Bbotin D.A.

Can a $3 \times 3 \times 3$ cube with a $1 \times 1 \times 1$ hole in the center be built using 13 bricks of size $1 \times 1 \times 2$?
Paint the small cubes $1 \times 1 \times 1$, from which the cube and bricks are composed, in white and black in a checkerboard pattern.

## Solution

We will paint the small cubes $1 \times 1 \times 1$, from which the cube and bricks are composed, in white and black in a checkerboard pattern. In 13 bricks, there are an equal number (13 each) of black and white cubes, while in the $3 \times 3 \times 3$ cube without the center, there are 12 of one color and 14 of the other.

## Answer

It is impossible.

On the lateral sides $AB$ and $CD$ of trapezoid $ABCD$, points $M$ and $N$ are located, respectively, such that $BM: AM = CN: ND = 3: 5$.

Find $MN$, if $BC = a$ and $AD = b$.

## Hint

Draw a diagonal of the trapezoid.

## Solution

From the theorem of proportional segments, it follows that $MN \parallel AD$.

First method. Draw the diagonal $AC$ and denote by $P$ the point of its intersection with $MN$. From the similarity of triangles $AMP$ and $ABC$, we find that

$MP = 5/8 a$, and from the similarity of triangles $CPN$ and $CAD - PN = 3/8 b$. Therefore, $MN = MP + PN = 1/8 (5a + 3b)$.

Second method. Assume that $ab$.

## Answer

$1/8 (5a + 3b)$.