The Occidentalia bank issues coins with denominations of $1$ peso, $8$ pesos, $27$ pesos... and any amount that is a perfect cube ($n^3$) of pesos. Determine what is the least amount $k$ of coins needed to give $2017$ pesos. For that amount, find all the possible ways to give $2017$ pesos using exactly $k$ currency.

From a point $P$, two tangent lines are drawn to a circle $\Gamma$, which touch it at points $A$ and $B$. A circle $\Phi$ is drawn with center at $P$ and passes through $A$ and $B$ and is taken a point $R$ that is on the circumference $\Phi$ and in the interior of $\Gamma$. The straight line $PR$ intersects $\Gamma$ at the points $S$ and $Q$. The straight lines $AR$ and $BR$ meet $\Gamma$ again at points $C$ and $D$, respectively. Prove that $CD$ passes through the midpoint of $SQ$.

In a building there are $119$ inhabitants who live in $120$ apartments (several inhabitants can live in the same apartment). We call an apartment overcrowded if $15$ or more people live in it. Every day in some overcrowded apartment (if there is one) its inhabitants have a fight and yes they all go to live in a different apartment (which may or may not be already inhabited). Should you always terminate this process?

Let $\vartriangle ABC$ be a triangle. Determine all points $P$ in the plane such that the triangles $\vartriangle ABP$, $\vartriangle ACP$ and $\vartriangle BCP$ all have the same area.

Laura and Daniel play with quadratic polynomials. First Laura says a nonzero real number $r$. Then Daniel says a nonzero real number $s$, and then again Laura says another nonzero real number $t$. Finally. Daniel writes the polynomial $P(x) = ax^2 + bx + c$ where $a,b$, and $c$ are $r,s$, and $t$ in some order Daniel chooses. Laura wins if the equation $P(x) = 0$ has two different real solutions, and Daniel wins otherwise. Determine who has a winning strategy and describe that strategy.

A change in a natural number $n$ consists of adding a pair of zeros between two digits or at the end of the decimal representation of $n$. A countryman of $n$ is a number that can be obtained from one or more changes in $n$. For example. $40041$, $4410000$ and $4004001$ are all countrymen from $441$. Determine all the natural numbers $n$ for which there is a natural number m with the property that $n$ divides $m$ and all the countrymen of $m$.