**A. 467.** Let *ABCD* be a circumscribed trapezoid such that the lines *AD* and *BC* intersect at point *R*. Denote by *I* the incenter of the trapezoid, and let the incircle touch the sides *AB* and *CD* at points *P* and *Q*, respectively. Let the line through *P*, which is perpendicular to *PR*, meet the lines angle bisector *AI* and *BI* at points *A*_{1} and *B*_{1}, respectively. Similarly, let the line through *Q*, perpendicular to *QR*, meet *CI* and *DI *at *C*_{1} and *D*_{1}, respectively. Show that *A*_{1}*D*_{1}=*B*_{1}*C*_{1}.

Proposed by: *Géza Bohner,* Budapest

(5 points)

**A. 468.** We are given two triangles. Their side lengths are *a*,*b*,*c* and *A*,*B*,*C* respectively, the areas are *t* and *T*, respectively. Prove that -*a*^{2}*A*^{2}+*a*^{2}*B*^{2}+*a*^{2}*C*^{2}+*b*^{2}*A*^{2}-*b*^{2}*B*^{2}+*b*^{2}*C*^{2}+*c*^{2}*A*^{2}+*c*^{2}*B*^{2}-*c*^{2}*C*^{2}16*tT*.

(5 points)

**C. 968.** As shown, the square *ABCD* is divided, with lines parallel to its sides, into the squares *NPLD* and *KBMP*, and two congruent rectangles. Let *P* denote the intersection of *KL* and *MN*, and let *Q* be the intersection of *BN* and *DK*. Show that the points *C*, *P* and *Q *are collinear.

(5 points)

**K. 188.** The base *AB* of an isosceles triangle *ABC* is extended beyond vertex *B* by the length of the leg. The endpoint obtained is *C*_{1}. A perpendicular is erected onto the base *AB* at vertex *A*. *C*_{2} is the point on the perpendicular that lies in the same half plane as vertex *C*, at a distance from *A* that equals the length of the leg. Given that the points *C*_{1}, *C* and *C*_{2} are collinear, find the angles of triangle *ABC*.

(6 points)

This problem is for grade 9 students only.

**K. 189.** In a mathematics lesson, the teacher gave the students five kinds of problems to practise. There were three problems of each kind. Students got 1 mark for solving a single problem of a kind, 4 marks per problem if they solved two of the same kind, and 9 marks per problem if they solved all three of a kind. The students worked on the problems and received marks in teams. At the end of the lesson, every team had a different number of marks, but all their numbers of marks were found to be divisible by 3. What was the maximum possible number of teams?

(6 points)

This problem is for grade 9 students only.

**K. 191.** Two kinds of windscreen wiper are investigated. Each of them has a 31-cm blade attached to an also 31-cm arm at the middle. The arm rotates through a 90^{o} angle when the wiper is in operation. One kind of wiper is rigid, the blade encloses 45^{o} with the lower edge of the windscreen and points in the direction of the arm throughout the wiping process. In the other wiper, the blade is attached to the arm by a hinge, and it remains perpendicular to the lower edge of the windscreen all the way long. Which type of wiper sweeps through a larger area?

(6 points)

This problem is for grade 9 students only.

**K. 192.** Ann, Beth and Cathy all filled out the same test of six true-or-false questions. Ann's answers (in this order) were F, F, T, T, T, T; Beth's answers were (in this order) T, F, F, T, T, T, and Cathy's answers (in this order) were T, T, F, F, T, T. Ann only got two answers wrong and Beth got only two of them right. How many correct answers may Cathy have got?

(6 points)

This problem is for grade 9 students only.