**A. 540.** Let *A*_{1}*A*_{2}*A*_{3} be a non-equilateral triangle, let point *M* be its orthocenter, let *F* be its Feuerbach point, and let *k* be the circumcircle of the triangle. For *i*=1,2,3 denote by *k*_{i} the circle that is internally tangent to *k* and tangent to the sides *A*_{i}*A*_{i+1} and *A*_{i}*A*_{i+2}. (The indices are considered modulo 3, i.e. *A*_{4}=*A*_{1} and *A*_{5}=*A*_{2}.) Let *T*_{i} be the point of tangency between *k* and *k*_{i}. Prove that the lines *A*_{1}*T*_{1}, *A*_{2}*T*_{2}, *A*_{3}*T*_{3} and *MF* are concurrent.

Proposed by: *Gábor Damásdi* and *Márton Mester, *Budapest

(5 points)

**B. 4373.** An international conference of polypodal creatures is organized on the top of the Glass Mountain. The numbers of legs of the participants, *a*_{1},...,*a*_{n}, all are positive even numbers. The side of the Glass Mountain is slippery. Each of the creatures gathering at the base of the mountain can only get to the top if they wear special climbing shoes on at least half their feet. At least how many shoes are needed altogether to get all participants up to the top if a shoe must always be worn on a foot when it travels up or down the mountain?

Suggested by *G. Mészáros,* Kemence

(4 points)

**B. 4377.** Regular triangles *ABD*, *BCE*, *CAF* are drawn to the sides of a triangle *ABC* on the outside. Let the midpoints of line segments *DE*, *EF*, *FD* be *G*, *H*, *I*, respectively. Prove that *BG*=*CH*=*IA*.

Suggested by *Sz. Miklós,* Herceghalom

(4 points)