road from A to B without passing through C. A deer and a hunter are standing at crossroads of different paths. Is it possible that they can exchange positions without their paths crossing at other points, that are not their initial positions?

Let $f$ be a function of the positive reals in the positive reals, such that $$f(x) \cdot f(y) - f(xy) = \frac{x}{y} + \frac{y}{x} \ \ for \ \ all \ \ x, y > 0 .$$(a) Find $f(1)$. (b) Find $f(x)$.

Let $ABCD$ be a convex quadrilateral and let $P$ be the intersection of the diagonals $AC$ and $BD$. The radii of the circles inscribed in the triangles $\vartriangle ABP$, $\vartriangle BCP$, $\vartriangle CDP$ and $\vartriangle DAP$ are the same. Prove that $ABCD$ is a rhombus,

Determine the smallest integer of the form $\frac{ \overline{AB}}{B}$ .where $A$ and $B$ are three-digit positive integers and $\overline{AB}$ denotes the six-digit number that is form by writing the numbers $A$ and $B$ consecutively.

Let $A = \overline{abcd}$ be a $4$-digit positive integer, such that $a\ge 7$ and $a > b >c > d > 0$. Let us consider a positive integer $B = \overline{dcba}$. If all digits of $A+B$ are odd, determine all possible values of $A$.

Let $a, b$ and $c$ be real numbers such that $0 < a, b, c < 1$. Prove that: $$\min \ \ \{ab(1 -c)^2, bc(1 - a)^2, ca(1 - b)^2 \} \le \frac{1}{16}.$$

Let $ABC$ be a triangle such that $AB > AC$, with a circumcircle $\omega$. Draw the tangents to $\omega$ at $B$ and $C$ and these intersect at $P$. The perpendicular to $AP$ through $A$ cuts $BC$ at $R$. Let $S$ be a point on the segment $PR$ such that $PS = PC$. (a) Prove that the lines $CS$ and $AR$ intersect on $\omega$. (b) Let $M$ be the midpoint of $BC$ and $Q$ be the point of intersection of $CS$ and $AR$. Circle $\omega$ and the circumcircle of $\vartriangle AMP$ intersect at a point $J$ ($J \ne A$), prove that $P$, $J$ and $Q$ are collinear.

If $p$ is a prime number and $x, y$ are positive integers, find in terms of $p$, all pairs $(x, y)$ that satisfy the equation: $$p(x -2) = x(y -1).$$If $x+y = 21$, find all triples $(x, y, p)$ that satisfy this equation.

Let $ABC$ be an acute triangle and $D$ be the foot of the altiutude from $A$ on $BC$, $E$ and $F$ are the midpoints of $BD$ and $DC$ respectively. $O$ and $Q$ are the circumcenters of the triangles $\vartriangle BF$ and $\vartriangle ACE$ respectively. $P$ is the intersection point of $OE$ and $QF$, show that $PB = PC$.

Determine the largest possible value of$ M$ for which it holds that: $$\frac{x}{1 +\dfrac{yz}{x}}+ \frac{y}{1 + \dfrac{zx}{y}}+ \frac{z}{1 + \dfrac{xy}{z}} \ge M,$$for all real numbers $x, y, z > 0$ that satisfy the equation $xy + yz + zx = 1$.