**A. 368.** The interior bisector of the angle *A* of the triangle *ABC* cuts the incircle *c* at two points; the one closer to *A* is denoted by *O*_{A}. Define the points *O*_{B} and *O*_{C} similarly on the internal bisectors of the angles *B* and *C*, respectively. The circle *c*_{A} is drawn about *O*_{A} and it is touching the sides *AB* and *CA*. Similarly, *c*_{B} is about *O*_{B} and it is touching the sides *BC* and *AB*, finally the circle *c*_{C} is about *O*_{C} and it is touching the sides *CA* and *BC*. Taken any two of the circles *c*_{A}, *c*_{B} and *c*_{C} consider their common external tangents different from the corresponding sides of the triangle, respectively. Prove that these three tangents are concurrent.

(5 points)

**B. 3809.** The triangle *ABC* is isosceles, namely *AB* = *BC*. The points *C*_{1}, *A*_{1}, *B*_{1} lie on the sides *AB*, *BC*, *CA*, respectively, and such that *BC*_{1}*A*_{1}=*CA*_{1}*B*_{1}=*CAB*. The lines *BB*_{1} and *CC*_{1} meet at *P*. Prove that the quadrilateral *AB*_{1}*PC*_{1} is cyclic.

(4 points)

**C. 802.** We want to prepare a folding rectangular dining table which when closed is in the position *ABCD*, when rotated about a pin by ninety degrees is in the position *A*'*B*'*C*'*D*' and it arrives to the position *B*_{1}*B*'*C*'*C*_{1} when unfolded. (See the *figure.*)

Where shall we put the pin about which the table rotates? How shall we set the dimensions of the table *ABCD* if, in case of a big fat dinner an even larger table could be produced from the position *B*_{1}*B*'*C*'*C*_{1} by repeating the previous procedure, i.e. rotating the table by right angle about the very same pin and unfolding it once more? (Proposed by *Papp Zoltán Dániel,* Szeged)

(5 points)

**K. 37.** At the confectioner's, we bought 5 pieces of candy in a paper bag, and 10 pieces in another, identical bag. (Each piece of candy has the same mass.) The first bag was weighed to be 85 grams and the second one 165 grams. How much did we pay for a paper bag if the candy cost 1200 forints (HUF) a kilo.

(6 points)

This problem is for grade 9 students only.