Let $r\ge 1$ be a real number that holds with the property that for each pair of positive integer numbers $m$ and $n$, with $n$ a multiple of $m$, it is true that $\lfloor nr \rfloor$ is multiple of $\lfloor mr \rfloor$. Show that $r$ has to be an integer number. Note: If $x$ is a real number, $\lfloor x \rfloor$ is the greatest integer lower than or equal to $x$.

In a triangle $ABC$, it is drawn a circumference with center in the incenter $I$ and that meet twice each of the sides of the triangle: the segment $BC$ on $D$ and $P$ (where $D$ is nearer two $B$); the segment $CA$ on $E$ and $Q$ (where $E$ is nearer to $C$); and the segment $AB$ on $F$ and $R$ ( where $F$ is nearer to $A$). Let $S$ be the point of intersection of the diagonals of the quadrilateral $EQFR$. Let $T$ be the point of intersection of the diagonals of the quadrilateral $FRDP$. Let $U$ be the point of intersection of the diagonals of the quadrilateral $DPEQ$. Show that the circumcircle to the triangle $\triangle{FRT}$, $\triangle{DPU}$ and $\triangle{EQS}$ have a unique point in common.

Let $n \geq2$ be an integer number and $D_n$ the set of all the points $(x,y)$ in the plane such that its coordinates are integer numbers with: $-n \le x \le n$ and $-n \le y \le n$. (a) There are three possible colors in which the points of $D_n$ are painted with (each point has a unique color). Show that with any distribution of the colors, there are always two points of $D_n$ with the same color such that the line that contains them does not go through any other point of $D_n$. (b) Find a way to paint the points of $D_n$ with 4 colors such that if a line contains exactly two points of $D_n$, then, this points have different colors.

Let $n$ be a positive integer. Consider the sum $x_1y_1 + x_2y_2 +\cdots + x_ny_n$, where that values of the variables $x_1, x_2,\ldots, x_n, y_1, y_2,\ldots, y_n$ are either 0 or 1. Let $I(n)$ be the number of 2$n$-tuples $(x_1, x_2,\ldots, x_n, y_1, y_2,\ldots, y_n)$ such that the sum of the number is odd, and let $P(n)$ be the number of 2$n$-tuples $(x_1, x_2,\ldots, x_n, y_1, y_2,\ldots, y_n)$ such that the sum is an even number. Show that: \[ \frac{P(n)}{I(n)}=\frac{2^n+1}{2^n-1} \]

In an acute triangle $\triangle{ABC}$, let $AE$ and $BF$ be highs of it, and $H$ its orthocenter. The symmetric line of $AE$ with respect to the angle bisector of $\sphericalangle{A}$ and the symmetric line of $BF$ with respect to the angle bisector of $\sphericalangle{B}$ intersect each other on the point $O$. The lines $AE$ and $AO$ intersect again the circuncircle to $\triangle{ABC}$ on the points $M$ and $N$ respectively. Let $P$ be the intersection of $BC$ with $HN$; $R$ the intersection of $BC$ with $OM$; and $S$ the intersection of $HR$ with $OP$. Show that $AHSO$ is a paralelogram.

Let $P = \{P_1, P_2, ..., P_{1997}\}$ be a set of $1997$ points in the interior of a circle of radius 1, where $P_1$ is the center of the circle. For each $k=1.\ldots,1997$, let $x_k$ be the distance of $P_k$ to the point of $P$ closer to $P_k$, but different from it. Show that $(x_1)^2 + (x_2)^2 + ... + (x_{1997})^2 \le 9.$