IT-CESENATICO 2001

In a basketball tournament every two teams play two matches. As usual, the winner of a match gets $2$ points, the loser gets $0$ , and there are no draws. A single team wins the tournament with $26$ points and exactly two teams share the last position with $20$ points. How many teams participated in the tournament?

Consider the equation $$x^{2001}=y^x .$$

• Find all pairs $(x,y)$ of solutions where $x$ is a prime number and $y$ is a positive integer.
• Find all pairs $(x,y)$ of solutions where $x$ and $y$ are positive integers.

(Remember that $2001=3 \cdot 23 \cdot 29$ .)

Let $ABC$ be a triangle and $\gamma$ the circle inscribed in $ABC$ . The circle $\gamma$ is tangent to side $AB$ at the point $T$ . Let $D$ be the point of $\gamma$ diametrically opposite to $T$ , and $S$ the intersection point of the line through $C$ and $D$ with side $AB$ . Prove that $AT=SB$ .

A panel contains $100$ light bulbs, arranged in a $10$ by $10$ square array. Some of them are on, the others are off. The electrical system is such that when the switch corresponding to a light bulb is pressed, all the light bulbs that are on the same row or column of it (including the bulb linked to the pressed switch) change their state (that is they are turned on or off).

• From which starting configurations, pressing the right sequence of switches, is it possible to achieve that all bulbs are on at the same time?
• What is the answer to the previous question if the bulbs are $81$ , arranged in a $9$ by $9$ panel?