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 X1X2 of k through point C, a chord Y1Y2 through D, a chord U1U2 through E, finally a chord V1V2 through F in such a way that X1, Y1, U1 and V1 lie on the same side of the line AB, and
holds. Let Z be the intersection of the lines X1X2 and Y1Y2, and let W be the intersection of U1U2 and V1V2. Show that the lines ZW obtained in this way are concurrent or they are parallel to each other.
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?
This problem is for grade 9 students only.
Solution (in Hungarian)