**A. 516.** In each of five boxes *B*_{1}, *B*_{2}, *B*_{3}, *B*_{4}, *B*_{5} there is initially one coin. There are two types of operation allowed:

*Type 1:* Choose a nonempty box *B*_{j} with 1*j*4. Remove one coin from *B*_{j} and add two coins to *B*_{j+1}.

*Type 2:* Choose a nonempty box *B*_{k} with 1*k*3. Remove one coin from *B*_{k} and exchange the contents of (possibly empty) boxes *B*_{k+1} and *B*_{k+2}.

Prove that for every integer 0*n*2^{2010} there is a finite sequence of such operations that results in boxes *B*_{1}, *B*_{2}, *B*_{3}, *B*_{4} being empty and box *B*_{5} containing exactly *n* coins.

(5 points)

**B. 4299.** Parallelogram *CDEF* is inscribed in triangle *ABC*, with vertices *D*, *E*, *F* lying on sides *CA*, *AB* and *BC*, respectively. Given the length of the line segment *DF*, construct the point *E*. For what point *E* will the length of diagonal *DF* be minimal?

(*Sándor Katz,* Bonyhád)

(5 points)

**K. 259.** *ABCD* and *EFGH* are coplanar rectangles with parallel sides. Given that *AB*=15 cm, *AD*=12 cm, *EF*=10 cm, *EH*=8 cm, *FI*=14 cm, calculate the difference of the shaded areas.

(6 points)

This problem is for grade 9 students only.

**K. 261.** There are four numbers listed on a sheet of paper. The mean of the first two, the last two, and the middle two is 7, 8.4, and 2.3, respectively. What is the mean of the first and last numbers?

(6 points)

This problem is for grade 9 students only.

**K. 263.** Charles writes the numbers 3, 5, 6 on three cards, and Charlotte writes the numbers 8, 9, 10 on three cards. Each of them selects two cards at random out of the three of their own. Charles multiplies his two numbers, and Charlotte adds hers. What is the probability that Charles will get a greater number than Charlotte?

(6 points)

This problem is for grade 9 students only.

**K. 264.** In a canning factory, four cylindrical tin cans are fixed together with a plastic tape wound about them in the ways shown in the *figures.*

What length of plastic tape is needed in each case if the diameter of the base of the cylinders is 10 cm, and 2-cm lengths of tape overlap at the ends?

(6 points)

This problem is for grade 9 students only.