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12.12. Two planes, parallel to a given plane $\Pi$, intersect the edges of a trihedral angle at points $A, B, C$ and $A_{1}, B_{1}, C_{1}$ (points denoted by the same letters lie on the same edge). Find the geometric locus of the points of intersection of the planes $A B C_{1}$, $A B_{1} C$ and $A_{1} B C$. | 12.12. The intersection of the planes $A B C_{1}$ and $A B_{1} C$ is the line $A M$, where $M$ is the point of intersection of the diagonals $B C_{i}$ in $B_{1} C$ of the face $B C C_{1} B_{1}$. The point $M$ lies on the line $l$, passing through the midpoints of the segments $B C$ and $B_{1} C_{1}$ and the vertex of t... | m | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,476 |
12.13. Find the geometric locus of points, the sum of the distances from which to the planes of the faces of a given trihedral angle is constant. | 12.13. On the edges of the given trihedral angle with vertex $O$, let's choose points $A, B$, and $C$, the distances from which to the planes of the faces are equal to a given number $a$. The area $S$ of each of the triangles $O A B, O B C$, and $O C A$ is equal to $3 V / a$, where $V$ is the volume of the tetrahedron ... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,477 |
12.14. A circle of radius $R$ touches the edges of a given trihedral angle, all plane angles of which are right angles. Find the geometric locus of all possible positions of its center.
## § 3. Various Loci | 12.14. Introduce a rectangular coordinate system, directing its axes along the edges of the given trihedral angle. Let $O_{i}$ be the center of the circle; П - the plane of the circle; $\alpha, \beta$ and $\gamma$ - the angles between the plane П and the coordinate planes. Since the distance from the point $O_{1}$ to t... | OO_{1}=\sqrt{2}R | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,478 |
12.15. On a plane, there is an acute-angled triangle $A B C$. Find the geometric locus of projections on this plane of all points $X$, for which the triangles $A B X, B C X$ and $C A X$ are acute-angled. | 12.15. If angles $X A B$ and $X B A$ are acute, then point $X$ lies between the planes passing through points $A$ and $B$ perpendicular to line $A B$ (for points $X$ not lying on segment $A B$, the converse is also true). Therefore, the desired locus of points lies inside (but not on the sides) of a convex pentagon, th... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,479 |
12.16. In the tetrahedron $ABCD$, the height $DP$ is the smallest. Prove that the point $P$ lies on the triangle, the sides of which pass through the vertices of the triangle $ABC$ parallel to its opposite sides. | 12.16. It is sufficient to verify that point $P$ is no farther from each side of triangle $A B C$ than its opposite vertex. Let's prove this statement, for example, for side $B C$. Consider the projection of this onto a plane perpendicular to the line $B C$; points $B$ and $C$ under this projection map to a single poin... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,480 |
12.17. Given a cube. The vertices of a convex polyhedron lie on its edges, with exactly one vertex on each edge. Find the set of points belonging to all such polyhedra. | 12.17. Each considered polyhedron is obtained from the given cube $A B C D A_{1} B_{1} C_{1} D_{1}$ by cutting off tetrahedra from each of its vertices. The tetrahedron cut off from vertex $A$ is contained in the tetrahedron $A A_{1} B D$. Thus, if tetrahedra are cut off from the cube, each of which is defined by three... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,481 |
12.18. Given a flat quadrilateral $A B C D$. Find the geometric locus of points $M$ such that the lateral surface of the pyramid $M A B C D$ can be intersected by a plane in such a way that the section is: a) a rectangle; b) a rhombus. | 12.18. Let $P$ and $Q$ be the points of intersection of the extensions of the opposite sides of quadrilateral $ABCD$. Then $MP$ and $MQ$ are the lines of intersection of the planes of the opposite faces of the pyramid $MABCD$. The intersection of a pair of planes intersecting along a line $l$ represents two parallel li... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,482 |
12.19. A broken line $a$ originates from the origin, and any plane parallel to a coordinate plane intersects the broken line in no more than one point. Find the geometric locus of the endpoints of such broken lines.
## § 4. Construction on Images | 12.19. Let $(x, y, z)$ be the coordinates of the end of the broken line, and $\left(x_{i}, y_{i}, z_{i}\right)$ be the coordinates of the $i$-th segment vector of the broken line. From the problem statement, it follows that the numbers $x_{i}, y_{i}$, and $z_{i}$ are non-zero and have the same sign as the numbers $x, y... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,483 |
12.21. Given the projection image of a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ onto a certain plane with points $P, Q, R$ marked on the edges $A A_{1}, B C$ and $C_{1} D_{1}$. Construct the section of the cube by the plane $P Q R$ on this image. | 12.21. In this case, the considerations used in the previous problem are no longer sufficient. Therefore, we will first construct the point $M$ of intersection of the line $P R$ and the plane of the base $A B C D$ as follows. The projection of point $P$ onto the plane of the base $A B C D$ is point $A$, and the project... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,485 |
12.22. a) Given the projection on a certain plane of a trihedral angle $O a b c$, on the faces $O b c$ and $O a c$ of which points $A$ and $B$ are marked. Construct on this image the point of intersection of the line $A B$ with the plane $O a b$.
b) Given the image of the projection on a certain plane of a trihedral a... | 12.22. a) Let $P$ be a point on the edge $c$. The plane $PAB$ intersects the edges $a$ and $b$ at the same points where the lines $PB$ and $PA$ intersect them, respectively. Let these points be $A_{1}$ and $B_{1}$. The desired point is the intersection of the lines $A_{1}B_{1}$ and $AB$ (Fig. 91).
b) Let points $A$, $... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,486 |
12.25. On a plane, six segments are given, equal to the edges of the tetrahedron $A B C D$. Construct a segment $r_{\text {r }}$ equal to the height $h_{a}$ of this tetrahedron. | 12.25. Drop a perpendicular \( A A_{\mathrm{i}} \) from vertex \( A \) of the tetrahedron \( A B C D \) to the plane \( B C D \), and drop perpendiculars \( A B^{\prime}, A C^{\prime} \), and \( A D^{\prime} \) to the lines \( C D, B D \), and \( B C \). By the theorem of three perpendiculars, \( A_{1} B^{\prime} \perp... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,489 |
12.26. On a plane, three angles equal to the face angles $\alpha, \beta$, and $\gamma$ of a trihedral angle are drawn. Construct on the same plane an angle equal to the dihedral angle opposite the face angle $\alpha$. | 12.26. Consider a trihedral angle with plane angles $\alpha, \beta$, and $\gamma$. Let $O$ be its vertex. On the edge opposite to angle $\alpha$, take a point $A$ and draw perpendiculars $A B$ and $A C$ to the edge $O A$ in the planes of the faces. This construction can be performed on the given plane for the developme... | BA^{\}C | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,490 |
12.27. Given a sphere. Using only a compass and a ruler, construct a segment on the plane equal to the radius of this sphere,
| 12.27. Construct with the help of a compass on a given sphere a circle with some center $A$ and take three arbitrary points on it. Using a compass, it is easy to construct on the plane a triangle equal to the triangle with vertices at those points. Then construct the circumscribed circle of that triangle and thereby fi... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,491 |
13.1. Prove that in any tetrahedron, there is an edge that forms acute angles with the edges emanating from its endpoints. | 13.1. If $AB$ is the largest side of triangle $ABC$, then $\angle C \geqslant \angle A$ and $\angle C \geqslant \angle B$; therefore, both angles $A$ and $B$ must be acute. Thus, to the largest edge of the tetrahedron, only acute angles are adjacent. | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,492 |
13.2. Prove that in any tetrahedron, there is a trihedral angle, all plane angles of which are acute. | 13.2. The sum of the angles of each face is $\pi$, and a tetrahedron has four faces. Therefore, the sum of all dihedral angles of the tetrahedron is $4\pi$. Since the tetrahedron also has four vertices, there will be a vertex where the sum of the planar angles is no more than $\pi$. Therefore, all planar angles at this... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,493 |
13.3. Prove that in any tetrahedron, there will be three edges emanating from one vertex, from which a triangle can be formed. | 13.3. Let $AB$ be the largest edge of the tetrahedron $ABCD$. Since $(AC + AD - AB) + (BC + BD - BA) = (AD + BD - AB) + (AC + BC - AB) > 0$, then $AC + AD - AB > 0$ or $BC + BD - BA > 0$. In the first case, a triangle can be formed from the edges emanating from vertex $A$, and in the second case - from the edges emanat... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,494 |
13.4. The base of the pyramid $A_{1} \ldots A_{n} S$ is a regular $n$-sided polygon $A_{1} \ldots A_{n}$. Prove that if $\angle S A_{1} A_{2}=\angle S A_{2} A_{3}=\ldots=\angle S A_{n} A_{1}$, then the pyramid is regular. regular. | 13.4. Construct on the plane an angle $B A C$, equal to $\alpha$, where $\alpha = \angle S A_{1} A_{2} = \ldots = \angle S A_{n} A_{1}$. We will assume that the length of the segment $A B$ is equal to the side of a regular polygon lying at the base of the pyramid. Then for each $i=1, \ldots, n$ on the ray $A C$ we can ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,495 |
13.5. Given a regular triangular prism
$A B C A_{1} B_{1} C_{1}$. Find all points on the face $A B C$ that are equidistant from the lines $A B_{1}, B C_{1}$ and $C A_{1}$. | 13.5. Let $O$ be a point on the face $ABC$, equidistant from the specified lines. We can assume that $A$ is the farthest vertex from point $O$ in the base $ABC$. Consider the triangles $AOB_1$ and $BOC_1$. The sides $AB_1$ and $BC_1$ of these triangles are equal, and these are the largest sides (see problem 10.5), i.e.... | OisthecenteroftheequilateraltriangleABC | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,496 |
13.9. Prove that for any convex polyhedron, there will be two faces with the same number of sides. | 13.9. Let the number of faces of a polyhedron be n. Then each of its faces can have from three to $n-1$ sides, i.e., the number of sides of each of the $n$ faces can take one of $n-3$ values. Therefore, there will be two faces with the same number of sides. | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,499 |
13.10. Inside a sphere of radius Z, there are several smaller spheres, the sum of whose radii is 25 (these spheres may intersect). Prove that for any plane, there exists a plane parallel to it that intersects at least 9 of the inner spheres. | 13.10. Consider the projection onto a line perpendicular to a given plane. The original sphere is projected in this case into a segment of length 3, while the inner spheres are projected into segments, the sum of whose lengths is 25. Suppose that the required plane does not exist, i.e., any plane parallel to the given ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,500 |
13.11. Given a convex polyhedron $P_{1}$ with nine vertices $A_{1}, A_{2}, \ldots, A_{9}$. Let $P_{2}, P_{3}, \ldots, P_{9}$ be polyhedra obtained from it by parallel translations by vectors $\overrightarrow{A_{1} A_{2}}, \ldots, \overrightarrow{A_{1} A_{9}}$ respectively. Prove that at least two of the 9 polyhedra $P_... | 13.11. Consider a polyhedron $P$, which is the image of the polyhedron $P_{1}$ under a homothety with center $A_{1}$ and coefficient 2. We will prove that all 9 polyhedra lie inside it. Let $A_{1}, A_{2}^{*}, \ldots, A_{9}^{*}$ be the vertices of the polyhedron $P$. We will prove, for example, that the polyhedron $P_{2... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,501 |
13.12. A projector illuminating a straight trihedral angle (octant) is located at the center of the cube. Can it be turned so that it does not illuminate any vertex of the cube? | 13.12. First, let's prove that the spotlight can be turned so that it illuminates adjacent vertices $A$ and $B$ of the cube. If $\angle A O B A B^{2}$
$$
Let's turn the spotlight so that it illuminates two vertices of the cube. The planes of the edges of the illuminated angle by the spotlight divide the space into 8 o... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,502 |
13.13. Consider a regular tetrahedron with unit-length edges. Prove the following statements:
a) On the surface of the tetrahedron, one can choose 4 points such that the distance from any point on the surface to one of these four points does not exceed 0.5;
b) On the surface of the tetrahedron, it is impossible to ch... | 13.13. a) It is easy to verify that the midpoints of the edges $AB, BC, CD, DA$ have the required property. Indeed, on two edges of each face, the chosen points lie. Now consider, for example, the face $ABC$. Let $B_i$ be the midpoint of the edge $AC$. Then triangles $ABB_1$ and $CBB_i$ are covered by circles of radius... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,503 |
13.14. Four pedestrians are walking along four straight roads, no two of which are parallel and no three of which pass through the same point, each at a constant speed. It is known that the first pedestrian met the second, third, and fourth, and the second met the third and fourth. Prove that then the third pedestrian ... | 13.14. Let's introduce a third coordinate axis - the time axis - in the plane where the pedestrians move, along with their coordinates. We will consider the graphs of the pedestrians' movements. It is clear that the pedestrians meet when their movement graphs intersect. From the condition of the problem, it follows tha... | proof | Logic and Puzzles | proof | Yes | Yes | olympiads | false | 24,504 |
13.15. Three lines intersect at point $O$. On the first of them, points $A_{1}$ and $A_{2}$ are taken, on the second $-B_{1}$ and $B_{2}$, on the third - $C_{1}$ and $C_{2}$. Prove that the points of intersection of the lines $A_{1} B_{1}$ and $A_{2} B_{2}, B_{1} C_{1}$ and $B_{2} C_{2}, A_{1} C_{1}$ and $A_{2} C_{2}$ ... | 13.15. Let's take points $C_{1}^{\prime}$ and $C_{2}^{\prime}$ in space such that they project onto points $C_{1}$ and $C_{2}$, and they themselves do not lie in the original plane. Then the points of intersection of the lines $A_{1} C_{1}^{\prime}$ and $A_{2} C_{2}^{\prime}, B_{1} C_{1}^{\prime}$ and $B_{2} C_{2}^{\pr... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,505 |
13.17. The common external tangents to three circles on a plane intersect at points $A, B$, and $C$. Prove that these points lie on a single straight line. | 13.17. Consider for each circle a cone whose base is the given circle and whose height is equal to its radius. We will assume that all these cones are located on one side of the initial plane. Let $O_{1}, O_{2}$, $O_{3}$ be the centers of the circles, and $O_{1}^{\prime}, O_{2}^{\prime}, O_{3}^{\prime}$ be the vertices... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,507 |
13.18. What is the least number of strips of width 1 required to cover a circle of diameter $d$? | 13.18. When solving this problem, we will use the fact that the area of a strip cut on a sphere of diameter $d$ by two parallel planes, the distance between which is $h$, equals $\pi d h$ (see problem 4.24).
Let a circle of diameter $d$ be covered by $k$ strips of width 1. Consider a sphere for which this circle serve... | k\geqslant | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,508 |
13.19. On the sides $B C$ and $C D$ of the square $A B C D$, points $M$ and $N$ are taken such that $C M + C N = A B$. The lines $A M$ and $A N$ divide the diagonal $B D$ into three segments.
. Then the projections of rectangles parallel to the faces of the cube cover the faces with one layer. Thus, in the original hexagon, the su... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,512 |
13.23. Quadrilateral $A B C D$ is circumscribed around a circle, and its sides $A B, B C, C D$ and $D A$ touch the circle at points $K, L, M$ and $N$ respectively.
Fig. 99 Prove that the l... | 13.23. Draw perpendiculars through the vertices of the quadrilateral $A B C D$ to the plane in which it is located. Lay off segments $A A^{\prime}, B B^{\prime}, C C^{\prime}$, and $D D^{\prime}$, equal to the tangents drawn from the corresponding vertices of the quadrilateral to a circle, such that points $A^{\prime}$... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,513 |
13.24. Prove that the lines connecting opposite vertices of a circumscribed hexagon intersect at one point, (Brianchon's Theorem.) | 13.24. Draw perpendiculars to the plane in which the hexagon $A B C D E F$ lies through its vertices, and lay off segments $A A^{\prime}, \ldots, F F^{\prime}$, equal to the tangents drawn from the corresponding vertices to a circle, such that points $A^{\prime}, C^{\prime}$, and $E^{\prime}$ lie on one side of the ori... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,514 |
13.25. On a plane, there is a system of points. Its triangulation is a set of non-intersecting segments with endpoints at these points, such that any other segment with endpoints at the given points intersects at least one of them (Fig. 99). Prove that there exists such a triangulation that none of the circumscribed ci... | 13.25. Let's take an arbitrary sphere tangent to the given plane and consider the stereographic projection of the plane onto the sphere. On the sphere, we obtain a finite system of points, and they are the vertices of some convex polyhedron. To obtain the required triangulation, we need to connect those original points... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,515 |
14.1. a) Prove that the center of mass of a system of points exists and is unique.
b) Prove that if $X$ is an arbitrary point in the plane, and $O$ is the center of mass of points $X_{1}, \ldots, X_{n}$ with masses $m_{1}, \ldots, m_{n}$, then $\overrightarrow{X O}=\frac{1}{m_{1}+\ldots+m_{n}}\left(m_{1} \overrightarr... | 14.1. Let $X$ and $O$ be arbitrary points in the plane. Then $m_{1} \overrightarrow{O X}_{1}+\ldots+m_{n} \overrightarrow{O X}_{n}=\left(m_{1}+\ldots+m_{n}\right) \overrightarrow{O X}+m_{1} \overrightarrow{X X}_{1}+\ldots$ $\ldots+m_{n} \overrightarrow{X X}_{n}$, therefore the point $O$ is the center of mass of the giv... | proof | Algebra | proof | Yes | Yes | olympiads | false | 24,518 |
14.2. Guess that the center of mass of the system of points $X_{1}, \ldots, X_{n}, Y_{1}, \ldots, Y_{m}$ with masses $a_{1}, \ldots, a_{n}, b_{1}, \ldots, b_{m}$ coincides with the center of mass of two points - the center of mass of the first system $X$ with mass $a_{1}+\ldots+a_{n}$ and the center of mass of the seco... | 14.2. Let $Z$ be an arbitrary point, $a=a_{1}+\ldots+a_{n}$, $b=b_{1}+\ldots+b_{m} . \quad$ Then $\overrightarrow{Z X}=\frac{1}{a}\left(a_{1} \overrightarrow{Z X}_{1}+\ldots+a_{n} \overrightarrow{Z X}_{n}\right)$ and $\overrightarrow{Z Y}=\frac{1}{b}\left(b_{1} \overrightarrow{Z Y}_{1}+\ldots+b_{m} \overrightarrow{Z Y}... | proof | Algebra | proof | Yes | Yes | olympiads | false | 24,519 |
14.3. a). Prove that the segments connecting the vertices of a tetrahedron with the points of intersection of the medians of the opposite faces intersect at one point, and each of them is divided by this point in the ratio $3: 1$, counting from the vertex (these segments are called medians of the tetrahedron).
b) Prov... | 14.3. Place unit masses at the vertices of a tetrahedron. The center of mass of these points is located on the segment connecting a vertex of the tetrahedron with the center of mass of the opposite face, and it divides this segment in the ratio $3:1$, counting from the vertex.
Thus, all medians of the tetrahedron pass... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,520 |
14.4. Given a parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$. The plane $A_{1} D B$ intersects the diagonal $A C_{1}$ at point $M$. Prove that $A M: A C_{1}=1: 3$. | 14.4. Placing unit masses at points $A_{1}, B$ and $D$. Let $O$ be the center of mass of this system. Then
i.e., point $O$ lies on the diagonal $A C_{1}$. On the other hand, the center of ma... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,521 |
14.5. Given triangle $A B C$ and line $l$; $A_{1}, B_{1}$, and $C_{1}$ are arbitrary points on line $l$. Find the geometric locus of the centroids of triangles with vertices at the midpoints of segments $A A_{1}, B B_{1}$, and $C C_{1}$. | 14.5. Place unit masses at points $A, B, C, A_{1}, B_{1}$, and $C_{1}$. On one side, the center of mass of this system coincides with the center of mass of the triangle with vertices at the midpoints of segments $A A_{1}, B B_{1}$, and $C C_{1}$. On the other side, it coincides with the midpoint of the segment connecti... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,522 |
14.6. On the edges $A B, B C, C D$ and $D A$ of the tetrahedron $A B C D$, points $K, L, M$ and $N$ are taken such that $A K: K B = D M: M C = p$ and $B L: L C = A N: N D = q$. Prove that the segments $K M$ and $L N$ intersect at a point $O$, and that $K O: O M = q$ and $N O: O L = p$. | 14.6. Place masses $1, p, p q$, and $q$ at points $A, B, C$, and $D$ respectively, and consider the center of mass $P$ of this system of points. Since $K$ is the center of mass of points $A$ and $B$, and $M$ is the center of mass of points $C$ and $D$, the point $P$ lies on the segment $K M$, and $K P: P M = (p q + q) ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,523 |
14.7. On the extensions of the altitudes of the tetrahedron \(ABCD\) from the vertices, segments \(AA_1, BB_1, CC_1,\) and \(DD_1\) are laid out, the lengths of which are inversely proportional to the altitudes. Prove that the centers of mass of the tetrahedrons \(ABCD\) and \(A_1B_1C_1D_1\) coincide. | 14.7. Let $M$ be the center of mass of the tetrahedron $ABCD$. Then $\overrightarrow{M A}_{1} + \overrightarrow{M B}_{1} + \overrightarrow{M C}_{1} + \overrightarrow{M D}_{1} = (\overrightarrow{M A} + \overrightarrow{M B} + \overrightarrow{M C} + \overrightarrow{M D}) + (\overrightarrow{A A}_{1} + \overrightarrow{B B}_... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,524 |
14.8. Two planes intersect the lateral edges of a regular $n$-sided prism at points $A_{1}, \ldots, A_{n}$ and $B_{1}, \ldots, B_{n}$ respectively, and these planes do not have common points inside the prism. Let $M$ and $N$ be the centers of mass of the polygons $A_{1} \ldots A_{n}$ and $B_{1} \ldots B_{n}$.
a) Prove... | 14.8. a) Since $\overrightarrow{M A}_{1}+\ldots+\overrightarrow{M A}_{n}=\overrightarrow{M B}_{1}+\ldots+\overrightarrow{M B}_{n}=\overrightarrow{0}$, by adding the equations $\overrightarrow{M A}_{i}+\overrightarrow{A_{i} B_{i}}+\overrightarrow{B_{i} N}=\overrightarrow{M N}$ for all $t=1, \ldots, n$, we get $\overrigh... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,525 |
14.9. Let $O$ be the center of mass of a system of points with total mass $m$. Prove that the moments of inertia of this system relative to the point $O$ and an arbitrary point $X$ are related by the equation $I_{X}=I_{o}+$ $+m X O^{2}$. | 14.9. Number the points of the given system. Let $\mathbf{x}_{i}$ be the vector from point $O$ to point $i$, and let the mass $m_{i}$ be assigned to this point. Then $\sum m_{i} \mathbf{x}_{i}=0$. Let, further, $\mathbf{a}=\overrightarrow{X O}$. Then $I_{O}=\sum m_{i} x_{i}^{2}, I_{X}=\sum m_{i}\left(\mathbf{x}_{i}+a\r... | I_{X}=I_{O}+^{2} | Algebra | proof | Yes | Yes | olympiads | false | 24,526 |
14.10. a) Prove that the moment of inertia relative to the center of mass of a system of points with unit masses is equal to $\frac{1}{n} \sum_{i<j} a_{i j}^{2}$, where $n-$ is the number of points, $a_{i j}$ - is the distance between points with indices $i$ and $j$.
b) Prove that the moment of inertia relative to the... | 14.10. a) Let $\mathbf{x}_{i}$ be a vector with its origin at the center of mass $O$ and its endpoint at the point with number $i$. Then $\sum_{i, j}\left(\mathbf{x}_{i}-\mathbf{x}_{j}\right)^{2}=\sum_{i, j}\left(x_{i}^{2}+x_{j}^{2}\right)-$ $-2 \sum_{i, j}\left(\mathbf{x}_{i}, \mathbf{x}_{j}\right)$, where the summati... | proof | Algebra | proof | Yes | Yes | olympiads | false | 24,527 |
14.11. Prove that the sum of the squares of the lengths of the medians of a tetrahedron is equal to $4 / 9$ of the sum of the squares of the lengths of its edges. | 14.11. Place unit masses at the vertices of the tetrahedron. Since their center of mass is the point of intersection of the medians of the tetrahedron, which divides each median in the ratio $3:1$, the moment of inertia of the tetrahedron relative to the center of mass is $\left(3 / 4 m_{a}\right)^{2}+\ldots$ $\ldots+\... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,528 |
14.12. Unit masses are placed at the vertices of a tetrahedron. Prove that the moment of inertia of this system relative to the center of mass is equal to the sum of the squares of the distances between the midpoints of opposite edges of the tetrahedron. | 14.12. The center of mass $O$ of the tetrahedron $ABCD$ is the point of intersection of the segments connecting the midpoints of opposite edges of the tetrahedron, and point $O$ divides each of these segments in half (Problem 14.3, b). If $K$ is the midpoint of edge $AB$, then $AO^2 + BO^2 = 20K^2 + AB^2 / 2$. Let's wr... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,529 |
14.13. Given a triangle $ABC$. Find the geometric locus of points $X$ in space such that $X A^{2}+$ $+X B^{2}=X C^{2}$. | 14.13. Place masses +1 at vertices $A$ and $B$, and a mass -1 at vertex $C$. The center of mass $M$ of this system of points is the diagonal of the parallelogram $A C B M$. According to the condition, $I_{X}=X A^{2}+X B^{2}-X C^{2}=0$, and since $I_{X}=(1+1-1) M X^{2}+I_{M I}$ (problem 14.9), then $M X^{2}=-I_{M}=a^{2}... | \sqrt{^{2}+b^{2}-^2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,530 |
14.14. Two triangles - an equilateral one with side $a$ and an isosceles right one with legs equal to $b,$ are positioned in space such that their centers of mass coincide. Find the sum of the squares of the distances from all vertices of one to all vertices of the other. | 14.14. If $M$ is the center of mass of triangle $ABC$, then $I_{M}=$ $=\left(AB^{2}+BC^{2}+AC^{2}\right) / 3$ (see problem 14.10, a), therefore for any point $X$ we have the equality $XA^{2}+XB^{2}+XC^{2}=I_{X}=$ $=3XM^{2}+I_{M}=3XM^{2}+\left(AB^{2}+BC^{2}+AC^{2}\right) / 3$. If $ABC$ is a given right triangle, $A_{1}B... | 3c^{2}+4b^{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,531 |
14.15. Inside a sphere of radius $R$, there are $n$ points. Prove that the sum of the squares of the spherical distances between them does not exceed $n^{2} R^{2}$. | 14.15. Place unit masses at the given points. It follows from the result of problem 14.10, a, that the sum of the squares of the pairwise distances between these points is $n I$, where $I$ is the moment of inertia of the system of points relative to the center of mass. Consider the moment of inertia of the system relat... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,532 |
14.16. Points $A_{1}, \ldots, A_{n}$ lie on a sphere, and $M$ is their center of mass. The lines $M A_{1}, \ldots, M A_{n}$ intersect this sphere at points $B_{1}, \ldots, B_{n}$ (distinct from $A_{1}, \ldots, A_{n}$). Prove that $M A_{1}+\ldots+M A_{n} \leqslant M B_{1}+\ldots+M B_{n}$.
## § 3. Barycentric Coordinate... | 14.16. Let $O$ be the center of a given sphere. If a chord $AB$ passes through the point $M$, then $AM \cdot BM = R^2 - d^2$, where $d = MO$. Let's denote by $I_X$ the moment of inertia of the system of points $A_1, \ldots, A_n$ relative to the point $X$. Then $I_O = I_M + n d^2$ (see problem 14.9). On the other hand, ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,533 |
14.17. Let a tetrahedron $A_{1} A_{2} A_{3} A_{4}$ be given in space.
a) Prove that any point $X$ has certain barycentric coordinates relative to it.
b) Prove that under the condition $m_{1}+m_{2}+m_{3}+m_{4}=1$, the barycentric coordinates of the point $X$ are uniquely defined. | 14.17. Let us introduce the following notations: $\mathbf{e}_{1}=\overrightarrow{A_{4} A_{1}}, \mathbf{e}_{2}=\overrightarrow{A_{6} A_{2}}, \mathbf{e}_{3}=\overrightarrow{A_{1} A_{3}}$ and $\mathbf{x}=\overrightarrow{X A}_{\overline{1}}$. The point $X$ is the center of mass of the vertices of the tetrahedron $A_{1} A_{... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,534 |
14.18. In the barycentric coordinate system associated with the tetrahedron $A_{1} A_{2} A_{3} A_{4}$, find the equation: a) of the line $A_{1} A_{2}$; b) of the plane $A_{1} A_{2} A_{3}$; c) of the plane passing through $A_{3} A_{4}$ parallel to $A_{1} A_{2}$. | 14.18. 'A point with barycentric coordinates $\left(x_{1}, x_{2}, x_{3}, x_{4}\right)$: a) lies on the line $A_{1} A_{2}$ if $x_{3}=x_{4}=0$; b) lies in the plane $A_{1} A_{2} A_{3}$ if $x_{4}=0$.
c) Using the notation from problem 14.17. The point $X$ lies in the specified plane if $\mathrm{x}=\lambda \cdot\left(\mat... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,535 |
14.19. Prove that if a point with barycentric coordinates $\left(x_{i}\right)$ and ( $y_{i}$ ) belongs to some line, then the point with coordinates $\left(x_{i}+y_{i}\right)$ also belongs to the same line, | 14.19. The point with barycentric coordinates $\left(x_{i}+y_{i}\right)$ is the center of mass of the points with coordinates ( $x_{i}$ ) and ( $y_{i}$ ). It is also clear that the center of mass of two points lies on the line passing through them. | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,536 |
14.20. Let $S_{a}, S_{b}, S_{c}$ and $S_{d}$ be the areas of the faces $BCD, ACD, ABD$ and $ABC$ of the tetrahedron $ABCD$. Prove that in the system of barycentric coordinates associated with the tetrahedron $ABCD$:
a) the center of the inscribed sphere has coordinates $(S_{a}, S_{b}, S_{c}, S_{d})$
b) the center of ... | 14.20. a) The center of the inscribed sphere is the point of intersection of the bisector planes of the dihedral angles of the tetrahedron. Let \( M \) be the point of intersection of the edge \( AB \) with the bisector plane of the dihedral angle at the edge \( CD \). Then \( AM: MB = S_b: S_a \) (Problem 3.32), so th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,537 |
14.21. Find the equation of the circumscribed sphere of the tetrahedron $A_{1} A_{2} A_{3} A_{4}$ in barycentric coordinates associated with it. | 14.21. Let $X$ be an arbitrary point, $O$ the center of the circumscribed sphere of a given tetrahedron, $\mathbf{e}_{i}=\overrightarrow{O A}_{i}$, and $\mathbf{a}=\overrightarrow{X O}$. If the point $\delta$ has barycentric coordinates $\left(x_{1}, x_{2}, x_{3}, x_{4}\right)$, then $\sum x_{i} \left( \mathbf{e}_{i} +... | \sum_{i<j}x_{i}x_{j}a_{ij}=0 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,538 |
14.22. a) Prove that if the centers $I_{1}, I_{2}, I_{3}$ and $I_{4}$ of the exinscribed spheres, touching the faces of a tetrahedron, are located on its circumscribed sphere, then this tetrahedron is equifacial.
b) Prove that the converse is also true: for an equifacial tetrahedron, the points $I_{1}, I_{2}, I_{3}$ a... | 14.22. a) Let $S_{1}, S_{2}, S_{3}$ and $S_{4}$ be the areas of the faces $A_{2} A_{3} A_{3}$, $A_{1} A_{3} A_{4}$, $A_{1} A_{2} A_{4}$, and $A_{1} A_{2} A_{3}$. The points $I_{1}, I_{2}, I_{3}$, and $I_{4}$ have barycentric coordinates $(-S_{1}, S_{2}, S_{3}, S_{4})$, $(S_{1}, -S_{2}, S_{3}, S_{4})$, $(S_{1}, S_{2}, -... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,539 |
15.1. a) Does there exist a quadrilateral pyramid in which two non-adjacent faces are perpendicular to the plane of the base?
b) Does there exist a hexagonal pyramid in which three (adjacent or not) lateral faces are perpendicular to the plane of the base? | 15.1. Yes, such pyramids do exist. As bases, we can take, for example, a quadrilateral and a non-convex pentagon, as shown in Fig. 106; the vertices of these pyramids lie on perpendiculars erected from points $P$ and $Q$ respectively,
. Then the triangle $A B S$ can be rotated around axis $A B$ so that segment $S C$ becomes perpendicular to the plane $A B C$. | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,544 |
15.6. Can any trihedral angle be intersected by a plane so that a regular triangle is obtained in the section? | 15.6. No, not any. Consider a trihedral angle $S A B C$, where $\angle B S C<60^{\circ}$ and the edge $A S$ is perpendicular to the plane $S B C$. Suppose that its section $A B C$ is an equilateral triangle. In the right triangles $A B S$ and $A C S$, the hypotenuses are equal, so $S B=S C$. In the isosceles triangle $... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,545 |
15.7. Find the plane angles at the vertices of a trihedral angle, given that any section of it is an acute-angled triangle. | 15.7. First, let's prove that any section of a trihedral angle with right plane angles is an acute-angled triangle. Let the cutting plane cut off segments of length \(a, b\) and \(c\) from the edges. Then the squares of the lengths of the sides of the section are \(a^2 + b^2\), \(b^2 + c^2\), and \(a^2 + c^2\). The sum... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,546 |
15.8. Can 6 pairwise non-parallel lines be arranged in space so that all pairwise angles between them are equal? | 15.8. Yes, we can. Let's draw lines connecting the center of the icosahedron with its vertices (see problem 9.4). It is easy to check that any two such lines pass through two points that are the endpoints of one edge. | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,547 |
15.9. Is a polyhedron necessarily a cube if all its faces are equal squares? | 15.9. No, not necessarily. Take a cube and attach a non-cube to each of its faces. The resulting (non-convex) polyhedron has all faces that are equal to each other as squares. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,548 |
15.10. All edges of a polyhedron are equal and touch one sphere. Must its vertices necessarily belong to one sphere? | 15.10. No, not necessarily. We construct externally on the faces of the cube as bases regular quadrangular pyramids with right angles at the base, equal to $45^{\circ}$. As a result, we get a 12-faced polyhedron, having 14 vertices, of which 8 are the vertices of the cube, and 6 are the vertices of the constructed pyra... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,549 |
15.11. Can a finite set of points in space, not lying in the same plane, have the following property: for any two points \( A \) and \( B \) from this set, there exist two other points \( C \) and \( D \) from it such that \( AB \parallel CD \) and these lines do not coincide? | 15.11. Yes, monkey. It is easy to check that the vertices of a regular hexagon have the required property. Consider now two regular hexagons with a common center $O$, lying in different planes. If $A$ and $B$ are vertices of different hexagons, then as $C$ and $D$ we can take the points symmetric to $A$ and $B$ with re... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,550 |
15.12. Can 8 non-intersecting tetrahedra be arranged so that any two of them touch along a surface segment with a non-zero area?
## § 2. Integer Lattices
A set of points in space, all three coordinates of which are integers, is called an integer lattice, and these points themselves are called nodes of the integer lat... | 15.12. It is possible. In Fig. 108, four triangles are depicted with a solid line (one lies inside the other three). Consider four triangular pyramids with a common vertex, the bases of which are these triangles. Similarly, four more triangular pyramids are constructed with a common vertex (located on the other side of... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,551 |
15.13. Nine vertices of a convex polyhedron lie at the nodes of an integer lattice. Prove that there is another node of the integer lattice inside it or on its surface. | 15.13. Each of the three coordinates of a node in an integer lattice can be either even or odd; in total, there are $2^{3}=8$ different variants. Therefore, among the nine vertices $17^{*}$ 259
of the polyhedron, there will be two vertices with coordinates of the same parity. The midpoint of the segment connecting thes... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,552 |
15.14. a) For which $n$ does there exist a regular $n$-gon with vertices at the nodes of a (spatial) integer lattice?
b) Which regular polyhedra can be arranged so that their vertices are nodes of an integer lattice? | 15.14. a) Let's first prove that for \( n=3,4,6 \) there exists a regular \( n \)-gon with vertices at the nodes of an integer lattice. Consider a cube \( A B C D A_{1} B_{1} C_{1} D_{1} \), whose vertices have coordinates \( (\pm 1, \pm 1, \pm 1) \).
. Consider a sphere ... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,554 |
15.16. Prove that the smallest area \( S \) of a parallelogram with vertices at integer points on the plane \( ax + by + cz = 0 \), where \( a, b \) and \( c \) are integers, is equal to the smallest length \( l \) of a vector with integer coordinates that is perpendicular to this plane. | 15.16. We can consider that the numbers $a, b$, and $c$ are mutually coprime, i.e., the greatest number by which all of them are divisible is 1. A vector perpendicular to this plane has coordinates $(\lambda a, \lambda b, \lambda c)$; these coordinates are integers only if $\lambda$ is an integer, so $l$ is the length ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,555 |
15.17. The vertices $A_{1}, B, C_{1}$ and $D$ of the cube $A B C D A_{1} B_{1} C_{1} D_{1}$ lie at the nodes of an integer lattice. Prove that the other vertices also lie at the nodes of an integer lattice. | 15.17. Let $\left(x_{i}, y_{i}, z_{i}\right)$ be the coordinates of the $i$-th vertex of a regular tetrahedron $A_{1} B C_{1} D$. Its center, coinciding with the center of the cube, has coordinates $\left(x_{1}+x_{2}+x_{3}+x_{9}\right) / 4$ and so on. The point symmetric to $(x_{i}, y_{1}, z_{1})$ with respect to the c... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,556 |
15.18. a) Given a parallelepiped (not necessarily rectangular) with vertices at nodes of an integer lattice, and such that inside it there are $a$ lattice nodes, inside the faces $-b$ nodes, and inside the edges $-c$ nodes. Prove that its volume is equal to $1+a+\frac{b}{2}+\frac{c}{4}$.
b) Prove that the volume of a ... | 15.18. a) We can assume that one of the vertices of the given parallelepiped is at the origin. Consider the cube $K_{1}$, the absolute values of the coordinates of whose points do not exceed some integer $n$. Divide the space into parallelepipeds equal to the given one by drawing planes parallel to its faces. Adjacent ... | 1++\frac{b}{2}+\frac{}{4} | Combinatorics | proof | Yes | Yes | olympiads | false | 24,557 |
15.19. a) Divide a tetrahedron with edge $2a$ into tetrahedra and octahedra with edge $a$.
b) Divide an octahedron with edge $2a$ into tetrahedra and octahedra with edge $a$. | 15.19. a) The midpoints of the edges of a tetrahedron with edge $2a$ are the vertices of an octahedron with edge $a$. If this octahedron is cut out of the tetrahedron, 4 tetrahedra with edge $a$ will remain.
b) If 6 octahedra with edge $a$, one vertex of each being a vertex of the original octahedron, are cut off from ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,558 |
15.21. Cut the cube into three equal pyramids. | 15.21. Let's take one vertex of the cube as the common vertex of these tetrahedra, and three non-adjacent faces of the cube as their bases. | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,560 |
15.22. What is the smallest number of tetrahedra into which a cube can be cut? | 15.22. If a tetrahedron \(A^{\prime} B C^{\prime} D\) is cut out from the cube \(A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}\), the remaining part of the cube splits into 4 tetrahedra, i.e., the cube can be cut into 5 tetrahedra.
We will prove that the cube cannot be cut into fewer than 5 tetrahedra. The face ... | 5 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,561 |
15.23. Prove that any tetrahedron can be cut by a plane into two parts such that they can be reassembled to form the same tetrahedron by applying them to each other in a different way. | 15.23. The sum of the angles of each of the four faces of a tetrahedron is $180^{\circ}$, so the sum of all planar angles of the tetrahedron is 4.180". Therefore, the sum of the planar angles at one of the four vertices of the tetrahedron does not exceed $180^{\circ}$, and hence, the sum of two planar angles at it is l... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,562 |
15.25. a) Prove that any convex polyhedron can be cut into tetrahedra.
b) Prove that any convex polyhedron can be cut into tetrahedra, the vertices of which are located at the vertices of the polyhedron.
* * ＊ | 15.25. a) Let's take an internal point \( P \) inside the polyhedron and cut it into triangles. Triangular pyramids with vertex \( P \) and bases which are these triangles give the desired partition.
b) We will prove the statement by induction on the number of vertices \( n \). For \( n=4 \), it is obvious. Suppose it ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,564 |
15.26. Into how many parts do the planes of the faces divide the space a) of a cube; b) of a tetrahedron? | 15.26. The planes of the faces of both polyhedra intersect only along the lines containing their edges. Therefore, the space divided into parts, into which the space is split, has a common point with the polyhedron. Moreover, to each vertex, each edge, and each face, one can correspond exactly one adjacent part to it, ... | 15 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,565 |
15.27. What is the maximum number of parts that $n$ circles can divide a sphere into? | 15.27. Let the required number be denoted by $S_{n}$. It is clear that $S_{1} = 2$. Now, let's express $S_{n+1}$ in terms of $S_{n}$. Consider a set of $n+1$ circles on a sphere and single out one of these circles. Let the remaining circles divide the sphere into $s_{n}$ parts ($s_{n} \leqslant S_{n}$), and let the num... | n^2-n+2 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,566 |
15.28. In space, there are $n$ planes, such that any three of them have exactly one common point and no four pass through the same point. Prove that they divide the space into $\left(n^{3}+5 n+6\right) / 6$ parts. | 15.28. First, let's prove that \( n \) lines, no two of which are parallel and no three of which pass through the same point, divide the plane into \( \frac{n^2 + n + 2}{2} \) parts. We will prove this by induction on \( n \). For \( n = 0 \), the statement is obvious.
Assume that the statement is true for \( n \) lin... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 24,567 |
15.29. In space, there are $n(n \geqslant 5)$ planes, such that any three of them have exactly one common point and no four pass through the same point. Prove that among the parts into which these planes divide the space, there are at least $(2 n-3) / 4$ tetrahedra.
* * * | 15.29. Let us consider all points of intersection of the given planes. We will prove that among these planes, there are no more than three that do not separate these points. Indeed, suppose there are 4 such planes. No plane can intersect all the edges of the tetrahedron \(ABCD\) defined by these planes; therefore, the ... | \frac{2n-3}{4} | Combinatorics | proof | Yes | Yes | olympiads | false | 24,568 |
15.30. The stone has the shape of a regular tetrahedron. Epo rolls it on a plane, flipping it over an edge. After several such flips, the stone returns to its original position. Could its faces have changed places in the process? | 15.30. No, they cannot. Let's divide the plane into triangles equal to the faces of the tetrahedron and number them as shown in Fig. 111. Cut out a triangle consisting of four such triangles and fold it into a tetrahedron. It is easy to check that if this tetrahedron is flipped over an edge and then unfolded back onto ... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,569 |
15.31. A rectangular parallelepiped of size $2 l \times 2 m \times 2 n$ is cut into cubes with a side of 1, and each of these cubes is painted in one of 8 colors, with any two cubes that share at least one vertex being painted in different colors. Prove that all corner cubes are painted in different colors.
## § 4. Lo... | 15.31. Let's cut a strip two cubes thick from the given parallelepiped and glue the remaining parts together. We will prove that the coloring of the new parallelepiped retains the previous property, i.e., adjacent cubes are colored in different colors. This needs to be verified only for the cubes adjacent to the cuttin... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 24,570 |
15.32. A plane intersects the lower base of a cylinder along its diameter and has only one common point with the upper base. Prove that the area of the cut-off part of the lateral surface of the cylinder is equal to the area of its axial section. | 15.32. Let 0 be the center of the lower base of the cylinder; $AB$ - the diameter along which the plane intersects the base; $\alpha$ - the angle between the base and the intersecting plane; $r$ - the radius of the cylinder. Consider an arbitrary generatrix $XY$ of the cylinder, having a common point $Z$ with the inter... | 2 | Geometry | proof | Yes | Yes | olympiads | false | 24,571 |
15.33. Inside a convex polyhedron with volume $V$, there are $3\left(2^{n}-1\right)$ points. Prove that it contains a polyhedron with volume $V / 2^{n}$, in the interior of which there are none of the given points. | 15.33. First, let's prove that through any two points lying inside a certain polyhedron, a plane can be drawn that divides it into two parts with equal volumes. Indeed, if a certain plane divides it into parts with a volume ratio of \(x\), then when this plane is rotated by \(180^{\circ}\) around a given line, the volu... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 24,572 |
15.34. In space, 4 points are given, not lying in the same plane. How many different parallelepipeds exist for which these points serve as vertices? | 15.34. Consider a parallelepiped for which the given points are vertices, and mark its edges connecting these points. Let \( n \) be the maximum number of marked edges of this parallelepiped emanating from one vertex; the number \( n \) can vary from 0 to 3. A simple enumeration shows that only the variants depicted in... | 29 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,573 |
16.1. a) Prove that under inversion with center $O$, a plane passing through point $O$ maps to itself,
b) Prove that under inversion with center $O$, a plane not containing point $O$ maps to a sphere passing through point $O$.
c) Prove that under inversion with center $O$, a sphere containing point $O$ maps to a plan... | 16.1. Let $R^{2}$ be the degree of the inversion considered.
a) Consider a ray with origin at point $O$ and introduce coordinates on it. Then the point with coordinate $x$ transitions under inversion to the point with coordinate $R^{2} / x$. Therefore, a ray with origin at point $O$ transitions under inversion to itse... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,574 |
16.2. Prove that under inversion with center $O$, a sphere not containing the point $O$ transforms into a sphere. | 16.2. Let $A$ and $B$ be the points where a given sphere $S$ is intersected by a line passing through point $O$ and the center of $S$, and let $X$ be an arbitrary point on $S$. It is sufficient to prove that $\angle A^{*} X^{*} B^{*} = 90^{\circ}$. From the equalities $O A \cdot O A^{*} = O X \cdot O X^{*}$ and $O B \c... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,575 |
16.3. Prove that under inversion, a line or circle transforms into a line or circle.
The angle between two intersecting spheres (a sphere and a plane) is defined as the angle between the tangent planes to the spheres, drawn through any point of intersection.
The angle between two intersecting circles in space (a circ... | 16.3. It is easy to verify that any straight line can be represented as the non-intersection of two planes, and any circle as the intersection of a sphere and a plane. In problems 16.1 and 16.2, it was shown that under inversion, a plane or sphere transforms into a plane or sphere. Therefore, under inversion, a straigh... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,576 |
16.4. a) Prove that the angle between intersecting spheres (planes) is preserved under inversion.
b) Prove that the angle between intersecting circles (lines) is preserved under inversion. | 16.4. a) First, let's prove that touching spheres transform under inversion either into touching spheres (a sphere and a plane), or into a pair of parallel planes. This easily follows from the fact that touching spheres are spheres that have a single common point (and from the fact that under inversion, a sphere transf... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,577 |
16.5. Let $O$ be the center of inversion, $R^{2}$ its power. Prove that then $A^{*} B^{*}=\frac{A B \cdot R^{2}}{O A \cdot O B}$. | 16.5. It is known that $O A \cdot O A^{*}=R^{2}=O B \cdot O B^{*}$. Therefore, $OA : O B^{*}=O B : O A^{*}$, i.e., $\triangle O A B \subset \triangle O B^{*} A^{*}$. Consequently,
$$
\frac{A^{*} B^{*}}{A B}=\frac{O B^{*}}{O A}=\frac{O B^{*}}{O A} \cdot \frac{O B}{O B}=\frac{R^{2}}{O A \cdot O B}
$$ | A^{*}B^{*}=\frac{AB\cdotR^{2}}{OA\cdotOB} | Geometry | proof | Yes | Yes | olympiads | false | 24,578 |
16.6. a) Let there be a sphere and a point $O$ lying on it. Prove that there exists an inversion with center $O$ that maps the given sphere onto itself.
b) Let there be a sphere and a point $O$ lying inside it. Prove that there exists an inversion with center $O$ that maps the given sphere onto a sphere symmetric to i... | 16.6. Let $X$ and $Y$ be the points of intersection of a given sphere with a line passing through point $O$. Consider the inversion with center $O$ and coefficient $R^{2}$. It is easy to verify that in both problems, it is essentially required to choose the coefficient $R^{2}$ such that for any line passing through poi... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,579 |
16.7. Let the inversion with center $O$ map the sphere $S$ to the sphere $S^{*}$: Prove that $O$ is the center of homothety that maps $S$ to $S^{*}$,
## § 2. Perform the inversion | 16.7. Let $A_{i}$ be a point on the sphere $S$, and $A_{2}$ be the second point of intersection of the line $O A_{i}$ with the sphere $S$ (if $O A_{i}$ is tangent to $S$, then $A_{2}=A_{1}$). It is easy to verify that the value $d=O A_{1} \cdot O A_{2}$ is the same for all lines not intersecting the sphere $S$. If $R^{... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,580 |
16.8. Prove that the angle between the circumscribed circles of two faces of a tetrahedron is equal to the angle between the circumscribed circles of its two other faces. | 16.8. Consider the inversion with the center at vertex $D$ of the tetrahedron $A B C D$. The circumscribed circles of the faces $D A B, D A C$, and $D B C$
Fig. 114 will transform into the... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,581 |
16.9. Given a sphere, a circle $S$ on it, and some point $P$ lying outside the sphere. Through point $P$ and each point of circle $S$, a line is drawn. Prove that the second points of intersection of these lines with the sphere lie on some circle, | 16.9. Let $X \cdot$ and $Y$ be the points of intersection of a sphere with a line passing through point $P$. It is not difficult to verify that the value $P X \cdot P Y$ does not depend on the choice of the line; denote it by $R^{2}$. Consider an inversion with center $P$ and power $R^{2}$. Then $X^{*}=$
$=Y$. Thus, th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,582 |
16.10. Let $C$ be the center of the circle along which a cone with vertex $X$ touches a given sphere. What is the geometric locus of points $C$ as $X$ traverses a plane P that has no points in common with the sphere? | 16.10. Let $O$ be the center of a given sphere, $X A$ be a tangent to the sphere. Since $A C$ is the height of the right triangle $O A X$, then $\triangle A C O \sim \triangle A O X$. Therefore, $A O: C O = X O: A O$, i.e., $C O \cdot X O = A O^{2}$. Thus, the point $C$ is the image of the point $X$ under the inversion... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,583 |
16.11. Prove that for any tetrahedron, there exists a triangle whose side lengths are equal to the products of the lengths of opposite edges of the tetrahedron.
Prove also that the area of this triangle is equal to \(6 V R\), where \(V\) is the volume of the tetrahedron, and \(R\) is the radius of its circumscribed sp... | 16.11. Let there be a tetrahedron \(ABCD\). Consider the inversion with center \(D\) and power \(r^2\). Then \(A^* B^* = \frac{AB r^2}{DA \cdot DB}\), \(B^* C^* = \frac{BC r^2}{BD \cdot DC}\), and \(A^* C^* = \frac{AC r^2}{DA \cdot DC}\). Therefore, if we take \(r^2 = DA \cdot DB \cdot DC\), then \(A^* B^* C^*\) is the... | 6VR | Geometry | proof | Yes | Yes | olympiads | false | 24,584 |
16.12. Consider a convex hexahedron, all faces of which are quadrilaterals. It is known that seven of its eight vertices lie on one sphere. Prove that the eighth vertex also lies on the same sphere.
## § 3. Chains of Tangent Spheres | 16.12. Let $A B C D A_{1} B_{1} C_{1} D_{i}$ be the given hexahedron, with only the vertex $C_{i}$ unknown as to whether it lies on the given sphere (Fig. 115, a). Consider the inversion with center $A$. Under this inversion, the given sphere transforms into a plane, and the circumscribed circles of the faces $A B C D,... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,585 |
16.15. Given $n$ spheres, each of which touches all the others, and no three spheres touch at the same point. Prove that $n \leqslant 5$.
untranslated part:
"шикакпе" should be "и никакие" in Russian, which translates to "and no" in English. Here is the corrected translation:
16.15. Given $n$ spheres, each of which... | 16.15. Consider an inversion with the center at one of the points of tangency of the spheres. These spheres transform into a pair of parallel planes, and the other $n-2$ spheres transform into spheres that are tangent to both of these planes. Clearly, the diameter of any sphere tangent to both planes is equal to the di... | n\leqslant5 | Geometry | proof | Yes | Yes | olympiads | false | 24,588 |
16.16. Given three pairwise tangent spheres $\Sigma_{1}$, $\Sigma_{2}$, $\Sigma_{3}$ and a set of spheres $S_{1}, S_{2}, \ldots, S_{n}$, such that each sphere $S_{i}$ is tangent to all three spheres $\Sigma_{1}$, $\Sigma_{2}$, $\Sigma_{3}$, as well as to the spheres $S_{i-1}$ and $S_{i+1}$ (it is meant that $S_{0}=S_{n... | 16.16. Consider the inversion with the center at the point of tangency of spheres $\Sigma_{1}$ and $\Sigma_{2}$. In this case, they transform into a pair of parallel planes, and the images of all other spheres touch these planes, and therefore their radii are equal. Thus, in a section by a plane equidistant from these ... | 6 | Geometry | proof | Yes | Yes | olympiads | false | 24,589 |
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