**A. 446.** The lengths of tangents drawn from the points *A*, *B* and *C* to a sphere *s* are *a*, *b *and *c*, respectively. *T* is a point on the sphere, not co-planar with *A*, *B* and *C*, such that .

Prove that the sphere passing through the points *A*, *B*, *C*, *T* touches *s*.

(5 points)

**A. 448.** The numbers 1,2,...,*N* are colored with 3 colors such that each color is used at most times. Let *A* be set of all (ordered) 4-tuples (*x*,*y*,*z*,*w*), consisting of such numbers, such that and *x*, *y*, *z*, *w* have the same color. Similarly, let *B* be the set of all (ordered) 4-tuples (*x*,*y*,*z*,*w*) such that , the numbers *x*, *y* have the same color, *z* and *w* have the same color, but these two colors are distinct. Prove that |*A*||*B*|.

(5 points)

**B. 4067.** Erect a perpendicular to each of the sides *AB*, *BC*, *CA* of a triangle of perimeter *k* at the vertices *A*, *B*, *C*, respectively. Let *K* denote the perimeter of the new triangle obtained from the perpendiculars. Prove that , where , and are the angles of the triangle *ABC*.

(3 points)

**B. 4068.** Let *A*, *B*, *C*, *a*, *b*, *c* denote positive integers, *a*^{.}*b*^{.}*c*>1. Prove that there exists a positive integer *n*, such that *A*^{.}*a*^{n}+*B*^{.}*b*^{n}+*C*^{.}*c*^{n} is a composite number.

(4 points)

**B. 4070.** The positive integers *a* and *b*, written in decimal system, can be obtained from each other by rearranging their digits. Prove that

*a*) the sum of the digits of 2*a* equals that of 2*b*;

*b*) if both *a* and *b* are even, then the sum of the digits of equals that of .

*Kvant*

(5 points)

**K. 158.** A square garden is formed in a park out of four flowerbeds (as shown in the *Figure*). A single type of flowers are planted in each flowerbed, so that adjacent beds contain different types. (It is allowed to plant the same type of flower in opposite beds.) The flowers are selected out of three types: roses, lilies or gladioluses. In how many different ways can the garden be organized if orientation matters, too (that is, if it makes a difference which kind of flowers are planted in the north, south, east and west flowerbed)?

(6 points)

This problem is for grade 9 students only.

**K. 160.** Jack loves playing the following memory game: There are 18 kinds of picture cards, two copies of each. The 36 cards are shuffled and all layed on the table face down. The game consists of turns. In each turn, two cards are turned face up. If they are a pair of identical cards, they are removed from the table. If not, they are turned face down again. The game continues until he has removed all cards from the table. Since Jack has excellent memory, he can always remember the position of any card that he has seen. With this condition, is it possible for a game to consist of 17 turns? Of 18 turns? Of 35 turns? If so, give an example for a possible game. If not, explain why. (Assume that Jack is always trying to finish the game in the lowest number of turns, that is, he will not turn up cards again that he has already seen.)

(6 points)

This problem is for grade 9 students only.