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6.35. a) Prove that the sum of the cosines of the dihedral angles of a regular tetrahedron is 2.
b) The sum of the plane angles of a trihedral angle is \(180^{\circ}\). Find the sum of the cosines of its dihedral angles.
## § 5. Orthocentric Tetrahedron
Definition. A tetrahedron is called orthocentric if all its alt... | 6.35. a) Let $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$ and $\mathbf{e}_{4}$ be unit vectors, non-perpendicular to the faces and directed outward. Since the areas of all faces are equal, then $\mathbf{e}_{1}+\mathbf{e}_{2}+\mathbf{e}_{3}+\mathbf{e}_{4}=0$ (see problem 7.19). Therefore, $0=\left|e_{1}+\mathbf{e}_{... | 2 | Geometry | proof | Yes | Yes | olympiads | false | 24,257 |
6.36. a) Prove that if $A D \perp B C$, then the altitudes dropped from vertices $B$ and $C$ (as well as the altitudes dropped from vertices $A$ and $D$) intersect at one point, and this point lies on the common perpendicular to $A D$ and $B C$.
b) Prove that if the altitudes dropped from vertices $B$ and $C$ intersec... | 6.36. a) Let $A D \perp B C$. Then there exists a plane $\Pi$, passing through $B C$ and perpendicular to $A D$. The height dropped from vertex $B$ is perpendicular to $A D$, so it lies in the plane $\Pi$. Similarly, the height dropped from vertex $C$ lies in the plane $\Pi$. Therefore, these heights intersect at one p... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,258 |
6.37. Prove that in an orthocentric tetrahedron, the common perpendiculars to pairs of opposite edges intersect at one point. | 6.37. From the solution of problem 6.36, a) it follows that the point of intersection of the heights belongs to each common perpendicular to the opposite pairs of edges. | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,259 |
6.38. Let $K, L, M$ and $N$ be the midpoints of the edges $A B, B C$, $C D$ and $D A$ of the tetrahedron $A B C D$.
a) Prove that $A C \perp B D$ if and only if $K M = L N$.
b) Prove that the tetrahedron is orthocentric if and only if the segments connecting the midpoints of opposite edges are equal. | 6.38. a) Quadrilateral $K L M N$ is a parallelogram, the sides of which are parallel to $A C$ and $B D$. Its diagonals $K M$ and $L N$ are equal if and only if it is a right-angled, i.e., $A C \perp B D$.
It should also be noted that the plane $K L M N$ is not perpendicular to the common perpendicular to $A C$ and $B ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,260 |
6.39. a) Prove that if $B C \perp A D$, then the altitudes dropped from vertices $A$ and $D$ to the line $B C$ intersect at one point.
b) Prove that if the altitudes dropped from vertices $A$ and $D$ to the line $B C$ intersect at one point, then $B C \perp A D$ (and hence, the altitudes dropped from vertices $B$ and ... | 6.39. a) Since $B C \perp A D$, there exists a plane $\Pi$, passing through the line $A D$ and perpendicular to $B C$; let $U$ be the point of intersection of the line $B C$ with the plane $\Pi$. Then $A U$ and $D U$ are perpendiculars dropped from points $A$ and $D$ to the line $B C$.
b) Let $A U$ and $D U$ be the al... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,261 |
6.40. Prove that a tetrahedron is orthocentric if and only if one of the following conditions is satisfied:
a) the sums of the squares of opposite edges are equal;
b) the products of the cosines of opposite dihedral angles are equal;
c) the angles between opposite edges are equal.
Note. There are other conditions t... | 6.40. a) Follows from problem 7.2.
b) Using the results of problems 6.6 and 6.10, we obtain that the products of the cosines of opposite dihedral angles are equal if and only if the sums of the squares of opposite edges are equal.
c) It is sufficient to check that if all angles between opposite edges are equal to $\a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,262 |
6.41. Prove that in an orthocentric tetrahedron:
a) all dihedral angles at one vertex are simultaneously either acute, right, or obtuse;
b) one of the faces is an acute-angled triangle. | 6.41. a) If $A B C D$ is an orthocentric tetrahedron, then $A B^{2}+C D^{2}=A D^{2}+B C^{2}$ (see problem 6.40, a). Therefore, $A B^{2}+$
+. $A C^{2}-B C^{2}=A D^{2}+A C^{2}-C D^{2}$, i.e., the cosines of angles $B A C$ and $D A C$ have the same sign.
b) Since a triangle cannot have two non-acute angles, taking into ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,263 |
6.42. Prove that in an orthocentric tetrahedron, the relation $O H^{2}=4 R^{2}-3 d^{2}$ holds, where $O-$ is the midpoint of opposite edges. | 6.42. Let $K$ and $L$ be the midpoints of edges $AB$ and $CD$. Point $H$ lies in the plane passing through $CD$ and perpendicular to $AB$, and point $O$ lies in the plane passing through $K$ and perpendicular to $AB$. These planes are symmetric with respect to the center of mass $M$ of the tetrahedron, which is the mid... | 4R^{2}-3d^{2} | Geometry | proof | Yes | Yes | olympiads | false | 24,264 |
6.43. a) Guess that the nine-point circles of triangles $A B C$ and $D B C$ belong to the same sphere if and only if $B C \perp A D$.
b) Prove that for an orthocentric tetrahedron, the nine-point circles of all faces belong to the same sphere (the 24-point sphere).
c) Prove that if $A D \perp B C$, then the sphere co... | 6.43. a) The circles of the triangles $ABC$ and $DBC$ belong to the same sphere if and only if the feet of the altitudes dropped from vertices $A$ and $D$ to the line $BC$ coincide. It remains to use the result of problem 6.39, b.
b) The segments connecting the midpoints of opposite edges intersect at one point, divid... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,265 |
6.44. Prove that in an orthocentric tetrahedron, the centroids of the faces, the points of intersection of the altitudes of the faces, and the points that divide the segments connecting the orthocenter with the vertices in the ratio \(2:1\), counting from the vertex, lie on one sphere (the sphere of 12 points). | 6.44. Let $O, M$ and $H$ be the center of the circumscribed sphere, the center of mass, and the orthocenter of a tetrahedron. From the solution to problem 6.42, it follows that $M$ is the midpoint of the segment $O H$. The centers of mass of the faces of the tetrahedron are the vertices of a tetrahedron that is similar... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,266 |
6.45. a) Let $H$ be the point of intersection of the altitudes of an orthocentric tetrahedron, $M'$ be the centroid of any face, and $N$ be the point of intersection of the ray $H M'$ with the circumscribed sphere of the tetrahedron. Prove that $H M' : M' N = 1 : 2$.
b) Let $M$ be the centroid of the orthocentric tetr... | 6.45. a) From the solution of problem 6.44, it follows that under a homothety with center \( H \) and coefficient 3, the point \( M^{\prime} \) transitions to a point on the circumscribed sphere of the tetrahedron.
b) From the solution of problem 6.44, it follows that under a homothety with center \( M \) and coeffici... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,267 |
6.46. Prove that in an orthocentric tetrahedron, the Monge point (see problem 7.32, a) coincides with the point of intersection of the altitudes.
## § 6. Completion of a Tetrahedron
By drawing a plane through each edge of the tetrahedron parallel to the opposite edge, the tetrahedron can be completed into a parallele... | 6.46. Since $A B \perp C D$, there exists a plane passing through $A B$ and perpendicular to $C D$. In this plane lies both the point of intersection of the heights dropped from vertices $A$ and $B$, and the Monge point. If such planes are drawn through all edges, they will have a unique common point. | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,268 |
6.48. Prove that all faces of a tetrahedron are equal if and only if one of the following conditions is satisfied:
a) when completing the tetrahedron, a rectangular parallelepiped is formed;
b) the segments connecting the midpoints of opposite edges are perpendicular;
c) the areas of all faces are equal;
d) the cen... | 6.48. a) The diagonals of the opposite faces of the obtained parallelepiped are two opposite edges of the tetrahedron. These faces will be rectangles if and only if the opposite edges are equal.
The result of this problem is used in solving problem 6-g.
b) It is sufficient to note that the given segments are parallel... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,270 |
6.49. Prove that in a regular tetrahedron all plane angles are acute.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 6.49. We will complete the equilateral tetrahedron to a parallelepiped. We will obtain a rectangular parallelepiped (problem 6.48, a). If its edges are equal to \(a\), \(b\), and \(c\), then the squares of the sides of the tetrahedron's face are \(a^{2}+b^{2}\), \(b^{2}+c^{2}\), and \(c^{2}+a^{2}\). Since the sum of th... | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 24,271 | |
6.50. Prove that the sum of the squares of the lengths of the edges of a tetrahedron is four times the sum of the squares of the distances between the midpoints of its opposite edges. | 6.50. Complete the tetrahedron to form a parallelepiped. The distances between the midpoints of the skew edges of the tetrahedron are equal to the lengths of the edges of this parallelepiped. It remains to use the fact that if $a$ and $b$ are the side lengths of a parallelogram, and $d_{1}$ and $d_{2}$ are the lengths ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,272 |
6.52. A line $l$ passes through the midpoints of edges $A B$ and $C D$ of the tetrahedron $A B C D$; a plane $\Pi$, containing $l$, intersects edges $B C$ and $A D$ at points $M$ and $N$. Prove that the line $l$ bisects the segment $M N$.
. The section of this parallelepiped by the plane П is a parallelogram; points $M$ and $N$ lie on its sides, and the line $l$ passes through the midpoints of the other two sides. | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,274 |
6.53. Prove that the lines connecting the midpoint of the height of a regular tetrahedron to the vertices of the face to which this height is dropped are pairwise perpendicular.
## § 7. Pyramid and Prism | 6.53. Let $A B_{1} C D_{1}$ be a tetrahedron inscribed in the cube $A B C D A_{1} B_{1} C_{1} D_{1} ; \quad H-$ the foot of the perpendicular from $A$ to the plane $B_{1} C D_{1} ; M$ - the midpoint of the segment $A H$, which is the height of the tetrahedron. Since $C_{1} H: H A=1: 2$ (Problem 2.1), the point $M$ is s... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,275 |
6.54. The planes of the lateral faces of a triangular pyramid form equal angles with the plane of the base. Prove that the projection of the vertex onto the plane of the base is the center of the inscribed or escribed circle of the base. | 6.54. If $\alpha$ is the angle between the planes of the lateral faces and the plane of the base, and $h$ is the height of the pyramid, then the distance from the projection of the vertex onto the plane of the base to any line containing an edge of the base is $h \operatorname{ctg} \alpha$.
Note also that if the dihed... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,276 |
6.55. In a triangular pyramid, the dihedral angles at the base edges are equal to $\alpha$. Find its volume if the lengths of the base edges are $a, b$ and $c$. | 6.55. Let $h$ be the height of the pyramid, $V$ its volume, and $S$ the area of the base. According to problem 6.54, $h = r \tan \alpha$, where $r$ is the radius of the inscribed circle of the base. Therefore, $V = Sh / 3 = Sr \tan \alpha / 3 = S^2 \tan \alpha / 3p = (p-a)(p-b)(p-c) \tan \alpha / 3$, where $p \Rightarr... | \frac{(p-)(p-b)(p-)\tan\alpha}{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,277 |
6.56. Based on the triangular pyramid $S A B C$, a point $M$ is taken and lines parallel to the edges $S A, S B$, and $S C$ are drawn through it, intersecting the lateral faces at points $A_{1}, B_{1}$, and $C_{1}$. Prove that
$$
\frac{M A_{1}}{S A}+\frac{M B_{1}}{S B}+\frac{M C_{1}}{S C}=1
$$ | 6.56. Let the line $A M$ intersect $B C$ at point $P$. Then $M A_{i}: S A=M P: A P=S_{M B C}: S_{A B C}$. Similarly $M B_{1}: S B \Longrightarrow$ $=S_{A M C}: S_{A B C}$ and $M C_{1}: S C=S_{A B M}: S_{A B C}$. Adding these equalities and considering that $S_{M B C}+S_{A M C}+S_{A B M}=S_{A B C}$, we obtain the requir... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,278 |
6.57. The vertex $S$ of the triangular pyramid $S A B C$ coincides with the vertex of a cone, and the points $A, B$ and $C$ lie on the circumference of its base. The dihedral angles at the edges $S A, S B$ and $S C$ are $\alpha, \beta$ and $\gamma$. Find the angle between the plane $S B C$ and the plane tangent to the ... | 6.57. Let $O$ be the center of the base of the cone. In the trihedral angles $S B O C, S C O A$, and $S A O B$, the dihedral angles at the edges $S B$ and $S C, S C$ and $S A, S A$ and $S B$ are equal. Denote these angles by $x, y$, and $z$. Then $\alpha=y+z, \beta=z+x$ and $\gamma=x+y$. Since the plane $SCO$ is perpen... | \frac{\pi+\alpha-\beta-\gamma}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,279 |
6.58. Vectors $\overrightarrow{A A}_{1}, \overrightarrow{B B}_{1}$ and $\overrightarrow{C C}_{1}$ are perpendicular to the plane $A B C$, and their lengths are equal to the corresponding altitudes of triangle $A B C$, the radius of the inscribed circle of which is $r$.
a) Prove that the distance from point $M$ of inte... | 6.58. a) Drop a perpendicular $M O$ from point $M$ to the plane $A B C$. Since the distance from point $A_{1}$ to the plane $A B C$ is equal to the distance from point $A$ to the line $B C$, the angle between the planes $A B C$ and $A_{1} B C$ is $45^{\circ}$. Therefore, the distance from point $O$ to the line $B C$ is... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,280 |
6.60. Through a point $M$ of the base of a regular pyramid, a perpendicular is drawn, intersecting the planes of the lateral faces at points $M_{1}, \ldots, M_{n}$. Prove that the sum of the lengths of the segments $M M_{1}, \ldots, M M_{n}$ is the same for all points $M$ of the base of the pyramid. | 6.60. Let $N_{i}$ be the base of the perpendicular dropped from point $M$ to the edge of the base (or its extension), and let point $M_{i}$ lie in the plane of the face passing through this edge. Then $M M_{i}=N_{i} M \operatorname{tg} \alpha$, where $\alpha$ is the angle between the base and the lateral face of the py... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,282 |
6.61. A sphere is inscribed in an $n$-sided pyramid. The lateral faces of the pyramid are rotated around the edges of the base and are laid in the plane of the base so that they lie on the same side of the corresponding edges together with the base. Prove that the vertices of these faces, different from the vertices of... | 6.61. If a sphere touches the sides of a dihedral angle, then when these sides coincide, the points of tangency coincide. Therefore, all points of tangency of the lateral faces with the inscribed sphere, when rotated around the edges, fall into one point - the point of tangency of the sphere with the base plane of the ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,283 |
6.62. Perpendiculars are drawn from the vertices of the base of an inscribed pyramid to the lateral faces. Prove that the lines connecting the bases of the altitudes in each face are parallel to one plane. (The planar angles at the vertex of the pyramid are not right angles.) | 6.62. We will prove that all the indicated lines are parallel to the plane tangent to the circumscribed sphere of the pyramid at its vertex. For this, it is sufficient to check that if $A A_{1}$ and $B B_{1}$ are the altitudes of triangle $A B C$, then the line $A_{1} B_{1}$ is parallel to the line tangent to the circu... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,284 |
6.63. At the base of a pyramid with vertex $S$ lies a parallelogram $A B C D$. Prove that its lateral edges form equal angles with some ray $S O$, lying inside the tetrahedral angle $S A B C D$, if and only if $S A+S C=S B+S D$. | 6.63. Suppose first that the lateral edges of the pyramid form equal angles with the given ray $S O$. Let the plane perpendicular to the ray $S O$ intersect the lateral edges of the pyramid at points $A_{1}, B_{1}, C_{1}$, and $D_{1}$. Since $S A_{1}=S B_{1}=S C_{1}=S D_{1}$, and the areas of triangles $B C D, A D B, A... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,285 |
6.64. The bases of the truncated quadrilateral pyramid $A B C D A_{1} B_{1} C_{1} D_{1}$ are parallelograms $A B C D$ and $A_{1} B_{1} C_{1} D_{1}$. Prove that any line intersecting three of the four lines $A B_{1}, B C_{1}, C D_{1}$, and $D A_{1}$ intersects the fourth line or is parallel to it. | 6.64. Let the line $l$ intersect the line $A B_{i}$ at point $K$. The statement of the problem is equivalent to the fact that the planes $K B C_{\text {i }}$, $K C D_{1}$, and $K D A_{1}$ have a common line, i.e., they have a common point different from $K$. Draw a plane through point $K$ parallel to the bases of the p... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,286 |
6.65. Find the area of the total surface of a prism described about a sphere, if the area of its base is $S$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 6.65. If $p$ is the semiperimeter of the base of the prism, and $r$ is the radius of the sphere, then the area of the base is $pr$, and the area of the lateral surface is $4pr$. Therefore, the area of the complete surface of the prism is $6 S$. | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 24,287 | |
6.66. On the lateral edges $B B_{1}$ and $C C_{1}$ of a regular prism $A B C A_{1} B_{1} C_{1}$, points $P$ and $P_{1}$ are taken such that $B P: P B_{1}=C_{1} P_{1}: P_{1} C=1: 2$.
a) Prove that the dihedral angles at the edges $A P_{1}$ and $A_{1} P$ of the tetrahedron $A A_{1} P P_{1}$ are right angles.
b) Prove t... | 6.66. a) Let $M$ and $N$ be the midpoints of the edges $P P_{1}$ and $A A_{1}$. It is clear that the tetrahedron $A A_{1} P P_{1}$ is symmetric with respect to the line $M N$. Further, let $P^{\prime}$ be the projection of point $P$ onto the plane of the face $A C C_{1} A_{1}$. Then $P^{\prime}$ lies on the projection ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,288 |
7.1. a) Given an arbitrary tetrahedron \(ABCD\). Prove that \((\overrightarrow{AB}, \overrightarrow{CD}) + (\overrightarrow{AC}, \overrightarrow{DB}) + (\overrightarrow{AD}, \overrightarrow{BC}) = 0\).
b) Prove that if in a tetrahedron two pairs of opposite edges are perpendicular, then the third pair of opposite edge... | 7.1. a) Let $\mathbf{a}=\overrightarrow{A B}, \mathbf{b}=\overrightarrow{B C}, \mathbf{c}=\overrightarrow{C D}$. Then $(\overrightarrow{A B}, \overrightarrow{C D})=$ $=(\mathbf{a}, \mathbf{c}),(\overrightarrow{A C}, \overrightarrow{D B})=(\mathbf{a}+\mathbf{b},-\mathbf{b}-\mathbf{c})=-(\mathbf{a}, \mathbf{b})-(\mathbf{... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,289 |
7.2. Prove that the sums of the squares of two opposite pairs of edges of a tetrahedron are equal if and only if the third pair of opposite edges is perpendicular. | 7.2. Let $\mathbf{a}=\overrightarrow{A B}, \mathbf{b}=\overrightarrow{B C}$ and $\mathbf{c}=\overrightarrow{C D}$. The equality $A C^{2}+$ $+B D^{2}=B C^{2}+A D^{2}$ means that $|\mathbf{a}+\mathbf{b}|^{2}+|\mathbf{b}+\mathbf{c}|^{2}=$ $=|\mathbf{b}|^{2}+|\mathbf{a}+\mathbf{b}+\mathbf{c}|^{2}$, i.e. (a, $\left.\mathbf{... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,290 |
7.3. The diagonal $A C_{1}$ of the rectangular parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$ is perpendicular to the plane $A_{1} B D$. Prove that this parallelepiped is a cube. | 7.3. Let $\mathbf{a}=\overrightarrow{A A_{1}}, \mathbf{b}=\overrightarrow{A B}$ and $\mathbf{c}=\overrightarrow{A D}$. Then $\overrightarrow{A C_{1}}=$ $=\mathbf{a}+\mathbf{b}+\mathbf{c}$, and therefore the vector $\mathbf{a}+\mathbf{b}+\mathbf{c}$ is perpendicular to the vectors $\mathbf{a}-\mathbf{b}, \mathbf{b}-\mat... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,291 |
7.4. In a regular truncated pyramid, $K$ is the midpoint of side $A B$ of the upper base, and $L$ is the midpoint of some side $C D$ of the lower base. Prove that the lengths of the projections of segments $A B$ and $C D$ onto line $K L$ are equal. | 7.4. If the vector x lies in the plane of the upper or lower base, we will denote by $R$ x the vector obtained from x by a $90^{\circ}$ rotation (in this plane) in the positive direction. Let $O_{1}$ and $O_{2}$ be the centers of the upper and lower bases; $\overrightarrow{O_{1} K}=\mathbf{a}$ and $\overrightarrow{O_{2... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,292 |
7.5. Given a trihedral angle with vertex $S$ and a point $N$. A sphere passing through points $S$ and $N$ intersects the edges of the trihedral angle at points $A, B$, and $C$. Prove that the centroids of triangle $A B C$ lie in the same plane. | 7.5. Let $O$ be the center of the sphere; $M$ be the center of mass of the triangle $ABC$; $\mathbf{u}=\overrightarrow{SO}$; $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ be unit vectors directed along the edges of the trihedral angle. Then $3 \overrightarrow{SM}=\overrightarrow{SA}+\overrightarrow{SB}+\overrightarrow{S... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,293 |
7.6. Prove that the sum of the distances from an internal point of a convex polyhedron to the planes of its faces does not depend on the position of the point if and only if the sum of the vectors of the unit external normals to the faces is zero. | 7.6. Let $\mathbf{n}_{1}, \ldots, \mathbf{n}_{k}$ be the unit outward normals to the faces; $M_{1}, \ldots, M_{h}$ be arbitrary points on these faces. The sum of the distances from an inner point $X$ of the polyhedron to all faces is
$$
\sum\left(\overrightarrow{X M}_{i}, \mathbf{n}_{i}\right)=\sum\left(\overrightarro... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,294 |
7.7. Prove that in an orthocentric tetrahedron, the center of mass is the midpoint of the segment connecting the orthocenter and the center of the circumscribed sphere.
## § 2. Scalar Product. Inequalities | 7.7. Let $O$ be the center of the circumscribed sphere of an orthocentric tetrahedron, $H$ its orthocenter, and $M$ its centroid. It is clear that $\overrightarrow{O M} = (\overrightarrow{O A} + \overrightarrow{O B} + \overrightarrow{O C} + \overrightarrow{O D}) / 4$. Therefore, it is sufficient to verify that $\overri... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,295 |
7.8. Prove that in space, it is impossible to choose more than 4 vectors, all angles between which are obtuse. | 7.8. First solution. Let several rays with a common origin $O$ be located in space, forming pairwise obtuse angles. Introduce a coordinate system, directing the axis $O . x$ along the first ray, and as the coordinate plane $O x y$ choosing the plane containing the first two rays. Each ray is defined by some vector $e$,... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,296 |
7.9. Prove that in space it is impossible to choose more than 6 vectors, all angles between which are acute. | 7.9. Suppose that the angles between the vectors $\mathbf{e}_{1}, \ldots, \mathbf{e}_{\text {}}$ are not acute. Direct the axis $O x$ along the vector $\mathbf{e}_{1}$. In the plane perpendicular to $\mathbf{e}_{1}$, there cannot be more than four vectors with angles between them that are not acute; together with the v... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,297 |
7.10. Prove that the sum of the cosines of the dihedral angles of a tetrahedron is positive and does not exceed 2. | 7.10. Let $e_{1}, e_{2}, e_{3}$ and $e_{4}$ be unit vectors perpendicular to the faces and directed outward; $\mathbf{n} = \mathbf{e}_{1} + \mathbf{e}_{2} + \mathbf{e}_{3} + \mathbf{e}_{4} ; s$ - the specified sum of cosines. Since $\left(\mathbf{e}_{i}, c_{j}\right) = -\cos \varphi_{i j}$, where $\varphi_{i j}$ is the... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,298 |
7.11. Inside the convex polyhedron $A_{1} \ldots A_{n}$, a point $A$ is taken, and inside the convex polyhedron $B_{1} \ldots B_{n}$, a point $B$ is taken. Prove that if $\angle A_{\imath} A A_{j} \leqslant \leqslant \angle B_{i} B B_{j}$ for all $i, j$, then in fact all these non-strict inequalities are equalities.
#... | 7.11. Let vectors $\mathbf{a}_{i}$ and $\mathbf{b}_{i}$ be collinear with rays $A A_{i}$ and $B B_{i}$ and have unit length. According to problem 7.16, there exist positive numbers $x_{1}, \ldots, x_{n}$, such that $x_{1} \mathbf{a}_{1}+\ldots+\ldots+x_{n} \mathbf{a}_{n}=0$. Consider the vector $\mathbf{b}=x_{1} \mathb... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,299 |
7.12. Points $O, A, B$ and $C$ do not lie in the same plane. Prove that point $X$ lies in the plane $ABC$ if and only if $\overrightarrow{O X}=p \overrightarrow{O A}+q \overrightarrow{O B}+r \overrightarrow{O C}$, where $p+q+r=1$. Moreover, if point $X$ belongs to triangle $ABC$, then $p: q: r=S_{\text {BXC }}: S_{C \t... | 7.12. Point $X$ lies in the plane $ABC$ if and only if $\overrightarrow{A X}=\lambda \overrightarrow{A B}+\mu \overrightarrow{A C}$, i.e., $\overrightarrow{O X}=\overrightarrow{O A}+\overrightarrow{A X}=\overrightarrow{O A}+\lambda \overrightarrow{A B}+$ $+\mu \overrightarrow{A C}=\overrightarrow{O A}+\lambda(\overrigh... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,300 |
7.13. On the edges $A B, A C$ and $A D$ of the tetrahedron $A B C D$, points $K, L$ and $M$ are taken such that $A B=\alpha A K, A C=\beta A L$ and $A D=\gamma A M$.
a) Prove that if $\gamma=\alpha+\beta+1$, then all planes $K L M$ contain a fixed point.
b) Prove that if $\beta=\alpha+1$ and $\gamma=\beta+1$, then al... | 7.13. Let $\mathbf{a}=\overrightarrow{A B}, \mathbf{b}=\overrightarrow{A C}$ and $\mathbf{c}=\overrightarrow{A D}$. Further, let $X$ be an arbitrary point and $\overrightarrow{A X}=\lambda \mathbf{a}+\mu \mathbf{b}+v \mathbf{c}$. The point $X$ belongs to the plane $K L M$ if $\overrightarrow{A X}=p \overrightarrow{A K}... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,301 |
7.14. Two regular pentagons $O A B C D$ and $O A_{1} B_{1} C_{1} D_{1}$ share a common vertex $O$ and do not lie in the same plane. Prove that the lines $A A_{1}, B B_{1}, C C_{1}$ and $D D_{1}$ are parallel to one plane. | 7.14. Let $\overrightarrow{O C}=\lambda \overrightarrow{O A}+\mu \overrightarrow{O B}$. Then, since the regular pentagons are similar, $\overrightarrow{O C_{1}}=\lambda \overrightarrow{O A_{1}}+\mu \overrightarrow{O B}_{1}$, and therefore, $\overrightarrow{C C}_{1}=$ $=\lambda \overrightarrow{A A}_{1}+\mu \overrightarr... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,302 |
7.15. a) Inside the tetrahedron $ABCD$, a point $O$ is taken. Prove that if $\alpha \overrightarrow{OA} + \beta \overrightarrow{OB} + \gamma \overrightarrow{OO} + \gamma \overrightarrow{OD} = \overrightarrow{0}_{2}$, then all numbers $\alpha, \beta, \gamma$, and $\delta$ are of the same sign.
b) From a point $O$ lying... | 7.15. a) In the equality $\alpha \overrightarrow{O A}+\beta \overrightarrow{O B}+\gamma \overrightarrow{O C}+\delta \overrightarrow{O D}=$ $=\overrightarrow{0}$, move all terms with negative numbers to the right side. If $p, q$ and $r$ are positive numbers, then the end of the vector $\overrightarrow{p O P}+\overrighta... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,303 |
7.16. Point $O$ lies inside the polyhedron $A_{1} \ldots \ldots A_{n}$. Prove that there exist such positive (and therefore all non-zero) numbers $x_{1} \ldots \ldots x_{n}$ that $x_{1} \overrightarrow{O A}_{1}+\ldots+x_{n} \overrightarrow{O A}_{n}=\overrightarrow{0}$.
## § 4. Various Problems | 7.16. Let the extension of the ray $O A_{i}$ beyond point $O$ intersect the polyhedron at point $M; \boldsymbol{P}$ - one of the vertices of the face containing point $M; QR$ - the edge of this face, intersecting the extension of the ray $M P$ beyond point $M$. Then $\overrightarrow{O M}=p \overrightarrow{O P}+q \overr... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,304 |
7.17. Let a, b, c, and d be unit vectors directed from the center of a regular tetrahedron to its vertices, and u be an arbitrary vector. Prove that $(\mathbf{a}, \mathbf{u}) \mathbf{a}+(\mathbf{b}, \mathbf{u}) \mathbf{b}+(\mathbf{c}, \mathbf{u}) \mathbf{c}+(\mathbf{d}, \mathbf{u}) \mathbf{d}=4 \mathbf{u} / 3$. | 7.17. First $p$ solution. Any vector $\mathbf{u}$ can be represented in the form $\mathbf{u}=\alpha \mathbf{a}+\beta \mathbf{b}+\gamma \mathbf{c}$; therefore, the proof is sufficient to conduct only for vectors $\mathbf{a}, \mathbf{b}$, and $\mathbf{c}$. Since the center of a regular tetrahedron divides its median in t... | proof | Algebra | proof | Yes | Yes | olympiads | false | 24,305 |
7.18. From point $M$, lying inside a regular tetrahedron, perpendiculars $M A_{i} (i=1,2$, $3,4)$ are dropped to its faces. Prove that $\overrightarrow{M A}_{1}+\overrightarrow{M A}_{2}+$ $+\overrightarrow{M A_{3}}+\overrightarrow{M A} A_{4}=4 \overrightarrow{M O} / 3$, where $O$ is the center of the tetrahedron. | 7.18. Drop perpendiculars $O B_{i}$ from point $O$ to the faces of the tetrahedron. Let $a_{i}$ be the unit vector in the direction of $\overrightarrow{O B}_{i}$. Then $\left(\overrightarrow{O M}, a_{i}\right) a_{i}+\overrightarrow{M A}_{i}=\overrightarrow{O B}_{i}$. Since the tetrahedron $B_{1} B_{2} B_{3} B_{i}$ is r... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,306 |
7.19. From a point $O$ lying inside a convex polyhedron, rays are drawn intersecting the planes of the faces and perpendicular to them. On these rays, from point $O$, vectors are laid off whose lengths are equal to the areas of the corresponding faces. Prove that the sum of these vectors is equal to zero. | 7.19. First solution. We will prove that the sum of the projections of all given vectors onto any line $l$ is zero. For this, consider the projection of the polyhedron onto a plane not perpendicular to the line $l$. The projection of the polyhedron is covered by the projections of its faces in two layers, as the faces ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,307 |
7.20. Given three mutually perpendicular lines, the distance between any two of which is a. Find the volume of the parallelepiped, the diagonal of which lies on one line, and the diagonals of two adjacent faces - on the other two lines. | 7.20. Consider the parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$. Let the diagonals of the faces sharing the edge $B C$ lie on given lines, and $A C$ is one of these diagonals. Then $B C_{1}$ is the second such diagonal, and $B_{1} D$ is the diagonal of the parallelepiped lying on the third given line. Introduce a r... | 9^{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,308 |
7.21. Let a, b, and c be arbitrary vectors. Prove that $|\mathbf{a}|+|\mathbf{b}|+|\mathbf{c}|+|\mathbf{a}+\mathbf{b}+\mathbf{c}| \geqslant$ $\geqslant|\mathbf{a}+\mathbf{b}|+|\mathbf{b}+\mathbf{c}|+|\mathbf{c}+\mathbf{a}|$.
## § 5. Vector Product
The vector product of two vectors a and b is a vector c, the length of... | 7.21. Let $a=|\mathrm{a}|, b=|\mathbf{b}|$ and $c=|\mathbf{c}|$. Let $x, y, z$ be the cosines of the angles between vectors $\mathbf{a}$ and $\mathbf{b}$, $\mathbf{b}$ and $\mathbf{c}$, $\mathbf{c}$ and $\mathbf{a}$. The difference between the left and right parts of the required inequality is $a+b+c+\sqrt{a^{2}+b^{2}+... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 24,309 |
7.22. Prove that
a) $[\mathbf{a}, \mathbf{b}]=-[\mathbf{b}, \mathbf{a}]$;
b) $[\lambda \mathbf{a}, \mu \mathbf{b}]=\lambda \mu[\mathbf{a}, \mathbf{b}]$
c) $[a, b+c]=[a, b]+[a, c]$, | 7.22. Assertions a) and b) easily follow from the definition.
c) First solution. Introduce a coordinate system $O x y z$, directing the axis $O x$ along the vector a. It can be verified that the vector product of vectors $\mathbf{2}=(a, 0,0)$ and $\mathbf{u}=(x, y, z)$ is the vector $(0, -a z, a y)$. Indeed, this vect... | proof | Algebra | proof | Yes | Yes | olympiads | false | 24,310 |
7.23. Vectors a and b have coordinates $\left(a_{1}, a_{2}, a_{3}\right)$ and ( $b_{1}, b_{2}, b_{3}$ ). Prove that the vector [a, b] has coordinates $\left(a_{2} b_{3}-a_{3} b_{2}, a_{3} b_{1}-a_{1} b_{3}, a_{1} b_{2}-a_{2} b_{1}\right)$. | 7.23. Let $\mathbf{a}=a_{1} \mathbf{e}_{1}+a_{2} \mathbf{e}_{2}+a_{3} \mathbf{e}_{3}$ and $\mathbf{b}=b_{1} \mathbf{e}_{1}+b_{2} \mathbf{e}_{2}+b_{3} \mathbf{e}_{3}$, where $\mathbf{e}_{1}, \mathbf{e}_{2}$ and $\mathbf{e}_{3}$ are unit vectors directed along the coordinate axes. To solve the problem, you can use the re... | proof | Algebra | proof | Yes | Yes | olympiads | false | 24,311 |
7.25. a) Prove that $[a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0$ (Jacobi identity).
b) Let point $O$ lie inside triangle $ABC$ and $\mathbf{a} = \overrightarrow{OA}$, $\mathbf{b} = \overrightarrow{OB}$, and $\mathbf{c} = \overrightarrow{OC}$. Prove that the Jacobi identity for vectors $\mathbf{a}$, $\mathbf{b}$, and ... | 7.25. a) According to problem 7.24, \(\mathbf{a} \quad [a,[b, c]]=b(c, a)-c(a, b)\), \([\mathbf{b},[\mathbf{c}, \mathbf{a}]]=\mathbf{c}(\mathbf{a}, \mathbf{b})-\mathbf{a}(\mathbf{b}, \mathbf{c})\) and \([\mathbf{c},[\mathrm{a}, \mathbf{b}]]=\mathrm{a}(\mathbf{b}, \mathbf{c})-b(a, c)\). By adding these equalities, we ob... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,313 |
7.26. The angles at the vertices of a spatial hexagon are right angles, and it has no parallel sides. Prove that the common perpendiculars to pairs of opposite sides of the hexagon are perpendicular to one line. | 7.26. Let a, b and c be vectors defining three non-adjacent sides of a hexagon; $a_{1}, b_{1}$ and $c_{1}$ - vectors of the opposite sides. Since vector $\mathbf{a}_{1}$ is perpendicular to vectors $\mathbf{b}$ and c, then $\mathbf{a}_{1}=\lambda[\mathbf{b}, \mathbf{c}]$. Therefore, the common perpendicular to vectors ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,314 |
7.27. Prove using the vector product the statement of problem 7.19 for the tetrahedron $A B C D$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 7.27. Let $\mathbf{a}=\overrightarrow{D A}, \mathbf{b}=\overrightarrow{D B}$ and $\mathbf{c}=\overrightarrow{D C} . \quad$ The statement of the problem is equivalent to the equality $[a, b]+[b, c]+[c, a]+[b-c, a-c]=0$. | Calculus | math-word-problem | Yes | Yes | olympiads | false | 24,315 | |
7.28. a) Prove that the planes passing through the bisectors of the faces of a trihedral angle $SABC$ and perpendicular to the planes of these faces intersect along a single line, and that this line is defined by the vector $[\mathbf{a}, \mathbf{b}]+[\mathbf{b}, \mathbf{c}]+[\mathbf{c}, \mathbf{a}]$, where $\mathbf{a},... | 7.28. a) Let's prove, for example, that the vector $[a, b]+[b, c]+[c, a]$ lies in the plane П, passing through the bisector of the edge $SAB$ and perpendicular to this edge. The plane II is perpendicular to the vector $a - b$, so it contains the vector $[c, a - b]$. In addition, the plane П contains the vector $[a, b]$... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,316 |
7.30. Prove that vectors with coordinates $\left(a_{1}, a_{2}, a_{3}\right),\left(b_{1}, b_{2}, b_{3}\right)$ and $\left(c_{1}, c_{2}, c_{3}\right)$ are parallel to one plane if and only if
$$
a_{1} b_{2} c_{3}+a_{2} b_{3} c_{1}+a_{3} b_{1} c_{2}=a_{1} b_{3} c_{2}+a_{2} b_{1} c_{3}+a_{3} b_{2} c_{1}
$$
For those fami... | 7.30. Three vectors are coplanar if and only if their scalar triple product is zero. Using the formula from problem 7.23, we get that the scalar triple product of the given vectors is
$$
\left(a_{2} b_{3}-a_{8} b_{2}\right) c_{1}+\left(a_{3} b_{1}-a_{1} b_{3}\right) c_{2}+\left(a_{1} b_{2}-a_{2} b_{1}\right) c_{3}
$$ | proof | Algebra | proof | Yes | Yes | olympiads | false | 24,318 |
7.31. Given a tetrahedron and a point $N$. Through each edge of the tetrahedron, a plane is drawn parallel to the segment connecting point $N$ with the midpoint of the opposite edge. Prove that all six of these planes intersect at one point. | 7.31. Let $M$ be the center of mass of the tetrahedron, $A$ be the midpoint of the edge through which the plane П passes, $B$ be the midpoint of the opposite edge, and $N^{\prime}$ be the point symmetric to $N$ with respect to point $M$. Since point $M$ is the midpoint of segment $A B$ (see problem 14.3), then $A N^{\p... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,319 |
7.32. a) Through the midpoint of each edge of a tetrahedron, a plane perpendicular to the opposite edge is drawn. Prove that all five of these planes intersect at one point (the Monge point).
b) Prove that if the Monge point lies in the plane of any face of the tetrahedron, then the foot of the altitude dropped to thi... | 7.32. a) Let $A$ be the midpoint of edge $a$, and $B$ be the midpoint of the opposite edge $b$. Further, let $M$ be the center of mass of the tetrahedron, $O$ be the center of its circumscribed sphere, and $O^{\prime}$ be the point symmetric to $O$ with respect to point $M$. Since point $M$ is the midpoint of segment $... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,320 |
7.34. Given two intersecting planes and a sphere tangent to them. All spheres tangent to these planes and the given sphere are considered. Find the geometric locus of the points of tangency of the spheres. | 7.34. Consider first,
Fig. 54 that both the given sphere and the sphere touching it are located in the same dihedral angle between the given planes. Then both spheres are symmetric relative... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,322 |
7.36. In a convex 5-sided pyramid $S A B C D E$, the lateral edges are equal and the dihedral angles at the lateral edges are equal. Prove that this pyramid is regular. | 7.36. Let $O$ be the projection of vertex $S$ onto the plane of the base of the pyramid. Since the vertices of the base of the pyramid are equidistant from point $S$, they are also equidistant from point $O$, which means they lie on a circle with center $O$. Now, let us assume that $B C = A E$. Let $M$ be the midpoint ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,324 |
7.37. What is the maximum number of planes of symmetry that a spatial figure consisting of three pairwise non-parallel lines can have?
Symmetry with respect to a line $l$ is a transformation of space that maps a point $X$ to a point $X^{\prime}$ such that the line $l$ passes through the midpoint of the segment $X X^{\... | 7.37. Let $P$ be a plane of symmetry of a figure consisting of three pairwise non-parallel lines. There are only two possible cases: 1) each given line is symmetric relative to $P$; 2) one line is symmetric relative to $P$, and the other two lines are symmetric to each other.
In the first case, either one line is perp... | 9 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,325 |
7.38. Prove that under symmetry with respect to a line defined by vector $\mathbf{b}$, vector $\mathbf{a}$ transforms into the vector
$$
2 \mathbf{b} \frac{(\mathbf{a}, \mathbf{b})}{(\mathbf{b}, \mathbf{b})} - \mathbf{a}
$$ | 7.38. Let $\mathbf{a}^{\prime}$ be the image of vector $\mathbf{a}$ under the considered symmetry; and $\mathbf{u}$ be the projection of vector $\mathbf{a}$ onto the given line. Then $\mathbf{a}^{\prime}+\mathbf{a}=2 \mathbf{u} \quad$ and $\mathbf{n}=\mathbf{b} \frac{(\mathbf{a}, \mathbf{b})}{(\mathbf{b}, \mathbf{b})}$... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,326 |
7.40. Prove that no body in space can have a non-zero even number of axes of symmetry.
## § 7. Homothety
Homothety is a transformation of space that maps a point $X$ to a point $X^{\prime}$, with the property that $\overrightarrow{O X^{\prime}}=k \overrightarrow{O X}$ (the point $O$ and the number $k$ are fixed). The... | 7.40. Let's fix some axis of symmetry $l$. We will prove that the other axes of symmetry are divided into pairs. First, note that under reflection in the line $l$, an axis of symmetry is transformed into an axis of symmetry. If an axis of symmetry $l^{\prime}$ does not intersect $l$ or intersects it at an angle other t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,328 |
7.41. Let $r$ and $R$ be the radii of the inscribed and circumscribed spheres of a tetrahedron. Prove that $R \geqslant 3 r$. | 7.41. Let $M$ be the center of mass of the tetrahedron. Under a homothety with center $M$ and coefficient $-1 / 3$, the vertices of the tetrahedron are transformed into the centers of mass of its faces, and thus, the circumscribed sphere of the tetrahedron is transformed into a sphere of radius $R / 3$, intersecting al... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,329 |
7.42. In the plane of a lateral face of a regular quadrilateral pyramid, an arbitrary figure Ф is taken. Let $\Phi_{1}$ be the projection of $Ф$ onto the base of the pyramid, and $\Phi_{2}$ be the projection of $\Phi_{1}$ onto the adjacent lateral face. Prove that the figures Ф and $\Phi_{2}$ are similar. | 7.42. Let $S A B$ be the original face of the pyramid $S A B C D$, and $S A D$ be the second face. Rotate the planes of these faces around the lines $A B$ and $A D$ so that they coincide with the plane of the base (the rotation is performed towards the smaller angle). Consider a coordinate system with the origin at poi... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,330 |
7.43. Prove that inside any convex polyhedron $M$, two polyhedra similar to it with a coefficient of $1 / 2$ can be placed so that they do not intersect. | 7.43. Let $A$ and $B$ be the points of the polyhedron that are farthest from each other. Then the images of the polyhedron $M$ under homotheties with centers at $A$ and $B$ and coefficient $1 / 2$ define the desired arrangement. Indeed, these polyhedra do not intersect, as they are located on opposite sides of the plan... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,331 |
7.44. Prove that a convex polyhedron cannot be covered by three polyhedra homothetic to it with a coefficient $k$, where $0<k<1$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 7.44. Consider a convex polyhedron $M$ and any three polyhedra $M_{1}, M_{2}$, and $M_{3}$, homothetic to it with a coefficient $k$. Let $O_{1}, O_{2}$, and $O_{3}$ be the centers of the corresponding homotheties. It is clear that if $A$ is the point of the polyhedron $M$ farthest from the plane containing the points $... | Calculus | math-word-problem | Yes | Yes | olympiads | false | 24,332 | |
7.45. On a plane, there is a triangle $ABC$. Find the geometric locus of such points $D$ in space that the segment $OM$, where $O$ is the center of the circumscribed sphere of the tetrahedron $ABCD$, and $M$ is the center of mass of this tetrahedron, is perpendicular to the plane $ADM$.
## § 8. Rotation. Compositions ... | 7.45. Let $N$ be the centroid of triangle $ABC$. Under a homothety with center $N$ and coefficient $1/4$, point $D$ transforms into $M$. We need to prove that point $M$ lies in the plane $\Pi$, passing through the center $O_{1}$ of the circumcircle of triangle $ABC$ and perpendicular to its median $AK$. Indeed, $OM \pe... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,333 |
7.46. Let $A_{i}^{\prime}$ and $A_{i}^{\prime \prime}$ be the projections of the vertices of the tetrahedron $A_{1} A_{2} A_{3} A_{4}$ onto the planes $\Pi^{\prime}$ and $\Pi^{\prime \prime}$. Prove that one of these planes can be moved in space so that the 4 lines $A_{i}^{\prime} A_{i}^{\prime \prime}$ become parallel... | 7.46. We can assume that the planes $\Pi^{\prime}$ and $\Pi^{\prime \prime}$ are not parallel, otherwise the statement is obvious. Let $l$ be the line of intersection of these planes, and $A_{i}^{*}$ be the point of intersection of the line $l$ with the plane $A_{i} A_{i}^{\prime} A_{i}^{\prime \prime}$. The plane $A_{... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,334 |
7.47. Prove that the composition of reflections with respect to two planes intersecting along a line $l$ is a rotation about the line $l$, and that the angle of this rotation is twice the angle of the rotation about the line $l$ that maps the first plane to the second. | 7.47. Consider a section by a plane perpendicular to the line $l$. The required statement then follows from the corresponding planimetric statement about the composition of two axial symmetries (see Prasolov, 8.14, b). | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,335 |
7.48. Prove that the composition of a symmetry with respect to a point $O$ and a rotation about a line $l$ passing through $O$ is also the composition of some rotation about the line $l$ and a symmetry with respect to a plane $\Pi$ passing through the point $O$ and perpendicular to $l$.
Definition. A movement is a tra... | 7.48. Let $A$ be some point, $B$ be its image under reflection about point $O$, $C$ be the image of point $B$ under rotation by angle $\varphi$ about line $l$, and $D$ be the image of point $C$ under reflection about plane $P$. Then point $D$ is the image of point $A$ under rotation by angle $180^{\circ}+\varphi$ about... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,336 |
7.49. a) Guess that any motion of space is a composition of no more than four symmetries with respect to planes.
b) Prove that any motion of space having a fixed point \( O \) is a composition of no more than three symmetries with respect to planes.
Definition. A motion that is a composition of an even number of symm... | 7.49. a) Let $T$ be some transformation that maps point $A$ to point $B$, different from $A$; $S$ - the symmetry with respect to the plane $\Pi$ passing through the midpoint of segment $AB$ and perpendicular to it. Then $S \circ T(A) = S(B) = A$, i.e., $A$ is a fixed point of the transformation $S \circ T$. Moreover, i... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,337 |
7.50. a) Prove that any motion of the first kind, having a fixed point, is a rotation about some axis.
b) Prove that any motion of the second kind, having a fixed point, is a composition of a rotation about some axis (possibly by a zero angle) and a reflection in a plane perpendicular to this axis. | 7.50. a) According to problem $7.49,5$, any motion of the first kind, having a fixed point, is a composition of two symmetries with respect to planes, i.e., a rotation about the line where these planes intersect (see problem 7.47$)$.
b) Let $T$ be the given motion of the second kind, and $I$ be the symmetry with respe... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,338 |
7.51. A ball lying in the corner of a rectangular box rolls along the side of the box to another corner, with the same point of the ball always touching the side wall. From the second corner, the ball rolls to the third, then to the fourth, and finally returns to the initial corner. During this process, a point $X$ on ... | 7.51. After rolling, any point $A$ on the surface of the pair moves to the point $T(A)$, where $T$ is a first-order motion having a fixed point - the center of the sphere. According to problem $7.50, \mathrm{a}$, the transformation $T$ is a rotation about some axis $l$. Consequently, the points $X_{1}, X_{2}$ and $X_{3... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,339 |
7.52. A ray of light enters a trihedral angle and is reflected from all three faces once. Prove that in this case, it simply changes direction. | 7.52. Let us associate a rectangular coordinate system Oxyz with the given trihedral angle. A ray of light moving in the direction of the vector ( $x, y, z$ ) will, after reflection from the plane $Oxy$, be traveling in the direction of the vector ( $x, y, -z$ ). Thus, after reflecting from all three faces, it will be ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,340 |
7.53. A light ray falls on a flat mirror at an angle $\alpha$. The mirror is rotated by an angle $\beta$ around the projection of the ray on the mirror. By what angle will the reflected ray deviate in this case? | 7.53. Let $B$ be the point of incidence of the ray on the mirror; $A$ - a point on the ray, different from $B$; $K$ and $L$ - the projections of $A$ on the mirror in its initial and new positions; $A_{1}$ and $A_{2}$ - points symmetric to $A$ relative to the initial positions of the mirror. The required angle is equal ... | \sin(\varphi/2)=\sin\alpha\sin\beta | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,341 |
7.54. The plane $\Pi$ passes through the vertex of the cone perpendicular to its axis; point $A$ lies in the plane $\Pi$. Let $M$ be a point on the cone such that a light ray going from $A$ to $M$, after being reflected off the surface of the cone, becomes parallel to the plane $\Pi$. Find the geometric locus of the pr... | 7.54. Let's introduce a coordinate system with the origin $O$ at the vertex of the cone and the axis $O x$ passing through the point $A$ (Fig. 55). Let $\overrightarrow{O M}=(x, y, z)$, then $\overrightarrow{A M}=(x-a, y, z)$, where $a=A O$. If
=0 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,342 |
8.1. a) The areas of all faces of a convex polyhedron are equal. Prove that the sum of the distances from its internal point to the planes of the faces does not depend on the position of the point.
b) The heights of a tetrahedron are \( h_{1}, h_{2}, h_{3} \) and \( h_{4} \); \( d_{1}, d_{2}, d_{3} \) and \( d_{4} \) ... | 8.1. a) Let \( V \) be the volume of a polyhedron, \( S \) be the area of its face, and \( h_{i} \) be the distance from a point \( X \), located inside the polyhedron, to the \( i \)-th face. By cutting the polyhedron into pyramids with apex \( X \) and bases being its faces, we get \( V = S h_{1} / 3 + \ldots + S h_{... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,343 |
8.2. a) Prove that a convex polyhedron cannot have exactly 7 edges.
b) Prove that a convex polyhedron can have any number of edges greater than 5 and different from 7. | 8.2. a) Suppose a polyhedron has only triangular faces, and their number is $\Gamma$. Then the number of edges of the polyhedron is $3 \Gamma / 2$, i.e., it is divisible by 3. If the polyhedron has a face with more than three sides, then the number of its edges is at least 8.
b) Let $n \geqslant 3$. Then $2 n$ edges h... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,344 |
8.3. A plane intersecting a circumscribed polyhedron divides it into two parts with volumes $V_{1}$ and $V_{2}$; it divides its surface into two parts with areas $S_{1}$ and $S_{2}$. Prove that $V_{1}: S_{1}=V_{2}: S_{2}$ if and only if the plane passes through the center of the inscribed sphere. | 8.3. For definiteness, assume that the center $O$ of the inscribed sphere belongs to the part of the polyhedron with volume $V_{1}$.
Consider a pyramid with vertex $O$, the base of which is the section of the polyhedron by the given plane. Let $V$ be the volume of this pyramid. Then $V_{1}-V=r S_{1} / 3$ and $V_{2}+V=... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,345 |
8.4. In a convex polyhedron, an even number of edges emanate from each vertex. Prove that any section of it by a plane, not containing vertices, is a polygon with an even number of sides. | 8.4. The number of lines connecting the vertices of a polyhedron is finite, so the given plane can be slightly moved so that during the movement it does not intersect any vertex and in its new position it is not parallel to any line connecting the vertices of the polyhedron. We will shift this plane parallel until it e... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,346 |
8.5. Prove that if any vertex of a convex polyhedron is connected by edges to all other vertices, then this polyhedron is a tetrahedron. | 8.5. If any vertex of a polyhedron is connected by edges to all other vertices, then all faces are triangular.
Consider two faces $A B C$ and $A B D$ with a common edge $A B$. Suppose the polyhedron is not a tetrahedron. Then it has another vertex $E$, different from the vertices of the considered faces. Since points ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,347 |
8.6. What is the maximum number of sides a coin can have as a projection of a convex polyhedron with $n$ faces? | 8.6. Answer: $2 n-4$. First, let's prove that the projection of a convex polyhedron with $n$ faces can have $2 n-4$ sides. Cut off the edge $C D$ of a regular tetrahedron $A B C D$ with a prismatic surface, the lateral edges of which are parallel to $C D$ (Fig. 57). The projection of the resulting polyhedron with $n$ f... | 2n-4 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,348 |
8.7. Each face of a convex polyhedron has a center of symmetry.
a) Prove that it can be dissected into parallelepipeds.
b) Prove that it has a center of symmetry. | 8.7. a) Let's take an arbitrary face of the given polyhedron and its edge $r_{1}$. Since the face is centrally symmetric, it contains an edge $r_{2}$, equal and parallel to $r_{1}$. The face adjacent to the edge $r_{2}$ also has an edge $r_{3}$, equal and parallel to $r_{1}$, and so on. In the end, we get a "belt" of f... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,349 |
8.8. Prove that if all faces of a convex polyhedron are parallelograms, then their number is the product of two consecutive natural numbers.
## §2. Criteria for Non-inscribability and Non-circumscribability of Polyhedra | 8.8. Let's use the solution from problem 8.7. Each "belt" divides the surface of the polyhedron into two "caps." Since the polyhedron is centrally symmetric, both "caps" contain an equal number of faces. Therefore, another "belt" cannot lie entirely within one "cap," i.e., any two "belts" intersect, and they intersect ... | (k-1)k | Geometry | proof | Yes | Yes | olympiads | false | 24,350 |
8.9. Some faces of a convex polyhedron are painted black, the others are white, and no two black faces share an edge. Prove that if the area of the black faces is greater than the area of the white faces, then a sphere cannot be inscribed in this polyhedron.
Can the area of the black faces equal the area of the white ... | 8.9. We will prove that if no two black faces of a circumscribed polyhedron share a common edge, then the area of the black faces does not exceed the area of the white ones. In the proof, we will use the fact that if two faces of a polyhedron touch a sphere at points $O_{1}$ and $O_{2}$, and $AB$ is their common edge, ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,351 |
8.10. Some faces of a convex polyhedron are painted black, and the rest are painted white, with no two black faces sharing an edge. Prove that if there are more black faces than half, then it is impossible to inscribe a sphere in this polyhedron. | 8.10. We will prove that if a sphere is inscribed in a polyhedron and no two black faces share a common edge, then the number of black faces is not greater than the number of white faces. In the proof, we will use the fact that if $O_{1}$ and $O_{2}$ are the points of tangency of the sphere with the faces sharing a com... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,352 |
8.11. Some vertices of a convex polyhedron are painted black, and the rest are white, with at least one end of each edge being white. Prove that if there are more black vertices than half, then this polyhedron cannot be inscribed in a sphere. | 8.11. Suppose a polyhedron is inscribed in a sphere and no two black vertices are connected by an edge, then the number of black vertices is not greater than the number of white vertices.
Let the planes tangent to the sphere with center $O$ at points $\boldsymbol{P}$ and $Q$ intersect along the line $A B$. Then any tw... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,353 |
8.14. Prove that $\mathrm{B}-\mathrm{P}+\Gamma=2$ (Euler's formula).
The translation is provided as requested, maintaining the original text's line breaks and format. | 8.14. First solution. Let $M$ be the projection of a polyhedron onto a plane that is not perpendicular to any of its faces; with such a projection, all faces are projected into polygons. Edges that project into sides of $M$ divide the polyhedron into two parts. Consider the projection of one of these parts (Fig. 58). L... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,355 |
8.15. a) Prove that the sum of the angles of all faces of a convex polyhedron is equal to twice the sum of the angles of a planar polygon with the same number of vertices,
b) Consider for each vertex of a convex polyhedron the difference between \(2 \pi\) and the sum of the planar angles converging at this vertex. Prov... | 8.15. Let $\Sigma$ be the sum of all the angles of the faces of a convex polyhedron. In part (a), it is required to prove that $\Sigma=2 (V - 2) \pi$, and in part (b), it is required to prove that $2 B \pi - \Sigma = 4 \pi$. Therefore, both problems are equivalent.
If a face contains $k$ edges, then the sum of its ang... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,356 |
8.16. Let $\Gamma_{k}$ be the number of $k$-sided faces of an arbitrary polyhedron, $\mathrm{B}_{h}$ be the number of its vertices where $k$ edges meet. Prove that $2 \mathrm{P}=3 \mathrm{~B}_{3}+$ $+4 \mathrm{~B}_{4}+5 \mathrm{~B}_{5} \ldots=3 \Gamma_{3}+4 \Gamma_{4}+5 \Gamma_{5}+\ldots$ | 8.16. Each edge can be associated with two vertices connected by it. In this case, a vertex where $k$ edges meet appears $k$ times. $\quad$ Therefore, $2 \mathrm{P}=3 \mathrm{~B}_{3}+4 \mathrm{~B}_{4}+5 \mathrm{~B}_{5}+\ldots$ On the other hand, each edge can be associated with two faces adjacent to it. In this case, a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,357 |
8.17. a) Prove that in any convex polyhedron, there will be either a triangular face or a trihedral angle.
b) Prove that for any convex polyhedron, the sum of the number of triangular faces and the number of trihedral angles is not less than 8. | 8.17. a) Suppose that a certain market polyhedron has neither triangular faces nor three-edged vertices. Then $B_{3}=\Gamma_{3}=0$, and therefore, $2 \mathrm{P}=4 \Gamma_{4}+5 \Gamma_{5}+\ldots \geqslant 4 \Gamma$ and $2 \mathrm{P}=4 \mathrm{~B}_{4}+5 \mathrm{~B}_{5}+\ldots \geqslant 4 \mathrm{~B}$ (see problem 8.16). ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,358 |
8.18. Prove that in any convex polyhedron, there is a face with fewer than six sides. | 8.18. Suppose that any face of a certain convex polyhedron has at least six sides. Then $\Gamma_{3}=\Gamma_{\overline{4}}=$ $=\Gamma_{5}=0$, and therefore $2 P=6 \Gamma_{6}+7 \Gamma_{8}+\ldots \geqslant 6 \Gamma$ (see problem 8.16), i.e., $P \geqslant 3$. Moreover, for any polyhedron $2 P=3 B_{3}+4 B_{\mathbf{4}}+\ldot... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,359 |
8.20. Given a convex polyhedron, all faces of which have 5, 6, or 7 sides, and all polyhedral angles are trihedral. Prove that the number of pentagonal faces is 12 more than the number of heptagonal faces.
## § 4. Traversals of Polyhedra | 8.20. Let $a, b$ and $c$ be the numbers of faces with 5, 6 and 7 sides respectively. Then $\mathrm{P}=(5 a+6 b+7 c) / 2, \Gamma=a+$ $+b+c$ and, since by condition three edges emanate from each vertex, $V=(5 a+6 b+7 c) / 3$. Multiplying all these expressions by 6 and substituting them into the formula $6(\mathrm{~V}+\Ga... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,361 |
8.21. A planet has the shape of a convex polyhedron, and cities are located at its vertices, with each edge being a road. Two roads are closed for repair. Prove that it is possible to travel from any city to any other city via the remaining roads. | 8.21. Let $A$ and $B$ be given cities. First, we will prove that it was possible to travel from $A$ to $B$ before the closure of two roads for repair. Consider the projection of the polyhedron onto some line that is not perpendicular to any of its edges (in such a projection, the vertices of the polyhedron do not merge... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 24,362 |
8.22. A direction is indicated on each edge of a convex polyhedron; at the same time, at least one edge enters and at least one edge exits from each vertex. Prove that there exist two faces that can be traversed, moving in accordance with the introduced orientation of the edges. | 8.22. Let's exit from some vertex of the polyhedron and walk along the edges in the direction indicated on them until we find a vertex that we have already visited before. The path from the first passage through this vertex to the second forms a "loop" that divides the polyhedron into two parts. Consider one of them an... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,363 |
8.24. A plane intersects the sides of a spatial polygon $A_{1} \ldots A_{n}$ (or their extensions) at points $B_{1}, \ldots, B_{n}$; the point $B_{i}$ lies on the line $A_{i} A_{i+1}$. Prove that $\frac{A_{1} B_{1}}{A_{2} B_{1}} \cdot \frac{A_{2} B_{2}}{A_{3} B_{2}} \cdot \ldots \cdot \frac{A_{n} B_{n}}{A_{1} B_{n}}=1$... | 8.24. Consider the projection onto a line perpendicular to a given plane. All points $B_{i}$ are projected onto a single point $B$, while points $A_{1}, \ldots, A_{n}$ are projected onto $C_{1}, \ldots, C_{n}$. Since the ratios of segments lying on the same line are preserved under projection, we have
$$
\frac{A_{1} B... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,365 |
8.25. Given four lines, no three of which are parallel to the same plane. Prove that there exists a spatial quadrilateral, the sides of which are parallel to these lines, and the ratio of the sides parallel to the corresponding lines is the same for all such quadrilaterals. | 8.25. Let a, b, c, and d be vectors parallel to given lines. Since any three vectors in space, not lying in the same plane, form a basis, there exist such non-zero numbers $\alpha, \beta$, and $\gamma$ that $\alpha \mathbf{a}+\beta \mathbf{b}+\gamma \mathbf{c}+\mathbf{d}=\mathbf{0}$. The vectors $\alpha \mathrm{a}, \be... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,366 |
8.26. a) How many pairwise non-equal spatial quadrilaterals exist with one and the same set of side vectors?
b) Prove that the volumes of all tetrahedra defined by these spatial quadrilaterals are equal. | 8.26. a) We fix one of the vectors of the sides. It can be followed by any of the three remaining vectors, and then by any of the two remaining ones. Therefore, there are exactly 6 different quadrilaterals.
b) Let $\mathbf{a}, \mathbf{b}, \mathbf{c}$, and $\mathbf{d}$ be the given vectors of the sides. Consider the pa... | 6 | Geometry | proof | Yes | Yes | olympiads | false | 24,367 |
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