**A. 606.** Prove that for every simple graph *G* with *n* vertices there exist some simple graphs *S*_{1}, ..., *S*_{k} with the following properties:

(*a*) every *S*_{i} is a complete bipartite graph;

(*b*) every edge of *G* is contained by an odd number of graphs *S*_{i};

(*c*) every edge of the complement of *G* is contained by an even number of graphs *S*_{i};

(*d*) .

(5 points)

**C. 1208.** The vertices of the parallelograms *PQRL* and *LSTA* in the plane are labelled in the same sense around the clock. The parallelograms do not have a point in common, except *L*. Prove that there exists a pentagon *ABCDE* (degenerated cases are allowed) in the plane such that the midpoints of the sides are *P*, *Q*, *R*, *S*, *T*, in this order.

(5 points)

This problem is for grade 11 - 12 students only.

**C. 1209.** The tangents drawn from a point *C* lying outside a circle touch the circle at points *A* és *B*. *M* is a point on the shorter arc *AB*. Let *MN*, *ME* and *MD *be the line segments drawn from *M*, perpendicular to the line segments *AB*, *BC *and *CA*, respectively. Given that *MN*=4, *MD*=2 and , find the area of triangle *MNE*.

(5 points)

This problem is for grade 11 - 12 students only.

**K. 405.** *a*) Find all sets of three integers such that their product is a positive prime and, if they are listed in increasing order, the differences of consecutive numbers are the same?

*b*) Find all sets of three integers such that their product is the double of a positive prime and, if they are listed in increasing order, the differences of consecutive numbers are the same?

(6 points)

This problem is for grade 9 students only.

**K. 406.** We say that an integer is a ``mountain number'' if it consists of digits distinct from one another and from 0, and its digits are increasing from the first one to the ``mountain top'', then decreasing from the top to the last digit. The mountain top cannot be the first or the last digit.

*a*) Determine the largest and the smallest mountain numbers.

*b*) How many mountain numbers with 4 digits are there?

(6 points)

This problem is for grade 9 students only.