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8.27. In space, points $A, B, C$ and $D$ are given, such that $A B=B C=C D$ and $\angle A B C=\angle B C D=\angle C D A=$ $=\alpha$. Find the angle between the lines $A C$ and $B D$. | 8.27. In triangles $A B C$ and $C D A$, sides $A B$ and $C D$ are equal, as are angles $B$ and $D$, and side $A C$ is common to both. If $\triangle A B C = \triangle C D A$, then $A C \perp B D$. Now consider the case when these triangles are not equal. Take a point $P$ on ray $B A$ such that $\triangle C B P = \triang... | \alpha | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,368 |
8.29. Prove that a pentagon, all sides and angles of which are equal, is flat.
```
***
``` | 8.29. G G e r v o e r e shenie. Suppose that the given pentagon $A_{1} \ldots A_{\text {b }}$ is not planar. The convex hull of its
Fig. 61
 / 2$ and $A P_{2}=(A D+A C-$ $-C D)^{2}$. Since $A B-B C=A D-C D$ by the condition, then $A P_{1}=A P_{2}$, i.e., points $P_{1}$ and $P_{2}$ c... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,371 |
8.31. A sphere touches all sides of a spatial quadrilateral. Prove that the points of tangency lie in the same plane. | 8.31. Let a sphere touch the sides $A B, B C, C D$ and $D A$ of the spatial quadrilateral $A B C D$ at points $K, L, M$ and $N$ respectively. Then $A N=A K, B K=B L, C L=C M$ and $D M=D N$. Therefore,
$$
\frac{A K}{B K} \cdot \frac{B L}{C L} \cdot \frac{C M}{D M} \cdot \frac{D N}{A N}=1
$$
Now consider the point $N^{... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,372 |
8.32. On the sides $AB, BC, CD$ and $DA$ of a spatial quadrilateral $ABCD$ (or on their extensions), points $K, L, M$ and $N$ are taken such that $AN = AK$, $BK = BL$, $CL = CM$ and $DM = DN$. Prove that there exists a sphere that is tangent to the lines $AB, BC, CD$ and $DA$. | 8.32. Since $A N = A K$, there exists a circle $S_{1}$ in the plane $D A B$ that is tangent to the lines $A D$ and $A B$ at points $N$ and $K$. Similarly, there exists a circle $S_{2}$ in the plane $A B C$ that is tangent to the lines $A B$ and $B C$ at points $K$ and $L$. We will prove that the sphere containing circl... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,373 |
8.33. Let $a, b, c$ and $d$ be the lengths of the sides $AB, BC$, $CD$ and $DA$ of a spatial quadrilateral $ABCD$.
a) Prove that if none of the three relations $a+b=c+d, a+c=b+d$ and $a+d=b+c$ are satisfied, then there exist exactly 8 distinct spheres that are tangent to the lines $AB, BC, CD$ and $DA$.
b) Prove that... | 8.33. a) Let's introduce coordinates on the lines $A B, B C, C D$ and $D A$, taking points $A, B, C$ and $D$ as the origins of the coordinates, respectively, and the directions of the rays $A B, B C, C D$ and $D A$ as the positive directions. According to the result of problem 8.32, we will look for points $K$, $L, M$ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,374 |
9.1. Prove that there are no other regular polyhedra except those listed above. | 9.1. Consider an arbitrary regular polyhedron. Let all its faces be regular $n$-gons, and all polyhedral angles contain $m$ faces. Each edge connects two vertices, and from each vertex, $m$ edges emanate. Therefore, $2P = mB$. Similarly, each edge belongs to two faces, and each face has $n$ edges. Hence, $2P = nG$. Sub... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,375 |
9.2. Prove that there exists a dodecahedron - a regular polyhedron with pentagonal faces and trihedral angles at the vertices. | 9.2. The proof will be based on the properties of a figure consisting of three identical regular pentagons with a common vertex, each two of
Fig. 66 which have a common edge. In the solutio... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,376 |
9.5. Prove that for any regular polyhedron there exists:
a) a sphere passing through all its vertices (circumscribed sphere);
b) a sphere touching all its faces (inscribed sphere). | 9.5. Let us draw perpendiculars to the faces through their centers. It is easy to see that for two adjacent faces, such perpendiculars intersect at one point, and this point is equidistant from each face at a distance of \(a \operatorname{ctg} \varphi\), where \(a\) is the distance from the center of the face to its si... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,379 |
9.6. Prove that the center of the circumscribed sphere of a regular polyhedron is its center of mass (i.e., the center of mass of a system of points with unit masses located at its vertices).
The center of the circumscribed sphere of a regular polyhedron, coinciding with the center of the inscribed sphere and the cent... | 9.6. We need to prove that the sum of the vectors connecting the center of the circumscribed sphere of a regular polyhedron with its vertices is zero. Let's denote this sum of vectors by x.
... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,380 |
9.7. a) Prove that it is possible to choose 4 vertices of a cube such that they will be the vertices of a regular tetrahedron. In how many ways can this be done?
b) Prove that it is possible to choose 4 planes of the faces of an octahedron such that they will be the planes of the faces of a regular tetrahedron. In how... | 9.7. a) If $A B C D A_{1} B_{1} C_{1} D_{1}$ is a cube, then $A B_{1} C D_{1}$ and $A_{1} B C_{1} D$ are regular tetrahedra.
b) It is easy to verify that the midpoints of the edges of a regular tetrahedron are the vertices of an octahedron. From this, it is clear that one can choose 4 faces of the octahedron such that... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,381 |
9.8. Prove that on the edges of a cube, 6 points can be chosen such that they will be the vertices of an octahedron. | 9.8. Let the edge of the cube $A B C D A_{1} B_{1} C_{1} D_{1}$ be equal to 4a. Take points on the edges emanating from vertex $A$, at a distance of $3 a$ from it. Similarly, take 3 points on the edges emanating from vertex $C_{1}$. Using the equality $3^{2}+3^{2}=1+$ $+4^{2}+1$, it is easy to verify that the lengths o... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,382 |
9.9. a) Prove that it is possible to choose 8 vertices of a dodecahedron such that they are the vertices of a cube. How many ways can this be done?
b) Prove that it is possible to choose 4 vertices of a dodecahedron such that they are the vertices of a regular tetrahedron. | 9.9. a) From the solution to problem 9.2, it is clear that there exists a cube whose vertices are located at the vertices of the dodecahedron. Moreover, one edge of the cube is located on each face of the dodecahedron. It is also clear that choosing any of the five diagonals of a certain face of the dodecahedron as an ... | 5 | Geometry | proof | Yes | Yes | olympiads | false | 24,383 |
9.10. a) Prove that it is possible to choose 8 faces of an icosahedron such that they will be the planes of the faces of an octahedron. How many ways can this be done?
b) Prove that it is possible to choose 4 planes of the faces of an icosahedron such that they will be the planes of the faces of a regular tetrahedron. | 9.10. a) From the solution to problem 9.4, it is clear that 8 faces of the icosahedron can be embedded such that they become faces of an octahedron. In this case, from each vertex of the icosahedron, exactly one edge does not lie in the plane of the octahedron's face. Therefore, the choice of any one of the three edges... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,384 |
9.11. Consider a convex polyhedron, the vertices of which are the centers of the faces of some regular polyhedron. Prove that this polyhedron is also regular. (It is called the polyhedron dual to the original.) | 9.11. When rotating about the line connecting the vertex of the original polyhedron with its center, which transforms the polyhedron into itself, the centers of the faces adjacent to this vertex map to themselves, i.e., they become vertices of a regular polyhedron. Similarly, considering a rotation about the line conne... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,385 |
9.12. a) Prove that the tetrahedron is dual to the tetrahedron.
b) Prove that the cube and octahedron are dual to each other.
c) Prove that the dodecahedron and icosahedron are dual to each other. | 9.12. To prove this, it is sufficient to note that if the original polyhedron has $m$-sided angles at the vertices and $n$-sided faces, then the dual polyhedron will have $n$-sided angles at the vertices and $m$-sided faces.
R e m a r k. The solutions to problems 9.2 and 9.4 are actually two different solutions to the... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,386 |
9.13. Prove that if the radii of the inscribed spheres of two dual regular polyhedra are equal, then: a) the radii of their circumscribed spheres are equal; b) the radii of the circumscribed circles of their faces are equal. | 9.13. a) Let $O$ be the center of the original polyhedron, $A$ one of its vertices, and $B$ the center of one of the faces with vertex $A$. Consider the face of the dual polyhedron, formed by the centers of the faces of the original polyhedron, adjacent to vertex $A$. Let $C$ be the center of this face, i.e., the point... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,387 |
9.14. A face of a dodecahedron and a face of an icosahedron lie in the same plane, and moreover, their opposite faces also lie in the same plane. Prove that all other vertices of the dodecahedron and icosahedron are located in two planes parallel to these faces.
## § 3. Projections and Sections of Regular Polyhedra | 9.14. If a dodecahedron and an icosahedron are inscribed in the same sphere, then the radii of their inscribed spheres are equal (Problem 9.13, a), i.e., they are equal to the distances between their opposite faces. We will call the "center of a spherical face" of the dodecahedron (or icosahedron) the point of intersec... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,388 |
9.15. Prove that the projections of a dodecahedron and an icosahedron onto planes parallel to their faces are regular polygons. | 9.15. To prove this, it is sufficient to note that the faces of the polyhedron coincide with themselves upon rotation, aligning the projection of the upper face with the projection of the lower face. Thus, the projection of the dodecahedron is a 10-sided polygon that coincides with itself upon rotation by \(36^{\circ}\... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,389 |
9.16. Prove that the projection of a dodecahedron onto a plane perpendicular to a line passing through its center and the midpoint of an edge is a pentagon (not a decagon). | 9.16. Consider a cube, the vertices of which are located at the vertices of a dodecahedron (see problem 9.2). In our problem, we are talking about the projection onto a plane parallel to a face of this cube. It is now easy to convince ourselves that the projection of the dodecahedron is indeed a pentagon (Fig. 70). | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,390 |
9.18. Does there exist a section of a cube that is a regular hexagon? | 9.18. Exists. The midpoints of the edges of the cube indicated in Fig. 72 are the vertices of a regular hexagon. This follows from the fact that the sides of this hexagon are parallel to the sides of the equilateral triangle \(P Q R\), and their lengths are half the lengths of the sides of this triangle.
.
. This section is a hexagon with pairwise parallel opposite sides. Upon... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,394 |
9.21. Two faces $A \dot{B} C$ and $A B D$ of an icosahedron share the common edge $A B$. A plane is drawn through vertex $D$ parallel to the plane $A B C$. Is it true that the section of the icosahedron by this plane is a regular pentagon?
## § 4. Self-superpositions of regular polyhedra | 9.21. No, it is not correct. Consider the projection of the icosahedron onto the plane $A B C$. It is a regular hexagon (see problem 9.15 and the figure to it). Therefore, the considered section would be a regular hexagon only if all 6 vertices, connected by edges to points $A, B$ and $C$ (and different from $A$, $B$ a... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,395 |
9.22. Which regular polyhedra have a center of symmetry? | 9.22. It is easy to verify that all correct polyhedra, except for the tetrahedron, have a center of symmetry. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,396 |
9.23. A convex polyhedron is symmetric with respect to some plane. Prove that it
either passes through the midpoint of its edge, or is a plane of symmetry of one of the polyhedral angles at the vertex. | 9.23. A plane of symmetry divides a polyhedron into two parts, but therefore it does not intersect at least one edge. Let's consider two cases.
1. The plane of symmetry passes through a vertex of the polyhedron. Then it is a plane of symmetry of the polyhedral angle at that vertex.
2. The plane of symmetry passes thro... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,397 |
9.24. a) Prove that for any regular polyhedron, the planes passing through the midpoints of its edges perpendicularly are planes of symmetry.
b) For which regular polyhedra are there other planes of symmetry? | 9.24. a) For the tetrahedron, cube, and octahedron, the statement of the problem is obvious. For the dodecahedron and icosahedron, we need to use the solutions of problems 9.2 and 9.4, respectively. For the dodecahedron, it is convenient to consider a plane passing through the midpoint of an edge parallel to the edge o... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,398 |
9.25. Find the number of planes of symmetry of each of the regular polyhedra. | 9.25. First, let's consider planes of symmetry passing through the midpoints of the edges perpendicularly. We need to determine how many midpoints such a plane passes through immediately. It is easy to verify that for a tetrahedron, each plane passes through the midpoint of one edge, for an octahedron, dodecahedron, an... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,399 |
9.26. Prove that any axis of rotation of a regular polyhedron passes through its center and either a vertex, the midpoint of an edge, or the center of a face. | 9.26. The axis of rotation intersects the surface of the polyhedron at two points. Consider one of them. There are three possible cases.
1. The point is a vertex of the polyhedron.
2. The point lies on an edge of the polyhedron but is not a vertex. Then this edge maps to itself under some rotation about it. Therefore,... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,400 |
9.27. a) How many axes of symmetry does each of the regular polyhedra have?
b) How many other axes of rotation does each of them have? | 9.27. a) For each regular polyhedron, the lines passing through the midpoints of opposite edges are their axes of symmetry. In a tetrahedron, there are 3 such axes, in a cube and octahedron - 6, in a dodecahedron and icosahedron - 15. In addition, in a cube, the lines passing through the centers of faces are axes of sy... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,401 |
9.28. How many symmetries (i.e., movements that translate a polyhedron into itself) are there for each of the regular polyhedra?
## § 5. Different Definitions of Regular Polyhedra | 9.28. Any face of a regular polyhedron can be translated into any other by a motion. If the faces of the polyhedron are $n$-sided, there are exactly $2n$ self-coincidences that preserve one of the faces: $n$ rotations and $n$ symmetries relative to planes. Therefore, the number of self-coincidences (including the ident... | 24,48,120 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,402 |
9.29. Guess that if all faces of a convex polyhedron are equal regular polygons, and all its dihedral angles are equal, then this polyhedron is regular. | 9.29. Need to prove that all polyhedral angles of a given polyhedron are equal. But its dihedral angles are equal by condition, and the plane angles are angles of equal polygons. | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,403 |
9.30. Prove that if all dihedral angles of a convex polyhedron are regular, and all faces are regular polygons, then this polyhedron is regular. | 9.30. We need to prove that all faces are equal and the polyhedral angles are also equal. Let's first prove the equality of the faces. Consider all faces meeting at some vertex. The polyhedral angle at this vertex is regular, so all its dihedral angles are equal, which means that the angles of the considered regular po... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,404 |
9.31. Prove that if all faces of a convex polyhedron are regular polygons, and the edges emanating from each vertex form a regular polygon, then this polyhedron is regular.
$$
\% * \%
$$ | 9.31. We need to prove that all polyhedral angles of our polyhedron are regular. Consider the ends of all edges emanating from a certain vertex. According to the problem statement, the polyhedron with vertices at these points and point $A$ is a pyramid, the base of which is a regular polygon, and all edges of the pyram... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,405 |
9.33. Is it necessarily correct that a convex polyhedron, in which: a) all edges and all dihedral angles are equal; b) all edges and all polyhedral angles are equal, is correct? | 9.33. No, not necessarily. Consider a convex polyhedron, the vertices of which are the midpoints of the edges of a cube. It can be verified that this polyhedron has equal edges, all dihedral angles are equal, and all polyhedral angles are equal, | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,407 |
10.1. Let $a, b$ and $c$ be the lengths of the edges of a parallelepiped, and $d$ be one of its diagonals. Prove that $a^{2}+b^{2}+c^{2} \geqslant d^{2} / 3$ | 10.1. Since $d \leqslant a+b+c$, then $d^{2} \leqslant a^{2}+b^{2}+c^{2}+$ $+2 a b+2 b c+2 c a \leqslant 3\left(a^{2}+b^{2}+c^{2}\right)$. | proof | Inequalities | proof | Yes | Yes | olympiads | false | 24,408 |
10.2. Given a cube with edge 1. Prove that the sum of distances from any point to all its vertices is not less than $4 \sqrt{\overline{3}}$. | 10.2. If $P Q$ is a diagonal of a cube with edge $1$, and $X$ is an arbitrary point, then $P X+Q X \geqslant P Q=\sqrt{3}$. Since a cube has 4 diagonals, the sum of the distances from point $X$ to all vertices of the cube is not less than $4 \sqrt{3}$. | 4\sqrt{3} | Geometry | proof | Yes | Yes | olympiads | false | 24,409 |
10.3. In the tetrahedron $ABCD$, the plane angles at vertex $A$ are equal to $60^{\circ}$. Prove that $AB + AC + AD \leqslant BC + CD + DB$. | 10.3. First, let's assume that if $\angle B A C=60^{\circ}$, then $A B + A C \leqslant 2 B C$. For this, consider points $B^{\prime}$ and $C^{\prime}$, which are symmetric to points $B$ and $C$ relative to the bisector of angle $A$. Since in any convex quadrilateral, the sum of the lengths of the diagonals is greater t... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 24,410 |
10.4. From points $A_{1}, A_{2}$ and $A_{3}$, lying on line $a_{8}$, perpendiculars $A_{i} B_{i}$ are dropped onto line $b$. Prove that if point $A_{2}$ lies between $A_{1}$ and $A_{3}$, then the length of segment $A_{2} B_{2}$ is between the lengths of segments $A_{1} B_{1}$ and $A_{3} B_{3}$. | 10.4. Let's draw a plane П through the line $b$ parallel to $a$. Let $C_{i}$ be the projection of the point $A_{i}$ onto the plane П. By the theorem of three perpendiculars, $C_{i} B_{i} \perp b$, so the length of the segment $B_{2} C_{2}$ is between $B_{1} C_{1}$ and $B_{3} C_{3}$; the lengths of all three segments $A... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,411 |
10.5. Inside a convex polyhedron, there is a segment. Prove that its length does not exceed the length of the largest segment with endpoints at the vertices of the polyhedron. | 10.5. In proving this, we will several times use the following planimetric statement: "If a point $X$ lies on the side $BC$ of triangle $ABC$, then either $AB \geqslant AX$ or $AC \geqslant AX$." (Indeed, one of the angles $BXA$ or $CXA$ is not less than $90^{\circ}$; if $\angle BXA \geqslant 90^{\circ}$, then $AB \geq... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,412 |
10.6. Let $P$ be the projection of point $M$ onto the plane containing points $A, B$, and $C$. Prove that if a triangle can be formed from the segments $P A, P B$, and $P C$, then a triangle can also be formed from the segments $M A, M B$, and $M C$. | 10.6. Let $a=P A, b=P B$ and $c=P C$. We can assume that $a \leqslant b \leqslant c$; then by condition $c<a+b$. Let, further, $h=P M$. It is required to prove that $\sqrt{c^{2}+h^{2}}<\sqrt{a^{2}+h^{2}}+\sqrt{b^{2}+h^{2}}$, i.e.
$$
c \sqrt{1+\left(\frac{h}{c}\right)^{2}}<a \sqrt{1+\left(\frac{h}{a}\right)^{2}}+b \sqr... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,413 |
10.7. Inside a convex polyhedron, points $P$ and $Q$ are taken. Prove that one of the vertices of the polyhedron is closer to $Q$ than to $P$. | 10.7. Consider the plane P passing through the midpoint of the segment $P Q$ and perpendicular to it. Suppose that all vertices of the polyhedron are at least as far from point $Q$ as from point $P$. Then all vertices of the polyhedron lie on the same side of the plane P as point $P$. Therefore, point $Q$ lies inside t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,414 |
10.8. Point $O$ is located inside the tetrahedron $A B C D$. Prove that the sum of the lengths of segments $O A, O B$, $O C$ and $O D$ does not exceed the sum of the lengths of the edges of the tetrahedron, | 10.8. Let $M$ and $N$ be the points of intersection of the planes $A O B$ and $C O D$ with the edges $C D$ and $A B$ respectively (Fig. 75). Since the angle $A O B$ lies inside the triangle $A M B$, then $A O + B O \leq A M + B M$. Similarly, $C O + D O \leq C N + D N$. Therefore, it is sufficient to prove that the sum... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,415 |
10.9. Inside a cube with edge 1, there are several segments, and any plane parallel to one of the faces of the cube intersects no more than
one segment. Prove that the sum of the lengths of these segments does not exceed 3. | 10.9. Let's number the segments and consider the segment with number $\ell$. Let $l_{i}$ be its length, and $x_{i}, y_{i}, z_{i}$ be the lengths of the projections onto the edges of the cube. It is easy to verify that $l_{i} \leqslant x_{i}+y_{i}+z_{i}$. On the other hand, if any plane parallel to a face of the cube in... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,416 |
10.10. A closed broken line passes over the surface of a cube with edge 1 and has common points with all its faces. Prove that its length is not less than $3 \sqrt{2}$. | 10.10. Let's consider projections onto 3 non-parallel edges of a cube. The projection of the given broken line onto any edge contains both ends of the edge, so it coincides with the edge itself. Therefore, the sum of the lengths of the projections of the segments of the broken line onto any edge is at least 2, and the ... | 3\sqrt{2} | Geometry | proof | Yes | Yes | olympiads | false | 24,417 |
10.13. Prove that the sum of the angles of a spatial quadrilateral does not exceed $360^{\circ}$. | 10.13. If the vertices of a spatial quadrilateral $A B C D$ do not lie in the same plane, then $\angle A B C < \angle A B D + \angle D B C$ and $\angle A D C < \angle A D B + \angle B D C$ (see problem 5.4). Adding these inequalities and then adding the angles $B A D$ and $B C D$ to both sides, we obtain the required r... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,420 |
10.14. Prove that no more than one vertex of a tetrahedron has the property that the sum of any two plane angles at this vertex is greater than $180^{\circ}$. | 10.14. Suppose that the indicated property is possessed by vertices $A$ and $B$ of the tetrahedron $A B C D$. Then $\angle C A B + \angle D A B >$ $>180^{\circ}$ and $\angle C B A + \angle D B A > 180^{\circ}$. On the other hand, $\angle C A B+$ $+\angle C B A=180^{\circ}-\angle A C B<180^{\circ}$ and $\angle D B A+\an... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,421 |
10.15. Point $O$ lies on the base of the triangular pyramid $S A B C$. Prove that the sum of the angles between the ray $S O$ and the lateral edges is less than the sum of the dihedral angles at vertex $S$ and greater than half of this sum. | 10.15. According to problem $5.4 \angle A S B<\angle A S O+\angle B S O$. And since the ray $S O$ lies inside the trihedral angle $S A B C$, then $\angle A S O+\angle B S O<\angle A S C+\angle B S C$ (see problem 5.6). Writing down two more such inequalities and adding them, we get the required. | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,422 |
10.16. a) Prove that the sum of the angles between the edges of a trihedral angle and the planes of the opposite faces does not exceed the sum of its dihedral angles.
b) Prove that if the dihedral angles of a trihedral angle are acute, then the sum of the angles between its edges and the planes of the opposite faces i... | 10.16. a) Let $\alpha, \beta$ and $\gamma$ be the angles between the edges $S A, S B$ and $S C$ and the planes of the opposite faces. Since the angle between a line $l$ and a plane $\Pi$ does not exceed the angle between the line $l$ and any line in the plane $\Pi$, then $\alpha \leqslant \angle A S B, \beta \leqslant ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,423 |
10.17. The diagonal of a rectangular parallelepiped forms angles $\alpha, \beta$ and $\gamma$ with its edges. Prove that $\alpha+\beta+\gamma<\pi$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 10.17. Let $O$ - the center of the rectangular parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$. The height $OH$ of the isosceles triangle $A O C$ is parallel to the edge $A A_{\mathbf{1}}$, hence $\angle A O C=2 \alpha$, where $\alpha-$ is the angle between the edge $A A_{1}$ and the diagonal $A C_{1}$. Similar reason... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,424 | |
10.18. All dihedral angles of a convex quadrilateral angle are equal to $60^{\circ}$. Prove that the angles between its opposite edges cannot be simultaneously acute or simultaneously obtuse. | 10.18. Let $S$ be the vertex of a given angle. From the solution of problem $5.16, b$, it follows that it can be intersected by a plane such that in the section a rhombus $A B C D$ is obtained, and $S A=S C$ and $S B=S D$, and the projection of the vertex $S$ onto the plane of the section coincides with the point $O$ o... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,425 |
10.19. Prove that the sum of the angles under which the faces of a tetrahedron are seen from an arbitrary point lying inside it is greater than \(3 \pi\). | 10.19. Let $O$ be a point inside the tetrahedron $ABCD$; $\alpha, \beta$, and $\gamma$ be the angles under which the edges $AD, BD$, and $CD$ are seen from it; $a, b$, and $c$ be the angles under which the edges $BC, CA$, and $AB$ are seen; $P$ be the point of intersection of the line $DO$ with the face $ABC$. Since th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,426 |
10.20. a) Prove that the sum of the dihedral angles at the four edges $A B, B C, C D$, and $D A$ of the tetrahedron $A B C D$ is less than $2 \pi$.
b) Prove that the sum of the dihedral angles of the tetrahedron is between $2 \pi$ and $3 \pi$. | 10.20. a) Apply the statement of problem 7.19 to the tetrahedron \(ABCD\). Let \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\), and \(\mathbf{d}\) be the vectors corresponding to the faces \(BCD\), \(ACD\), \(ABD\), and \(ABC\). The sum of these vectors is zero, so there exists a spatial quadrilateral whose consecutive ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,427 |
10.21. The space is completely covered by a finite set of right circular cones (infinite in one direction) with apex angles $\varphi_{1}, \ldots, \varphi_{n}$. Prove that $\varphi_{1}^{2}+\ldots+\varphi_{n}^{2} \geqslant 16$.
## § 3. Areas | 10.21. The bases of all cones can be enclosed within a pair of radii $r$. Consider a sphere of radius $R$ with the same center $O$. If $R / r$ tends to infinity, then the fraction of the surface of this sphere included inside the given cones tends to the fraction of its surface included inside cones with the same angle... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,428 |
10.22. Prove that the area of any face of a tetrahedron is less than the sum of the areas of the other three faces. | 10.22. The projections of the edges of a tetrahedron onto the planes of its three other faces completely cover that face. It is also clear that the area of the projection of a triangle onto a plane not parallel to it is less than the area of the triangle itself (see problem 2.13). | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,429 |
10.23. A convex polyhedron lies inside another. Prove that the surface area of the outer polyhedron is greater than the surface area of the inner one. | 10.23. Postroim vnevnim obrazom na granakh vnutrennego mnogogrannika kak na osnovaniyakh pravilnykh prizm, rebra kotorykh dostatochno veliki: vse oni dolzhny peresecti poverkhnost' vneshnego mnogogrannika. Ety prizmy vysyekayut na poverkhnosti vneshnego mnogogrannika poparno neperesekayushchiesya figury, ploshchad' kaz... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,430 |
10.24. Prove that for any tetrahedron, there exist two such planes that the ratio of the areas of the projections of the tetrahedron onto them is not less than $\sqrt{2}$. | 10.24. Let the plane II be parallel to two skew edges of the tetrahedron. We will prove that the required two planes can be found even among planes perpendicular to P. The projection of the tetrahedron onto any such plane is a trapezoid (or a triangle) with a constant height equal to the distance between the chosen ske... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,431 |
10.25. a) Prove that the area of any triangular section of a tetrahedron does not exceed the area of one of its faces.
b) Prove that the area of any quadrilateral section of a tetrahedron does not exceed the area of one of its faces. | 10.25. a) If a triangular section does not pass through the vertex of the tetrahedron, then there exists a parallel triangular section passing through the vertex; the area of the latter section is larger. Therefore, it is sufficient to consider the case when the section passes through the vertex or the base of the tetr... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,432 |
10.26. A plane tangent to a sphere inscribed in a cube cuts off a triangular pyramid from it. Prove that the surface area of this pyramid does not exceed the area of a face of the cube.
## § 4. Volumes | 10.26. Let the given plane intersect the edges $AB, AD$ and $AA'$ at points $K, L$ and $M; P, Q$ and $R$ - the centers of the faces $ABB'A'$, $ABCD$ and $ADD'A'$; $O$ - the point of tangency of the plane with the sphere. The planes $KOM$ and $KPM$ are tangent to the sphere at points $O$ and $P$, so $\triangle KOM = \tr... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,433 |
10.27. On each edge of the tetrahedron, a point is marked. Consider four tetrahedra, one of the vertices of each of which is a vertex of the original tetrahedron, and the other vertices are the marked points lying on the edges emanating from this vertex. Prove that the volume of one of them does not exceed one eighth o... | 10.27. If two tetrahedra have a common trihedral angle, then the ratio of their volumes is equal to the product of the ratios of the lengths of the edges lying on the edges of this trihedral angle (see problem 3.1). Therefore, the product of the ratios of the volumes of the four considered tetrahedra to the volume of t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,434 |
10.28. The lengths of five edges of a tetrahedron do not exceed 1. Prove that its volume does not exceed $1 / 8$. | 10.28. Let the lengths of all edges of the tetrahedron \(ABCD\), except for the edge \(CD\), not exceed 1. If \(h_{1}\) and \(h_{2}\) are the heights dropped from vertices \(C\) and \(D\) to the line \(AB\), and \(a = AB\), then the volume \(V\) of the tetrahedron \(\triangle BCD\) is \(a h_{1} h_{2} \sin \varphi / 6\)... | \frac{1}{8} | Geometry | proof | Yes | Yes | olympiads | false | 24,435 |
10.29. The volume of a convex polyhedron is $V$, and the surface area is $S$.
a) Prove that if a sphere of radius $r$ is placed inside it, then $V / S \geqslant r / 3$.
b) Prove that inside it, a sphere of radius $V / S$ can be placed.
c) One convex polyhedron is located inside another. Let $V_{1}$ and $S_{1}$ be the... | 10.29. a) Let $O$ be the center of the given sphere. Divide the given polyhedron into pyramids with vertex $O$, the bases of which are its faces. The heights of these pyramids are not less than $r$, so the sum of their volumes is not less than $S r / 3$, and therefore, $v \geqslant S r / 3$.
b) Construct rectangular p... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,436 |
10.30. Inside a cube, there is a convex polyhedron, the projection of which onto each face of the cube coincides with that face. Prove that the volume of the polyhedron is not less than \(1 / 3\) of the volume of the cube. | 10.30. On each edge of the polyhedron, there is a point of multitraninka, since otherwise its projection along that edge would not coincide with the face. Let's take one point of multitraninka on each edge of the cube and consider a new convex polyhedron with vertices at these points. Since it is contained in the origi... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,437 |
10.31. The areas of the projections of a body onto the coordinate planes are $S_{1}, S_{2}$, and $S_{3}$. Prove that its volume does not exceed $\sqrt{S_{1} S_{2} S_{3}}$.
## § 5. Various Problems | 10.31. Let us conduct planes parallel to the coordinate planes at a distance of $n \varepsilon$, where $n$ runs through integers, and $\varepsilon$ is some fixed number. These planes divide the space into cubic cells with edge $\varepsilon$. The proof is sufficient to conduct for bodies consisting of these cubes. Indee... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,438 |
10.32. Prove that the radius of the inscribed circle of any face of a tetrahedron is greater than the radius of its inscribed sphere. | 10.32. Consider the section of the tetrahedron by a plane parallel to the face $A B C$ and passing through the center of its inscribed sphere. This section is a triangle $A_{1} B_{1} C_{1}$, similar to triangle $A B C$, with a similarity coefficient less than 1. Triangle $A_{1} B_{1} C_{1}$ contains a circle of radius ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,439 |
10.33. Based on the triangular pyramid $O A B C$ with vertex $O$, a point $M$ is taken. Prove that
$$
O M \cdot S_{A B C} \leqslant O A \cdot S_{M B C}+O B \cdot S_{M A C}+O C \cdot S_{M A B}
$$ | 10.33. According to problem $7.12 \overrightarrow{O M}=p \overrightarrow{O A}+q \overrightarrow{O B}+r \overrightarrow{O C}$, where $p=S_{M B C}: S_{A B C}, \quad q=S_{M A C}: S_{A B C}$ and $\quad r=S_{M A B}: S_{A B C}$. It remains to note that $O M \leqslant \mu O A+q O B+r O C$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,440 |
10.34. Let $r$ and $R$ be the radii of the inscribed and circumscribed spheres of a regular tetrahedral pyramid. Prove that $R / r \geqslant 1+\sqrt{2}$. | 10.34. Let $2a$ be the base of the isosceles triangle, $h$ be its height. Then $r$ is the radius of the inscribed circle of the isosceles triangle with height $h$ and base $2a$; $R$ is the radius of the circumscribed circle of the isosceles triangle with height $h$ and base $2\sqrt{2}a$. Therefore, $r\left(a+\sqrt{a^{2... | 1+\sqrt{2} | Geometry | proof | Yes | Yes | olympiads | false | 24,441 |
10.35. Can a hole be cut in a cube through which a cube of the same size can pass? | 10.35. Mommno. The projection of a cube with edge a onto a plane perpendicular to the diagonal is a regular hexagon with side $b=a \sqrt{2} / \sqrt{3}$. Inscribing a square in the resulting hexagon as shown in Fig. 80, it is easy to verify that the side of this square is $2 V / 3 b /(1+V \overline{3})=$ $=2 \sqrt{2} a ... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,442 |
10.36. Sections $M_{1}$ and $M_{2}$ of a convex centrally symmetric polyhedron are parallel, with $M_{1}$ passing through the center of symmetry.
a) Is it true that the area of $M_{1}$ is not less than the area of $M_{2} ?$
б) Is it true that the radius of the smallest circle containing $M_{1}$ is not less than the r... | 10.36. a) Yes, it is correct. Let \( O \) be the center of symmetry of the given polyhedron; \( M_{2}^{\prime} \) be the polygon symmetric to \( M_{2} \) with respect to point \( O \). Consider the smallest convex polyhedron \( P \) containing \( M_{2} \) and \( M_{2}^{\prime} \). We will prove that the area of the par... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,443 |
10.37. Inside a sphere of radius $R$ there is a convex polyhedron. The length of its $i$-th edge is $l_{i}$, and the dihedral angle at this edge is $\varphi_{i}$. Prove that
$$
\sum l_{i}\left(\pi-\varphi_{i}\right) \leqslant 8 \pi R .
$$
## Problems for independent solving | 10.37. Consider a body consisting of points removed from a given polyhedron by a distance not greater than $d$. The surface area of this body is equal to $S+d \sum l_{i}\left(\pi-\varphi_{i}\right)+4 \pi d^{2}$, where $S$ is the surface area of the polyhedron (Problem 3.13). Since this body is contained within a sphere... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,444 |
11.1. The ends of the segment $A B$ move along the given lines $a$ and $b$. Prove that its length will be the smallest when it is perpendicular to both lines. | 11.1. Let's draw a plane П through the line $b$ parallel to $a$. Let $A^{\prime}$ be the projection of point $A$ onto plane П. Then $A B^{2}=A^{\prime} B^{2}+A^{\prime} A^{2}=A^{\prime} B^{2}+h^{2}$, where $h$ is the distance between the line $a$ and the plane П. The point $A^{\prime}$ coincides with $B$ if $A B \perp$... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,445 |
11.2. Find the smallest area of the section of a cube with edge a by a plane passing through its two-
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
It seems there is a typo or missing part in the original text, as it cuts ... | 11.2. Let a plane pass through the diagonal $A C_{1}$ of the cube $A B C D A_{1} B_{1} C_{1} D_{1}$ and intersect its edges $B B_{1}$ and $D D_{1}$ at points $P$ and $Q$ respectively. The area of the parallelogram $A P C_{1} Q$ is equal to the product of the length of the segment $A C_{1}$ and the distance from point $... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,446 | |
11.3. All edges of a regular triangular prism $A B C A_{1} B_{1} C_{1}$ have length $a$. Points $M$ and $N$ lie on the lines $B C_{1}$ and $C A_{1}$, respectively, and the line $M N$ is parallel to the plane $A A_{1} B$. What is the minimum length of such a segment $M N$? | 11.3. If $M^{\prime}$ and $N^{\prime}$ are the projections of points $M$ and $N$ onto the plane $A B C$, then $M^{\prime} N^{\prime} \| A B$. Let $C M^{\prime}=x$. Then $M^{\prime} N^{\prime}=x$, and the length of the projection of segment $M N$ onto line $C C_{1}$ is $|a-2 x|$. Therefore,
$$
M N^{2}=x^{2}+(a-2 x)^{2}... | \frac{}{\sqrt{5}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,447 |
11.4. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with edge length a. The ends of the segment intersecting edge $C_{1} D_{1}$ lie on the lines $A A_{1}$ and $B C$. What is the minimum length that this segment can have | 11.4. Let points $M$ and $N$ lie on the lines $A A_{1}$ and $B C$ respectively, and the segment $M N$ intersects the line $C_{1} D_{1}$ at point $L$. Then points $M$ and $N$ lie on the rays $A A_{1}$ and $B C$, with $x = A M > a$ and $y = B N > a$. Considering projections onto the planes $A A_{1} B$ and $A B C$, we get... | 3a | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,448 |
11.5. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with edge length a. The ends of a segment forming an angle of $60^{\circ}$ with the plane of the face $A B C D$ lie on the lines $A B_{1}$ and $B C_{1}$. What is the minimum length that this segment can have?
## § 2. Area and Volume | 11.5. Let's introduce a coordinate system, directing the axes $O x, O_{y}$, and $O_{z}$ along the rays $B C, B A$, and $B B_{\mathbf{i}}$ respectively. Suppose point $M$ on the line $B C_{1}$ has coordinates $(x, 0, x)$, and point $N$ on the line $B_{1} A$ has coordinates $(0, y, a-y)$. Then the square of the length of... | 2(\sqrt{3}-\sqrt{2}) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,449 |
11.6. What is the minimum value of the ratio of the volumes of a cone and a cylinder circumscribed about the same sphere? | 11.6. Let $r$ be the radius of the given sphere. If the axial section of the cone is an isosceles triangle with height $h$ and base $2a$, then $ah = S = r\left(a + \sqrt{h^2 + a^2}\right)$. Therefore, $a^2(h - r)^2 = r^2(h^2 + a^2)$, i.e., $a^2 = r^2 h^2 / (h - 2r)$. Hence, the volume of the cone is $\pi r^2 h^2 / 3(h ... | \frac{4}{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,450 |
11.7. The area of the surface of a spherical segment is $S$ (the spherical part of its surface is meant). What is the greatest possible volume of such a segment? | 11.7. Let $V$ be the volume of a spherical segment, $R$ be the radius of the sphere. Since $S=2 \pi R h$ (problem 4.24) and $V=\pi h^{2}(3 R-$ - $h$ ) $/ 3$ (problem 4.27), then $V=S h / 2-\pi h^{3} / 3$. Therefore, the derivative of $V$ with respect to $h$ is $S / 2-\pi h^{2}$. The maximum volume will be at $h=$ $=\sq... | S\sqrt{S/18\pi} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,451 |
11.8. Prove that among all regular $n$-sided pyramids with a fixed area of the total surface, the pyramid with the largest volume is the one where the dihedral angle at the base edge is equal to the dihedral angle at the edge of a regular tetrahedron. | 11.8. Let $h$ be the height of a regular pyramid, $r$ be the radius of the inscribed circle of its base. Then the volume and the area of the full surface of the pyramid are
$$
\frac{n}{3} \operatorname{tg} \frac{\pi}{n}\left(r^{2} h\right) \text { and } n \operatorname{tg} \frac{\pi}{n}\left(r^{2}+r \sqrt{h^{2}+r^{2}}... | \cos\varphi=\frac{1}{3} | Geometry | proof | Yes | Yes | olympiads | false | 24,452 |
11.9. Through a point $M$, lying inside a given trihedral angle with right dihedral angles, all possible planes are drawn. Prove that the volume of the tetrahedron cut off by such a plane from the trihedral angle will be the smallest when $M$ is the point of intersection of the medians of the triangle that is the secti... | 11.9. Let's introduce a coordinate system, directing its axes along the edges of the given trihedral angle. Suppose point $M$ has coordinates ( $\alpha, \beta, \gamma$ ); the plane intersects the edges of the trihedral angle at points that are at distances $a, b$, and $c$ from its vertex. Then this plane is given by th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,453 |
11.10. What is the maximum area of the projection of a regular tetrahedron with edge length a onto a plane? | 11.10. The projection of a tetrahedron can be a triangle or a quadrilateral. In the first case, it is the projection of one of the faces, so its area does not exceed $\sqrt{3} a^{2} / 4$. In the second case, the diagonals of the quadrilateral are projections of the edges of the tetrahedron, so the area of the projectio... | \frac{^2}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,454 |
11.11. What is the maximum area of the projection of a rectangular parallelepiped with edges $a, b$, and $c$ onto a plane? | 11.11. The area of the projection of a parallelepiped is at least twice the area of the projection of one of the triangles with vertices at the ends of three edges of the parallelepiped emanating from one point; for example, if the projection of the parallelepiped is a hexagon, then as such a vertex, one should take th... | \sqrt{^{2}b^{2}+b^{2}^{2}+^{2}^{2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,455 |
11.12. A cube with edge $a$ lies on a plane. A light source is located at a distance $b$ from the plane, where $b>a$. Find the minimum value of the area of the shadow cast by the cube on the plane.
## § 3. Distances | 11.12. Let $A B C D$ be a square with side $a$; point $X$ is removed from the line $A B$ by a distance $b$, with $b > a$; $C'$ and $D'$ are the points of intersection of the extensions of segments $X C$ and $X D$ beyond points $C$ and $D$ with the line $A B$. Since $\triangle C' D' X$ is similar to $\triangle C D X$, t... | (\frac{}{b-})^2 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,456 |
11.13. a) Consider for each internal point of a regular tetrahedron the sum of distances from it to its vertices. Prove that this sum will be the smallest for the center of the tetrahedron.
b) Two opposite edges of the tetrahedron are equal to $b$ and $c$, while the other edges are equal to $a$. What is the minimum va... | 11.13. a) Let's draw planes through the vertices of a regular tetrahedron \(ABCD\) parallel to the opposite faces. These planes also form a regular tetrahedron. It follows that the sum of the distances from any internal point \(X\) of the tetrahedron \(ABCD\) to these planes is constant (Problem 8.1, a). The distance f... | \sqrt{4a^2+2bc} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,457 |
11.14. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with edge $a$. On the rays $A_{1} A, A_{1} B_{1}$, and $A_{1} D_{1}$, points $E$, $F$, and $G$ are taken such that $A_{1} E=A_{1} F=A_{1} G=b$. Let $M$ be a point on the circle $S_{1}$ inscribed in the square $A B C D$, and $N$ be a point on the circle $S_{2}$ passi... | 11.14. Let 0 be the center of a cube. Consider two spheres with center $O$, containing circles $S_{1}$ and $S_{2}$ respectively. Let $R_{1}$ and $R_{2}$ be the radii of these spheres. The distance between points on circles $S_{1}$ and $S_{2}$ cannot be less than $\left|R_{1}-R_{2}\right|$. If two cones with a common ve... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,458 | |
11.15. In a truncated cone, the angle between the axis and the generatrix is $30^{\circ}$. Prove that the shortest path on the surface of the cone, connecting a point on the boundary of one base with the diametrically opposite point on the boundary of the other base, has a length of $2 R$, where $R$ is the radius of th... | 11.15. Let us show that the path from the point \(A\) on the boundary of the larger base to the diametrically opposite point \(C\) of the other base consists of the generatrix \(AB\) and the diameter \(BC\); the length of this path is \(2R\). Let \(r\) be the radius of the smaller base, \(O\) be its center. Consider th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,459 |
11.16. The lengths of three pairwise perpendicular segments $O A, O B$, and $O C$ are $a, b$, and $c$, respectively, with $a \leqslant b \leqslant 208 \leqslant c$. What is the greatest and the least value that the sum of the distances from points $A, B$, and $C$ to a line $l$ passing through point $O$ can take?
## § ... | 11.16. Let the angles between the line $l$ and the lines $O A, O B$ and $O C$ be $\alpha, \beta$ and $\gamma$. Then $\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1$ (Problem 1.21), and therefore, $\quad \sin ^{2} \alpha+\sin ^{2} \beta+$ $+\sin ^{2} \gamma=2$. The sum of the distances from points $A, B$ and $C$ to... | \sqrt{2(^2+b^2+^2)} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,460 |
11.17. The line $l$ lies in the plane of one face of a given dihedral angle. Prove that the angle between the line $l$ and the plane of the other face is greatest when $l$ is perpendicular to the edge of the given dihedral angle. | 11.17. Let $A$ be the point of intersection of line $l$ with the edge of a dihedral angle. On line $l$, we lay off the segment $A B$ of length 1. Let $B^{\prime}$ be the projection of point $B$ onto the plane of the other face, and $O$ be the projection of point $B$ onto the edge of the dihedral angle. Then $\sin B A B... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,461 |
11.18. The height of a regular quadrilateral prism $A B C D A_{1} B_{1} C_{1} D_{1}$ is half the length of the side of the base. Find the maximum value of the angle $A_{1} M C_{1}$, where $M$ is a point on the edge $A B$. | 11.18. Let $A A_{1}=1, A M=x$. Introduce a coordinate system, the axes of which are parallel to the edges of the prism. The vectors $\overrightarrow{M A}_{\text {}}$ and $\overrightarrow{M C}_{1}$ have coordinates $(0,1,-x)$ and ( $2,1,2-x$ ); their scalar product is equal to $1-2 x+x^{2}=(1-x)^{2} \geqslant 0$. Theref... | 90 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,462 |
11.19. Three identical cylindrical surfaces of radius $R$ with mutually perpendicular axes touch each other pairwise.
a) What is the radius of the smallest sphere that touches these cylinders?
b) What is the radius of the largest cylinder that touches the three given ones, the axis of which passes inside the triangle... | 11.19. There exists a parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$, the edges $A A_{1}, D C$ and $B_{1} C_{1}$ of which lie on the axes of the given cylinders (problem 1.19); it is clear that this parallelepiped is a cube with edge $2 R$.
a) The center of this cube is at a distance of $\sqrt{2} R$ from all edges, ... | (\sqrt{2}-1)R | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,463 |
11.20. Can a regular tetrahedron with an edge of 1 pass through a circular hole with a radius of: a) 0.45; b) 0.44? (The thickness of the hole can be neglected.)
## Problems for Independent Solving | 11.20. In the process of a tetrahedron passing through an opening, there will inevitably be a moment when vertex \( B \) is on one side of the plane of the opening, vertex \( A \) is in the plane of the opening, and vertices \( C \) and \( D \) are on the other side of the plane of the opening (or in the plane of the o... | 0.4478 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,464 |
12.1. Find the geometric locus of the midpoints of segments parallel to a given plane and whose ends lie on two given skew lines. | 12.1. Let the lines $l_{1}$ and $l_{2}$ intersect the given plane $\Pi$ at points $P$ and $Q$ (if $l_{1} \| \Pi$ or $l_{2} \| \Pi$, then the desired segments do not exist). Draw through the midpoint $M$ of segment $PQ$ the lines $l_{1}^{\prime}$ and $l_{2}^{\prime}$, parallel to the lines $l_{1}$ and $l_{2}$ respective... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,465 |
12.3. Given three pairwise intersecting lines. Find the geometric locus of the points of intersection of the medians of triangles parallel to a given plane and whose vertices lie on the given lines. | 12.3. The geometric locus of the midpoints of side $AB$ of the indicated triangles is the line $l$ (see problem 12.1). The desired geometric locus of points (GML) consists of points that divide in the ratio $1:2$ segments parallel to a given plane, with their ends lying on the line $l$ and on a third given line. By sli... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,467 |
12.4. In space, there are two skew lines and a point $A$ on one of them. Through these lines, two perpendicular planes are drawn, forming a right dihedral angle. Find the geometric locus of the projections of point $A$ on the edges of such angles. | 12.4. Let $\pi_{1}$ and $\pi_{2}$ be perpendicular planes passing through lines $l_{1}$ and $l_{2}$; $l$ is the line of their intersection; $X$ is the projection of point $A$, lying on line $l_{1}$, onto line $l$. Draw
. Find the locus of points $M$. | 12.5. Let's draw a plane through point $A$ perpendicular to line $l$. Let $M^{\prime}$ and $N^{\prime}$ be the projections of points $M$ and $N$ onto this plane. Since $M N \perp l$, then $M^{\prime} N^{\prime} \| M N$. The line $M N$ is perpendicular to the plane $A M M^{\prime}$, because $N M \perp M M^{\prime}$ and ... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,469 |
12.6. Pairwise intersecting lines $l_{1}, l_{2}$ and $l_{3}$ are perpendicular to a line $p$ and intersect it at points $A_{1}, A_{2}$ and $A_{3}$ respectively. Let $M$ and $N$ be points on lines $l_{1}$ and $l_{2}$ such that lines $M N$ and $l_{3}$ intersect. Find the geometric locus of the midpoints of segments $M N$... | 12.6. The projection of the line $l_{\mathrm{s}}$ onto the plane perpendicular to $l_{3}$ passes through the point $A_{3}$; in the projection $M^{\prime} N^{\prime}$ of the line $M N$, it passes through this point; moreover, the projections of the lines $l_{1}$ and $l_{2}$ are parallel. Therefore, $\overrightarrow{A_{1... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,470 |
12.7. Given two intersecting perpendicular lines. The ends of segments $A_{1} A_{2}$, parallel to a given plane, lie on these lines. Prove that all spheres with diameters $A_{1} A_{2}$ have a common circle. | 12.7. Let $B_{1} B_{2}$ be the common perpendicular to the given lines (points $A_{i}$ and $B_{1}$ lie on one of the given lines). Since $A_{2} B_{1} \perp A_{1} B_{1}$, point $B_{1}$ belongs to the sphere with diameter $A_{1} A_{2}$. Similarly, point $B_{2}$ belongs to this sphere. The geometric locus of the midpoints... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,471 |
12.8. Points $A$ and $B$ move along two intersecting lines with constant but unequal speeds; the ratio of their speeds is $k$. Let $M$ and $N$ be points on the line $AB$ such that $AM: BM = AN: BN = k$ (point $M$ lies on the segment $AB$). Prove that points $M$ and $N$ move along two perpendicular lines.
## § 2. Spher... | 12.8. Let $A_{i}$ and $B_{1}$ be the positions of points $A$ and $B$ at another moment in time; P - a plane parallel to the given skew lines. Consider the projection onto plane P parallel to line $A_{1} B_{1}$. Let $A^{\prime}, B^{\prime}, M^{\prime}$ and $N^{\prime}$ be the projections of points $A, B, M$ and $N$; $C^... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,472 |
12.9. Lines $l_{1}$ and $l_{2}$ are tangent to a sphere. Segment $M N$ with endpoints on these lines is tangent to the sphere at point $X_{\text {, }}$ Find the locus of $X$.
untranslated part:
әтих
Note: The word "әтих" seems to be a non-standard or possibly incorrectly transliterated term. If it is meant to be "t... | 12.9. Let the line $l_{1}$, containing point $M$, be tangent to the sphere at point $A$, and the line $l_{2}$ at point $B$. Draw a plane through the line $l_{1}$ parallel to $l_{2}$, and consider the projection onto this plane parallel to the line $A B$. Let $N^{\prime}$ and $X^{\prime}$ be the images of points $N$ and... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,473 |
12.10. Points $A$ and $B$ lie on the same side of plane $P$, and line $AB$ is not parallel to $P$. Find the geometric locus of the centers of spheres passing through the given points and touching the given plane. | 12.10. Let $C$ be the point of intersection of the line $AB$ with a given plane, $M$ be the point of tangency of one of the sought spheres with plane II. Since $CM^2 = CA \cdot CB$, point $M$ lies on a circle of radius $\sqrt{CA \cdot CB}$ centered at $C$. Therefore, the center $O$ of the sphere belongs to the lateral ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,474 |
12.11. The centers of two spheres of different radii lie in the plane ПІ. Find the geometric locus of points $X$ in this plane, through which a plane can be drawn that touches the spheres: a) internally; b) externally. (Internal tangency - the spheres lie on opposite sides of the plane, external - on the same side.)
#... | 12.11. a) Let the given spheres intersect the plane П along the circumferences $S_{1}$ and $S_{2}$. The internal tangents to these circumferences divide the plane into 4 parts. Consider a right circular cone, the axial section of which consists of those lines that contain $S_{1}$ and $S_{2}$. Planes that internally tou... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,475 |
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