**A. 613.** There is given a convex quadrilateral \(\displaystyle ABCD\). The points \(\displaystyle E\) and \(\displaystyle F\) lie on the line segment \(\displaystyle AB\), \(\displaystyle G\) and \(\displaystyle H\) lie on \(\displaystyle BC\), \(\displaystyle I\) and \(\displaystyle J\) lie on \(\displaystyle CD\), and \(\displaystyle K\) and \(\displaystyle L\) lie on \(\displaystyle DA\) in such a way that \(\displaystyle AE<AF<AB\), \(\displaystyle BG<BH<BC\), \(\displaystyle CI<CJ<CD\), and \(\displaystyle DK<DL<DA\). The line \(\displaystyle EJ\) meets \(\displaystyle GL\) and \(\displaystyle HK\) at \(\displaystyle P\) and \(\displaystyle S\), and \(\displaystyle FI\) meets \(\displaystyle GL\) and \(\displaystyle HK\) at \(\displaystyle Q\) and \(\displaystyle R\), respectively. The points \(\displaystyle P\) and \(\displaystyle R\) lie on the diagonal \(\displaystyle AC\) and the points \(\displaystyle Q\) and \(\displaystyle S\) lie on \(\displaystyle BD\) (see the first cover). Suppose that each of the quadrilaterals \(\displaystyle AEPL\), \(\displaystyle BGQF\) and \(\displaystyle CIRH\) has an inscribed circle.

Show that the quadrilateral \(\displaystyle DKSJ\) also has an inscribed circle.

Based on a problem of the International *Zhautykov Olympiad*

(5 points)

**B. 4613.** The rhombus *A*_{1}*B*_{1}*C*_{1}*D*_{1} lies in the interior of a parallelogram *ABCD*. The vectors and have the same direction, and the vectors and also have the same direction. Show that *ABCD* is a rhombus if and only if the sum of the areas of quadrilaterals *AA*_{1}*D*_{1}*D* and *BCC*_{1}*D*_{1} is equal to the sum of the areas of quadrilaterals *ABB*_{1}*A*_{1} and *CDD*_{1}*C*_{1}.

(Problem by *L. Longáver,* Matlap, Kolozsvár)

(3 points)

**B. 4615.** Each angle of the triangle *ABC* is smaller than 120^{o}. The isogonal point of the triangle is *P*. Draw lines through point *P* parallel to the sides. The parallels intersect side *AB* at *D* and *E*, side *BC* at *F* and *G*, and side *CA* at *H* and *I*. Let *K*, *L* and *M* denote the isogonal points of triangles *DEP*, *FGP*, *HIP*, respectively. Show that triangle *KLM* is equilateral.

Suggested by: *Sz. Miklós,* Herceghalom

(5 points)

**C. 1218.** On a circular table of radius 80 cm, the square tablecloth has been pulled aside, such that one corner lies exactly on the edge of the table, and the two sides that meet at the opposite vertex are tangent to the edge of the table. Determine the length of the side of the tablecloth to the nearest millimetre.

(5 points)

This problem is for grade 1 - 10 students only.

**K. 415.** Students played with dice in the maths club. 40% of the dice were red, the rest of them were green. Next time, the number of red dice was increased by 10%, and the number of green dice was reduced by 5%. By what percentage did the number of dice change?

(6 points)

This problem is for grade 9 students only.

**K. 416.** Given that , determine the digits *a*, *b*, *c*. ( is a six-digit number, and is a four-digit number, where identical letters denote identical digits.)

(6 points)

This problem is for grade 9 students only.

**K. 417.** A confectioner's shop sells four kinds of cakes: cheese, walnut, poppy seed and chocolate. The number of cakes on stock is 162 without the cheese cakes, 158 without the walnut cakes, 150 without the poppy seed cakes, and 160 without the chocolate cakes. How many cakes of each kind are there on stock?

(6 points)

This problem is for grade 9 students only.