Day 1
We have $98$ cards, in each one we will write one of the numbers: $1, 2, 3, 4,...., 97, 98$. We can order the $98$ cards, in a sequence such that two consecutive numbers $X$ and $Y$ and the number $X - Y$ is greater than $48$, determine how and how many ways we can make this sequence!!
Let $C$ be a circunference, $O$ is your circumcenter, $AB$ is your diameter and $R$ is any point in $C$ ($R$ is different of $A$ and $B$) Let $P$ be the foot of perpendicular by $O$ to $AR$, in the line $OP$ we match a point $Q$, where $QP$ is $\frac{OP}{2}$ and the point $Q$ isn't in the segment $OP$. In $Q$, we will do a parallel line to $AB$ that cut the line $AR$ in $T$. Denote $H$ the point of intersections of the line $AQ$ and $OT$. Show that $H$, $B$ and $R$ are collinears.
Show that, exist infinite triples $(a, b, c)$ where $a, b, c$ are natural numbers, such that: $2a^2 + 3b^2 - 5c^2 = 1997$
Day 2
Consider a board with $n$ rows and $4$ columns. In the first line are written $4$ zeros (one in each house). Next, each line is then obtained from the previous line by performing the following operation: one of the houses, (that you can choose), is maintained as in the previous line; the other three are changed: * if in the previous line there was a $0$, then in the down square $1$ is placed; * if in the previous line there was a $1$, then in the down square $2$ is placed; * if in the previous line there was a $2$, then in the down square $0$ is placed; Build the largest possible board with all its distinct lines and demonstrate that it is impossible to build a larger board.
Let $n$ be a natural number $n>3$. Show that in the multiples of $9$ less than $10^n$, exist more numbers with the sum of your digits equal to $9(n - 2)$ than numbers with the sum of your digits equal to $9(n - 1)$.
Let $ABC$ be a acute-angle triangle and $X$ be point in the plane of this triangle. Let $M,N,P$ be the orthogonal projections of $X$ in the lines that contains the altitudes of this triangle Determine the positions of the point $X$ such that the triangle $MNP$ is congruent to $ABC$