CONO-SUR 1998

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.