# Problem 5.

The segments of two lines, enclosed between two parallel planes, are in the ratio $5: 9$, and the acute angles between these lines and one of the planes are in the ratio $2: 1$. Find the cosine of the smaller angle.
Answer: 0.9

Solution. If $h$ is the distance between the planes and $\alpha$ is the smaller of the angles, then the lengths of the segments are $h / \sin 2 \alpha$ and $h / \sin \alpha$, respectively. Therefore,

$$
\frac{\sin \alpha}{\sin 2 \alpha}=\frac{5}{9}, \quad \text { from which } \quad \cos \alpha=\frac{\sin 2 \alpha}{2 \sin \alpha}=\frac{9}{10}
$$

## B-2

The segments of two lines enclosed between two parallel planes are in the ratio $5: 8$, and the acute angles between these lines and one of the planes are in the ratio $2: 1$. Find the cosine of the smaller angle.

Answer: 0.8

## B-3

The segments of two lines enclosed between two parallel planes are in the ratio $2: 3$, and the acute angles between these lines and one of the planes are in the ratio $2: 1$. Find the cosine of the smaller angle.

Answer: 0.75

## B-4

The segments of two lines enclosed between two parallel planes are in the ratio $4: 7$, and the acute angles between these lines and one of the planes are in the ratio $2: 1$. Find the cosine of the smaller angle.

Answer: $0.875 \approx 0.88$

B-5 Three mutually tangent spheres touch the plane of a triangle with sides $5, 6, 7$ at the vertices of this triangle. Find the product of the radii of these three spheres.

Answer: 26.25

Solution. Let the three spheres touch the plane and each other. We need to find the relationship between the product of the lengths of the sides of the triangle formed by the points of tangency of the spheres with the plane and the product of the radii of the spheres.

Let $O_{a}, O_{b}, O_{c}$ be the centers of the spheres, their radii $r_{A}, r_{B}, r_{C}$, and they touch the plane at points $A, B, C$ with $A B=c, B C=a, C A=b$. The points $A B O_{B} O_{A}$ form a right trapezoid with sides $A B=c, B O_{B}=r_{B}, A O_{A}=r_{A}, O_{B} O_{A}=r_{A}+r_{B}$, the bases of which are $B O_{B}$ and $A O_{A}$, and $B O_{B} \perp A B, A O_{A} \perp A B$. Let, for definiteness, $r_{B} \geqslant r_{A}$. Draw a perpendicular from $O_{A}$ to the side $B O_{B}$, denoted as $O_{A} D$. The points $A O_{A} D B$ form a rectangle, and $O_{A} D=A B=c$. $O_{B} D=r_{B}-r_{A}$, and for the right triangle $O_{A} D O_{B}$, we can write the Pythagorean theorem: $c^{2}=\left(r_{A}+r_{B}\right)^{2}-\left(r_{A}-r_{B}\right)^{2}$, and thus, $c=2 \sqrt{r_{A} r_{B}}$. Considering other pairs of spheres, we similarly obtain $b=2 \sqrt{r_{A} r_{C}}$ and $a=2 \sqrt{r_{B} r_{C}}$. Therefore, $a b c=8 r_{A} r_{B} r_{C}$.

B-6 Three mutually tangent spheres touch the plane of a triangle with sides $7, 8, 9$ at the vertices of this triangle. Find the product of the radii of these three spheres.

Answer: 63

B-7 Three mutually tangent spheres with radii 3, 5, and 6 touch the plane of a triangle at the vertices of this triangle. Find the product of the sides of this triangle.

Answer: 720

B-8 Three mutually tangent spheres with radii 2, 4, and 7 touch the plane of a triangle at the vertices of this triangle. Find the product of the sides of this triangle.

Answer: 448

## Lomonosov Olympiad in Mathematics

Preliminary Round 2020/21 academic year for $10-11$ grades

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