4. At a point $R$ on a line, there is a robot that moves along this line to the left or right as it wishes. It is programmed to take 2 steps on the first move, 4 steps on the second move, 6 steps on the third move, and in general, $2n$ steps on the $n$-th move.

a) Describe a variant of the robot's movement such that it starts from point $R$ and ends its movement at point $R$ after exactly 4 moves.

b) What is the minimum number of moves the robot must make so that it starts from point $R$ and ends up back at $R$ at the end of its movement?

c) Prove that there is a variant of the robot's movement such that it starts from $R$ and ends up at $R$ after 179 moves.

## NOTE

- All questions are mandatory;
- Each question is worth 7 points;
- No points are awarded by default;
- The actual working time is 2 hours from the moment the question is received.


## Mathematics Olympiad  Regional Phase - February 16, 2013

## Grade 6 - grading rubric
4. a) If $S$ represents a move to the left and $D$ to the right, then one variant would be S-D-D-S

$2 p$

b) The minimum number of moves is 3, for example S-S-D

$2 p$

c) The idea is that 4 consecutive moves in the pattern S-D-D-S leave the robot in the same place. 173 gives a remainder of 3 when divided by 4, so the first three moves are made as in point b), and then every four moves follow the pattern above.

NOTE

- Any correct solution is graded accordingly.