**A. 529.** There is given a circle *k* on the plane, a chord *AB* of *k*, furthermore four interior points, *C*, *D*, *E* and *F*, on the line segment *AB*. Draw an arbitrary chord *X*_{1}*X*_{2} of *k* through point *C*, a chord *Y*_{1}*Y*_{2} through *D*, a chord *U*_{1}*U*_{2} through *E*, finally a chord *V*_{1}*V*_{2} through *F* in such a way that *X*_{1}, *Y*_{1}, *U*_{1} and *V*_{1} lie on the same side of the line *AB*, and

holds. Let *Z* be the intersection of the lines *X*_{1}*X*_{2} and *Y*_{1}*Y*_{2}, and let *W *be the intersection of *U*_{1}*U*_{2} and *V*_{1}*V*_{2}. Show that the lines *ZW* obtained in this way are concurrent or they are parallel to each other.

(5 points)

**B. 4334.** Given a line and a point on it, let *Z* denote the set of those points of the line that are at integer distances from the given point. Prove that if *H* is any three-element subset of *Z*, then it is possible to partition *Z* into subsets congruent to *H*.

(Suggested by *B. Bodor* and *A. Éles,* Budapest)

(5 points)

**C. 1069.** The first three elements of a *sequence of figures* made up of squares are shown. Students were asked how many squares there were in the *n*th figure of the sequence, expressed in terms of *n*.

The following answers were given:

*a*) ,

*b*) 1+(*n*-1)^{.}4,

*c*) 1+(1+2+...+(*n*-1))^{.}4,

*d*) (*n*-1)^{2}+*n*^{2}.

Which answer is correct?

(5 points)

**K. 287.** Ann's grandmother made a birthday cake for her grandchild. Before decorating it with almond paste figurines, she weighed the cake on her digital scales that display weights rounded to the nearest tenth of a kilogram. The reading was 3.4 kg. When she put as many identical almond paste figurines on the cake as the age of Ann in years, the reading changed to 3.6 kg. The true weight of each figurine is an integer multiple of 10 grams, but any one of them alone reads 0.1 kg on the scales. How old may Ann be and what may be the weight of the figurines?

(6 points)

This problem is for grade 9 students only.