B4. A cube of size $9 \mathrm{~cm} \times 9 \mathrm{~cm} \times 9 \mathrm{~cm}$ is composed of smaller cubes of size $1 \mathrm{~cm} \times 1 \mathrm{~cm} \times 1 \mathrm{~cm}$. The surfaces of the cube are painted red.

(A) How many smaller cubes remain unpainted?

(B) How many smaller cubes have only one face painted?

## PROBLEMS FOR THE SECOND YEAR

Before you are two sets of problems. You solve the problems from 1 to 6 of the first set by selecting the correct answer from the five proposed answers on this sheet and writing it in the table under the appropriate number. Only one answer is correct. A correct answer will be scored with two points, while a wrong answer will result in the deduction of one point.

Problems 1 to 4 of the second set are solved on the attached paper. The solution to each of these problems will be rated from 0 to 6 points. On the sheets where you will solve the problems, do not sign, write only your code. Write your solution clearly and legibly with a pen. If you solve a problem in multiple ways, it must be clearly marked which one should be considered.

The time for solving is 90 minutes.

THE NATIONAL COMPETITION COMMITTEE WISHES YOU GREAT SUCCESS.

## I. PART
B4. (A) The unpainted cubes are all those inside, that is, in the cube whose edge is 2 units shorter than the edge of the large cube. This means there are $7^{3}=343$ of them.

(B) On each face of the large cube, exactly one face of those cubes that are not common to two faces is painted. There are $(9-2) \cdot(9-2)=49$ such cubes on each face, so in total $49 \cdot 6=294$.

## Second Year

I. PART

| 1 | 2 | 3 | 4 | 5 | 6 |
| :--- | :--- | :--- | :--- | :--- | :--- |
| $\mathrm{B}$ | $\mathrm{B}$ | $\mathrm{D}$ | $\mathrm{C}$ | $\mathrm{A}$ | $\mathrm{E}$ |