**A. 494.** Let *p*_{1},...,*p*_{k} be prime numbers, and let *S* be the set of those integers whose all prime divisors are among *p*_{1},...,*p*_{k}. For a finite subset *A* of the integers let us denote by the graph whose vertices are the elements of *A*, and the edges are those pairs *a*,*b**A* for which *a*-*b**S*. Does there exist for all *m*3 an *m*-element subset *A* of the integers such that (*a*) is complete? (*b*) is connected, but all vertices have degree at most 2?

*Miklós Schweitzer Memorial Competition,* 2009

(5 points)

**A. 495.** In the acute triangle *ABC* we have *BAC*=. The point *D* lies in the interior of the triangle, on the bisector of *BAC*, and points *E* and *F* lie on the sides *AB *and *BC*, respectively, such that *BDC*=2, *AED*=90^{o}+, and *BEF*=*EBD*. Determine the ratio *BF*:*FC*.

(5 points)

**A. 496.** Let *a*_{1},*a*_{2},...,*a*_{2k} be distinct integers and let *M* be a set of *k* integers not containing 0 and *s*=*a*_{1}+*a*_{2}+...+*a*_{2k}. A grasshopper is to jump along the real axis, starting at the point 0 and making 2*k* jumps with lengths *a*_{1},*a*_{2},...,*a*_{2k} in some order. If *a*_{i}>0 then the grasshopper jumps to the right; while if *a*_{i}<0 then the grasshopper jumps to the left, to the point in the distance |*a*_{i}| in the respective steps. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in *M*.

(5 points)

**B. 4223.** Consider the number pairs (1,36), (2,35), ..., (12,25). Is it possible to select one number from each given pair, such that the sum of the numbers selected equals the sum of the numbers not selected? Will the answer change if the last two pairs are left out?

(Suggested by *J. Pataki,* Budapest)

(4 points)

**K. 224.** There are dice of six and four faces (cubes and tetrahedra) on a table. Their faces are numbered with dots, 1 to 6 and 1 to 4, respectively. The number of all dots on the dice is 323. If we had as many six-sided dice as we have of the four-sided dice and vice versa, the number of dots would be 185. How many dice of each kind are there on the table?

(6 points)

This problem is for grade 9 students only.

**K. 230.** The traditional way of numbering houses in a street uses odd numbers on one side and even numbers on the other side. Numberless Street is bounded by identical plots of land on both sides. The odd side is fully developed, there is one house on each plot of land. On the even side, there are a few consecutive plots that have no houses on them yet, but there are two houses on the first plot. The owners decided to use number plates of the same design in the whole street. The number plates they bought are sold for 50 forints a digit (HUF, Hungarian currency). They spent 4250 forints altogether. The numbers for the odd side cost 550 forints more than the numbers for the even side. The owner of the first house on the odd side considered 1 an unlucky number, so they started the numbering by 3. On the even side, the two houses on the first plot got the numbers 2 and 4. They did not buy number plates for the houses that would be built on the vacant plots, but they did take them into consideration in numbering: they did not give their numbers to other buildings. What is the largest number on the odd side and how many vacant plots are there on the even side?

(6 points)

This problem is for grade 9 students only.

**K. 231.** If the last digits of the products 1^{.}2, 2^{.}3, 3^{.}4, ..., *n*(*n*+1) are added, the result is 2010. How many products are used?

(6 points)

This problem is for grade 9 students only.