**A. 383.** The diameter *AB* of the circle *k* is perpendicular to the line , which does not pass through either *A* or *B*. Let *C* be a point of outside *k*, and let *D* denote the other intersection of the line *AC* with the circle. Draw one of the tangents from *C* to *k *and denote the point of contact by *E*. Let *F* be the intersection of the lines *BE* and , and finally let *G* denote the intersection of the line segment *AF* with *k*. Show that the line *DF* passes through the reflection of the point *G* about *AB*.

*Based on a problem from Germany, suggested for the IMO*

(5 points)

**A. 384.** *a*_{0},*a*_{1},...,*a*_{n} and *b*_{0},*b*_{1},...,*b*_{k} are non-negative real numbers, such that *a*_{0}=*b*_{0}=1 and (*a*_{0}+*a*_{1}*x*+...+*a*_{n}*x*^{n})(*b*_{0}+*b*_{1}*x*+...+*b*_{k}*x*^{k})=1+*x*+...+*x*^{n+k}. Prove that each of the numbers *a*_{i} and *b*_{i} is either 0 or 1.

*IMC 2001, Prague*

(5 points)

**B. 3859.** Find all natural numbers *n* greater than 1, for which there is an order *a*_{1},*a*_{2},...,*a*_{n} of the numbers 1,2,3,...,*n*, such that the products *a*_{1}, *a*_{1}*a*_{2}, *a*_{1}*a*_{2}*a*_{3}, ..., *a*_{1}*a*_{2}^{...}*a*_{n} divided by *n *all leave different remainders.

(5 points)

**C. 829.** The five numbers drawn in the lottery of 10 September, 2005 were as follows: 4, 16, 22, 48, 88. All the five numbers are even, exactly four of them are divisible by four, three are divisible by 8 and two by 16. In how many different ways is it possible to select five numbers out of the whole numbers 1 to 90?

(5 points)

**K. 55.** There are two milk bars in Cowton, and both of them have hot chocolate with frothed chocolate topping on their menus. Each milk bar serves in a cylindrical glass of the same height. (On serving, one half of the volume of the drink is liquid chocolate, and the other half is frothed chocolate.) In a short time, the froth turns into liquid chocolate of one quarter as much volume. In the Jolly Cowboy, chocolate is served in glasses of radius 6 cm, and sold for 12 Cowton cents a glass. In the Happy Cowgirl, the radius of the glasses is 5 cm, but they fill up the glass again when the froth of the first filling has settled. They charge 11 Cowton cents for a glass. In which milk bar is chocolate cheaper? [To obtain the volume of a cylinder, the area of its base is multiplied by its height.] (Based on a problem of the *24th József Öveges Memorial Competition*)

(6 points)

This problem is for grade 9 students only.