**A. 440.** Given an isosceles triangle *ABC* such that the midpoint of base *BC* is *M*, points *D* and *E* lie on sides *AB* and *AC*, respectively, and *DE* is parallel to *BC*.

Choose two points, *P* and *Q* on the extensions of segment *BC* beyond *B* and *C*, respectively, such that . Let *PD* and *QE* meet at *R*. What is the locus of point *R*?

(5 points)

**A. 441.** For an arbitrary sequence of numbers , let be the sequence of partial sums of the series . Is there any sequence *A*, not constant zero, for which the sequences *A*, *SA*, *SSA*, *SSSA*,... are all convergent?

*Miklós Schweitzer Competition,* 2007

(5 points)

**C. 921.** The average of Steve's mathematics marks was between 4 and 5 before the last maths test of the first semester was given back to the class. (Marks in Hungarian schools are 1 to 5, 5 means excellent and 1 means fail. Teachers sometimes give half marks such as 3 and a half, too, to indicate that it is between a 3 and a 4, but final marks are whole numbers.) The teacher said to Steve, ``16 of you took this test. I also gave half marks. The range of your marks is 2, the mode is 4.5, and the median is 4. The average of the marks is the worst possible average that a group of 16 may have with these conditions. If you can tell me whether you may or may not have got a 5 on this test, then you will get a 5 for the first semester.'' Steve did get his 5. What was his answer to the question?

(5 points)

**K. 147.** In a sale, a -forint (HUF, Hungarian currency) car was sold for forints. In the same sale (that is, with a price reduced by the same percentage), another car was sold for forints less than 3/5 of its original price. What was the original price?

(6 points)

This problem is for grade 9 students only.

**K. 148.** Anna, Bea, Cecilia, Dora, Emma and Fiona are going to the cinema. Their tickets are for six consecutive seats in a row. Anna and Bea insist on sitting next to each other, and Cecilia and Dora are not willing to sit next to each other at all. How many different seatings are there with these restrictions?

(6 points)

This problem is for grade 9 students only.