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class | __index_level_0__ int64 0 742k |
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158. Given three mutually perpendicular lines, the distance between any two 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. | 158. Introduce a rectangular coordinate system such that the first line coincides with the $x$-axis, the second is parallel to the $y$-axis and passes through the point $(0,0, a)$, and the third is parallel to the $z$-axis and passes through the point $(a, a, 0)$. Let $A B C D A_{1} B_{1} C_{1} D_{1}$ be a parallelepip... | 9^{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,864 |
160. Given a triangle $A B C$, with area $S$, and the radius of the circumscribed circle is $R$. Perpendiculars are erected at vertices $A$, $B$, and $C$ to the plane of the triangle, and points $A_{1}, B_{1}$, and $C_{1}$ are taken on these perpendiculars such that the segments $A A_{1}, B B_{1}, C C_{1}$ are equal in... | 160. Let's introduce the usual notations: $a, b, c$ - the sides of the triangle, $h_{a}, h_{b}, h_{c}$ - its altitudes, $p$ - the semiperimeter, $r$ - the radius of the inscribed circle. Let $M$ be the point of intersection of the planes $A_{1} B_{1} C, A_{1} B C_{\hat{1}}$ and $A B_{1} C_{\hat{1}}, O_{a}, O_{b}, O_{c}... | \frac{4}{3}SR | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,866 |
162. Does there exist a triangular pyramid such that the bases of all its altitudes lie outside the corresponding faces? | 162. The specified property is possessed by a pyramid in which two opposite dihedral angles are obtuse. | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,868 |
164. What regular polygons can be obtained by intersecting a cube with a plane? | 164. Triangle, quadrilateral, and hexagon. A section of a cube cannot be a regular pentagon, since in a section having more than three sides, there will be at least one pair of parallel sides, while a regular pentagon has no parallel sides. | Triangle,quadrilateral,hexagon | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,870 |
165. Prove that the sum of the plane angles of a trihedral angle is less than $2 \pi$, and the sum of the dihedral angles is greater than $\pi$. | 165. On the edges of a trihedral angle from the vertex $S$, we mark equal segments $S A, S B, S C$. Let $O$ be the projection of $S$ onto the plane $A B C$. Triangles $A S B$ and $A O B$ are isosceles with a common base $A B$, and the lateral sides of triangle $A O B$ are smaller than the lateral sides of triangle $A S... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,871 |
166. Let the plane angles of a trihedral angle be $\alpha$, $\beta$, and $\gamma$, and the dihedral angles opposite to them be $A$, $B$, and $C$. Prove that the following equalities hold:
1)
$$
\frac{\sin \alpha}{\sin A}=\frac{\sin \beta}{\sin B}=\frac{\sin \gamma}{\sin C}
$$
(Sine theorem for a trihedral angle),
2)... | 166. 167) Let $S$ be the vertex of the angle, $M$ a point on the edge, $M_{1}$ and $M_{2}$ the projections of $M$ onto the other two edges, and $N$ the projection of $M$ onto the opposite face. Suppose the edge $SM$ corresponds to the dihedral angle $C$. If $|SM| = a$, then, finding sequentially $\left|SM_{1}\right|$, ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,872 |
169. Prove that in an arbitrary tetrahedron, there exists a trihedral angle, all plane angles of which are acute. | 169. The sum of all dihedral angles of a tetrahedron is $4 \pi$. Therefore, there exists a vertex where the sum of the dihedral angles is not more than $\pi$. All dihedral angles at this vertex are acute. Otherwise, one angle would be greater than the sum of the other two. | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,875 |
171. Prove that a triangular prismatic surface can be intersected by a plane in such a way that the intersection will be an equilateral triangle. | 171. Let $A B C$ be a perpendicular section, $|B C|=a$, $|C A|=b,|A B|=c$. Draw a section $A B_{1} C_{\mathrm{i}}\left(B\right.$ and $B_{i}$, $C$ and $C_{\overline{1}}$ - on the corresponding edges). Let, further, $\left|B B_{\hat{1}}\right|=|x|$, $\left|C C_{\hat{1}}\right|=|y|$. (If $B_{1}$ and $C_{1}$ are on the sam... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,877 |
172. In a triangular pyramid, all plane angles at vertex $A$ are right angles, edge $A B$ is equal to the sum of the other two edges emanating from $A$. Prove that the sum of the plane angles at vertex $B$ is $\pi / 2$. | 172. Let's denote the two remaining vertices of the tetrahedron as $C$ and $D$. By the condition, $|A C| + |A D| = |A B|$. Consider a square $K L M N$ with side length equal to $|A B|$. Take points $P$ and $Q$ on its sides $L M$ and $M N$ such that $|P M| = |A D|$, $|Q M| = |A C|$. Then $|L P| = |A C|$, $|N Q| = |A D|$... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,878 |
173. Can any trihedral angle be intersected by a plane in such a way that a regular triangle is obtained in the section? | 173. No, not any. For example, if one plane angle of a trihedral angle is sufficiently small, and the other two plane angles are right angles, it is easy to verify that no section of this trihedral angle is an equilateral triangle. | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,879 |
174. Find the plane angles at the vertex of a trihedral angle, given that any section of it by a plane is an acute-angled triangle. | 174. Show that if at least one dihedral angle of a given trihedral angle is not a right angle, then it can be intersected by a plane so that the section is an obtuse triangle. If, however, all dihedral angles of the trihedral angle are right angles, then any section is an acute triangle. For this, it is sufficient to e... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,880 |
175. Prove that in any tetrahedron there is a vertex such that from the segments equal to the edges emanating from this vertex, a triangle can be constructed. | 175. Let $a$ be the length of the longest edge, $b$ and $c$ be the lengths of the edges adjacent to one end of edge $a$, and $e$ and $f$ be the lengths of the edges adjacent to the other end.
We have: $(b+c-a)+(e+f-a)=b+c+e+f-2a>0$. From this, it follows that at least one of the two inequalities holds: $b+c-a>0$ or $e... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,881 |
176. Prove that any tetrahedron can be cut by a plane into two parts such that the resulting pieces can be reassembled to form the same tetrahedron by applying them to each other in a different way. | 176. In any tetrahedron, there exists a vertex for which the sum of any two dihedral angles is less than $180^{\circ}$. (In fact, a stronger statement is true: there exists a vertex for which the sum of all dihedral angles is not greater than $180^{\circ}$.) Let this property be possessed by vertex $A$. Take points $K,... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,882 |
177. Find the plane angles at the vertex of a trihedral angle, given that there exists another trihedral angle with the same vertex, the edges of which lie in the planes forming the faces of the given angle and are perpendicular to the opposite edges of the given angle. | 177. Suppose that no plane angle of the given trihedral angle is a right angle. Let \( S \) be the vertex of this angle. Translate the second trihedral angle parallel to itself so that its vertex coincides with some point \( A \) on one of the edges of the given angle (Fig. 37). \( A B, A C \) and \( A D \) are paralle... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,883 |
178. The line $l$ forms acute angles $\alpha, \beta$ and $\gamma$ with three mutually perpendicular lines. Prove that $\alpha+\beta+\gamma<\pi$. | 178. It can be considered that the line $l$ is the diagonal of a rectangular parallelepiped and forms angles $\alpha, \beta$ and $\gamma$ with its edges. Then, by combining three equal parallelepipeds as shown in Fig. 38, we obtain that the angles between the three diagonals of these parallelepipeds, emanating from a c... | 2\alpha+2\beta+2\gamma<2\pi | Geometry | proof | Yes | Yes | olympiads | false | 29,884 |
179. Prove that the sum of the angles formed by the edges of a trihedral angle with the opposite faces is less than the sum of its dihedral angles.
Prove also that if the dihedral angles of a trihedral angle are acute, then the sum of the angles formed by its edges with the opposite faces is greater than half the sum ... | 179. Let $S$ be the vertex of an angle, and $A, \quad B$ and $C$ be some points on its edges. We will prove that the angle between any edge and the plane of the opposite face is always less than any of the two plane angles enclosing this edge. Since the angle between a line and a plane cannot be obtuse, it is sufficien... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,885 |
180. Prove that the sum of four dihedral angles of a tetrahedron (excluding any two opposite angles) is less than \(2 \pi\), and the sum of all dihedral angles of the tetrahedron is between \(2 \pi\) and \(3 \pi\). | 180. Let $\alpha$ and $\alpha_{1}, \beta$ and $\beta_{1}, \gamma$ and $\gamma_{i}$ be the dihedral angles of a tetrahedron (angles denoted by the same letters correspond to opposite edges). Consider four vectors $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}$ and $\boldsymbol{d}$, perpendicular to the faces of the tet... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,886 |
181. From an arbitrary point on the base of a regular pyramid, a perpendicular is erected. Prove that the sum of the segments from the base of the perpendicular to the points of intersection with the lateral faces or their extensions is a constant value. | 181. The statement of the problem follows from the fact that for a regular polygon, the sum of the distances from any point inside it to its sides is a constant value. | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,887 |
182. Prove that if $x_{1}, x_{2}, x_{3}, x_{4}$ are the distances from an arbitrary point inside a tetrahedron to its faces, and $h_{1}$, $h_{2}$, $h_{3}$, $h_{4}$ are the corresponding heights of the tetrahedron, then
$$
\frac{x_{1}}{h_{1}}+\frac{x_{2}}{h_{2}}+\frac{x_{3}}{h_{3}}+\frac{x_{4}}{h_{4}}=1
$$ | 182. If $S_{1}, S_{2}, S_{3}, S_{4}$ are the areas of the corresponding faces of a tetrahedron, and $V$ is its volume, then
$$
\begin{aligned}
\frac{x_{1}}{h_{1}}+\frac{x_{2}}{h_{2}}+\frac{x_{3}}{h_{3}}+\frac{x_{4}}{h_{4}}=\frac{S_{1} x_{1}}{S_{1} h_{1}} & +\frac{S_{2} x_{2}}{S_{2} h_{2}}+\frac{S_{3} x_{3}}{S_{3} h_{3... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,888 |
183. Prove that a plane passing through the midpoints of two skew edges of a tetrahedron divides it into two parts of equal volume. | 183. Let $M$ and $K$ be the midpoints of the edges $AB$ and $DC$ of the tetrahedron $ABCD$. The plane passing through $M$ and $K$ intersects the edges $AD$ and $BC$ at points $L$ and $N$ (Fig. $39, a$). Since the plane $DMC$ divides the volume of the tetrahedron in half, it is sufficient to prove that the pyramids
. Prove that one of the three numbers $a a_{1} \cos \alpha, b b_{1} \cos \beta, c c_{1} \cdot ... | 185. Let's pass through each edge of the tetrahedron a plane parallel to the opposite edge. The three pairs of resulting planes form a parallelepiped. The opposite edges of the tetrahedron will be the diagonals of a pair of opposite faces of the parallelepiped. Let, for example, \(a\) and \(a_{1}\) be the diagonals of ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,891 |
186. In the tetrahedron $A B C D$, the edges $D A, D B$, and $D C$ are equal to the corresponding altitudes of triangle $A B C$ (i.e., $D A$ is equal to the altitude from vertex $A$, and so on). Prove that the sphere passing through three vertices of the tetrahedron intersects the edges emanating from the fourth vertex... | 186. Let the sphere pass through the vertices $A, B$ and $C$ and intersect the edges $D A, D B$ and $D C$ at points $K, L$ and $M$. From the similarity of triangles $D K L$ and $A B D$, we find: $|L K|=|A B| \frac{|D L|}{|D A|}$, and from the similarity of triangles $D M L$ and $D B C$, we find: $|M L|=|B C| \frac{|D L... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,892 |
187. Given a pyramid $M A B C D$ with a convex quadrilateral $A B C D$ as its base. A plane intersects the edges $M A, M B, M C$, and $M D$ at points $K, L, P$, and $N$ respectively. Prove that the following relation holds:
$$
S_{B C D} \frac{|M A|}{|M K|}+S_{A D B} \frac{|M C|}{|M P|}=S_{A B C} \frac{|M D|}{|M N|}+S_... | 187. The points $K, L, P$ and $N$ belonging to the same plane means that
$$
V_{M K L P}+V_{M P N K}=V_{M N K L}+V_{M L P N}
$$
From problem 9, it follows that
$$
\begin{aligned}
V_{M K L P} & =\frac{|M K| \cdot|M L| \cdot|M P|}{|M A| \cdot|M B| \cdot|M C|} V_{M A B C} \\
V_{M P N K} & =\frac{|M P| \cdot|M N| \cdot|M... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,893 |
189. Given three parallel lines. $A, B$ and $C$ are fixed points on these lines. Let $M, N$ and $L$ be the corresponding points on these same lines, located on one side of the plane $A B C$. Prove that if: a) the sum of the lengths of segments $A M, B N$ and $C L$ is constant, or b) the sum of the areas of trapezoids $... | 189. Both points follow from the following general statement: if the sum $\alpha|A M|+\beta|B N|+\gamma|C L|$, where $\alpha, \beta, \gamma$ are given coefficients, is constant, then the plane $M N L$ passes through a fixed point. This statement, in turn, follows from the equality
$$
\alpha|A M|+\beta|B N|=(\alpha+\be... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,895 |
190. The sum of the lengths of two skew edges of a tetrahedron is equal to the sum of the lengths of two other skew edges. Prove that the sum of the dihedral angles, whose edges are the first pair of edges, is equal to the sum of the dihedral angles, whose edges are the second pair of edges of the tetrahedron. | 190. If in the tetrahedron $A B C D$ the equality $|A B| + |C D| = |B C| + |D A|$ holds, then just as in the planar case, it can be proven that there exists a sphere that touches the edges $A B, B C, C D, D A$, and all points of tangency are located inside the segments $A B, B C, C D$, and $D A$: If we draw a plane thr... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,896 |
191. Let $O$ be the center of a regular tetrahedron. From an arbitrary point $M$, located on one of the faces of the tetrahedron, perpendiculars are dropped to the other three faces. $K, L, N$ are the bases of these perpendiculars. Prove that the line $O M$ passes through the centroid of the triangle $K L N$. | 191. Let $R$ be the point of intersection of $OM$ with the plane $KLN$ (Fig. 41). The statement that $R$ is the centroid of $\triangle KLN$ is equivalent to the statement that the volumes of the tetrahedra MKLO, MLNO, and MNKO are equal. Let $x$,
 and the center of the inscribed sphere intersects the edges \(AB\) and \(CD\), then \(|AC|=|BD|\), \(|AD|=|BC|\). | 201. Let $K$ and $M$ be the midpoints of edges $AB$ and $CD$: From the condition, it follows that the line $KM$ passes through the point $O$ - the center of the inscribed sphere: $O$ is equidistant from the faces $ACD$ and $BCD$. Therefore, the point $K$ is also equidistant from these faces. From this, it follows that ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,907 |
202. Given a cube $A B C D A_{1} B_{1} C_{1} D_{\mathrm{i}}$. A plane is drawn through vertex $A$, tangent to the sphere inscribed in the cube. Let $M$ and $N$ be the points of intersection of this plane with the lines $A_{1} B$ and $A_{1} D$. Prove that the line $M N$ is tangent to the sphere inscribed in the cube. | 202. Let's rotate the cube around the diagonal $A C_{\mathbf{i}}$ by some angle. Since the plane of the triangle $A_{1} B D$ is perpendicular to $A C_{\mathrm{i}}$, and its sides touch the inscribed sphere of the cube, the sides of the triangle resulting from the rotation of $A_{1} B D$ will also touch the inscribed sp... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,908 |
203. Prove that for a tetrahedron, where all dihedral angles at one of the vertices are right angles, the following statement is true: the sum of the squares of the areas of the rectangular faces is equal to the square of the area of the fourth face (the Pythagorean theorem for a rectangular tetrahedron). | 203. Let $\alpha, \beta, \gamma$ be the angles formed by the rectangular faces with the fourth face. If $S_{1}, S_{2}, S_{3}, S_{4}$ are the areas of the faces, then $S_{i}=S_{4} \cos \alpha, \quad S_{2}=S_{4} \cos \beta, S_{3}=S_{4} \cos \gamma$. After this, we can use the fact that $\cos ^{2} \alpha+\cos ^{2} \beta+\... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,909 |
204. Prove that the sum of the squares of the projections of the edges of a cube onto an arbitrary plane is constant. | 204. Let's take a line perpendicular to a given plane and denote by $\alpha, \beta$ and $\gamma$ the angles formed by this line with the edges of the cube. The projections of the edges onto the plane take the values $\sin \alpha$, $\sin \beta, \sin \gamma$. Since $\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1$, t... | 8a^2 | Geometry | proof | Yes | Yes | olympiads | false | 29,910 |
205. Prove that the sum of the squares of the projections of the edges of a regular tetrahedron onto an arbitrary plane is constant. | 205. Let's pass through each edge of the tetrahedron a plane parallel to the opposite edge. We will obtain a cube in which the tetrahedron is inscribed. If the edge of the tetrahedron is \( b \), then the edge of the cube will be \( b / \sqrt{2} \). The projection of each face of the cube is a parallelogram, the diagon... | 4b^2 | Geometry | proof | Yes | Yes | olympiads | false | 29,911 |
206. Two bodies in space move along two straight lines with constant and unequal speeds. Prove that there exists a fixed circle in space such that the ratio of the distances from any point on it to these bodies is constant and equal to the ratio of their speeds. | 206. Let's first consider the case when the lines intersect. Denote by $A$ and $B$ the positions of the points at some moment in time, $k$ - the ratio of their speeds (the speed of the body located at point $A$ is $k$ times greater than the speed of the other body). $M$ and $N$ - two points on the line $A B$ such that ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,912 |
207. Given a sphere and two points $A$ and $B$ outside it. From $A$ and $B$, two intersecting tangents are drawn to the sphere. Prove that the point of their intersection lies in one of two fixed planes. | 207. Let $O$ be the center of the sphere, $r$ its radius, $AP$ and $BQ$ the tangents to the sphere ($P$ and $Q$ being the points of tangency), and $M$ the point of intersection of the lines $AP$ and $BQ$. Denote: $|OA|=a, |OB|=b, |PM| = |QM| = x$. Then $|OM|^2 = r^2 + x^2$, $|AM|^2 = (\sqrt{a^2 - r^2} \pm x)^2$, $|BM|^... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,913 |
208. Three spheres touch the plane of a given triangle at its vertices and touch each other. Prove that, if the triangle is scalene, there exist two spheres that touch the three given spheres and the plane of the triangle, and if $r$ and $\rho(\rho>r)$ are the radii of these spheres, $R$ is the radius of the circumcirc... | 208. Let $ABC$ be a given triangle, the sides of which, as usual, are equal to $a, b$ and $c$. The radii of three spheres touching each other and the plane of the triangle at points $A, B$, and $C$ are respectively $\frac{bc}{2a}, \frac{ca}{2b}, \frac{ab}{2c}$. Let $x$ be the radius of the sphere touching the three giv... | \frac{1}{r}-\frac{1}{\rho}=\frac{2\sqrt{3}}{R} | Geometry | proof | Yes | Yes | olympiads | false | 29,914 |
209. Given a tetrahedron \(ABCD\). One sphere touches the edges \(AB\) and \(CD\) at points \(A\) and \(C\), and the other sphere touches the edges \(B\) and \(D\) at points \(B\) and \(D\). Prove that the projections of \(AC\) and \(BD\) onto the line passing through the centers of these spheres are equal. | 209. Let $M$ be the midpoint of $AB$, $O_{1}$ and $O_{2}$ be the centers of the spheres, and $R_{1}$ and $R_{2}$ be their radii, then
$$
\left|M O_{1}\right|^{2}-\left|M O_{2}\right|^{2}=\left(R_{1}^{2}+\frac{|A B|^{2}}{4}\right)-\left(R_{2}^{2}+\frac{|A B|^{2}}{4}\right)=R_{1}^{2}-R_{2}^{2}
$$
This means that the mi... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,915 |
211. Prove that a pentagon, all sides of which are equal to each other and the angles of which are also equal to each other, is planar. | 211. Let $A_{1} A_{2} A_{3} A_{4} A_{5}$ be the given pentagon. From the condition, it follows that all diagonals of the pentagon are equal to each other. We choose three vertices of the pentagon in such a way that the two remaining vertices are on the same side of the plane defined by the three taken vertices. Let, fo... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,917 |
212. Given a parallelepiped $A B C D A_{1} B_{1} C_{1} D_{i}$, the diagonal $A C_{1}$ of which is equal to $d$, and the volume $V$. Prove that from the segments equal to the distances from the vertices $A_{i}, B$ and $D$ to the diagonal $A C_{1}$, a triangle can be constructed and that if $s$ is the area of this triang... | 212. Let $M$ be the point of intersection of the diagonal $A C_{i}$ with the plane $A_{1} B D$. Then $M$ is the point of intersection of the medians of triangle $A_{1} B D$, and, moreover, $M$ divides the diagonal $A C_{1}$ in the ratio $1: 2$, i.e., $|A M|=\frac{1}{3}d$.
Consider the pyramid $A B A_{1} D$ (Fig. 45). ... | 2 | Geometry | proof | Yes | Yes | olympiads | false | 29,918 |
213. Given a tetrahedron \(ABCD\), \(A_1, B_1, C_1, D_1\) are the points of intersection of the medians of the faces \(BCD\), \(CDA\), \(DAB\), and \(ABC\). Prove that there exists a tetrahedron \(A_2B_2C_2D_2\) such that the edges \(A_2B_2\), \(B_2C_2\), \(C_2D_2\), and \(D_2A_2\) are equal and parallel to the segment... | 213. Let $M$ be the centroid of the tetrahedron $ABCD$. The volume of the pyramid $MABC$ is $\frac{1}{4}$ of the volume of the given tetrahedron. We extend the pyramid $MABC$ to a parallelepiped such that the segments $MA$, $MB$, and $MC$ are its edges. In Fig. 46, this parallelepiped is shown separately. Clearly, the ... | \frac{16}{27}V | Geometry | proof | Yes | Yes | olympiads | false | 29,919 |
214. Given a tetrahedron. Prove that there exists another tetrahedron $K L M N$, the edges $K L, L M, M N$ and $N K$ of which are perpendicular to the corresponding faces of the given tetrahedron and their lengths are numerically equal to the areas of these faces. Find the volume of the tetrahedron $K L M N$, if the vo... | 214. In solving problem 180, we proved that the sum of vectors perpendicular to the faces of a tetrahedron, directed outward from the tetrahedron and equal in length to the areas of the corresponding faces, is zero. From this, it follows that a tetrahedron $K L M N$ exists.
When finding the volume of the tetrahedron, ... | \frac{3}{4}V^{2} | Geometry | proof | Yes | Yes | olympiads | false | 29,920 |
215. Given three intersecting spheres. Through a point located on the common chord of all three spheres, three chords belonging to different spheres are drawn. Prove that the ends of these three chords lie on one sphere. | 215. The statement of the problem follows from the fact that the products of the segments into which each of these chords is divided by the point of intersection are equal to each other. | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,921 |
218. Prove that the line of intersection of two conical surfaces with parallel axes and equal angles in axial sections is a plane curve. | 218. The equation
$$
(x-a)^{2}+(y-b)^{2}=b^{2}(z-c)^{2}
$$
describes a conical surface with its vertex at the point $S(a, b, c)$, the axis parallel to the $z$-axis, and $b=\operatorname{tg} \alpha$, where $\alpha$ is the angle between the axis of the cone and the generatrix. Subtracting the equations of two conical s... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,923 |
219. 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, located in the same plane. Let $P$ be an arbitrary point in space. The lines $P K, P L, P M$ and $P N$ intersect for the second time the circumcircles of triangles $P A B, P B C, P C D$ and $P D A$ at points $Q... | 219. Let $F$ be the point of intersection of the lines $K L$ and $M N$, and let $E$ be the point of intersection of the line $P F$ with the sphere passing through the points $P, A, B$ and $C$ (assuming that $P$ does not lie in the plane of the face $A B C$).
The points $P, Q, R$ and $E$ belong to the same circle, whic... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,924 |
220. Prove that the edges of a convex quadrilateral angle are the generators of a cone, the vertex of which coincides with the vertex of this angle, if and only if the sums of the opposite dihedral angles of the quadrilateral angle are equal to each other. | 220. Let the edges $S A, S B, S C$ and $S D$ of a tetrahedral angle be the generators of a cone, the axis of which is $S O$. Then, in the trihedral angle formed by the lines $S O, S B$ and $S C$, the dihedral angles with edges $S A$ and $S B$ are equal. Considering three other such angles, it is easy to see that the su... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,925 |
221. Given a hexahedron, all faces of which are quadrilaterals. It is known that seven of its eight vertices lie on the surface of one sphere. Prove that the eighth vertex also lies on the surface of the same sphere. | 221. Let all vertices of the hexagon $A B C D E F K L$, except for $C$, be located on the surface of a sphere with center $O$ (Fig. 47). Denote by $C_{\hat{1}}$ the point of intersection of the line $K C$ with the surface of the sphere.
For brevity, we will denote by $\Varangle F E L$ the dihedral angle between the pl... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,926 |
222. On each edge of a tetrahedron, an arbitrary point different from the vertex of the tetrahedron is taken. Prove that the four spheres, each passing through one vertex of the tetrahedron and the three points taken on the edges emanating from this vertex, intersect at one point.
## § 2. Problems on Maximum-Minimum. ... | 222. Let $ABCD$ be the given tetrahedron, and $K, L, M, N, P$, and $Q$ be the points on the edges $AB, AC, AD, BC, CD$, and $DB$ respectively.
Fig. 47. The given points are respectively on ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,927 |
224. In a convex quadrilateral angle, all plane angles are equal to $60^{\circ}$. Prove that the angles between opposite edges cannot be simultaneously acute or simultaneously obtuse. | 224. Let $S$ be the vertex of an angle. Suppose we intersect the angle with a plane in such a way that a pyramid $S A B C D$ is formed, where $A B C D$ is the base, and the opposite lateral edges are equal:
$|S A|=|S C| \cdot |S B|=|S D|$.
(Prove that this is always possible.) Since the plane angles at the vertices a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,928 |
226. Find the maximum value of the volume of a tetrahedron inscribed in a cylinder, the radius of the base of which is $R$, and the height is $h$. | 226. The tetrahedron with the greatest volume has two opposite edges that are perpendicular and are diameters of the bases. Its volume is $\frac{2}{3} R^{2} h$. | \frac{2}{3}R^{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,929 |
228. In a regular quadrilateral prism $A B C D A_{1} B_{1} C_{1} D_{1}$, the height is half the length of the side of the base. Find the maximum value of the angle $A_{1} M C_{1}$, where $M$ is a point on the edge $A B$. | 228. Let the height of the prism be $1, |A M|=x$. Circumscribe a circle around triangle $A_{1} M C_{1}$. Consider the solid obtained by rotating the arc $A_{1} M C_{1}$ of this circle around the chord $A_{1} C_{1}$. The angle $A_{1} M C_{1}$ will be the largest if the line $A B$ is tangent to the surface of the resulti... | \frac{\pi}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,931 |
229. The length of the edge of the cube $A B C D A_{1} B_{1} C_{1} D_{1}$ is 1. On the extension of the edge $A D$ beyond point $D$, a point $M$ is chosen such that $|A M| = 2 \sqrt{2 / 5}$. Point $E$ is the midpoint of the edge $A_{1} B_{1}$, and point $F$ is the midpoint of the edge $D D_{1}$. What is the maximum val... | 229. The lines $A E$ and $C F$ are perpendicular. Let $Q_{1}$ be the projection of $Q$ onto the plane $A B B_{1} A_{1} \cdot Q_{1}$ lies on the segment $B L$, where $L$ is the midpoint of $A A_{1}$. Let $N$ be the intersection point of $A E$ and $L B$. It is not difficult to find that $|A N|=\frac{1}{\sqrt{5}}$. Denote... | \sqrt{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,932 |
230. The edge length of the cube $A B C D A_{1} B_{1} C_{1} D_{1}$ is $a$. Points $E$ and $F$ are the midpoints of edges $B B_{1}$ and $C C_{1}$, respectively. Triangles are considered, whose vertices are the points of intersection of a plane parallel to the plane $A B C D$ with the lines $A C_{1}, C E$ and $D F$. Find... | 230. Consider $\triangle K L M$, representing the section of the given triangle on the plane $A B C D, K$ - on the line $C B$, $L$ - on $C D, M$ - on $C A$. If $|C K|=x$, then $|C L|=|a-x|$, $|C M|=\sqrt{\overline{2}}\left|a-\frac{x}{2}\right|$
It is not difficult to obtain that
$$
S_{K L M}=\frac{1}{2}\left|x(a-x)-a... | \frac{7^{2}}{32} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,933 |
231. A right rectangular parallelepiped is inscribed in a regular quadrilateral pyramid with the side of the base and the height both equal to 1, such that the base of the parallelepiped lies in the plane of the base of the pyramid, and the vertex of the opposite face is on the lateral surface of the pyramid. The area ... | 231. Let $x$ be the height of the parallelepiped. Consider the section of the pyramid by a plane at a distance $x$ from its base. In the section, a square with side $(1-x)$ is obtained, in which a rectangle of area $s$ is inscribed, which is a face of the parallelepiped. There are two cases:
1) The base of the parallel... | \begin{aligned}& | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,934 |
232. The bases of a truncated pyramid are regular triangles $ABC$ and $A_{1}B_{1}C_{1}$ with sides 3 and 2 respectively. The segment connecting vertex $C_{1}$ to the center $O$ of the base $ABC$ is perpendicular to the bases; $\left|C_{1}O\right|=3$. A plane is drawn through vertex $B$ and points $M$ and $N$ - the midp... | 232. Let's consider the section of the polyhedron $A B C A_{1} \pi \uparrow N C_{1}$ by a plane at a distance $h$ from the plane $A_{1} B_{1} C_{1}$, and project the resulting section onto the plane $A_{1} B_{1} C_{1}$ (Fig. 48). In the figure, the projection of this section is denoted by dashed lines. It is clear that... | \frac{8\pi}{27} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,935 |
233. All edges of a regular triangular prism $A B C A_{1} B_{1} C_{1}$ have length $a$. Segments with endpoints on the diagonals $B C_{1}$ and $C A_{1}$ of the lateral faces, parallel to the plane $A B B_{1} A_{1}$, are considered. Find the smallest length of such segments. | 233. If a plane, drawn through our segment parallel to the face $A B B_{1} A_{1}$, intersects $C B$ at point $K$ such that $|C K|=x$, then the projection of the segment on the face $A B C$ has a length of $x$, and its projection on the edge $C C_{\mathbf{1}}$ is $|a-2 x|$; thus, the length of the segment will be
$$
\s... | \frac{}{\sqrt{5}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,936 |
234. Given a trihedral angle and a point inside it. A plane is drawn through this point. Prove that the volume of the tetrahedron formed by the given angle and the drawn plane will be minimal in the case when the given point is the centroid of the triangle that is the section of the trihedral angle by the drawn plane. | 234. The plane analog of this problem is the following statement. Given an angle and a point $N$ inside it, consider all possible triangles formed by the sides of the angle and a line passing through point $N$. The triangle with the smallest area among such triangles is the one where the side passing through $N$ is bis... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,937 |
235. The surface area of a spherical segment is $S$ (the spherical part of the segment is considered). Find the maximum value of the volume of this segment. | 235. If $h$ is the height of the segment, then its volume is $\frac{1}{2} S h - \frac{1}{3} \pi h^{3}$. The maximum volume will be at $h = \sqrt{\frac{S}{2 \pi}}$; it will be equal to $\frac{S}{3} \sqrt{\frac{S}{2 \pi}}$ | \frac{S}{3}\sqrt{\frac{S}{2\pi}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,938 |
236. A cube with edge $a$ is standing on a plane. The light source is located at a distance $b(b>a)$ from the plane. Find the minimum value of the area of the shadow cast by the cube on the plane. | 236. Note that the shadow cast by only the upper face of the cube (assuming all other faces are transparent) is a square with side $\frac{a b}{b-a}$. From this, it follows that the area of the shadow cast by the cube will be the smallest when the light source is positioned above the upper face (only the upper face of t... | (\frac{}{b-})^{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,939 |
237. Given a convex centrally symmetric polyhedron. Consider sections of this polyhedron parallel to a given plane. Are the following statements true:
1) the section passing through the center has the largest area
2) for each section, consider the circle of the smallest radius containing it. Is it true that the section... | 237. Statement 1) is correct, let's prove it. Denote by $M_{1}$ the polygon obtained by intersecting our polyhedron with a plane not passing through its center, $S$ - the area of this polygon. $M_{2}$ - the polygon symmetric to $M_{1}$ with respect to the center of the polyhedron. Denote by П the smallest convex polyhe... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,940 |
239. Two cones have a common base and are located on opposite sides of it. The radius of the base is $r$, the height of one cone is $h$, and the other is $H (h \leqslant H)$. Find the greatest distance between two generatrices of these cones. | 239. Let $A$ and $B$ be the vertices of the cones, $M$ and $N$ be two points on the circumferences of the bases, and $L$ be the point diametrically opposite to point $M\left(|A M|=\sqrt{r^{2}+H^{2}},|B M|=\sqrt{r^{2}+h^{2}}\right)$. Draw a plane through $M$ perpendicular to $A M$, and denote by $B_{1}, N_{1}$, and $L_{... | \frac{(+H)r}{\sqrt{r^{2}+H^{2}}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,941 |
240. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with edge $a$. Find the radius of the smallest sphere that touches the lines $A B_{1}, B_{1} C$, $C D$ and $D A$. | 240. Extend the edge $B_{1} B$ beyond point $B$ and take a point $K$ on the extension such that $|B K|=a$. It is not difficult to see that $K$ is equidistant from all sides of the quadrilateral $A B_{1} C D$. Now, take a point $L$ on the diagonal $B_{1} D$ such that $\frac{\left|B_{1} L\right|}{|L D|}=\sqrt{2}$. Point ... | \sqrt{1-\frac{\sqrt{2}}{2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,942 |
241. The diagonal of a cube with an edge length of 1 lies on the edge of a dihedral angle of size $\alpha\left(\alpha<180^{\circ}\right)$. Within what limits can the volume of the part of the cube enclosed by this angle vary? | 241. Let the diagonal $A C_{1}$ of a cube lie on the edge of a dihedral angle, and the faces of the angle intersect the edges of the cube at points $M$ and $N$. It is not hard to notice that if the volume of the part of the cube inside this angle reaches its maximum or minimum value, then the areas of triangles $A C_{1... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,943 |
242. The lengths of the edges of a rectangular parallelepiped are $a, b$, and $c$. What is the maximum value of the area of the rectangular projection of this parallelepiped onto a plane? | 242. Note that the area of the projection of any parallelepiped is always twice the area of the projection of any triangle with vertices at the ends of three edges of the parallelepiped, emanating from one of its vertices. For a rectangular parallelepiped, all such triangles are equal. The largest area of the projectio... | \sqrt{^{2}b^{2}+b^{2}^{2}+^{2}^{2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,944 |
244. The vertex $E$ of the pyramid $A B C E$ is located inside the pyramid $A B C D$. Are the following statements true:
1) the sum of the edges $A E, B E$ and $C E$ is less than the sum of the edges $A D$, $B D$ and $C D$;
2) at least one of the edges $A E, B E, C E$ is less than the corresponding edge $A D, B D, C D$... | 244. 245) This statement is incorrect. For example, take two points \(D_1\) and \(E_1\) inside triangle \(ABC\) such that the sum of the distances from \(D_1\) to the vertices of the triangle is less than the sum of the distances from \(E_1\) to the vertices. Now, take a point \(D\) sufficiently close to \(D_1\) so tha... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,946 |
245. Let $r$ and $R$ be the radii of the inscribed and circumscribed spheres of a regular quadrilateral pyramid, respectively. Prove that
$$
\frac{R}{r} \geqslant \sqrt{\overline{2}}+1
$$ | 245. Let $2a$ be the side of the base, $h$ be the height of the pyramid. Then $R$ is the radius of the circle circumscribed around an isosceles triangle with base $2a\sqrt{2}$ and height $h$, $R=\frac{2a^{2}+h^{2}}{2h}$; $r$ is the radius of the circle inscribed in an isosceles triangle with base $2a$ and height $h$,
... | \sqrt{2}+1 | Geometry | proof | Yes | Yes | olympiads | false | 29,947 |
246. Let $R$ and $r$ be the radii of the circumscribed and inscribed spheres of a certain tetrahedron, respectively. Prove that $R \geqslant 3 r$. | 246. The centroids of the faces of a tetrahedron serve as the vertices of a tetrahedron similar to the given one with a similarity coefficient of $1 / 3$. Therefore, the radius of the sphere passing through the centroids of the faces of the given tetrahedron is $R / 3$. Obviously, this radius cannot be less than the ra... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,948 |
247. Two opposite edges of a tetrahedron have lengths $b$ and $c$, while the others are equal to $a$. What is the minimum value of the sum of the distances from an arbitrary point in space to the vertices of this tetrahedron? | 247. Let in tetrahedron $ABCD$, $|AB| = b$, $|CD| = c$, and the other edges are equal to $a$. If $N$ is the midpoint of $AB$ and $M$ is the midpoint of $CD$, then the line $MN$ is the axis of symmetry of the tetrahedron $ABCD$ (Fig. 49, a).
}{12}$. | \frac{(4b^{2}-^{2})}{12} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,953 |
252. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with edge $a$. Find the length of the shortest segment, the ends of which are located on the lines $A B_{1}$ and $B C_{1}$, forming an angle of $60^{\circ}$ with the plane of the face $A B C D$. | 252. Let $M$ be a point on the line $A B_{1}, N$ be on the line $B C_{1}$, $M_{1}$ and $N_{1}$ be the projections of $M$ and $N$ onto the plane $A B C D$. Denote: $\left|B M_{1}\right|=x,\left|B N_{1}\right|=y$, then
$$
\left|M_{1} N_{1}\right|=\sqrt{x^{2}+y^{2}}, \quad|M N|=\sqrt{x^{2}+y^{2}+(a-x-y)^{2}}
$$
By the c... | 2(\sqrt{3}-\sqrt{2}) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,954 |
253. Three identical cylindrical surfaces of radius $R$ with mutually perpendicular axes touch each other pairwise.
a) What is the radius of the smallest sphere that touches these cylindrical surfaces?
b) What is the radius of the largest cylinder that touches the three given ones, the axis of which passes inside the... | 253. Consider a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with edge $2 R$. Place the axes of the given cylinders on the lines $A A_{1}, D C, B_{1} C_{1}$.
a) The center of the cube is at a distance of $\dot{R} \sqrt{2}$ from all edges of the cube. Any point in space is at a distance greater than $R \sqrt{2}$ from at leas... | R(\sqrt{2}-1) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,955 |
254. Two vertices of a tetrahedron are located on the surface of a sphere with radius $\sqrt{10}$, the other two vertices - on the surface of a sphere with radius 2, concentric with the first. What is the greatest volume of such tetrahedra? | 254. Let $ABCD$ be a tetrahedron of maximum volume, and $O$ be the center of the given spheres. Each segment connecting $O$ with a vertex of the tetrahedron must be perpendicular to the face opposite that vertex. For example, if $AO$ is not perpendicular to the plane $BCD$, then on the surface of the sphere on which po... | 6\sqrt{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,956 |
255. Two trihedral angles are arranged in such a way that the vertex of one is equidistant from the faces of the other and vice versa, the distance between the vertices is \(a\). What is the minimum volume of the hexahedron bounded by the faces of these angles, if all the plane angles of one of them are \(60^{\circ}\),... | 255. Let $A$ be the vertex of a trihedral angle, the plane angles of which are right angles, and $B$ be the vertex of another angle. Take a point $M$ on the segment $AB$ such that $2|AM| = |MB|$. Draw a plane through point $M$ perpendicular to $AB$. The drawn plane will intersect each of these two trihedral angles in a... | \frac{^3\sqrt{3}}{20} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,957 |
257. Given a regular tetrahedron with edge $a$. Find the radius of a sphere centered at the center of the tetrahedron, such that the combined volume of the part of the tetrahedron outside the sphere and the part of the sphere outside the tetrahedron is minimized. | 257. Let $x$ be the radius of the sphere, $V(x)$ be the sum of the volume of the part of the sphere located outside the tetrahedron and the part of the tetrahedron located outside the sphere. It is easy to see that $V^{\prime \prime}(x)=$ $=S_{1}(x)-S_{2}(x)$, where $S_{1}(x)$ is the surface area of the part of the sph... | \frac{1}{3}\sqrt{\frac{2}{3}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,958 |
258. Prove that among triangular pyramids with a given base and equal heights, the one with the smallest lateral surface area is the one where the vertex projects onto the center of the circle inscribed in the base. | 258. Let $a, b, c$ be the sides of the base, $p=\frac{a+b+c}{2}, r-$ the radius of the inscribed circle, $x, y, z$ - the distances from the base of the height of the pyramid to the sides $a, b, c, \quad h$ - the height of the pyramid. Then
$$
S_{\text {side }}=\frac{1}{2} a \sqrt{h^{2}+x^{2}}+\frac{1}{2} b \sqrt{h^{2}... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,959 |
259. Given a cube with edge $a$. Let $N$ be a point on the diagonal of a side face, $M$ be a point on the circle lying in the plane of the base, with the center at the center of the base and radius $\frac{5}{12} a$. Find the minimum value of the magnitude $|M N|$. | 259. If 0 is the center of the circle, $L$ is the projection of $N$ onto the plane of the base, then the point $M$, since $M$ is the closest point to $N$ on the circle, must lie on the segment LO. On the other hand, since $N$ is the closest point to $M$ on the diagonal of the face, $MN$ is perpendicular to this diagona... | \frac{\sqrt{34}}{24} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,960 |
260. a) In the pyramid $S A B C$, the base is a triangle $A B C$, where $\widehat{B A} C=A, \widehat{C B A}=B$, and the radius of the circumscribed circle around it is $R$. The edge $S C$ is perpendicular to the plane $A B C$. Find $| S C |$, if it is known that $\frac{1}{\sin \alpha}+\frac{1}{\sin \beta}-\frac{1}{\sin... | 260. a) Let $|S C|=d, a, b, c$ be the sides of $\triangle A B C, h_{a}, h_{b}, h_{c}$ the altitudes of $\triangle A B C, s$ its area. Then
$$
\sin \alpha=\frac{h_{a}}{\sqrt{a^{2}+b^{2}}}, \quad \sin \beta=\frac{h_{b}}{\sqrt{d^{2}+a^{2}}}, \quad \sin \gamma=\frac{h_{c}}{\sqrt{d^{2}+h_{c}^{2}}}
$$
Thus, we obtain the e... | |SC|=2R\sqrt{\cos(A+B)\cos(A-B)} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,961 |
261. Can a regular tetrahedron with an edge of 1 pass through a circular hole with a radius of: a) 0.45; b) 0.44? The thickness of the hole can be neglected.
## § 3. Geometric Loci
## § 3. Geometric Places of Points | 261. Let $A B C D$ be a given tetrahedron. Take points $M$ and $N$ on the edges $B C$ and $B D$ and solve the following problem: for what positions of points $M$ and $N$ does the radius of the smallest circle containing the triangle $A M N$ (considering circles located in the plane $A M N$) reach its minimum value? (Ob... | 0.4478 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,962 |
262. Prove that in an arbitrary trihedral angle, the bisectors of two dihedral angles and the angle adjacent to the third dihedral angle lie in the same plane. | 262. Let $S$ be the vertex of an angle. Take points $A$, $B$, and $C$ on the edges such that $|S A|=|S B|=|S C|$. The bisectors of angles $A S B$ and $B S C$ pass through the midpoints of segments $A B$ and $B C$, and the bisector of the angle adjacent to angle $C S A$ is parallel to $C A$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,963 |
265. Three mutually perpendicular lines pass through the point $O, A, B$ and $C$ - points on these lines such that
$$
|O A|=|O B|=|O C|
$$
Let $l$ be an arbitrary line passing through $O$; $A_{1}, B_{1}$ and $C_{1}$ - points symmetric to $A, B$ and $C$ with respect to $l$. Through $A_{1}, B_{1}$ and $C_{1}$, three pl... | 265. We will consider the given straight lines as coordinate axes. Let the straight line form angles $\alpha, \beta$ and $\gamma$ with these axes. Then the projections of vectors $\overrightarrow{O A_{i}}, \overrightarrow{O B_{1}}, \overrightarrow{O C} \vec{O}_{i}$ on the axes $O A, O B$ and $O C$ respectively will be ... | (-,-,);(-,,-);(,-,-) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,965 |
266. Find the geometric locus of the midpoints of segments parallel to a given plane, the ends of which are on two skew lines. | 266. Let us denote these lines by $l_{1}$ and $l_{2}$. We will draw through $l_{1}$ a plane $p_{1}$ parallel to $l_{2}$, and through $l_{2}$ a plane $p_{2}$ parallel to $l_{1}$. It is clear that the midpoints of segments with endpoints on $l_{1}$ and $l_{2}$ belong to a plane $p$, parallel to $p_{1}$ and $p_{2}$ and eq... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,966 |
267. Given three pairwise intersecting lines. Find:
a) the geometric locus of the centroids of triangles $ABC$ with vertices on these lines;
b) the geometric locus of the centroids of triangles $ABC$ with vertices on these lines, the planes of which are parallel to a given plane. | 267. a) The entire space.
b) Just as was done in problem 266, it can be proven that the geometric locus of points dividing all possible segments in a given ratio, which are parallel to a given plane and have their endpoints on given intersecting lines, is a straight line. By applying this statement twice (first, we fi... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,967 |
268. Three pairwise intersecting lines $l_{1}, l_{2}, l_{3}$ are perpendicular to the same line and intersect with it. Let $N$ and $M$ be two points on the lines $l_{1}$ and $l_{2}$, respectively, such that the line $N M$ intersects the line $l_{3}$. Find the geometric locus of the midpoints of the segments $N M$. | 268. Let's draw a plane $p$ through the common perpendicular to the lines, perpendicular to $l_{3}$. Let the line $N M$ intersect $l_{3}$ at point $L, N_{1}, M_{1}, L_{1}$ - points of intersection of the lines $l_{1}, l_{2}, l_{3}$ with the common perpendicular, $N_{2}, M_{2}$ - projections of $N$ and $M$ onto the draw... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,968 |
269. In space, several arbitrary lines and a point $A$ are given. Through $A$, a line is drawn such that the sum of the cosines of the acute angles formed by this line with the given ones is equal to a given number. Find the geometric locus of such lines. | 269. Introduce a rectangular coordinate system, choosing its origin at point $A$. Let $\boldsymbol{e}_{1}\left(a_{1}, b_{1}, c_{1}\right), \boldsymbol{e}_{2}\left(a_{2}, b_{2}, c_{2}\right), \ldots, \boldsymbol{e}_{n}\left(a_{n}, b_{n}\right.$, $\left.c_{\boldsymbol{n}}\right)$ be unit vectors parallel to the given lin... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,969 |
270. Given a triangle $A B C$ and a line $l$. $A_{1}, B_{1}, C_{1}$ are three arbitrary points on the line $l$. Find the geometric locus of the centroids of triangles with vertices at the midpoints of segments $A A_{1}, B B_{1}, C C_{1}$. | 270. Place equal weights at points $A, B, C, A_{1}, B_{1}$ and $C_{1}$. Then the center of gravity of the resulting system of weights will coincide with the center of gravity of the triangle with vertices at the midpoints of segments $A A_{1}$, $B B_{1}$, $C C_{1}$.
On the other hand, the center of gravity of this sys... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,970 |
271. Given a line $l$ and a point $A$. Through $A$, an arbitrary line intersecting $l$ is drawn. Let $M N$ be the common perpendicular to this line and to $l$ ( $M$ - on the line passing through $A$ ). Find the geometric locus of points $M$. | 271. Draw a line $t$ through $A$, parallel to $l$. The desired geometric locus of points represents a cylindrical surface, in which $l$ and $t$ are diametrically opposite generators, excluding the lines $l$ and $t$ themselves. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,971 |
272. Two spheres $\alpha$ and $\beta$ touch the sphere $\omega$ at points $A$ and $B$. On the sphere $\alpha$, a point $M$ is taken, the line $M A$ intersects the sphere $\omega$ again at point $N$, the line $N B$ intersects the sphere $\beta$ again at point $K$. Find the geometric locus of such points $M$, for which t... | 272. First, let us prove that if the line $M K$ is tangent to the sphere $\beta$, then it is also tangent to the sphere $\alpha$. Consider the section of the given spheres by the plane passing through the points $M, K, A, B$, and $N$ (Fig. 55). $\widehat{M K B}$ is measured by half the arc $\overline{K B}$ enclosed wit... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,972 |
273. Given a plane and two points on one side of it. Find the geometric locus of the centers of spheres passing through these points and touching the plane. | 273. Let $A$ and $B$ be given points, $C$ be the point of intersection of the line $A B$ with a given plane, and $M$ be the point of tangency of some sphere with the plane. Since $|C M|^{2}=|C A| \cdot|C B|, \quad M$ lies on a circle with center at point $C$ and radius $\sqrt{|C A| \cdot|C B|}$. Therefore, the center o... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,973 |
274. Find the geometric locus of the midpoints of the common tangents to two given spheres. | 274. Let $O_{1}, O_{2}$ and $R_{1}, R_{2}$ be the centers and radii of the given spheres, respectively; $M$ - the midpoint of some common tangent. Then it is easy to see that
$$
\left|O_{\mathrm{i}} M\right|^{2}-\left|O_{2} M\right|^{2}=R_{1}^{2}-R_{2}^{2}
$$
and, consequently, $M$ lies in a plane perpendicular to th... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,974 |
275. Lines $l_{1}$ and $l_{2}$ are tangent to a certain sphere. Let $M$ and $N$ be two points on $l_{1}$ and $l_{2}$, respectively, such that the line $M N$ is also tangent to the same sphere. Find the geometric locus of the points of tangency of the line $M N$ with this sphere. | 275. Let $A$ and $B$ be the points of tangency of the lines $l_{1}$ and $l_{2}$ with the sphere, and let $K$ be the point of tangency of the line $MN$ with the sphere. We will have
$$
|A M|=|M K|, \quad|B N|=|N K| .
$$
Project $l_{\text {і }}$ and $l_{2}$ onto a plane perpendicular to $A B$. Let $A_{1}, M_{1}, N_{1}$... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,975 |
279. Given a triangle $A B C$. On the line perpendicular to the plane $A B C$ and passing through $A$, a random point $D$ is taken. Find the geometric locus of the points of intersection of the altitudes of triangles $D B C$. | 279. Let $B K$ be the altitude of $\triangle A B C$, $H$ the orthocenter of $\triangle A B C$, $B M$ the altitude of $\triangle D B C$, and $N$ the orthocenter of $\triangle D B C$. We will prove that $N$ is the projection of point $H$ onto the plane of $\triangle B C$.
Indeed, $K M$ is perpendicular to $D C$ since $B... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,976 |
283. Given a planar quadrilateral $A B C D$. Find the geometric locus of points $M$ such that the lateral surface of the pyramid $A B C D M$ can be intersected by a plane so that the intersection is: a) a rectangle, b) a rhombus, c) a square, d) in the previous case, find the geometric locus of the centers of the squar... | 283. Let $P$ and $Q$ be the points of intersection of the opposite sides of quadrilateral $ABCD$. If the section of the lateral surface of the pyramid $ABCDM$ by a plane is a parallelogram, then the section plane must be parallel to the plane $PQM$, and the sides of the parallelogram will be parallel to the lines $PM$ ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,977 |
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