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 $H$ be the orthocenter of the triangle $ABC$, $M$ is the midpoint of the segment $BC$. Let $X$ be the point of the intersection of the line $HM$ with arc $BC$(without $A$) of the circumcircle of $ABC$, let $Y$ be the point of intersection of the line $BH$ with the circle, show that $XY = BC$.

Prove that, least $30$% of the natural numbers $n$ between $1$ and $1000000$ the first digit of $2^n$ is $1$.

Find all functions $R-->R$ such that: $f(x^2) - f(y^2) + 2x + 1 = f(x + y)f(x - y)$

In Terra Brasilis there are $n$ houses where $n$ goblins live, each in a house. There are one-way routes such that: - each route joins two houses, - in each house exactly one route begins, - in each house exactly one route ends. If a route goes from house $A$ to house $B$, then we will say that house $B$ is next to house $A$. This relationship is not symmetric, that is: in this situation, not necessarily house $A$ is next to house $B$. Every day, from day $1$, each goblin leaves the house where he is and arrives at the next house. A legend of Terra Brasilis says that when all the goblins return to the original position, the world will end. a) Show that the world will end. b) If $n = 98$, show that it is possible for elves to build and guide the routes so that the world does not end before $300,000$ years.

The mayor of a city wishes to establish a transport system with at least one bus line, in which: - each line passes exactly three stops, - every two different lines have exactly one stop in common, - for each two different bus stops there is exactly one line that passes through both. Determine the number of bus stops in the city.