problem stringlengths 1 13.6k | solution stringlengths 0 18.5k ⌀ | answer stringlengths 0 575 ⌀ | problem_type stringclasses 8
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class | __index_level_0__ int64 0 742k |
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38. The center of the sphere is located in the plane of the base of a regular triangular pyramid. The vertices of the base lie on the surface of the sphere. Find $l$ - the length of the line of intersection of the surfaces of the sphere and the pyramid, if the radius of the sphere is $R$, and the plane angle at the ver... | 38. If $0<\alpha<\arccos \frac{1}{4}$,
$$
l=R \sqrt{27+3 \operatorname{tg}^{2} \frac{\alpha}{2}}\left[\operatorname{arctg}\left(3 \operatorname{ctg} \frac{\alpha}{2}\right)-\alpha\right] ;
$$
if $\alpha \geqslant \arccos \frac{1}{4}, l=0$. | =R\sqrt{27+3\operatorname{tg}^{2}\frac{\alpha}{2}}[\operatorname{arctg}(3\operatorname{ctg}\frac{\alpha}{2})-\alpha];if\alpha\geqslant\arccos\frac{1}{4},=0 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,747 |
39. In a regular hexagonal pyramid $S A B C D E F$ ( $S$ - the vertex) on the diagonal $A D$ three points are taken, dividing the diagonal into four equal parts. Through these points, sections are drawn parallel to the plane $S A B$. Find the ratios of the areas of the resulting sections. | 39. $25: 20: 9. | 25:20:9 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,748 |
41. At the base of a triangular pyramid, all lateral edges of which are pairwise perpendicular; lies a triangle with an area of $S$. The area of one of the lateral faces is $Q$. Find the area of the projection of this face onto the base. | 41. $\frac{Q^{2}}{S}$. | \frac{Q^{2}}{S} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,750 |
42. $A B C A_{1} B_{1} C_{1}$ - a regular triangular prism, all edges of which are equal to each other. $K$ is a point on edge $A B$, different from $A$ and $B, M$ is on line $B_{1} C_{1}, L$ is in the plane of face $A C C_{1} A_{1}$. Line $K L$ forms equal angles with planes $A B C$ and $A B B_{1} A_{1}, L M$ forms eq... | 42. Let the side of the base and the height of the prism be denoted by $a$, $|K B|=x$. From the problem statement, it follows that the projection of $K M$ onto the base plane is parallel to the bisector of angle $C$ of triangle $A B C$, i.e., $\left|B_{1} M\right|=2 x,\left|M C_{1}\right|=a-2 x$. Let $L_{1}$ be the pro... | \frac{7}{\sqrt{97}},\frac{\sqrt{6}+\sqrt{14}}{8} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,751 |
43. In a regular quadrilateral pyramid, the angle between a lateral edge and the plane of the base is equal to the angle between the lateral edge and the plane of the lateral face that does not contain this edge. Find this angle. | 43. $\operatorname{arctg} \sqrt{\frac{3}{2}}$.
43. $\operatorname{arctan} \sqrt{\frac{3}{2}}$. | \operatorname{arctan}\sqrt{\frac{3}{2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,752 |
44. Find the dihedral angle between the base and the lateral face of a regular truncated triangular pyramid, given that a sphere can be inscribed in it and, moreover, there exists a sphere that touches all its edges. | 44. Extend the lateral faces until they intersect. In this case, we obtain two similar pyramids, the bases of which are the larger and smaller bases of the given truncated pyramid. Let $a$ be the side of the larger base of the truncated pyramid, $\alpha$ be the dihedral angle at this base: We can find: the height of th... | 2\operatorname{arctg}(\sqrt{3}-\sqrt{2}) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,753 |
45. Three edges of a triangular pyramid are equal to 1, while the other three are equal to $a$. No face is an equilateral triangle. Within what limits can $a$ vary? What is the volume of this pyramid? | 45. $\frac{-1+\sqrt{5}}{2}<a<\frac{1+\sqrt{5}}{2}, a \neq 1$;
$$
V=\frac{1}{12} \sqrt{\left(a^{2}+1\right)\left(3 a^{2}-1-a^{4}\right)}
$$ | \frac{1}{12}\sqrt{(^{2}+1)(3^{2}-1-^{4})} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,754 |
46. The lateral faces of a triangular pyramid are equal in area and are inclined to the base plane at angles $\alpha, \beta$, and $\gamma$. Find the ratio of the radius of the sphere inscribed in this pyramid to the radius of the sphere that touches the base of the pyramid and the extensions of the three lateral faces. | 46. $\frac{3-\cos \alpha-\cos \beta-\cos \gamma}{3+\cos \alpha+\cos \beta+\cos \gamma}$. | \frac{3-\cos\alpha-\cos\beta-\cos\gamma}{3+\cos\alpha+\cos\beta+\cos\gamma} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,755 |
47. All edges of a regular hexagonal prism are equal to $a$. Find the area of the section made through the side of the base at an angle $\alpha$ to the base plane. | 47. If $0<\alpha<\frac{\pi}{6}$, then $S=\frac{3 a^{2} \sqrt{3}}{2 \cos \alpha} ;$ if $\frac{\pi}{6} \leqslant \alpha<$ $<\operatorname{arctg} \frac{2}{\sqrt{3}}$, then $S=\frac{a^{2}}{6 \cos \alpha}\left(18 \operatorname{ctg} \alpha-3 \sqrt{3}-2 \sqrt{3} \operatorname{ctg}^{2} \alpha\right)$; if $\operatorname{arctg} ... | S=\frac{3^{2}\sqrt{3}}{2\cos\alpha}if0<\alpha<\frac{\pi}{6};\,S=\frac{^{2}}{6\cos\alpha}(18\operatorname{ctg}\alpha-3\sqrt{3}-2\sqrt{3}\operatorname{ctg}^{2}\alpha) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,756 |
48. In a rectangular parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$, it is known that $|A B|=a,|A D|=b,\left|A A_{1}\right|=c$. Find the angle between the planes $A B_{1} D_{1}$ and $A_{1} C_{1} D$. | 48. $\arccos \left(\frac{a^{2} b^{2}+b^{2} c^{2}-c^{2} a^{2}}{a^{2} b^{2}+b^{2} c^{2}+c^{2} a^{2}}\right)$. | \arccos(\frac{^{2}b^{2}+b^{2}^{2}-^{2}^{2}}{^{2}b^{2}+b^{2}^{2}+^{2}^{2}}) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,757 |
49. At the base of the pyramid $A B C D M$ lies a square $A B C D$ with side $a$, the lateral edges $A M$ and $B M$ are also equal to $a$, the lateral edges $C M$ and $D M$ have length $b$. On the face $C D M$, as the base, a triangular pyramid $C D M N$ is constructed outward, with lateral edges of length $a$. Find th... | 49. The polyhedron $A B M D C N$ is a triangular prism with base $A B M$, lateral edges $A D, B C', M N$,
$$
\text { Answer: } \frac{b}{2 a} \sqrt{4 a^{2}-b^{2}}
$$ | \frac{b}{2}\sqrt{4^{2}-b^{2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,758 |
51. At the base of a triangular pyramid lies a triangle with sides $a, b$ and $c$, the opposite lateral edges of the pyramid are equal to $m, n$ and $p$ respectively. Find the distance from the vertex of the pyramid to the centroid of the base. | 51. $\frac{1}{3} \sqrt{3 m^{2}+3 n^{2}+3 p^{2}-a^{2}-b^{2}-c^{2}}$. | \frac{1}{3}\sqrt{3^{2}+3n^{2}+3p^{2}-^{2}-b^{2}-^{2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,760 |
52. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$; a plane is drawn through the edge $A A_{1}$, forming equal angles with the lines $B C$ and $B_{1} D$. Find these angles. | 52. Let's take a point \( K \) on the extension of edge \( C C_{1} \) such that \( B_{1} K \| B C_{1} \), and through edge \( B B_{1} \) we draw a plane parallel to the given one (Fig. 1). This plane should pass either through the internal or the external bisector of angle \( D B_{1} K \). Since the ratio in which the ... | \arcsin\frac{\sqrt{6}\1}{5} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,761 |
53. The lateral edges of a triangular pyramid are pairwise perpendicular, and one of them is equal to $a$ and is equal to the sum of the other two. Find the radius of the sphere that touches the base of the pyramid and the extensions of its lateral faces. | 53. Let $ABCD$ be a given pyramid, the lateral edges of which $|DA|=a$, $|DB|=x$, $|DC|=y$; by condition, these edges are perpendicular and $x+y \doteq a$. It is not difficult to find that
$$
S_{ABC}=\frac{1}{2} \sqrt{a^{2}\left(x^{2}+y^{2}\right)+x^{2} y^{2}}, \quad V_{ABCD}=\frac{1}{6} a x y
$$
On the other hand, i... | \frac{}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,762 |
55. At the base of the triangular pyramid $S A B C$ lies an isosceles right triangle $A B C$ ( $\widehat{A}=$ $=90^{\circ}$ ). The angles $\widehat{S A B}, S \widehat{C A}, S \widehat{A C}, S \widehat{B A}$ (in the given order) form an arithmetic progression with a non-zero common difference. The areas of the faces $S ... | 55. Let the angles $\widehat{S A B}, \widehat{S C A}, \widehat{S A C}, \widehat{S B A}$ be $\alpha-2 \varphi, \alpha-\varphi$, $\alpha, \alpha+\varphi$. From the Law of Sines in $\triangle S A B$ we find:
$$
|S A|=|A B| \frac{\sin (\alpha+\varphi)}{\sin (2 \alpha-\varphi)}
$$
and from $\triangle S A C$ we find:
$$
|... | \frac{\pi}{2}-\arccos(\sqrt{2}-1),\frac{\pi}{2}-\frac{1}{2}\arccos(\sqrt{2}-1),\frac{\pi}{2},\frac{\pi}{2}+\frac{1}{2}\arccos(\sqrt{2}-1) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,764 |
56. At the base of the triangular pyramid $S A B C$ lies an equilateral triangle $A B C$ with side $a$. Find the volume of this pyramid, given that $\widehat{A S C}=\widehat{A S B}=\alpha$, $\widehat{S A B}=\beta$. | 56. Let $|S A|=l, \quad l$ can be easily expressed through $a, \alpha$ and $\beta$. If $l \leqslant a$, then $\triangle A S C=\triangle A S B$. (We will construct $\triangle A S C:$ take an angle with vertex $S$ of size $\alpha$, lay off $|S A|=l$ on one side, construct a circle of radius $a$ centered at $A$; since $a ... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,765 | |
57. Point $K$ is the midpoint of edge $A A_{1}$ of the cube $A B C D A_{1} B_{1} C_{1} D_{1}$, point $L$ lies on edge $B C$. Segment $K L$
is tangent to the sphere inscribed in the cube. In what ratio does the point of tangency divide segment $K L$? | 57. $\frac{4}{5}$, starting from point $K$. | \frac{4}{5} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,766 |
58. In the tetrahedron $ABCD$, it is given that $A \widehat{BC} = \widehat{BAD} = 90^{\circ}$, $|AB| = a$, $|DC| = b$, and the angle between the edges $AD$ and $BC$ is $\alpha$. Find the radius of the circumscribed sphere. | 58. Let's take $C_{1}$ such that $A B C C_{1}$ is a rectangle (Fig. 2). $D_{1}$ is the midpoint of $A C_{1}$, $O_{1}$ and $O_{2}$ are the centers of the circumcircles of triangles $A C_{1} D$ and $A B C$, respectively, and $O$ is the center of the sphere circumscribed around $A B C D$. Clearly, $O_{2}$ is the midpoint ... | \frac{1}{2\sin\alpha}\sqrt{b^{2}-^{2}\cos^{2}\alpha} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,767 |
59. The edge of a cube and the edge of a regular tetrahedron lie on the same line, and the midpoints of the opposite edges of the cube and the tetrahedron coincide. Find the volume of the common part of the cube and the tetrahedron, if the edge of the cube is equal to \(a\). | 59. Let $K$ be the midpoint of edge $AB$ of the cube $ABCD A_{1} B_{1} C_{1} D_{1}$, and $M$ be the midpoint of edge $D_{1} C_{1}$. $K$ and $M$ are simultaneously the midpoints of edges $PQ$ and $RS$ of the regular tetrahedron $PQRS$. $D_{1} C_{1}$ lies on $RS$. If the edge of the tetrahedron is $b$, then $|MK| = b \sq... | \frac{^{3}\sqrt{2}}{12}(16\sqrt{2}-17) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,768 |
60. In what ratio does a plane, parallel to two skew edges of a triangular pyramid and dividing one of the other edges in the ratio $2: 1$, divide the volume of the pyramid? | 60. Let the lengths of these intersecting edges be denoted by $a$ and $b$, the distance between them by $d$, and the angle by $\varphi$. Using the formula from problem 15, we find the volumes of the resulting parts:
$$
V_{1}=\frac{10}{81} a b d \sin \varphi, \quad V_{2}=\frac{7}{162} a b d \sin \varphi
$$
Answer: $\f... | \frac{20}{7} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,769 |
61. In a regular truncated quadrilateral pyramid, a section is made through the diagonals of the bases and a section passing through the side of the lower base and the opposite side of the upper base. The angle between the cutting planes is $\alpha$. Find the ratio of the areas of the sections. | 61. The area of the projection of the second section onto the first plane is half the area of the first section. On the other hand (see problem 8), the ratio of the area of the projection of the second section to the area of the section itself is $\cos \alpha$.
Answer: $2 \cos \alpha$. | 2\cos\alpha | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,770 |
63. Given a sphere and a point inside it. Through the point, three mutually perpendicular planes are drawn, intersecting the sphere in three circles. Prove that the sum of the areas of these three circles is constant, and find this sum if the radius of the sphere is \( R \), and the distance from the point of intersect... | 63. If $x, y$ and $z$ are the distances from the center of the sphere to the planes, then $x^{2}+y^{2}+z^{2}=d^{2}$, and the sum of the areas of the three circles will be
$$
\pi\left[\left(R^{2}-x^{2}\right)+\left(R^{2}-y^{2}\right)+\left(R^{2}-z^{2}\right)\right]=\pi\left(3 R^{2}-d^{2}\right)
$$ | \pi(3R^2-^2) | Geometry | proof | Yes | Yes | olympiads | false | 29,772 |
64. In a sphere of radius $R$, a diameter $A B$ is drawn. Two lines are tangent to the sphere at points $A$ and $B$ and form an angle $\alpha\left(\alpha<90^{\circ}\right)$ between them. Points $C$ and $D$ are taken on these lines such that $C D$ is also tangent to the sphere and the angle between $A B$ and $C D$ is $\... | 64. Let $|A C|=x,|B D|=y$ (where $A C$ and $B D$ are tangent to the sphere). $D_{1}$ is the projection of $D$ onto the plane passing through $A C$ parallel to $B D$. We have
$$
|C D|=x+y=\frac{2 R}{\cos \varphi}, \quad\left|C D_{1}\right|=2 R \operatorname{tg} \varphi
$$
In $\triangle C A D_{1}$, the angle $\widehat{... | \frac{2}{3}R^{3}\operatorname{tg}\frac{\alpha}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,773 |
65. In a tetrahedron, two opposite edges are perpendicular, their lengths are $a$ and $b$, and the distance between them is $c$. A cube is inscribed in the tetrahedron, four of its edges are perpendicular to these two edges of the tetrahedron, and on each face of the tetrahedron lie exactly two vertices of the cube. Fi... | 65. Let the common perpendicular to the given edges be divided by the cube into segments $y, x$ and $z, y+x+z=c(x-$ edge of the cube, $y$ adjacent to the edge $a$). The faces of the cube, parallel to the given edges, intersect the tetrahedron in two rectangles, the sides of the first being $\frac{x+z}{c} a, \frac{y b}{... | \frac{}{++} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,774 |
66. Two equal triangles $K L M$ and $K L N$ have a common side $K L, \quad \widehat{K L M}=\widehat{L K N}=\pi / 3, \quad|K L|=a$, $|L M|=|K N|=6 a$. The planes $K L M$ and $K L N$ are mutually perpendicular. A sphere touches the segments $L M$ and $K N$ at their midpoints. Find the radius of the sphere. | 66. Let $O_{1}$ and $O_{2}$ be the projections of the center of the sphere $O$ onto the planes $K L M$ and $K L N$, and let $P$ be the midpoint of $M L$.
The projections of $O_{1}$ and $O_{2}$ onto $K L$ should coincide. It can be proven that these projections fall on the midpoint of $K L$ - point $Q$ (Fig. 4). Since ... | \frac{}{2}\sqrt{\frac{137}{3}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,775 |
67. A sphere of radius $R$ touches all the lateral faces of a triangular pyramid at the midpoints of the sides of its base. The segment connecting the vertex of the pyramid with the center of the sphere is bisected by the point of intersection with the base of the pyramid. Find the volume of the pyramid. | 67. Using the equality of tangents emanating from one point, we will prove that at the base lies a regular triangle and the medians of the lateral faces, drawn to the sides of the base, are equal. From this it will follow that the pyramid is regular.
Answer: $\frac{R^{3} \sqrt{6}}{4}$. | \frac{R^{3}\sqrt{6}}{4} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,776 |
68. In a tetrahedron, three dihedral angles are right angles. One of the segments connecting the midpoints of opposite edges of the tetrahedron is equal to \(a\), and the other is \(b (b > a)\). Find the length of the greatest edge of the tetrahedron. | 68. Three right angles cannot adjoin one face; furthermore, they cannot adjoin one vertex, since in this case all segments connecting the midpoints of opposite edges would be equal. The remaining case is when the three edges corresponding to the right angles form an open broken line. Let these be the edges $A B, B C$, ... | \sqrt{\frac{2}{b^{2}+3^{2}}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,777 |
69. A right circular cone with vertex $S$ is inscribed in a triangular pyramid $S P Q R$ such that the circle of the cone's base is inscribed in the base $P Q R$ of the pyramid. It is known that $\widehat{P S} R=\pi / 2, \widehat{S Q R}=\pi / 4, \widehat{P S} Q=7 \pi / 12$. Find the ratio of the lateral surface area of... | 69. $\pi \frac{4 \sqrt{3}-3}{13}$. | \pi\frac{4\sqrt{3}-3}{13} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,778 |
70. The base of the pyramid $A B C D E$ is a parallelogram $A B C D$. No lateral face is an obtuse triangle. There exists a point $M$ on the edge $D C$ such that the line $E M$ is perpendicular to $B C$. In addition, the diagonal of the base $A C$ and the lateral edges $E D$ and $E B$ are related by the following ratio... | 70. First, we will prove that $A B C D$ is a rectangle and the plane $D E C$ is perpendicular to the plane $A B C D$. For this, we will draw a section through $E$ perpendicular to $B C$. This section must intersect the base along a line passing through $M$ and intersecting the segments $B C$ and $A D$ (possibly at thei... | \frac{3}{\delta}\sqrt{\frac{65}{14}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,779 |
71. A segment $AB$ of unit length, which is a chord of a sphere with radius 1, is positioned at an angle of $\pi / 3$ to the diameter $CD$ of this sphere. The distance from the end $C$ of the diameter to the nearest end $A$ of the chord $AB$ is $\sqrt{2}$. Determine the length of the segment $BD$. | 71. Draw a line through $C$ parallel to $A B$, and take a point $E$ on it such that $|C E|=|A B|$, making $A B E C$ a parallelogram. If $O$ is the center of the sphere, then since $\widehat{O C E}=\pi / 3$ and $|C E|=1$ (as follows from the condition), $\triangle O C E$ is equilateral. Therefore, point $O$ is equidista... | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,780 |
72. In the triangular pyramid $A B C D$, the faces $A B C$ and $A B D$ have areas $p$ and $q$ and form an angle $\alpha$ between them. Find the area of the section of the pyramid passing through the edge $A B$ and the center of the sphere inscribed in the pyramid. | 72. If $x$ is the area of the desired section, $|A B|=a$, 10, using the formula for the volume of the pyramid $A B C D$ and its parts from problem 11, we get
$$
\frac{2}{3} \frac{p x \sin \frac{\alpha}{2}}{a}+\frac{2}{3} \frac{q x \sin \frac{\alpha}{2}}{a}=\frac{2}{3} \frac{p q \sin \alpha}{a}
$$
from which
$$
x=\fr... | \frac{2pq\cos\frac{\alpha}{2}}{p+q} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,781 |
74. $A B C D$ is a regular tetrahedron with edge $a$. Let $M$ be the center of the face $A D C$, and $N$ be the midpoint of the edge $B C$. Find the radius of the sphere inscribed in the trihedral angle $A$ and tangent to the line $M N$. | 74. When the sphere is intersected by the plane $A M N$, we obtain a circle inscribed in the triangle $A M N$. In this triangle, $|A N|=a \frac{\sqrt{\overline{3}}}{2}, \quad|A M|=a \frac{\sqrt{\overline{3}}}{3}, \quad|M N|=\frac{a}{2} \quad$ (determined from $\triangle C M N$). Therefore, if $L$ is the point of tangen... | \frac{5\sqrt{\overline{6}}-3\sqrt{2}}{48} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,783 |
75. At the base of the triangular pyramid $A B C D$ lies an equilateral triangle $A B C$. The face $B C D$ forms an angle of $60^{\circ}$ with the base plane. On the line passing through point $D$ perpendicular to the base, lies the center of a sphere with a unit radius, which touches the edges $A B$, $A C$, and the fa... | 75. $\frac{9 \sqrt{3}}{8}$. | \frac{9\sqrt{3}}{8} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,784 |
78. A sphere touches the base plane $ABCD$ of a regular quadrilateral pyramid $SABCD$ at point $A$ and, in addition, touches the inscribed sphere of the pyramid. A cutting plane is drawn through the center of the first sphere and the side of the base $BC$. Find the angle of inclination of this plane to the base plane, ... | 78. $\operatorname{arctg} \frac{1}{2 \sqrt{3}}$.
The arctangent (arctg) of $\frac{1}{2 \sqrt{3}}$ is the angle whose tangent is $\frac{1}{2 \sqrt{3}}$. This can be simplified to $\frac{\sqrt{3}}{6}$. The angle whose tangent is $\frac{\sqrt{3}}{6}$ is $\frac{\pi}{6}$ or 30 degrees. However, the exact value of $\operato... | \operatorname{arctg}\frac{1}{2\sqrt{3}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,787 |
79. On a sphere with a radius of 2, there are three circles with a radius of 1, each touching the other two. Find the radius of the circle, smaller than the given ones, which is also located on the given sphere and touches each of the given circles. | 79. Notations: $O$ - center of the sphere, $O_{1}, O_{2}, O_{3}$ - centers of the given circles, $O_{4}$ - center of the required circle. Clearly, $\triangle O_{1} O_{2} O_{3}$ is equilateral. Let's find its sides ( $M$ - the point of tangency of the circles with centers $O_{1}$ and $\left.O_{2}\right) \cdot\left|O_{1}... | 1-\sqrt{\frac{2}{3}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,788 |
80. In a rectangular parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$, the lengths of the edges $A B, B C$, and $B B_{1}$ are equal to $2 a, a$, and $a$, respectively. Point $E$ is the midpoint of edge $B C$. Vertices $M$ and $N$ of a regular tetrahedron $M N P Q$ lie on the line $C_{1} E$, and vertices $P$ and $Q$ lie... | 80. Since opposite edges of a regular tetrahedron are perpendicular, the lines \( C_{1} E \) and \( B_{1} F \) (Fig. 5) must be perpendicular.
Fig. 5,
If \( K \) is the midpoint of \( C_{1... | \frac{4}{3\sqrt{5}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,789 |
81. The edge of the cube $A B C D A_{1} B_{1} C_{1} D_{1}$ has length $a$. Points $M$ and $N$ lie on segments $B D$ and $C C_{1}$, respectively. The line $M N$ forms an angle $\pi / 4$ with the plane $A B C D$ and an angle $\pi / 6$ with the plane $B B_{1} C_{1} C$. Find: a) the length of segment $M N$; b) the radius o... | 81. a) $a$; б) $\frac{a(2-\sqrt{2})}{2}$. | a | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,790 |
82. The vertex $A$ of a regular prism $A B C A_{1} B_{1} C_{1}$ coincides with the vertex of a cone, vertices $B$ and $C$ lie on the lateral surface of this cone, and vertices $B_{1}$ and $C_{1}$ lie on the circumference of its base. Find the ratio of the volumes of the cone and the prism, if $\left|A A_{1}\right|=2.4|... | 82. Let $|A B|=a$, then $\left|A B_{1}\right|=\left|A C_{1}\right|=2.6 a$. Take points $K$ and $L$ on the lines $A B$ and $A C$ such that $|A K|=|A L|=\left|A B_{1}\right|=$ $=\left|A C_{1}\right|=2.6 a$. The isosceles trapezoid $K L C_{1} B_{1}$ is inscribed in the base circle of the cone. All sides of this trapezoid ... | \frac{15379\pi}{4800\sqrt{3}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,791 |
83. The length of the edge of the cube $A B C D A_{1} B_{1} C_{1} D_{1}$ is $a$. Points $P, K, L$ are the midpoints of the edges $A A_{1}, A_{1} D_{1}, B_{1} C_{1}$ respectively, and point $Q$ is the center of the face $C C_{1} D_{1} D$. The segment $M N$ has endpoints on the lines $A D$ and $K L$ and intersects the li... | 83. Note that the segment $M N$ is bisected by its intersection with the line $P Q$. Let's project this segment onto the plane $A B C D$. If $N_{1}$ is the projection of $N$, $K_{1}$ is the midpoint of $A D$, and $Q_{1}$ is the midpoint of $D C$ (where $K_{1}$ and $Q_{1}$ are the projections of $K$ and $Q$), then $N_{1... | \frac{}{3}\sqrt{14} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,792 |
84. In a regular prism $A B C A_{1} B_{1} C_{1}$, the length of the lateral edge and the height of the base are equal to $a$. Through vertex $A$, two planes are drawn: one perpendicular to the line $A B_{1}$, and the second perpendicular to the line $A C_{1}$. Through vertex $A_{1}$, two more planes are drawn: one perp... | 84. Let's draw a plane through the edge $A A_{\mathrm{i}}$ perpendicular to the plane $B C C_{1} B_{1}$ (Fig. 6). $M$ and $N$ are the points of intersection of this plane with $C_{1} B_{1}$ and $C B_{*}$. Take a point $K$ on $M N$ such that $|N K|=|M N|$. According to the condition, $A A_{1}{ }^{*} M N$ is a square, so... | \frac{9^{3}\sqrt{3}}{4} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,793 |
85. Point $O$ is the common vertex of two equal cones located on the same side of plane $\alpha$ such that only one generatrix of each cone ( $O A$ for one cone and $O B$ for the other) lies in plane $\alpha$. It is known that the angle between the heights of the cones is $\beta$, and the angle between the height and t... | 85. The sought angle complements to $\pi / 2$ the angle between the line $O A$ and the axis of the second cone. Let $P$ and $Q$ be the centers of the bases of the given cones, and $S$ be the point where the planes of the bases of the cones intersect the perpendicular erected to the plane $O A B$ at point $O$ (Fig. 7). ... | \frac{\pi}{2}-\arccos(\cos\varphi-\frac{2\sin^{2}\frac{\beta}{2}}{\cos\varphi}) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,794 |
87. In a regular quadrilateral pyramid $S A B C D$ ( $A B C D$ - base), the side of the base is equal to $a$, and the angle between the lateral edge and the plane of the base is $\alpha$. A plane, parallel to the diagonal of the base $A C$ and the lateral edge $B S$, intersects the pyramid such that a circle can be ins... | 87. If a plane intersects the edges $A D$ and $C D$, then the section will be a triangle, and the radius of the inscribed circle will vary from 0 to $\frac{a}{\sqrt{2}\left(2 \cos \alpha+\sqrt{4 \cos ^{2} \alpha+1}\right)}$.
Now let the plane intersect the edges $A B$ and $B C$ at points $P$ and $N$, $S A$ and $S C$ a... | \frac{\sqrt{2}}{1+2\cos\alpha+\sqrt{4\cos^{2}\alpha+1}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,796 |
88. The edge of a regular tetrahedron is $a$. The plane $\boldsymbol{P}$ passes through the vertex $B$ and the midpoints of the edges $A C$ and $A D$. A sphere touches the lines $A B, A C, A D$ and that part of the plane $P$ which is enclosed within the tetrahedron. Find the radius of the sphere. | 88. Let's conduct a section with a plane passing through edge $A B$ and point $L$ - the midpoint of $C D, K$ - the intersection point of plane $P$ and $A L$, the height dropped from $A$ to $B L$ intersects $B K$ at point $N$, and $B L-$ at point $Q$ (Fig. 9). It is not difficult to prove that the center of the sphere l... | \frac{\sqrt{2}}{5\\sqrt{11}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,797 |
89. In a regular tetrahedron, points $M$ and $N$ are the midpoints of opposite edges. The projection of the tetrahedron onto a plane parallel to $M N$ is a quadrilateral with area $S$, one of whose angles is $60^{\circ}$. Find the surface area of the tetrahedron. | 89. Let $x$ be the edge of the tetrahedron, $|M N|=-\frac{x}{\sqrt{2}}$. If the edge, the midpoint of which is $M$, forms an angle $\alpha$ with the given plane, then the opposite edge forms an angle $\frac{\pi}{2}-\alpha$. The projection of the tetrahedron onto this plane is an isosceles trapezoid with bases $x \cos \... | 3S\sqrt{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,798 |
90. In the cube $A B C D A_{1} B_{1} C_{1} D_{1}$, point $M$ is taken on $A C$ and point $N$ is taken on the diagonal $B D_{1}$ of the cube such that $\widehat{N M C}=$ $=60^{\circ}, \widehat{M N B}=45^{\circ}$. In what ratio do points $M$ and $N$ divide the segments $A C$ and $B D_{1}$? | 90. Let the edge of the cube be 1. Denote by $O$ the center of the face $A B C D$. From the fact that $\widehat{N M C}=60^{\circ}$ and $\widehat{N O C}=90^{\circ}$, it follows that $O$ is between $M$ and $C$. Denote $|O M|=x,|N B|=y$. Then $|M N|=$ $=2 x,|N O|=x \sqrt{3},|M B|=\sqrt{\frac{1}{2}+x^{2}}$. Applying the co... | |AM|:|MC|=2-\sqrt{3},|BN|:|ND_{1}|=2 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,799 |
91. The base of the right prism $A B C D A_{1} B_{1} C_{1} D_{1}$ is an isosceles trapezoid $A B C D$, in which $A D$ is parallel to $B C, |A D| / |B C| = n, n > 1$. Planes are drawn through the edges $A A_{1}$ and $B C$ parallel to the diagonal $B_{1} D$; planes are drawn through the edges $D D_{1}$ and $B_{1} C_{1}$ ... | 91. A plane passing through $A A_{1}$ parallel to $B_{1} D$ will be parallel to the plane $D D_{1} B_{1} B$. Similarly, a plane passing through $D D_{1}$ parallel to $A_{1} C$ will be parallel to the plane $A A_{1} C_{1} C$.
On the other hand, planes passing through the edges $B C$ and $B_{1} C_{1}$ will be parallel t... | \frac{(5n+3)^{3}}{12(n+1)^{3}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,800 |
92. The side of the base $A B C$ of a regular triangular prism $A B C A_{1} B_{1} C_{1}$ is equal to $a$. Points $M$ and $N$ are the midpoints of edges $A_{1} B_{1}$ and $A A_{1}$, respectively. The projection of segment $B M$ onto line $C_{1} N$ is $a / 2 \sqrt{5}$. Determine the height of the prism. | 92. Let the height of the prism be $x$. Take a point $K$ on the extension of the edge $B_{1} B$ such that $|B K|=\frac{3}{2} x,\left|B_{1} K\right|=\frac{5}{2} x$. Since $K N$ is parallel to $B M$ and $|K N|=2|B M|$, the projection of $K N$ onto $C N$ is twice the projection of $B M$ onto $C N$, i.e., it is equal to $\... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,801 | |
93. Two spheres touch each other and the faces of a dihedral angle, the magnitude of which is $\alpha$. Let $A$ and $B$ be the points of contact of these spheres with the faces ( $A$ and $B$ belong to different spheres and different faces). In what ratio is the segment $A B$ divided by the points of intersection with t... | 93. Let $A_{1}$ and $B_{1}$ be the other two points of tangency, $R$ and $r$ be the radii of the circles. In the trapezoid $A A_{1} B B_{1}$, we find the bases: $\left|A A_{1}\right|=$ $=2 R \cos \frac{\alpha}{2},\left|B B_{1}\right|=2 r \cos \frac{\alpha}{2}$ and the lateral sides $\left|A B_{1}\right|=$ $=\left|A_{1}... | \cos^{2}\frac{\alpha}{2}:\sin^{2}\frac{\alpha}{2}:\cos^{2}\frac{\alpha}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,802 |
94. The base of the pyramid $ABCD$ is an equilateral triangle $ABC$ with a side length of 12. The edge $BD$ is perpendicular to the plane of the base and equals 10 $\sqrt{3}$. All vertices of this pyramid lie on the lateral surface of a right circular cylinder, whose axis intersects the edge $BD$ and the plane $ABC$. D... | 94. It can be proven that the axis of the cylinder should pass through the midpoint of edge $B D$ and belong to the plane $B D L$, where $L$ is the midpoint of $A C$. Let the axis of the cylinder form an acute angle $\alpha$ with $B D$. Project the pyramid onto a plane perpendicular to the axis of the cylinder, obtaini... | \frac{5\sqrt{6}}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,803 |
95. The base of the pyramid is a square $ABCD$ with side $a$, the lateral edge $SC$ is perpendicular to the base plane and equals $b$. $M$ is a point on the edge $AS$. Points $M, B$, and $D$ lie on the lateral surface of a right circular cone with vertex at point $A$, and point $C$ lies in the base plane of this cone. ... | 95. Let's take a point $K$ on the edge $A S$ such that $|A K|=a$. Then the points $B$; $D$ and $K$ belong to the section of the cone by a plane parallel to the base of the cone ( $|A B|=|A D|=|A K|)$. From the fact that $C$ lies in the plane of the base, it follows that the plane $B D K$ bisects the height of the cone.... | \frac{4\pi\sqrt{2}^{2}(\sqrt{b^{2}+2^{2}}-)}{\sqrt[4]{b^{2}+2^{2}}\cdot\sqrt{3\sqrt{b^{2}+2^{2}}-4}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,804 |
96. Inside a right circular cone, a cube is positioned such that one edge of the cube lies on the diameter of the cone's base, the vertices of the cube not belonging to this edge lie on the lateral surface of the cone, and the center of the cube lies on the height of the cone. Find the ratio of the volume of the cone t... | 96. Let the radius of the base of the cone be $R$, the height be $h$, and the edge of the cube be $a$. The section of the cone by a plane parallel to the base and passing through the center of the cube is a circle with radius $R \frac{2 h - a \sqrt{2}}{2 h}$, in which a rectangle (the section of the cube) with sides $a... | \frac{\pi(53-7\sqrt{3})\sqrt{2}}{48} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,805 |
98. In the tetrahedron $ABCD$, edge $AB$ is perpendicular to edge $CD$, $\widehat{ACB} = \widehat{ADB}$, the area of the section passing through edge $AB$ and the midpoint of edge $DC$ is $S$, $|DC| = a$. Find the volume of the tetrahedron $ABCD$. | 98. From the equality $\widehat{A C B}=\widehat{A D} B$ and the perpendicularity of $A B$ and $D C$, it can be derived that points $C$ and $D$ are symmetric with respect to the plane passing through $A B$ and perpendicular to $C D$.
Answer: $\frac{a S}{3}$. | \frac{}{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,807 |
99. Given a regular triangular pyramid $S A B C$ (with $S$ as its vertex). The edge $S C$ of this pyramid coincides with the lateral edge of a regular triangular prism $A_{1} B_{1} C A_{2} B_{2} S\left(A_{1} A_{2}, B_{1} B_{2}\right.$ and $C S$ are lateral edges, and $A_{1} B_{1} C$ is one of the bases). The vertices $... | 99. Let $K$ be the midpoint of $AB$, and $P$ be the foot of the perpendicular dropped from $K$ to $CS$. Take points $M$ and $N$ on $AB$ such that $\triangle PMN$ is equilateral (Fig. 11). The pyramid $SPMN$ can be extended to a regular prism $PMNSM_1N_1$ such that $PMN$ and $SM_1N_1$ are its bases, and $PS, MM_1, NN_1$... | \frac{\sqrt{3}}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,808 |
100. In a regular truncated quadrilateral pyramid with lateral edges $A A_{1}, B B_{1}, C C_{1}, D D_{1}$, the side of the upper base $A_{1} B_{1} C_{1} D_{1}$ is 1, and the side of the lower base is 7. A plane passing through the edge $B_{1} C_{1}$ perpendicular to the plane $A D_{1} C$ divides the pyramid into two pa... | 100. Let the plane passing through $B_{1} C_{1}$ intersect $A B$ and $D C$ at points $K$ and $L$ (Fig. 12). By the condition, the volumes of the polyhedra $A K L D A_{1} B_{1} C_{1} D_{1}$ and $K B C L B_{1} C_{1}$ are equal. Applying the Simpson's formula (Problem 15) to them, denoting $|A K|=|D L|=a$. Since the heigh... | \frac{38\sqrt{5}}{5} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,809 |
104. The centers of three spheres, with radii of 3, 4, and 6, are located at the vertices of an equilateral triangle with a side length of 11. How many planes exist that are tangent to all three spheres simultaneously? | 104. Any tangent plane divides space into two parts, and either all three spheres are located on one side, or two are on one side and one on the other. It is obvious that if a certain plane is tangent to the spheres, then the plane symmetric to it relative to the plane passing through the centers of the spheres is also... | 6 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,813 |
105. All plane angles of the trihedral angle $N K L M (N-$ vertex) are right angles. On the face $L N M$, a point $P$ is taken at a distance of 2 from the vertex $N$ and at a distance of 1 from the edge $M N$. A light ray is directed from some point $S$, located inside the trihedral angle $N K L M$, to the point $P$. T... | 105. The solution of the problem is based on the fact that the continuation of the falling ray is symmetric to the reflected ray relative to the edge from which the ray is reflected. We will introduce a coordinate system naturally, taking its origin at point $N$, and as the $x, y$ and $z$ axes, the edges $N K, N L$ and... | 2\sqrt{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,814 |
106. The base of the triangular pyramid $A B C D$ is the triangle $A B C$, in which $\widehat{A}=\pi / 2, \widehat{C}=\pi / 6$, $|B C|=2 \sqrt{2}$. The lengths of the edges $A D, B D$ and $C D$ are equal to each other. A sphere of radius 1 touches the edges $A D, B D$, the extension of the edge $C D$ beyond point $D$, ... | 106. Let $K$ be the point of tangency of the sphere with the extension of $CD$, and $M$ and $L$ be the points of tangency with the edges $AD$ and $BD$, and $N$ be the midpoint of $BC$ (Fig. 14). Since $|CD| = |DB| = |DA|$, $DN$ is perpendicular to the plane $ABC$, $|DK| = |DM| = |DL|$, $KL$ is parallel to $DN$, $ML$ is... | \sqrt{3}-1 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,815 |
107. Three spheres, two of which are identical, touch the plane $P$ and, in addition, touch each other pairwise. The vertex of a right circular cone lies on the plane $P$, and the axis of the cone is perpendicular to this plane. All three spheres are located outside the cone, and each touches its lateral surface. Find ... | 107. Let $O_{1}, O_{2}, O_{3}$ be the points of tangency of the spheres with the plane $P$, where in point $O_{1}$ the sphere has radius $r$, and in $O_{2}$ and $O_{3}$ the radii are $\boldsymbol{R}, \boldsymbol{O}$ - the vertex of the cone (Fig. 15), $\varphi$ - the angle between the generatrix of the cone and the pla... | \cos\varphi=\frac{1}{7} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,816 |
108. The volume of the tetrahedron \(ABCD\) is 5. A plane is drawn through the midpoints of the edges \(AD\) and \(BC\), intersecting the edge \(CD\) at point \(M\). The ratio of the length of segment \(DM\) to the length of segment \(CM\) is \(2/3\). Calculate the area of the section of the tetrahedron by the specifie... | 108. Let $K$ and $L$ be the midpoints of edges $AD$ and $BC$, and let $N$ and $P$ be the points of intersection of the plane with lines $AB$ and $AC$ (Fig. 16). We need to find the ratios $\frac{|PA|}{|PC|}$ and $\frac{|PK|}{|PM|}$. Draw $KQ$ and $AR$ parallel to $DC$, with $Q$ being the midpoint of $AC$.
$$
|AR|=|DM|... | 3 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,817 |
109. A sphere of radius 2 is inscribed in a regular triangular pyramid $S A B C$ with vertex $S$ and base $A B C$; the height of the pyramid $S K$ is 6. Prove that there exists a unique plane intersecting the edges of the base $A B$ and $B C$ at some points $M$ and $N$ such that $|M N|=$ $=7$, touching the sphere at a ... | 109. Given the radius of the sphere inscribed in a regular triangular pyramid and the height of the pyramid, it is not difficult to find the side of the base. It is equal to $12, |M K|=|K N|$ (by the condition, the tangents to the sphere from points $M$ and $N$ are equal).
Let $|B M|=x, |B N|=y$. By finding $|M N|$ us... | 6\frac{12}{13} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,818 |
110. All edges of the triangular pyramid $A B C D$ touch a certain sphere. Three segments connecting the midpoints of the skew edges are equal. The angle $A B C$ is $100^{\circ}$. Find the ratio of the heights of the pyramid dropped from vertices $A$ and $B$. | 110. From the fact that the edges of the pyramid $ABCD$ touch the sphere, it follows that the sums of opposite edges of the pyramid are equal. Extend the pyramid $ABCD$ to a parallelepiped by drawing a plane through each edge of the pyramid, parallel to the opposite edge. The edges of the pyramid will be the diagonals ... | \sqrt{3}\operatorname{tg}50 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,819 |
111. In the pyramid $S A B C$, the products of the lengths of the edges of each of the four faces are equal to the same number. The length of the height of the pyramid, dropped from $S$ to the face $A B C$, is $2 \sqrt{\frac{102}{55}}$, and the measure of the angle $C A B$ is $\arccos \left(\frac{1}{6} \sqrt{\frac{17}{... | 111. Equality of the products of the lengths of the edges at each face means that the opposite edges of the pyramid are equal. Let's complete the pyramid $S A B C$ in the usual way to form a parallelepiped by drawing a plane through each edge parallel to the opposite edge. Due to the equality of the opposite edges of t... | \frac{34\sqrt{6}}{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,820 |
112. In the plane $P$, there is an isosceles triangle $ABC (|AB| = |BC| = l, |AC| = 2a)$. A sphere of radius $r$ touches the plane $P$ at point $B$. Two skew lines pass through points $A$ and $C$ and are tangent to the sphere. The angle between each of these lines and the plane $P$ is $\alpha$. Find the distance betwee... | 112. Let $M$ and $N$ be the points of tangency of the tangents drawn from $A$ and $B$, and $M_{1}$ and $N_{1}$ be the projections of points $M$ and $N$ onto the plane $ABC$ (Fig. 19, $a$; the figure shows one of the two equivalent cases of the arrangement of the tangents, in which the tangents intersect, in the other t... | \frac{2\tan\alpha\sqrt{2r\sin\alpha-(^{2}+r^{2})\sin^{2}\alpha}}{\sqrt{^{2}-^{2}\cos^{2}\alpha}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,821 |
113. The base of the pyramid $A B C E H$ is a convex quadrilateral $A B C E$, which is divided into two equal-area triangles by the diagonal $B E$. The length of the edge $A B$ is 1, the lengths of the edges $B C$ and $C E$ are equal. The sum of the lengths of the edges $A H$ and $E H$ is $\sqrt{2}$. The volume of the ... | 113. Let $|E A|=x$, the area of $\triangle E M A$ will be the largest if $|E H|=|H A|=\frac{\sqrt{2}}{2}$, and in this case, it is equal to $\frac{x}{2} \sqrt{\frac{1}{2}-\frac{x^{2}}{4}}$. The distance from $B$ to the plane $E A H$ is not greater than $|A B|=1$. Since $S_{A E B}=S_{E B C}$
$$
\begin{aligned}
\frac{1}... | \frac{3-\sqrt{5}}{4} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,822 |
114. In the pyramid $S A B C$, a line intersecting the edges $A C$ and $B S$ and perpendicular to them passes through the midpoint of the edge $B S$. The face $A S B$ is equal in area to the face $B S C$, and the area of the face $A S C$ is twice the area of the face $B S C$. Inside the pyramid, there is a point $M$, t... | 114. From the fact that a line perpendicular to the edges $AC$ and $BS$ passes through the midpoint of $BS$, it follows that the faces $ACB$ and $ACS$ are equal in area.
Let $S_{ASB} = S_{BSC} = Q$, then $S_{ACB} = S_{ACS} = 2Q$. Denote by $A_1, B_1, C_1, S_1$ the projections of $M$ onto the faces $BCS, ACS, ABS$, and... | \frac{\sqrt{10}}{6} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,823 |
116. In a triangular pyramid $S A B C$ with base $A B C$ and equal lateral edges, the sum of the dihedral angles
with edges $S A$ and $S C$ is $180^{\circ}$. It is known that $|A B|=a_{s}$ $|B C|=b$. Find the length of the lateral edge. | 116. Extend the edge $S A$ beyond point $S$ and take a point $A_{i}$ on the extension such that $\left|S A_{1}\right|=|S A|$. In $S A_{1} B C$, the dihedral angles at the edges $S A_{i}$ and $S C$ will be equal, and since $\left|S A_{1}\right|=|S C|$, then $\left|A_{1} B\right|=|C B|=b$. Triangle $A B A_{1}$ is a right... | \frac{1}{2}\sqrt{^{2}+b^{2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,825 |
117. Given a regular tetrahedron with edge $a$. A sphere touches the three edges of the tetrahedron emanating from one vertex at their ends. Find the area of the part of the spherical surface located inside the tetrahedron. | 117. Consider a tetrahedron with edge $2a$. The surface of the sphere that touches all its edges is divided by the surface of the tetrahedron into 4 equal segments and 4 equal curved triangles, each of which is equal to the desired triangle. The radius of the sphere is $\frac{a \sqrt{2}}{2}$, the height of each segment... | \frac{\pi^{2}}{6}(2\sqrt{3}-3) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,826 |
118. On the surface of a sphere with radius 2, there are three pairwise tangent circles with radius $\sqrt{2}$. The part of the sphere's surface located outside the circles forms two curvilinear triangles. Find the areas of these triangles. | 118. Consider a cube with edge $2 \sqrt{2}$. A sphere centered at the center of the cube, touching its edges, has a radius of 2. The surface of the sphere is divided by the surface of the cube into 6 segments and 8 curved triangles, ${ }^{\text {p }}$ equal to the smaller of the sought triangles.
Answer: $\pi(3 \sqrt{... | \pi(3\sqrt{2}-4)\pi(9\sqrt{2}-4) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,827 |
119. Three dihedral angles of a tetrahedron, not belonging to the same vertex, are equal to $\pi / 2$. The remaining three dihedral angles are equal to each other. Find these angles. | 119. $\arccos \frac{\sqrt{5}-1}{2}$.
119. $\arccos \frac{\sqrt{5}-1}{2}$. | \arccos\frac{\sqrt{5}-1}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,828 |
120. Two spheres are inscribed in the lateral surface of a cone and touch each other. A third sphere passes through the two circles where the first two spheres touch the surface of the cone. Prove that the volume of the part of the third sphere located outside the cone is equal to the volume of the part of the cone enc... | 120. Let's conduct a section through the axis of the cone. Consider the trapezoid $ABCD$ formed in this section, where $A$ and $B$ are the points of tangency with the surface of the cone of one sphere, and $C$ and $D$ are those of another. It can be proven that if $F$ is the point of tangency of the spheres, then $F$ i... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,829 |
122. 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 gravity coincide. Find the sum of the squares of the distances from all vertices of one to all vertices of the other. | 122. Using the Leibniz formula (see (1) problem 153) *) $3|M G|^{2}=|M A|^{2}+|M B|^{2}+|M C|^{2}-\frac{1}{3}\left(|A B|^{2}+|B C|^{2}+|C A|^{2}\right)$, where $G$ is the centroid of triangle $A B C$.
If now $A B C$ is a given right triangle, $A_{1} B_{1} C_{1}$ is a given equilateral triangle, and $G$ is their common... | 3^{2}+4b^{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,831 |
123. In a regular triangular pyramid $S A B C$ ( $S$ - vertex), point $E$ is the midpoint of the apothem of face $S B C$, and points $F, L$, and $M$ lie on edges $A B, A C$, and $S C$ respectively, such that $|A L|=\frac{1}{10}|A C|$. It is known that $\boldsymbol{E F L M}$ is an isosceles trapezoid and the length of i... | 123. Let the side of the base of the pyramid be $a$, and the lateral edge be $b$. Draw a plane through $F E$ parallel to $A S C$, and denote by $K$ and $N$ the points of intersection of this plane with $B C$ and $S B$. Since $E$ is the midpoint of the apothem of the face $S C B$, we have $|A F|=|C K| = a / 4, |S N|=b /... | \frac{16}{3}\sqrt{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,832 |
124. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with edge $a$. The bases of the cylinder are inscribed in the faces $A B C D$ and $A_{1} B_{1} C_{1} D_{1}$. Let $M$ be a point on the edge $A B$ such that $|A M|=a / 3$, and $N$ be a point on the edge $B_{1} C_{1}$ such that $\left|N C_{1}\right|=a / 4$. A plane pass... | 124. Prove that a plane intersecting the lateral surface of a cylinder divides its volume in the same ratio as it divides the axis of the cylinder.
Answer: $\frac{\pi a^{3}}{24}$.
*) Here and below (1) denotes: Sharygin I. F. Problems in Geometry: Planimetry. - M.: Nauka, 1982 (Library "Kvant", issue 17).
125 Each l... | \frac{\pi^{3}}{24} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,833 |
127. A truncated cone is described around a sphere. The total surface area of this cone is $S$. A second sphere touches the lateral surface of the cone along the circumference of the cone's base. Find the volume of the truncated cone, given that the part of the surface of the second sphere that is inside the first has ... | 127. When solving the problem, the following facts are used:
1) The center of the sphere inscribed in the cone lies on the surface of the second sphere (consider the corresponding statement from planimetry);
2) From the fact that the center of the inscribed sphere lies on the surface of the second, it follows that the ... | \frac{1}{3}S\sqrt{\frac{Q}{\pi}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,835 |
129. A section of maximum area is made through the vertex of a right circular cone. It is known that the area of this section is twice the area of the axial section. Find the angle at the vertex of the axial section of the cone. | 129. Any of the considered sections represents an isosceles triangle, the lateral sides of which are equal to the generatrix of the cone. Therefore, the section with the greatest area is the one where the sine of the angle at the vertex takes the greatest value. If the angle at the vertex of the axial section of the co... | \frac{5}{6}\pi | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,836 |
130. A triangular pyramid $S A B C$ is inscribed in a cone ( $S$ coincides with the vertex of the cone, $A, B$ and $C$ lie on the circumference of the base of the cone), the dihedral angles at the edges $S A, S B$ and $S C$ are equal to $\alpha, \beta$ and $\gamma$ respectively. Find the angle between the plane $S B C$... | 130. Let's draw $S O$ - the height of the pyramid. This creates three pyramids $S A B O, S B C O, S C A O$. In each of these pyramids, the dihedral angles at the lateral edges $S A$ and $S B, S B$ and $S C, S C$ and $S A$ are right angles. Denote these angles by $x, y$, and $z$. We obtain the system
$$
\left\{\begin{a... | \frac{\pi-\alpha+\beta-\gamma}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,837 |
131. Three points $A, B$, and $C$, located on the surface of a sphere with radius $R$, are connected by arcs of great circles, each less than a semicircle. Another great circle is drawn through the midpoints of the arcs $\overrightarrow{A B}$ and $\widehat{A C}$, intersecting the extension of $\widetilde{B} C$ at point... | 131. Chord $B C$ is parallel to any plane passing through the midpoints of chords $A B$ and $A C$. Therefore, chord $B C$ is parallel to the plane passing through the center of the sphere and the midpoints of arcs $\overrightarrow{A B}$ and $\overrightarrow{A C}$. From this, it follows that the great circle passing thr... | \frac{\piR}{2}\\frac{}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,838 |
132. Find the volume of the body obtained by rotating an equilateral triangle with side $a$ around a line parallel to its plane and such that the projection of this line onto the plane of the triangle contains some height of the triangle. | 132. It is easy to see that the cross-section of the given body by a plane perpendicular to the axis of rotation is a ring, the area of which does not depend on the distance from the axis of rotation to the plane of the triangle.
$$
\text { Answer: } \frac{\pi a^{3} \sqrt{3}}{24} \text {. }
$$ | \frac{\pi^{3}\sqrt{3}}{24} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,839 |
133. Consider a body consisting of points that are at a distance of no more than $d$ from some point inside or on the boundary of a planar convex figure with perimeter $2 p$ and area $S$. Find the volume of this body. | 133. If the given plane figure represents a convex polygon, then the considered body consists of a prism of volume $2 d S$, semicylinders with a total volume of $\pi p d^{2}$, and a collection of spherical sectors that together form a sphere of volume $\frac{4}{3} \pi d^{3}$. Therefore, in this case, the volume of the ... | 2+\pip^{2}+\frac{4}{3}\pi^{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,840 |
135. Inside a regular triangular pyramid, there is a vertex of a trihedral angle, all plane angles of which are right angles, and the bisectors of the plane angles pass through the vertices of the base. In what ratio does the surface of this angle divide the volume of the pyramid, if each face of the pyramid is divided... | 135. In Fig. 21, \( SABC \) is a given pyramid, \( SO \) is its height, and \( G \) is the vertex of the trihedral angle. From the condition, it follows that \( G \) is located on \( SO \). Moreover, the faces of the trihedral angle, when intersecting the plane of the base \( ABC \), form an equilateral triangle, the s... | 3:11 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,842 |
137. Two identical triangular pyramids of volume $V$ are arranged in space symmetrically relative to point $O$. Find the volume of their common part if point $O$ lies on the segment connecting the vertex of the pyramid with the centroid of the base and divides this segment in the ratio 1) $1: 1$, 2) $3: 1$, 3) $2: 1$, ... | 137. In figures $22, a-\varepsilon$ the common parts of these two pyramids are depicted for all four cases.
1) The common part is a parallelepiped (Fig. $22, a$ ). To determine the volume, one needs to subtract the volumes of three pyramids similar to the original pyramid with a similarity coefficient of $2 / 3$, and a... | \frac{2}{3}V,\frac{V}{2},\frac{110}{243}V,\frac{12}{25}V | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,843 |
138. A regular tetrahedron of volume \( V \) is rotated about the line connecting the midpoints of its skew edges by an angle \( \alpha \). Find the volume of the common part of the given tetrahedron and the rotated one \((0<\alpha<\pi)\). | 138. Let the edge of a regular tetrahedron \(ABCD\) be \(a\), and let \(R\) and \(L\) be the midpoints of the edges \(CD\) and \(AB\) (Fig. 23). We take a point \(M\) on the edge \(CB\),
F... | \frac{1+\tan^2\frac{\alpha}{2}}{(1+\tan\frac{\alpha}{2})^2}V | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,844 |
139. The edge of a cube is $a$. The cube is rotated about a diagonal by an angle $\alpha$. Find the volume of the common part of the original cube and the rotated one. | 139. Let the cube $A B C D A_{1} B_{1} C_{1} D_{\text {i }}$ rotate by an angle $\alpha$ around the diagonal $A C_{j}$ (Fig. 24). Take points $K$ and $L$ on the edges $A_{1} B_{1}$ and $A_{1} D_{1}$ such that $\left|A_{1} K\right| = \left|A_{1} L\right| = x$, and drop perpendiculars from $K$ and $L$ to the diagonal $A ... | \frac{3^{3}(1+\operatorname{ctg}^{2}\frac{\alpha}{2})}{(1+\sqrt{3}\operatorname{ctg}\frac{\alpha}{2})^{2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,845 |
140. A light ray falls on a flat mirror at an angle $\alpha$. The mirror is rotated by an angle $\beta$ around the projection of the ray on the mirror. By what angle will the reflected ray deviate? | 140. Let $A$ be some point on the ray, $B$ be the point of incidence of the ray on the mirror, $K$ and $L$ be the projections of $A$ onto the given mirror and the reflected one, $A_{1}$ and $A_{2}$ be the points symmetric to $A$ relative to these mirrors, respectively. The required angle is equal to the angle $\widehat... | 2\arcsin(\sin\alpha\sin\beta) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,846 |
141. In space, points $A, B, C$ and $D$ are given, such that $|A B|=|B C|=|C D|, \quad \widehat{A B} C=\widehat{B C D}=\widehat{C D A}=\alpha$. Find the angle between the lines $A C$ and $B D$. | 141. Let's fix $\triangle A B C$, then in $\triangle A D C$ two of its sides $|A C|$ and $|D C|$ and the angle $\widehat{A D C}=\alpha$ are known. Construct in the plane of $\triangle A D C$ a circle of radius $|A C|$ with center at $C$ (Fig. 25, a). If
Fig. 26. where $|A B|=1,|B C|=2,|C D|=$ $=a, A B C D$ is a trapezoid in which $\frac{|A D|}{|B C|}=\lambda=\frac{... | \frac{\sqrt{5}-1}{4}\sqrt{14+10\sqrt{5}-2(\sqrt{5}+1)^{2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,849 |
144. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}, \quad M$ is the center of the face $A B B_{1} A_{1}, N$ is a point on the edge $B_{1} C_{1}, L$ is the midpoint of $A_{1} B_{1}$; $K$ is the foot of the perpendicular dropped from $N$ to $B C_{1}$. In what ratio does the point $N$ divide the edge $B_{1} C_{1}$, if $\w... | 144. Let the edge of the cube be $a, |N C_{1}|=x$. Find:
$$
\begin{array}{r}
|L M|=\frac{a}{2}, \quad|N K|=\frac{x}{\sqrt{2}} \\
|L N|^{2}=\left|L B_{1}\right|^{2}+\left|B_{1} N\right|^{2}=\frac{a^{2}}{4}+(a-x)^{2}=\frac{5}{4} a^{2}-2 a x+x^{2} \\
|L K|^{2}=\left|L B_{1}\right|^{2}+\left|B_{1} K\right|^{2}=\left|L B_{... | \frac{|B_{1}N|}{|NC_{1}|}=\sqrt{2}+1 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,850 |
145. In a regular hexagonal pyramid, the center of the circumscribed sphere lies on the surface of the inscribed sphere. Find the ratio of the radii of the circumscribed and inscribed spheres. | 145. There are two cases: 1) the center of the circumscribed sphere coincides with the center of the base, and 2) the center of the circumscribed sphere is located at a point on the surface of the inscribed sphere, diametrically opposite to the center of the base.
In the second case, denoting the radii of the inscribe... | \frac{3+\sqrt{21}}{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,851 |
146. In a regular quadrilateral pyramid, the center of the circumscribed sphere lies on the surface of the inscribed sphere. Find the measure of the plane angle at the vertex of the pyramid. | 146. There are two cases: 1) the center of the circumscribed sphere coincides with the center of the base, 2) the center of the circumscribed sphere is located at a point on the surface of the inscribed sphere, diametrically opposite to the center of the base. In the first case, the plane angle at the vertex is $\pi / ... | \frac{\pi}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,852 |
147. At the base of the quadrilateral pyramid $S A B C D$ lies a square $A B C D$ with side $a$. Both angles between opposite lateral faces are right angles. The dihedral angle at edge $S A$ is $\alpha$. Find the volume of the pyramid. | 147. Let $K$ be the projection of vertex $S$ onto the plane $ABCD$, and $L, M, N$, and $P$ be the projections of $S$ onto the sides $AB, BC, CD$, and $DA$.
From the condition, it follows that triangles $LSN$ and $MSP$ are right triangles with right angles at vertex $S$. Therefore,
}}{6} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,853 |
148. A plane intersecting the surface of a triangular pyramid divides the medians of the faces emanating from one vertex in the ratios $2: 1, 1: 2, 4: 1$ respectively (counting from the vertex). In what ratio does this plane divide the volume of the pyramid? | 148. Let's first solve the following problem. In $\triangle ABC$, points $L$ and $K$ are taken on sides $AB$ and $AC$ such that $\frac{|AL|}{|LB|}=m, \quad \frac{|AK|}{|KC|}=n$. In what ratio does the line $KL$ divide the median $AM$?
Let $N$ be the intersection point of $KL$ and $AM$, $Q$ be the intersection point of... | \frac{7123}{16901} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,854 |
149. $n$ equal cones have a common vertex. Each touches two others along a generatrix, and all touch one plane. Find the angle at the vertex of the axial section of these cones. | 149. Consider the pyramid $S A B C$ (Fig. 28), in which $|C A| = |A B|, \quad \widehat{B A C} = \frac{2 \pi}{n}, S A$ is perpendicular to the plane $A B C$,
Fig. -8. II such that the vertex... | 2\arcsin\frac{\operatorname{tg}\frac{\pi}{n}}{\sqrt{1+2\operatorname{tg}^{2}\frac{\pi}{n}}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,855 |
150. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$. A plane passing through $A$ and tangent to the sphere inscribed in the cube intersects the edges $A_{1} B_{1}$ and $A_{1} D_{1}$ at points $K$ and $N$. Determine the measure of the dihedral angle between the planes $A C_{1} K$ and $A C_{1} N$. | 150. Let the plane $A K N$ touch the sphere at point $P$, and the line $4 P$ intersect $N K$ at point $M$ (Fig. 29). Then the plane $C_{1} N A$ is the bisector plane of the dihedral angle formed by the planes $D_{1} C_{1} A$ and $C_{1} M A$ (the planes $D_{1} A N$ and $A N M$ touch the sphere, and the planes $D_{1} C_{... | \pi/3 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,856 |
151. Given a tetrahedron \(ABCD\). Another tetrahedron \(A_1B_1C_1D_1\) is positioned such that its vertices \(A_1, B_1, C_1\), and \(D_1\) lie respectively in the planes \(BCD\), \(CDA\), \(DAB\), and \(ABC\), and the planes of its faces \(A_1B_1C_1\), \(B_1C_1D_1\), \(C_1D_1A_1\), and \(D_1A_1B_1\) contain respective... | 151. Let $K, L$ and $M$ be the midpoints of $AB, AC$ and $AD$ (Fig. 30). From the problem statement, it follows that the tetrahedron $A_{1} B_{1} C_{1} D_{1}$ is bounded by the planes $D K A_{1}, B L A_{1}, C M A_{1}$, and the plane passing through $A$ parallel to $BCD$. In this case, the vertices $B_{1}, C_{1}$, and $... | \frac{3}{8}V | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,857 |
152. In the tetrahedron $ABCD$, it is given that $|BC|=|CD|=$ $=|DA|, |BD|=|AC|, |BD|>|BC|$, and the dihedral angle at edge $AB$ is $\pi / 3$. Find the sum of the other dihedral angles. | 152. First, let's prove that the dihedral angles at the edges $DB$ and $AC$ are equal to $\pi / 2$. Let $|AD| = |CD| = |BC| = a$, $|BD| = |AC| = b$, $|AB| = c$, and $b > a$. Drop perpendiculars $DK$ and $CL$ from $D$ and $C$ to the edge $AB$ (Fig. 31, a). Denote
$|AK| = |BL| = |x|, \quad |KL| = |c - 2x|, \quad |DK| = ... | 2\pi | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,858 |
154. In a regular tetrahedron \(ABCD\) with edge \(a\), points \(A_1, B_1, C_1,\) and \(D_1\) are taken in the planes \(BCD, CDA, DAB,\) and \(ABC\) respectively, such that the line \(A_1 B_1\) is perpendicular to the plane \(BCD\), \(B_1 C_1\) is perpendicular to the plane \(CDA\), \(C_1 D_1\) is perpendicular to the ... | 154. Drop perpendiculars $A_{1} M$ and $B_{1} M$ to $C D$, $B_{1} N$ and $C_{1} N$ to $A D$, $C_{1} K$ and $D_{1} K$ to $A B$, $D_{1} L$ and $A_{1} L$ to $C B$.
Since
$$
\frac{\left|A_{1} M\right|}{\left|B_{1} M\right|}=\frac{\left|B_{1} N\right|}{\left|N C_{1}\right|}=\frac{\left|C_{1} K\right|}{\left|K D_{1}\right|... | \frac{^{3}\sqrt{2}}{162} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,860 |
155. $n$ equal spheres of radius $R$ touch the lateral surface from the inside and the plane of the base of a cone, and each sphere touches two adjacent ones; $n$ spheres of radius $2R$ are arranged similarly, touching the lateral surface from the outside. Find the volume of the cone. | 155. Without limiting the generality, we will assume that all the generators of the cone, touching the spheres, touch two spheres simultaneously - the inner and the outer one. Let's make a section through the vertex of the cone $S$ and the centers of two spheres touching one generator
^{3}(1+\sqrt{1-8\sin^{2}\frac{\pi}{n}})}{12\sin^{2}\frac{\pi}{n}(1-6\sin^{2}\frac{\pi}{n}+} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,861 |
156. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$. Points $M$ and $N$ are taken on segments $A A_{1}$ and $B C_{1}$ such that the line $M N$ intersects the line $B_{1} D$. Find
$$
\frac{\left|B C_{1}\right|}{|B N|}-\frac{|A M|}{\left|A A_{1}\right|}
$$ | 156. Projecting a cube onto a plane perpendicular to $B_{i} D_{\text {, }}$ we obtain a regular hexagon $A B C C_{1} D_{1} A_{1}$ (Fig. 35) with side
Fig. $35_{s}$ equal to $\sqrt{\frac{2}... | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,862 |
157. It is known about the tetrahedron that all its faces are similar, but not all are equal to each other triangles. In addition, any two faces have at least one pair of equal edges, not counting the common edge. Find the volume of this tetrahedron if the lengths of two edges lying in the same face are 3 and 5. | 157. If two unequal and similar triangles have two equal sides, it is easy to see that the sides of each of them form a geometric progression, and the sides of one can be denoted as $a, \lambda a, \lambda^{2} a$, and the other as $\lambda a, \lambda^{2} a, \lambda^{3} a$.
Further, if the sides of a triangle form a geo... | \frac{55\sqrt{6}}{18} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,863 |
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