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A sphere is inscribed in a regular quadrilateral pyramid, touching the base and all lateral faces. The sphere divides the height of the pyramid in the ratio $1: 8$, measured from the apex of the pyramid. Find the volume of the pyramid if the apothem of the pyramid is $a$.

#
Let $2b$ be the side of the base of the given pyramid with vertex $P$, and $V$ be the volume of the pyramid. Since the pyramid is regular, the center of the inscribed sphere lies on the height, and the sphere touches the lateral faces of the pyramid at points lying on the apothems, and the base at its center $M$. Consider the section of the pyramid by a plane passing through the apothems $PA$ and $PB$ lying in two opposite lateral faces. We obtain an isosceles triangle $APB$ with sides $PA = PB = a$, $AB = 2b$, and the inscribed circle with center $O$ lying on the height $PM$. If the circle intersects the height $PM$ at a point $K$ different from $M$, then by the problem's condition, $PK / MK = 1 / 3$. Let $PK = x$. Then

$$
MK = 8x, \quad OM = \frac{1}{2} MK = 4x, \quad OP = OK + PK = 4x + x = 5x
$$

The center of the circle inscribed in an angle lies on the angle bisector, so $BO$ is the bisector of triangle $PBM$. By the angle bisector theorem,

$$
b / a = BM / BP = OM / OP = 4x / (5x) = 4 / 5
$$

Thus, $b = 4a / 5$. By the Pythagorean theorem in the right triangle $PBM$,

$$
PM = \sqrt{PB^2 - BM^2} = \sqrt{a^2 - 16a^2 / 25} = 3a / 5
$$

Therefore,

$$
V = \frac{1}{3} AB^2 \cdot PM = \frac{1}{3} (2b)^2 \cdot 3a / 5 = \frac{1}{3} 4 \cdot (16 / 25) a^2 \cdot 3a / 5 = 64a^3 / 125
$$

## Answer

## [ Tiling with dominoes and tiles ]  $\left.\begin{array}{rl}\text { Problem } \underline{88042} \text { Topics: } & \text { Parity and oddity }\end{array}\right]$

All the cells of an $8 \times 8$ chessboard are covered with 32 dominoes (each domino covers exactly two cells).

Prove that the number of vertically placed dominoes is even.

## Solution

Color the even rows white and the odd rows black. Each horizontal domino covers an even number of white cells (0 or 2), while each vertical domino covers exactly one white cell. Since the total number of white cells is even, the number of vertical dominoes must be even.