**A. 485.** Let *ABCD* be a tetrahedron with circumcenter *O*. Suppose that the points *P*, *Q* and *R* are interior points of the edges *AB*, *AC* and *AD*, respectively. Let *K*, *L*, *M* and *N* be the centroids of the triangles *PQD*, *PRC*, *QRB* and *PQR*, respectively. Prove that if the plane *PQR* is tangent to the sphere *KLMN* then *OP*=*OQ*=*OR*.

(5 points)

**A. 487.** Let *x*, *y*, *z* be positive numbers satisfying *xyz*1. Prove that .

Proposed by *Tuan Le,* Anaheim, California, USA

(5 points)

**B. 4200.** For a finite point set \(\displaystyle A\) in the plane, let \(\displaystyle v(A)\) denote the number of triangles in a triangulation of set \(\displaystyle A\), and let \(\displaystyle A+A=\{x+y\mid x,y\in A\}\), where the sum of two points is defined as the point whose position vector is the sum of the position vectors of the terms. Prove that \(\displaystyle v(A+A)\ge
4v(A)\).

Suggested by *I. Ruzsa,* Budapest

(5 points)

**C. 998.** The points *A*, *B*, *C* and *D* lie on a line in this order, and *AB*=*BC*. Draw perpendiculars to *AD* at *B *and *C*. The perpendicular drawn at *B* intersects the circle of diameter *AD* at *P* and *Q*. The perpendicular drawn at *C *intersects the circle of diameter *BD* at *K* and *L*. Show that the centre of the circle passing through points *P*, *K*, *L* and *Q* is point *B*.

(5 points)

**K. 212.** Four football teams organize a championship. Every team plays every other team once. The winning teams scores 3 points, the losing team scores 0 points and each team scores 1 point if there is a draw. The Albatrosses did not lose any game and won the tournament. The Black Sheep came out second. They did not have any draws. The third place went to the Cheerful Devils who did not win any game. The Dragonflies were in the fourth place. (In the case of equal scores, the order of teams is determined by the game played against each other. If it is a draw, a tie is declared.) Show that under the given conditions the scores of the four teams were different. List who was the winner in which game and which games ended with a draw. Justify your answers.

(6 points)

This problem is for grade 9 students only.