**B. 4348.** The diagonals of a cyclic quadrilateral *ABCD* are not perpendicular. The feet of the perpendiculars dropped from the vertices *A*, *B*, *C*, *D* onto the diagonals not passing through them are *A*', *B*', *C*', *D*', respectively. The intersections of lines *AA*' and *DD*', *DD*' and *CC*', *CC*' and *BB*', and finally, *BB*' and *AA*' are *E*, *F*, *G*, *H*, respectively. Prove that *A*'*B*'*C*'*D*' is a cyclic quadrilateral with the centre of its circumscribed circle lying at the intersection of the line segments *EG* and *FH*.

(Suggested by *B. Bíró,* (Eger)

(5 points)

**B. 4350.** In a tetrahedron *A*_{1}*A*_{2}*A*_{3}*A*_{4}, for every interior point *P* and for any order *i*, *j*, *k*, *l* of the numbers 1, 2, 3, 4, the following inequality is true: *PA*_{i}+*PA*_{j}+*PA*_{k}<*A*_{l}*A*_{i}+*A*_{l}*A*_{j}+*A*_{l}*A*_{k}. Does it follow that the tetrahedron is regular?

(5 points)

**K. 290.** On my wife's birthday cake, her age was written in two digits, made of almond paste. We noticed that the same two digits would be suitable for my birthday cake, too, but they should then form a power expression. Given that the difference between our ages is equal to the sum of the two digits, find our ages.

(6 points)

This problem is for grade 9 students only.

**K. 291.** 125 small cubes, 1 cm on edge, are glued together to form one big solid cube. Then holes of square cross section are bored through the cube, perpendicular to its faces. As a result, each face of the cube is as shown in the *diagram* (black squares indicating the position of the holes.) The resulting solid is dipped in red paint.

*a*) Find the volume of the resulting solid in cm^{3}.

*b*) Find the total area in cm^{2} that is painted red.

(6 points)

This problem is for grade 9 students only.

**K. 292.** Consider the points *A*(0,0), *B*(*b*,2), *C*(*b*,5), *D*(0,*d*) on the coordinate plane. Given that the points form a trapezium *ABCD* of area 25 units, and that *b* and *d *are positive integers, find the values of the missing coordinates of the vertices.

(6 points)

This problem is for grade 9 students only.

**K. 293.** Fred's cows are grazing on a field. Each cow eats the same daily amount of grass, independently of the actual number of cows on the field. One day, Fred took 6 cows to the field. It took the cows 3 days to eat all the grass on the field, so at the end of the third day Fred had to withdraw his cows from the field to let the grass grow back. When the original amount of grass was restored, Fred took 3 cows to the field. He was surprised to observe that this time it took 7 days for the cows to eat up all the grass. He was so puzzled that he asked Ben who was good at maths. Ben reminded him that he had forgotten about something while watching the cows graze and counting the days. What did Fred forget, and how long would it take a single cow to eat all the grass in the field?

(6 points)

This problem is for grade 9 students only.