3

Part of 2004 May Olympiad

Problems(2)

8x2 pool table, a ball that traveled 29 m starting from the center, no of hits

Source: May Olympiad (Olimpiada de Mayo) 2004 L2

We have a pool table

8

meters long and

2

meters wide with a single ball in the center. We throw the ball in a straight line and, after traveling

29

meters, it stops at a corner of the table. How many times did the ball hit the edges of the table?
Note: When the ball rebounds on the edge of the table, the two angles that form its trajectory with the edge of the table are the same.

changing numbers in 5x5 board, 2004 steps

Source: X May Olympiad (Olimpiada de Mayo) 2004 L1 P3

In each square of a

5\times 5

board is written

1

-1

. In each step, the number of each of the

25

cells is replaced by the result of the multiplication of the numbers of all its neighboring cells. Initially we have the board of the figure. https://cdn.artofproblemsolving.com/attachments/2/d/29b8e5df29526630102ac400a3a2b2f8fee4f7.gif Show how the board looks after

2004

steps.
Clarification: Two squares are neighbors if they have a common side.