**A. 476.** Let *n*3 be an odd integer, and let *A*={0,1,...,*n*-1} denote the set of residual classes modulo *n*. Call a non-empty subset *B**A* a *Dutch set,* if for every *a**A* and for every *b**B* at least one of *b*+*a* and *b*-*a* lies in *B*. Determine the smallest possible cardinality of a Dutch set in terms of *n*.

Proposed by: *Gerhard Woeginger,* Amsterdam

(5 points)

**A. 477.** Let *G* be the complete graf with 2*n* vertices, and suppose that *S*_{1},...,*S*_{k} are subgraphs of *G* 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 subgraphs *S*_{i}.

Show that *k**n*.

(5 points)

**B. 4169.** Prove that if *a*, *b*, *c* are pairwise distinct positive integers then

*S*=(42*a*+43*b*+43*c*)^{3}+(43*a*+42*b*+43*c*)^{3}+(43*a*+43*b*+42*c*)^{3}

-3(42*a*+43*b*+43*c*)(43*a*+42*b*+43*c*)(43*a*+43*b*+42*c*)

is divisible by 128 but is not a perfect power of 2.

Suggested by *D. Nagy*

(5 points)

**K. 206.** Steve and Charlie are brothers. Their mom bought them petits fours at the confectioner's shop. There are 8 larger cubes of edge 3 cm and 27 smaller cubes 2 cm on edge. Five faces of each cake are covered with icing that has the same uniform thickness on all of them. The bottom of the cakes is not iced. Steve and Charlie would like to divide the petits fours between them so that each gets the same total volume, the cakes are not cut in pieces, and each of them gets both kinds.

*a*) Show that these conditions are impossible to meet.

*b*) When they see that they cannot divide the petits fours equally by volume, they decide to have equal quantities of icing instead. (They still do not want to cut the cakes, and each of them should get both kinds.) Find all possible ways to divide the petits fours.

(6 points)

This problem is for grade 9 students only.

**K. 207.** The floor of a rectangular shaped hallway of area 14.4 m^{2} is tiled with rectangular tiles. In every fifth row of tiles along the length of the hallway, the tiles are rotated through a right angle. In this way, 15 rows of tiles are needed and no tiles need to be cut. If all tiles were laid in the way they are laid in the fifth rows, the hallway could also be tiled without cutting any tiles. In that case, there would be 18 rows of tiles. What may be the dimensions of the tiles if their sides in centimetres are whole numbers?

(6 points)

This problem is for grade 9 students only.

**K. 208.** *a*) In the coordinate plane, a circle of radius 5 is drawn about the origin. How many lattice points lie on the circumference of the circle? (A lattice point is a point whose coordinates are both integers.)

*b*) Find an integer *r*, such that there are more than 14 lattice points on a circle of radius *r* centred at the origin. Justify your answer.

(6 points)

This problem is for grade 9 students only.

**K. 210.** The *Figure* shows a part of little Dorothy's bicycle. The centre of the rear wheel is *A* and the pedal lever *TK* (with one of the pedals at its endpoint *K*) rotates about point *T*. Assume that the points *A*, *K*, *T* are coplanar. The length of *TK* is 20 cm and the length of *AT* is 48 cm. As the lever turns around point *T*, how many positions does the point *K* have in which the distance *AK* in centimeters is a whole number and the triangle *AKT* is acute-angled?

(6 points)

This problem is for grade 9 students only.