**B. 3869.** Point *M* lies on the angle bisector drawn from vertex *A*, inside an acute-angled triangle *ABC*. The other intersections of the lines *AM*, *BM*, *CM* with the circumscribed circle are *A*_{1}, *B*_{1} and *C*_{1}, respectively. The lines *AB* and *C*_{1}*A*_{1} intersect at *L*, and the lines *AC* and *B*_{1}*A*_{1} intersect at *N*. Prove that *LN* is parallel to *BC*.

(5 points)

**C. 830.** Lord Moneybag said to his grandson, ``Listen, Bill. Christmas is coming up. I have got some money between 300 and 500 pounds on me. It is an integer multiple of 6 pounds. I will give you 5 pounds out of it in one-pound coins. As I hand you the coins one by one, the amount that remains on me will first be divisible by 5, then by 4, then by 3, then by 2 and finally only by 1. If you can tell me how many pounds I have got on me, you'll get an extra ten-pound note.'' How much money did the lord have on him?

(5 points)

**K. 62.** Sophie has a toy that contains 9 wooden rods of different colours sticking out of a square wooden board, as shown in the *figure.* It is a logical puzzle. The task is to place nine red, nine green, and nine blue rings of the same shape on the rods. Each rod should have three rings of different colours on it, and at each level, the colours of the three rings along the directions indicated by the arrows in the diagram should also be different. In how many arrangements can the rings be placed on the rods? (Two arrangements count as different if there is at least one rod on which the order of the rings is different.)

(6 points)

This problem is for grade 9 students only.