**A. 431.** Triangle *ABC* is not isosceles. The incenter is *I*, the excenter is *O*. The incircle touches the sides *BC*, *CA*, *AB* at points *D*, *E*, *F*, respectively. Lines *FD* and *AC *intersect at *P*, lines *DE* and *AB* intersect at *Q*. The midpoints of segments *EP* amd *FQ* are *M* and *N*, respectively. Prove that *MN* and *OI* are perpendicular.

(*Chinese *Mathematical Olympiad, 2007)

(5 points)

**K. 128.** The starting position of a logical game is shown below:

The pieces marked with arrows are moving. Every piece may move in the direction indicated by the arrow on it: it may either move to the adjoining vacant field or skip to the next field over any piece in the adjoining field, provided that the next field is free. The task is to make the counters marked with the two kinds of arrows exchange places. Draw the sequence of moves to achieve the exchange.

(6 points)

This problem is for grade 9 students only.

**K. 129.** At the beginning of the new schoolyear Pete started to print the sequence

200720082007200820072008...

on his computer. The printer however ran out of ink after the 2007^{th} digit has been printed. The amount of ink to print a digit 2 is 2/3 of the amount to print a 0 or 8, while printing a digit 7 requires half the amount of ink to print a 2. How many 7's could have been printed using the original amount of ink in the printed?

(6 points)

This problem is for grade 9 students only.