problem stringlengths 1 13.6k | solution stringlengths 0 18.5k ⌀ | answer stringlengths 0 575 ⌀ | problem_type stringclasses 8
values | question_type stringclasses 4
values | problem_is_valid stringclasses 1
value | solution_is_valid stringclasses 1
value | source stringclasses 8
values | synthetic bool 1
class | __index_level_0__ int64 0 742k |
|---|---|---|---|---|---|---|---|---|---|
162. Denote
$$
1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n-1}-\ln n=\gamma_{n}
$$
Prove that:
a) for any $n$, the number $\gamma_{n}$ is between 0 and 1;
b) as $n \rightarrow \infty$, the number $\gamma_{n}$ tends to a certain limit $\gamma$ (also, of course, between 0 and 1).
Thus, for large values of $n$, the ap... | 162. a) $\ln n$ is equal to the area of the curvilinear trapezoid $A B C D$ bounded by the hyperbola $y=\frac{1}{x}$, the x-axis, and the lines $x=1$ and $x=n$.
Now let's construct a stepped figure consisting of $n$ rectangles, all of which have a base of one unit, and heights equal to $1, \frac{1}{2}, \frac{1}{3}, \l... | proof | Calculus | proof | Yes | Yes | olympiads | false | 24,018 |
164. Prove that for $l>1$ the sum
$$
1+\frac{1}{2^{l}}+\frac{1}{3^{l}}+\ldots+\frac{1}{n^{l}}
$$
as $n$ increases without bound, tends to a certain limit $C$, which lies between $\frac{1}{l-1}$ and $\frac{l}{l-1}$.
Thus, for large $n$, the approximate equality holds, with its accuracy increasing as $n$ increases:
$... | 164. This problem is solved similarly to problem 162a). Let \(ABCD\) be a curvilinear trapezoid bounded by the curve \(y = \frac{1}{x^l}\), the x-axis, and the lines \(x = 1\) and \(x = n\). Consider a step figure composed of \(n-1\) rectangles with a base of 1, inscribed in this curvilinear trapezoid, and a step figur... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 24,020 |
170***. The Third Mertens' Theorem. Let 2, 3, 5, 7, 11, ..., p be all prime numbers not exceeding the integer N. Prove that there exists a number c such that as \( N \rightarrow \infty \), the ratio of the product
\[
\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{5}\right)\left(1-\frac{1}{7}\righ... | 170. We need to evaluate the product
$$
\pi^{(N)}=\left(1-\frac{1}{p_{1}}\right)\left(1-\frac{1}{p_{2}}\right)\left(1-\frac{1}{p_{3}}\right) \ldots\left(1-\frac{1}{p_{r}}\right)
$$
where \( p_{1}, p_{2}, p_{3}, \ldots, p_{r} \) are all prime numbers not exceeding the integer \( N \). By taking the logarithm of this p... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 24,025 |
0.10. Can 6 pencils be arranged so that any two of them touch? The same question for 7 pencils. | 0.10. Fig. 11 shows how to stack three pencils. Three more pencils can be placed on top of them in a similar manner (Fig. 11, b); in this case, the pencils can be arranged so that the diameter of the inscribed circle of the triangle formed by the points of contact of the three lower (or three upper) pencils is equal to... | notfound | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 24,035 |
0.11. A person walked a kilometer to the north, then a kilometer to the west, and a kilometer to the south. Could he have returned to the initial position in this way? | 0.11. Yes, he could. Suppose a person left point $A$ and, walking along the meridian 1 km to the north, ended up at point $B$. By walking 1 km along the parallel, he could return to point $B$ again if he circled the North Pole one or more times.
 Vertices $M$ and $N$ of a cube with edge length are asymmetric relative to the center of the cube. Find the length of the shortest path from $M$ to $N$ along the surface of the cube.
b) A box has the shape of a rectangular parallelepiped with dimensions $30 \times 12 \times 12$. Point $A$ is on the $12 \times... | 0.13. a) Any path going along the surface of a cube can be unfolded onto a plane. In Fig. 13, a, six shortest paths from $M$ to $N$ are shown unfolded; the length of each is $\sqrt{5}$.
Fig.... | 40 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,038 |
0.17. a) Is it possible to connect 3 rubber rings in such a way that they cannot be separated, but after cutting any one of them, they would come apart?
b) The same question for 10 rings. | 0.17. It is possible. Fig. 15 shows how to link 5 rings. Similarly, any number of rings can be linked. | proof | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 24,042 |
1.1. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with edge $a$. Find the angle and distance between the lines $A_{1} B$ and $A C_{1}$. | 1.1. To prove that the triangle $A_{1} B D$ is equilateral. Moreover, point $A$ is equidistant from its vertices. Therefore, it projects onto the center of this triangle. Similarly, point $C_{1}$ projects onto the center of triangle $A_{1} B D$. Consequently, the lines $A_{1} B$ and $A C_{1}$ are perpendicular, and the... | \frac{}{\sqrt{6}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,043 |
1.2. Given a cube with an edge of 1. Find the angle and distance between the skew diagonals of two of its adjacent faces. | 1.2. Consider the diagonals $A B_{1}$ and $B D$ of the cube $A B C D A_{1} B_{1} C_{1} D_{1}$. Since $B_{1} D_{1} \| B D$, the angle between the diagonals $A B_{1}$ and $B D$ is equal to the angle $A B_{1} D_{1}$. But the triangle $A B_{1} D_{1}$ is equilateral, so $\angle A B_{1} D_{1}=60^{\circ}$.
It is easy to chec... | 60 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,044 |
1.4. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with edge $1, K$ - the midpoint of edge $D D_{1}$. Find the angle and the distance between the lines $C K$ and $A_{1} D$. | 1.4. Let's calculate the angle between the lines $C$ and $A_{1} D$. Let $M$ be the midpoint of edge $B B_{1}$. Then $A_{1} M \| K C$, so the angle between the lines $C$ and $A_{1} D$ is equal to the angle $M A_{1} D$. This angle can be calculated using the cosine theorem, since $A_{1} D=\sqrt{2}, A_{1} M=\sqrt{5 / 2}$ ... | \cosMA_{1}D=\frac{1} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,046 |
1.5. Edge $C D$ of the tetrahedron $A B C D$ is perpendicular to the plane $A B C$; $M$ is the midpoint of $D B$, $N$ is the midpoint of $A B$, and point $K$ divides edge $C D$ in the ratio $C K: K D = 1: 2$. Prove that the line $C N$ is equidistant from the lines $A M$ and $B K$. | 1.5. Consider the projection onto a plane perpendicular to the line $C N$. The projection of any point $\mathbb{X}$ will be denoted as $X_{1}$. The distance from the line $C N$ to the line $A M$ (respectively $B K$) is equal to the distance from the point $C_{1}$ to the line $A_{1} M_{1}$ (respectively $\left.B_{1} K_{... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,047 |
1.6. Find the distance between two intersecting medians of the faces of a regular tetrahedron with edge 1. (Consider all possible arrangements of the medians.)
## § 2. Angles between lines and planes | 1.6. Let $A B C D$ be a given regular tetrahedron, $K$ - the midpoint of $A B$, $M$ - the midpoint of $A C$. Consider the projection onto the plane $\mathrm{b}$, perpendicular to the face $A B C$ and passing through the line $A B$. Let $D_{1}$ be the projection of vertex $D$, $M_{1}$ - the projection of point $M$, i.e.... | \sqrt{2/35} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,048 |
1.7. The plane is given by the equation $a x+b y+c z+$ $+d=0$. Prove that the vector ( $a, b, c$ ) is perpendicular to this plane. | 1.7. Let $\left(x_{1}, y_{1}, z_{1}\right)$ and ( $\left.x_{2}, y_{2}, z_{2}\right)$ be points on the given plane. Then $a x_{1}+b y_{1}+c z_{1}-\left(a x_{2}+b y_{2}+c z_{2}\right)=0$, which means that the vectors ( $x_{1}-x_{2}, y_{1}-y_{2}, z_{1}-z_{2}$ ) are perpendicular to ( $a, b, c$ ). Therefore, any line passi... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,049 |
1.8. Find the cosine of the angle between the vectors with coordinates $\left(a_{1}, b_{1}, c_{1}\right)$ and $\left(a_{2}, b_{2}, c_{2}\right)$. | 1.8. Since $( \mathbf{u}, \mathbf{v})=|\mathbf{u}| \cdot|\mathbf{v}| \cos \varphi$, where $\varphi-$ is the angle between vectors $\mathbf{u}$ and $\mathbf{v}$, the sought cosine of the angle is
$$
\frac{a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}} \sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}} .
$... | \frac{a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 24,050 |
1.9. In a rectangular parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$, the lengths of the edges are known: $A B=a$, $A D=b, A A_{1}=c$.
a) Find the angle between the planes $B B_{1} D$ and $A B C_{1}$.
b) Find the angle between the planes $A B_{1} D_{1}$ and $A_{1} C_{1} D$.
c) Find the angle between the line $B D_{... | 1.9. a) First solution. Let's take point $A$ as the origin and direct the axes $O x, O y$ and $O z$ along the rays $A B$, $A D$ and $A A_{1}$. Then the vector with coordinates ( $b, a, 0$ ) is perpendicular to the plane $B B_{1} D$, and the vector ( $0, c, -b$ ) is perpendicular to the plane $A B C_{1}$. Therefore, the... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,051 |
1.10. At the base of a regular triangular prism lies a triangle $ABC$ with side $a$. Points $A_{1}, B_{1}$, and $C_{1}$ are taken on the lateral edges, the distances from which to the plane of the base are $a / 2$, $a$, and $3 a / 2$. Find the angle between the planes $ABC$ and $A_{1} B_{1} C_{1}$.
## § 3. Lines formi... | 1.10. Let $O$ be the point of intersection of the lines $A B$ and $A_{1} B_{1}$, $M$ be the point of intersection of the lines $A C$ and $A_{1} C_{1}$. First, we will prove that $M O \perp O A$. For this, we will take points $B_{2}$ and $C_{2}$ on the segments $B B_{1}$ and $C C_{1}$ such that $B B_{3}=C C_{2}=A A_{1}$... | 45 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,052 |
1.11. The line $l$ forms equal angles with two intersecting lines $l_{1}$ and $l_{2}$, and it is not perpendicular to the plane $\Pi$ containing these lines. Prove that the projection of the line $l$ onto the plane $\Pi$ also forms equal angles with the lines $l_{1}$ and $l_{2}$. | 1.11. It is sufficient to prove the case when the line $l$ passes through the point $O$ of intersection of the lines $l_{1}$ and $l_{2}$. Let $A$ be some point on the line $l$, different from $O$; $P$ - the projection of the point $A$ onto the plane П; $B_{1}$ and $B_{2}$ - the bases of the perpendiculars dropped from ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,053 |
1.12. Prove that a line $l$ forms equal angles with two intersecting lines if and only if it is perpendicular to one of the two bisectors of the angles between these lines. | 1.12. Let П be the plane containing the given lines. The case when $l \perp$ is obvious. If the line $l$ is not perpendicular to the plane П, then $l$ forms equal angles with the given lines if and only if its projection on П is the bisector of one of the angles between them (see problem 1.11); this means that $l$ is p... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,054 |
1.13. Given two skew lines $l_{1}$ and $l_{2}$; on $l_{1}$ points $O_{1}$ and $A_{1}$ are taken, on $l_{2}$ points $O_{2}$ and $A_{2}$ are taken, and $O_{1} O_{2}$ is the common perpendicular to the lines $l_{1}$ and $l_{2}$, while the line $A_{1} A_{2}$ forms equal angles with the lines $l_{1}$ and $l_{2}$. Prove that... | 1.13. Draw a line $l_{1}^{\prime}$ through point $O_{2}$, parallel to $l_{1}$. Let $\Pi$ be the plane containing the lines $l_{2}$ and $l_{1}^{\prime} ; A_{1}^{\prime}$ - the projection of point $A_{1}$ onto the plane $\Pi$. As follows from problem 1.11, the line $A_{1}^{\prime} A_{2}$ forms equal angles with the lines... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,055 |
1.14. Points $A_{1}$ and $A_{2}$ belong to planes $\Pi_{1}$ and $1_{2}$, intersecting along line $l$. Prove that the line $A_{1} A_{2}$ forms equal angles with planes $\Pi_{1}$ and $\Pi_{2}$ if and only if points $A_{1}$ and $A_{2}$ are equidistant from line $l$. | 1.14. Consider the projection onto the plane П, perpendicular to the line $l$. Points $A_{1}$ and $A_{2}$ are projected to $A_{1}^{\prime}$ and $A_{2}^{\prime}$, the line $l$ is projected to the point $L$, and the planes $\Pi_{1}$ and $\Pi_{2}$ are projected to the lines $p_{1}$ and $p_{2}$. As follows from the solutio... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,056 |
1.15. Prove that a line forming equal angles with three pairwise intersecting lines lying in plane П is perpendicular to plane II. | 1.15. If a line is not perpendicular to a plane П and forms equal angles with two intersecting lines of this plane, then its projection onto the plane П is parallel to the bisector of one of the two angles formed by these lines (problem 1.12). It can be assumed that all three lines intersect at one point. If line $l$ i... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,057 |
1.16. Given three lines not parallel to the same plane. Prove that there exists a line forming equal angles with them; moreover, through any point, exactly four such lines can be drawn.
## § 4. Skew Lines | 1.16. It can be assumed that these lines pass through one point. Let $a_{1}$ and $a_{2}$ be the bisectors of the angles between the first and second line, $b_{1}$ and $b_{2}$ - between the second and the third. A line forms equal angles with the three given lines if and only if it is perpendicular to the lines $a_{i}$ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,058 |
1.17. Given two intersecting lines. Prove that there exists a unique perpendicular segment whose ends lie on these lines. | 1.17. First solution. Let the line $l$ be perpendicular to the given lines $l_{1}$ and $l_{2}$. Draw a plane through the line $l_{1}$ parallel to $l$. The point of intersection of this plane with the line $l_{2}$ is one end of the desired segment.
Second solution. Consider the projection of the given lines onto a plan... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,059 |
1.18. In space, there are two skew lines $l_{1}$ and $l_{2}$ and a point $O$, not belonging to either of them. Does there always exist a line passing through point $O$ and intersecting both given lines? Can there be two such lines? | 1.18. The line $l$ passes through the point $O$ and intersects the lines $l_{1}$ and $l_{2}$. Consider the planes $\Pi_{1}$ and $\Pi_{2}$, containing the point $O$ and the lines $l_{1}$ and $l_{2}$, respectively. The line $l$ belongs to both planes $\Pi_{1}$ and $\Pi_{2}$. The planes $\Pi_{1}$ and $\Pi_{2}$ are not par... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,060 |
1.19. In space, there are three pairwise skew lines. Prove that there exists a unique parallelepiped, three edges of which lie on these lines. | 1.19. To obtain the desired parallelepiped, it is necessary to draw two planes through each of the given lines: a plane parallel to one of the remaining lines, and a plane parallel to the other of the remaining lines. | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,061 |
1.20. On the common perpendicular to the skew lines $p$ and $q$, a point $A$ is taken. A point $M$ moves along the line $p$; $N$ is the projection of the point $M$ onto the line $q$. Prove that all planes $AMN$ have a common line.
## § 5. Pythagorean Theorem in Space | 1.20. Let $P Q$ be a common non-perpendicular to the lines $p$ and $q$, with points $P$ and $Q$ lying on lines $p$ and $q$ respectively. Draw lines $q^{\prime}$ and $p^{\prime}$ through points $P$ and $Q$, parallel to lines
$, where $x= \pm \cos \alpha, y= \pm \cos \beta, z= \pm \cos \gamma$. Therefore, $\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=x^{2... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,063 |
1.22. The plane angles at vertex $D$ of the tetrahedron $ABCD$ are right angles. Prove that the sum of the squares of the areas of its three rectangular faces is equal to the square of the area of face $ABC$.
| 1.22. First solution. Let $\alpha, \beta$ and $\gamma$ be the angles between the plane $ABC$ and the planes $DBC, DAC$ and $DAB$ respectively. If the area of the face $ABC$ is $S$, then the areas of the faces $DBC, DAC$ and $DAB$ are $S \cos \alpha, S \cos \beta$ and $S \cos \gamma$ (see problem 2.13). It remains to ch... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,064 |
1.23. Inside a sphere of radius $R$, a point $A$ is taken at a distance $a$ from its center. Through point $A$, three pairwise perpendicular chords are drawn.
a) Find the sum of the squares of the lengths of these chords.
b) Find the sum of the squares of the lengths of the segments into which point $A$ divides these... | 1.23. Consider a rectangular parallelepiped, the edges of which are parallel to the given chords, and points $A$ and the center $O$ of the sphere are its opposite vertices. Let $a_{1}, a_{2}$, and $a_{3}$ be the lengths of its edges; it is clear that $a_{1}^{2}+a_{2}^{2}+a_{3}^{2}=a^{2}$.
a) If a chord is at a distanc... | 6R^{2}-2^{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,065 |
1.24. Prove that the sum of the squares of the lengths of the projections of the edges of a cube onto any plane is $8 a^{2}$, where $a$ is the length of the edge of the cube. | 1.24. Let $\alpha, \beta$ and $\gamma$ be the angles between the edges of a cube and a line perpendicular to a given plane. Then the lengths of the projections of the cube's edges onto this plane take the values $a \sin \alpha$, $a \sin \beta$ and $a \sin \gamma$, and each value is taken exactly 4 times. Since $\cos ^{... | 8^{2} | Geometry | proof | Yes | Yes | olympiads | false | 24,066 |
1.25. Prove that the sum of the squares of the lengths of the projections of the edges of a regular tetrahedron onto any plane is $4 a^{2}$, where $a$ is the length of the edge of the tetrahedron. | 1.25. Let's draw through each edge of the tetrahedron a plane parallel to the opposite edge. As a result, we will obtain a cube in which the given tetrahedron is inscribed, and the edge of the cube is equal to \( a / \sqrt{2} \). The projection of each face of the cube is a parallelogram, the diagonals of which are equ... | 4^{2} | Geometry | proof | Yes | Yes | olympiads | false | 24,067 |
1.26. Given a regular tetrahedron with edge $a$. Prove that the sum of the squares of the lengths of the projections (onto any plane) of the segments connecting its center with the vertices is equal to $a^{2}$.
## §ิ 6. Method of Coordinates | 1.26. As in the previous problem, we will assume that the vertices of the tetrahedron $A B_{1} C D_{i}$ are located in the vertices of the cube $\triangle B C D A_{1} B_{1} C_{1} D_{1}$; the length of the edge of this cube is a/ $\sqrt{2}$. Let 0 be the center of the tetrahedron. Segments $O A$ and $O D_{1}$ are half-d... | ^{2} | Geometry | proof | Yes | Yes | olympiads | false | 24,068 |
1.27. Prove that the distance from the point with coordinates ( $x_{0}, y_{0}, z_{0}$ ) to the plane given by the equation $a x+b y+c z+d=0$ is
$$
\frac{\left|a x_{0}+b y_{0}+c z_{0}+d\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}
$$ | 1.27. Let $\left(x_{1}, y_{1}, z_{1}\right)$ be the base of the perpendicular dropped from a given point to a given plane. Since the vector ( $a, b, c$ ) is perpendicular to the given plane (Problem 1.7), then $x_{1}=x_{0}+\lambda a, y_{1}=y_{0}+\lambda b$ and $z_{1}=z_{0}+\lambda c$, and the desired distance is equal ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,069 |
1.28. Given two points $A$ and $B$ and a positive number $k \neq 1$. Find the geometric locus of points $M$ (GML) such that $A M: B M=k$. | 1.28. Introduce a coordinate system such that points $A$ and $B$ have coordinates $( -a, 0,0 )$ and $( a, 0,0)$ respectively. If point $M$ has coordinates $(x, y, z)$, then $\frac{A M^{2}}{B M^{2}}=\frac{(x+n)^{2}+y^{2}+z^{2}}{(x-a)^{2}+y^{2}+z^{2}}$. The equation $A M: B M=k$ can be transformed into the form
$$
\left... | (x+\frac{1+k^{2}}{1-k^{2}})^{2}+y^{2}+z^{2}=(\frac{2k}{1-k^{2}})^{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,070 |
1.29. Find the geometric locus of points $X$ such that $p A X^{2}+q B X^{2}+r C X^{2}=d$, where $A, B$ and $C$ are given points, $p, q, r$ and $d$ are given numbers, and $p+q+$ $+r=0$. | 1.29. Let's introduce a coordinate system, directing the $C z$ axis perpendicular to the plane $A B C$. Suppose point $X$ has coordinates ($x$, $y$, $z$). Then, for example, $A X^{2}=\left(x-a_{1}\right)^{2}+\left(y-a_{2}\right)^{2}+z^{2}$. Therefore, for the coordinates of point $X$, we obtain an equation of the form ... | \alphax+\betay+\delta=0 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,071 |
1.30. The axes of two cones, for which the angles between the axis and the generatrix are equal, are parallel. Prove that all points of intersection of their surfaces lie in one plane. | 1.30. The axis of the cone is parallel to the axis $O_{z}$; its vertex has coordinates ( $a, b, c$ ); $\alpha$ is the angle between the axis of the cone and the generatrix. Then the points on the surface of the cone satisfy the equation
$$
(x-a)^{2}+(y-b)^{2}=k^{2}(z-c)^{2}
$$
where $k=\operatorname{tg} \alpha$. The ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,072 |
1.31. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with edge $a$. Prove that the distance from any point in space to one of the lines $A A_{1}, B_{1} C_{1}, C D$ is not less than $a / \sqrt{2}$. | 1.31. Let's introduce a coordinate system, directing the axes $O x, O y$ and $O z$ along the rays $A B, A D$ and $A A_{1}$. The line $A A_{1}$ is defined by the equations $x=0, y=0$; the line $C D$ - by the equations $y=a, z=0$; the line $B_{1} C_{1}$ - by the equations $x=a, z=a$. Therefore, the squares of the distanc... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,073 |
1.32. On three mutually perpendicular lines intersecting at point $O$, points $A, B$, and $C$ are given, equally distant from $O$. Let $l$ be an arbitrary line passing through $O$; points $A_{1}, B_{1}$, and $C_{1}$ are symmetric to $A, B$, and $C$ with respect to $l$. Planes passing through points $A_{1}, B_{1}$, and ... | 1.32. Let's direct the coordinate axes along the rays $O A, O B$ and $O C$. Suppose the line $l$ forms angles $\alpha, \beta$ and $\gamma$ with these axes, respectively. The coordinates of point $M$ are equal to the coordinates of the projections of points $A_{1}, B_{1}$ and $C_{1}$ onto the axes $O x, O y$ and $O z$ r... | x+y+- | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,074 |
2.1. Given a parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1} . M-$ is the point of intersection of the diagonal $A C_{1}$ with the plane $A_{1} B D$. Prove that $A M=A C_{1} / 3$. | 2.1. Let's consider the projection of this parallelepiped onto the plane $A B C$ parallel to the line $A_{1} D$ (Fig. 17). It is clear that in this figure $A M: M C_{1}=A D: B C_{1}=1: 2$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,075 |
2.2. a) In the cube $A B C D A_{1} B_{1} C_{1} D_{1}$, a common perpendicular $M N$ is drawn to the lines $A_{1} B$ and $B_{1} C$ (point $M$ lies on the line $A_{1} B$). Find the ratio $A_{1} M: M B$.
b) Given the cube $A B C D A_{1} B_{1} C_{1} D_{1}$. Points $M$ and $N$ are taken on the segments $A A_{1}$ and $B C_{... | 2.2. a) The first solution. Consider the projection of this cube onto a plane perpendicular to the line $B_{1} C$ (Fig. 18, a). On this drawing, the line $B_{1} C$ is represented by one point, and the segment $M N$ is a perpendicular dropped from this point to the line $A_{1} B$. It is also clear that on this drawing, ... | 2:1 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,076 |
2.4. At the base of the pyramid lies a polygon with an odd number of sides. Is it possible to place arrows on its edges so that the sum of the resulting vectors is equal to zero? | 2.4. No, it cannot. Consider the projection onto a line perpendicular to the base. The projections of all base vectors are zero, while the projection of the sum of the side vectors cannot be zero, since the sum of an odd number of numbers $\pm 1$ is odd. | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,077 |
2.5. The plane passing through the midpoints of edges $AB$ and $CD$ of the tetrahedron $ABCD$ intersects edges $AD$ and $BC$ at points $L$ and $N$. Prove that $BC: CN = AD: DL$. | 2.5. Consider the projection of a tetrahedron onto a plane perpendicular to the line connecting the midpoints of edges $A B$ and $C D$. This plane is projected onto a line $L N$ passing through the intersection point of the diagonals of the parallelogram $\triangle D B C$. It is clear that for the projections $B^{\prim... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,078 |
2.6. In space, points $A, A_{1}, B, B_{1}, C,$ and $C_{1}$ are given, not lying in the same plane, such that vectors $\overrightarrow{A A}_{1}, \overrightarrow{B B}_{1}$, and $\overrightarrow{C C}_{1}$ are collinear. Planes $A B C_{1}$, $A B_{1} C$, and $A_{1} B C$ intersect at point $P$, while planes $A_{1} B_{1} C$, ... | 2.6. Let $K$ be the point of intersection of the segments $B C_{1}$ and $B_{1} C$. Then the planes $A B C_{1}$ and $A B_{1} C$ intersect along the line $A K$, and the planes $A_{1} B_{1} C$ and $A_{1} B C_{1}$ intersect along the line $A_{1} K$. Consider the projection onto the plane $A B C$ parallel to $A A_{i}$. Both... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,079 |
2.7. Given a plane $\Pi$ and points $A$ and $B$ outside it. Find the geometric locus of points $X$ in the plane $\Pi$ for which the lines $A X$ and $B X$ form equal angles with the plane $\Pi$. | 2.7. Let $A_{1}$ and $B_{1}$ be the projections of points $A$ and $B$ onto plane П. The lines $A X$ and $B X$ form equal angles with plane П if and only if the right triangles $A A_{1} X$ and $B B_{1} X$ are similar, i.e., $A_{1} X: B_{1} X = A_{1} A: B_{1} B$. The geometric locus of points in a plane, the ratio of who... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,080 |
2.8. Prove that the sum of the lengths of the edges of a convex polyhedron is greater than $3d$, where $d$ is the greatest distance between its vertices.
## § 2. Theorem of Three Perpendiculars | 2.8. Let $d=AB$, where $A$ and $B$ are vertices of the polyhedron. Consider the projection of the polyhedron onto the line $AB$. If some point $C$ projects not onto the segment $AB$, but onto its extension, for example, beyond point $B$, then $AC > AB$. Therefore, all points of the polyhedron project onto points of the... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,081 |
2.9. Line $l$ is not perpendicular to plane $\Pi$, $l^{\prime}$ - its projection on plane $\Pi$. Let $l_{1}$ be some line in plane $\Pi$. Prove that $l \perp l_{1}$ if and only if $l^{\prime} \perp l_{1}$ (Theorem of Three Perpendiculars). | 2.9. Let $O$ be the point of intersection of line $l$ and plane П (the case when line $l$ is parallel to plane П is obvious); $A$ - an arbitrary point on line $l$, different from point $O$; $A^{\prime}$ its projection on plane П. Line $A A^{\prime}$ is perpendicular to any line in plane П, therefore $A A^{\prime} \perp... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,082 |
2.10. a) Prove that the opposite edges of a regular tetrahedron are perpendicular.
b) In the base of a regular pyramid with vertex \( S \) lies the polygon \( A_{1} \ldots A_{2 n-1} \). Prove that the edges \( S A_{1} \) and \( A_{n} A_{n+1} \) are perpendicular. | 2.10. We immediately solve problem b), a particular case of which is problem a). The projection of vertex $S$ onto the plane of the base is the center $O$ of the regular polygon $A_{1} \ldots A_{2}$ p-1, and the projection of the line $S A_{1}$ onto this plane is the line $O A_{1}$. Since $O A_{1} \perp A_{n} A_{n+1}$,... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,083 |
2.12. Edge $A D$ of the tetrahedron $A B C D$ is perpendicular to the face $A B C$. Prove that the projection onto the plane $B C D$ of the orthocenter of triangle $A B C$ coincides with the orthocenter of triangle $B C D$.
## § 3. Area of the Projection of a Polygon
| 2.12. Let $B K$ and $B M$ be the altitudes of triangles $A B C$ and $D B C$ respectively. Since $B K \perp A C$ and $B K \perp A D$, the line $B K$ is perpendicular to the plane $A D C$, and therefore, $B K \perp D C$. According to the theorem of three perpendiculars, the projection of the line $B K$ onto the plane $B ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,085 |
2.13. The area of a polygon is $S$. Consequently, the area of its projection on plane П is $S \cos \varphi$, where $\varphi$ is the angle between plane П and the plane of the polygon. | 2.13. The assertion is obvious for a triangle, one of whose sides is parallel to the line of intersection of plane P with the plane of the polygon. Indeed, the length of this side does not change under projection, while the length of the height dropped to it is reduced by a factor of $\cos \varphi$.
Now let us prove t... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,086 |
2.14. Calculate the dihedral angle cosine of a regular tetrahedron. | 2.14. Let $\varphi$ be the dihedral angle along the edge of a regular tetrahedron; $O$ - the projection of the vertex $D$ of a regular tetrahedron $A B C D$ onto the opposite face. Then $\cos \varphi = S_{A B O} : S_{A B D} = 1 / 3$ | \frac{1}{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,087 |
2.15. The dihedral angle at the base of a regular $n$-sided pyramid is $\alpha$. Find the dihedral angle between adjacent lateral faces. | 2.15. Let $S$ be the area of a lateral face, $h$ the height of the pyramid, $a$ the side of the base, $\varphi$ the required angle. The area of the projection on the bisector plane of the dihedral angle between adjacent lateral faces for each of these faces is $S \cos (\varphi / 2)$; on the other hand, it is equal to $... | \cos(\varphi/2)=\sin\alpha\sin(\pi/n) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,088 |
2.17. The dihedral angles at the edges of the base of a triangular pyramid are $\alpha, \beta$, and $\gamma$; the areas of the corresponding lateral faces are $S_{a}, S_{b}$, and $S_{c}$. Prove that the area of the base is
$$
S_{c} \cos \alpha + S_{b} \cos \beta + S_{c} \cos \gamma
$$
## § 4. Problems on Projections | 2.17. Let $D^{\prime}$ be the projection of vertex $D$ of the pyramid $A B C D$ onto the plane of the base. Then $S_{A B C}= \pm S_{B C D^{\prime}} \pm S_{A C D^{\prime}} \pm S_{A B D^{\prime}}=S_{a} \cos \alpha+S_{b} \cos \beta+S_{c} \cos \gamma$. The area of triangle $B C D^{\prime}$ is taken with a minus sign if poi... | S_{}\cos\alpha+S_{b}\cos\beta+S_{}\cos\gamma | Geometry | proof | Yes | Yes | olympiads | false | 24,090 |
2.18. Projections of a spatial figure on two intersecting planes are straight lines. Is this figure necessarily a straight line? | 2.18. Not necessarily. Consider a plane perpendicular to two given planes. Any figure located in this plane will have the required property, provided that its projections on the given planes are not limited. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,091 |
2.21. Given an arbitrary triangle $ABC$. Prove that a regular triangle can be orthogonally projected onto some plane so that this projection is similar to the given triangle. | 2.21. Draw lines through vertices $A$ and $B$ perpendicular to the plane $ABC$, and take points $A_{\mathbf{i}}$ and $B_{\mathbf{i}}$ on them. Let $A A_{1}=x$ and $B B_{1}=y$ (if points $A_{i}$ and $B_{\mathbf{i}}$ lie on opposite sides of the plane $ABC$, then we consider that the numbers $x$ and $y$ have opposite sig... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,094 |
2.23. In space, there are two parallel planes and two spheres, where the first sphere touches the first plane at point $A$, the second sphere touches the second plane at point $B$, and the spheres touch each other at point $C$. Prove that points $A, B$, and $C$ lie on the same line. | 2.23. In any case, points $A, B$ lie in the same plane, and therefore one can consider the section by a plane containing these points. Since the plane of section passes through the point of tangency of the spheres (spheres and the plane), in the section, we get touching circles (a circle and a line). Let $O_{1}$ and $O... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,096 |
2.25. Two opposite edges of a tetrahedron are perpendicular, and their lengths are $a$ and $b$; 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, exactly two vertices of the cube lie. ... | 2.25. The common perpendicular to the given edges is divided by planes parallel to them and the faces of the cube into segments of length $y, x$ and $z$ ( $x$ - the length of the edge of the cube; the segment of length $y$ is adjacent to edge a). The planes of the cube's faces, parallel to the given edges, intersect th... | \frac{abc}{++ca} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,098 |
2.26. What regular polygons can result from the intersection of a cube with a plane? | 2.26. Each side of the obtained polygon belongs to one of the faces of the cube, so the number of its sides does not exceed 6. Moreover, sides belonging to opposite faces of the cube are parallel, since the lines of intersection of the plane with two parallel planes are parallel. Therefore, the section of the cube cann... | regulartriangle,,regularhexagon | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,099 |
2.27. All sections of a certain body by planes are circles. Prove that this body is a sphere. | 2.27. Consider some circle, which is a cross-section of the given body, and draw a line $l$ through its center, perpendicular to its plane. This line intersects the given body along some segment $A B$. All sections passing through the line $l$ are circles with diameter $A B$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,100 |
2.28. Through the vertex $A$ of a right circular cone, a section of maximum area is drawn. Its area is twice the area of the section passing through the axis of the cone. Find the angle at the vertex of the axial section of the cone. | 2.28. Consider an arbitrary section passing through the vertex $A$. This section is a triangle $A B C$, and its sides $A B$ and $A C$ are the generators of the cone, i.e., they have a constant length. Therefore, the area of the section is proportional to the sine of the angle $B A C$. The angle $B A C$ varies from $0^{... | 120 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,101 |
2.29. A plane divides the medians of the faces $ABC, ACD$, and $ADB$ of the tetrahedron $ABCD$, emanating from vertex $A$, in the ratios $2:1, 1:2$, and $4:1$, counting from the vertex. Let $P, Q$, and $R$ be the points of intersection of this plane with the lines $AB, AC$, and $AD$. Find the ratios $AP:PB$, $AQ:QC$, a... | 2.29. Let's restate the following problem. On the sides $AB$ and $AC$ of triangle $ABC$, points $L$ and $K$ are taken such that $AL : LB = m$ and $AK : KC = n$. Let $N$ be the intersection point of line $KL$ and median $AM$. We need to find the ratio $AN : NM$. For this, consider points $S$ and $T$, where line $KL$ int... | p=-\frac{4}{5},\frac{4}{9},r=\frac{4}{7} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,102 |
2.33. Prove that if the sum of the planar angles at the vertex of a pyramid is greater than $180^{\circ}$, then each of its lateral edges is less than half the perimeter of the base. | 2.33. Let $S A_{\mathbf{i}} \ldots A_{n}$ be a given pyramid. Cut its lateral surface along the edge $S A_{1}$ and unfold it onto a plane (Fig. 23). By the condition, point $S$ lies inside the polygon $A_{1} \ldots A_{n} A_{1}^{\prime}$. Let $B$ be the point of intersection of the extension of the segment $A_{1} S$ bey... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,105 |
2.34. Let $S_{A}, S_{B}, S_{C}$ and $S_{D}$ be the sums of the dihedral angles of the tetrahedron $ABCD$ at vertices $A, B, C$ and $D$. Prove that if $S_{A}=S_{B}$ and $S_{C}=S_{D}$, then $\triangle ABC = \triangle BAD$ and $\triangle ACD = \triangle BDC$.
## Problems for independent solving | 2.34. Since the sum of the angles of each face of the tetrahedron is $180^{\circ}$, then $S_{A}+S_{B}+S_{C}+S_{D}=4 \cdot 180^{\circ}$. Let for definiteness $S_{A} \leqslant S_{C}$. Then $360^{\circ}-S_{C}=S_{A} \leqslant 180^{\circ}$. Consider the unfolding of this tetrahedron on the plane $A B C$ (Fig. 24). Since $\a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,106 |
3.2. Prove that the volume of the tetrahedron $ABCD$ is equal to $AB \cdot AC \cdot AD \cdot \sin \beta \sin \gamma \sin D / 6$,
where $\beta$ and $\gamma$ are the plane angles at vertex $A$ opposite to the edges $AB$ and $AC$, and $D$ is the dihedral angle at the edge $AD$. | 3.2. The height of triangle $ABD$, drawn from vertex $B$, is equal to $AB \sin \gamma$, so the height of the tetrahedron, dropped onto the plane $ACD$, is equal to $AB \sin \gamma \sin D$. It is also clear that the area of triangle $ACD$ is equal to $AC \cdot AD \sin \beta / 2$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,108 |
3.3. The areas of two faces of a tetrahedron are $S_{1}$ and $S_{2}$, $a$ is the length of their common edge, and $\alpha$ is the dihedral angle between them. Prove that the volume $V$ of the tetrahedron is equal to $2 S_{1} S_{2} \sin \alpha / 3 a$. | 3.3. Let $h_{1}$ and $h_{2}$ be the heights of the given faces dropped to their common side. Then $V=\left(h_{1} \sin \alpha\right) S_{2} / 3=a h_{1} h_{2} \sin \alpha / 6$. It remains to note that $h_{1}=2 S_{1} / a, h_{2}=2 S_{2} / a$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,109 |
3.4. Prove that the volume of the tetrahedron $ABCD$ is equal to $d AB \cdot CD \sin \varphi / 6$, where $d$ is the distance between the lines $AB$ and $CD$, and $\varphi$ is the angle between them. | 3.4. Consider a parallelepiped formed by planes passing through the edges of a tetrahedron parallel to the opposite edges. The planes of the faces of the original tetrahedron cut off 4 tetrahedra from the parallelepiped, each with a volume of $1 / 6$ of the volume of the parallelepiped. Therefore, the volume of the tet... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,110 |
3.5. Point $K$ belongs to the base of a pyramid with vertex $O$. Prove that the volume of the pyramid is equal to $S \cdot K O / 3$, where $S$ is the area of the projection of the base onto a plane perpendicular to $K O$. | 3.5. The angle $\alpha$ between the line $K O$ and the height $h$ of the pyramid is equal to the angle between the plane of the base and the plane perpendicular to $K O$. Therefore, $h=K O \cos \alpha$ and $S=S^{\prime} \cos \alpha$, where $S^{\prime}$ is the area of the base (see problem 2.13). Consequently, $S \cdot ... | S\cdotKO=S^{\} | Geometry | proof | Yes | Yes | olympiads | false | 24,111 |
3.6. In the parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$, the diagonal $A C_{1}$ is equal to $d$. Prove that there exists a triangle, the lengths of whose sides are equal to the distances from the vertices $A_{1}, B$ and $D$ to this diagonal, and that the volume of the parallelepiped is $2 d S$, where $S$ is the ar... | 3.6. Consider the projection of this parallelepiped onto a plane perpendicular to the line $A C_{1}$ (Fig. 25). In the further course of the solution, the notations of Fig. 25 are used.
On this figure, the lengths of the segments $A A_{1}, A B$ and $A D$ are equal to the distances from the vertices $A_{1}, B$ and $D$ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,112 |
3.8. Prove that the ratio of the volumes of a sphere and a frustum of a cone circumscribed about it is equal to the ratio of the areas of their complete surfaces. | 3.8. Both the cone and the sphere itself can be considered as a certain limit of polyhedra circumscribed around a given sphere. It remains to note that for each of these polyhedra
Fig. 25, ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,114 |
3.9. A sphere of radius $R$ touches one base of a truncated cone and touches its lateral surface along a circumference, which is the circumference of the other base of the cone. Find the volume of the body consisting of the cone and the sphere, if the area of the complete surface of this body is $S$.
A sphere of radiu... | 3.9. The same considerations as in problem 3.8 show that the volume of this body is $S R / 3$. | SR/3 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,115 |
3.10. a) The radius of a right circular cylinder and its height are equal to $R$. Consider a sphere of radius $R$ with its center at the center $O$ of the lower base of the cylinder and a cone with vertex $O$, whose base is the upper base of the cylinder. Prove that the volume of the cone is equal to the volume of the ... | 3.10. a) Consider an arbitrary section parallel to the bases. Let $M P$ be the radius of the section of the cone, $M C$ - the radius of the section of the sphere, $M B$ - the radius of the section of the cylinder. It is required to check that $\pi M P^{2}=\pi M B^{2}-$ $-\pi M C^{2}$, i.e., $M B^{2}=M P^{2}+$ $+M C^{2}... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,116 |
3.11. Find the volume $V$ of a truncated cone with height $h$ and base radii $R$ and $r$. | 3.11. The given cone is obtained by cutting a cone with height $x$ and base radius $r$ from a cone with height $x+h$ and base radius $R$. Therefore, $V=\pi\left(R^{2}(x+h)-r^{2} x\right) / 3$. Since $x: r=(x+h): R$, then $x=r h /(R-r)$ and $x+h=R h /(R-$ $-r)$. Consequently, $V=\pi\left(r^{2}+r R+R^{2}\right) h / 3$. | \pi(r^{2}+rR+R^{2})3 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,117 |
3.12. Given a plane convex figure with perimeter $2 p$ and area $S$. Consider the body consisting of points that are at a distance of no more than $d$ from this figure. Find the volume of this body. | 3.12. Let's preliminarily assume that the given plane figure is a convex $n$-polygon. Then the considered body consists of a prism with volume $2 d S$, $n$ semi-cylinders with a total volume of $\pi p d^{2}$, and $n$ bodies from which a sphere with volume $4 \pi d^{3} / 3$ can be formed. Let's consider these last $n$ b... | 2+\pip^{2}+\frac{4\pi^{3}}{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,118 |
3.13. The volume of a convex polyhedron is $V$, the surface area is $S$; the length of the $i$-th edge is $l_{i}$, the dihedral angle at this edge is $\varphi_{i}$. Consider the body consisting of points that are at a distance not greater than $d$ from the polyhedron. Find the volume and surface area of this body. | 3.13. As in the previous problem, we will divide the obtained body into the original polyhedron, prisms corresponding to the faces, parts of cylinders corresponding to the edges, and parts of a sphere of radius $d$ corresponding to the vertices. Now it is easy to check that the volume of the obtained body is equal to $... | V+S+\frac{1}{2}^{2}\sum_{i}(\pi-\varphi_{i})l_{i}+\frac{4}{3}\pi^{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,119 |
3.14. All vertices of a convex polyhedron are located in two parallel planes. Prove that its volume is equal to $h\left(S_{1}+S_{2}+4 S\right) / 6$, where $S_{1}$ and $S_{2}$ are the areas of the faces lying in these planes, $S$ is the area of the section of the polyhedron by a plane equidistant from these, and $h$ is ... | 3.14. First solution. Let $O$ be an internal point of the polyhedron, equidistant from the given planes. The surface of the polyhedron, enclosed between the given planes, can be divided into triangles with vertices at the vertices of the polyhedron. Consequently, the polyhedron can be divided into two pyramids with ver... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,120 |
3.15. In space, there are two skew lines. The opposite edges of the tetrahedron move along these lines, with their lengths remaining constant. Prove that the volume of the tetrahedron does not change in this process. | 3.15. The volume of such a tetrahedron is equal to $a b d \sin \varphi / 6$, where $a$ and $b$ are the lengths of the edges, $d$ is the distance between the skew lines, and $\varphi$ is the angle between them (Problem 3.4). | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,121 |
3.16. In space, there are three parallel lines $a, b$, and $c$. An edge of a tetrahedron moves along line $a$, while its length remains constant, and the two remaining vertices move along lines $b$ and $c$. Prove that the volume of the tetrahedron does not change in this process. | 3.16. When projected onto a plane perpendicular to the given lines, the lines $a, b$ and $c$ transform into points $A, B$ and $C$. Let $s$ be the area of triangle $ABC$; $K S$ be the height of the tetrahedron, which does not move along the line $a$. According to problem 3.5, the volume of the considered tetrahedron is ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,122 |
3.17. Prove that a plane intersecting only the lateral surface of a cylinder divides its volume in the same ratio as it divides the axis of the cylinder. | 3.17. Let the plane П intersect the axis of the cylinder at point O. Draw through point O a plane $\Pi^{\prime}$ parallel to the bases of the cylinder. These two planes divide the cylinder into 4 parts, and the 2 parts enclosed between the planes have equal volume. Therefore, the volumes of the parts into which the cyl... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,123 |
3.18. Prove that the plane passing through the midpoints of two skew edges of a tetrahedron divides it into two parts of equal volume. | 3.18. Let $M$ and $K$ be the midpoints of the edges $AB$ and $CD$ of the tetrahedron $ABCD$. For definiteness, the plane passing through $M$ and $K$ intersects the edges $AD$ and $BC$ at points $L$ and $N$ (Fig. 27). The plane $DMC$ divides the tetrahedron into two parts of equal volume, so it is sufficient to prove th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,124 |
3.19. Parallel lines $a, b, c$ and $d$ intersect one plane at points $A, B, C$ and $D$, and another plane at points $A^{\prime}, B^{\prime}, C^{\prime}$ and $D^{\prime}$. Prove that the volumes of the tetrahedra $A^{\prime} B C D$ and $A B^{\prime} C^{\prime} D^{\prime}$ are equal. | 3.19. According to problem $3.16 V_{A^{\prime} A B C}=V_{A A^{\prime} B^{\prime} C^{\prime}}$. Writing similar equalities for the volumes of tetrahedrons $A^{\prime} A D C$ and $A^{\prime} A B D$ and expressing $V_{A^{\prime} B C D}$ and $V_{A B^{\prime} C^{\prime} D^{\prime}}$ through these volumes, we obtain the requ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,125 |
3.20. In the planes of the faces of the tetrahedron \(ABCD\), points \(A_1, B_1, C_1\), and \(D_1\) are taken such that the lines \(AA_1, BB_1, CC_1\), and \(DD_1\) are parallel. Find the ratio of the volumes of the tetrahedra \(ABCD\) and \(A_1B_1C_1D_1\).
## § 4. Calculation of Volume | 3.20. Let $A_{2}$ be the intersection point of the line $A A_{1}$ with the plane $B_{1} C_{1} D_{1}$. We will prove that $A_{1} A_{2} = 3 A_{1} A$. Then $V_{A B C D} : V_{A_{2} B C D} = 1: 3$; using the result of problem 3.19, we finally get $V_{A B C D} : V_{A_{1} B_{1} C_{1} D_{1}} = V_{A B C D} : V_{A_{2} B \sim D} ... | 1:3 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,126 |
3.21. The planes $A B C_{1}$ and $A_{1} B_{1} C$ divide the triangular prism $A B C A_{1} B_{1} C_{1}$ into four parts. Find the ratio of the volumes of these parts. | 3.21. Let $P$ and $Q$ be the midpoints of segments $A C_{1}$ and $B C_{1}$, i.e., $P Q$ is the line of intersection of the given planes. The ratio of the volumes of tetrahedra $C_{1} P Q C$ and $C_{1} A B C$ is $\left(C_{1} P: C_{1} A\right)\left(C_{1} Q: C_{1} B\right)=$ $=1: 4$ (see problem 3.1). It is also clear tha... | 1:3:3:5 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,127 |
3.22. The volume of the parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$ is $V$. Find the volume of the common part of the tetrahedra $A B_{1} C D_{1}$ and $A_{1} B C_{1} D$. | 3.22. The common part of the specified tetrahedra is a convex polyhedron with vertices at the centers of the faces of the parallelepiped. A plane equidistant from two opposite faces of the parallelepiped divides this polyhedron into two quadrilateral pyramids, the volume of each of which is equal to $V / 12$. | \frac{V}{6} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,128 |
3.23. In what ratio does a plane divide the volume of a tetrahedron if the plane is parallel to two of its skew edges and divides one of the other edges in the ratio $2: 1?$ | 3.23. The section of the tetrahedron by a given plane is a parallelogram. Each of the two resulting parts of the tetrahedron can be cut into a pyramid, the base of which is this parallelogram, and a tetrahedron. The volumes of these pyramids and tetrahedrons can be expressed in terms of the lengths $a$ and $b$ of the i... | \frac{20}{7} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,129 |
3.24. On three parallel lines, vectors $\overrightarrow{A A}_{1}, \overrightarrow{B B}_{1}$, and $\overrightarrow{C C}_{1}$ are taken. Prove that the volume of the convex polyhedron $A B C A_{1} B_{1} C_{1}$ is equal to $S\left(A A_{1}+B B_{1}+C C_{1}\right) / 3$, where $S$ is the area of the triangle formed by the int... | 3.24. Extend the edge $B B_{1}$ beyond point $B_{1}$ to point $B_{2}$, such that the segment $B_{1} B_{2}$ is equal to the edge $A A_{1}$. Let $K$ be the midpoint of the segment $A_{1} B_{1}$, i.e., the point of intersection of the segments $A_{1} B_{1}$ and $A B_{2}$. Since the volumes of the tetrahedra $A_{1} K C_{1}... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,130 |
3.25. Let $M$ be the point of intersection of the medians of the tetrahedron $ABCD$ (see p. 244). Prove that there exists a quadrilateral whose sides are equal and parallel to the segments connecting $M$ with the vertices of the tetrahedron. Calculate the volume of the tetrahedron defined by this spatial quadrilateral ... | 3.25. Complete the pyramid $MABC$ to a parallelepiped (Fig. 29). Let $MK$ be its diagonal. Since $\overrightarrow{MA} + \overrightarrow{MB} + \overrightarrow{MC} + \overrightarrow{MD} = \overrightarrow{0}$ (see problem 14.3, a), then $\overrightarrow{KM} = \overrightarrow{MD}$.
 about a point lying on the perpendicular bisector of this segment and at a distance $x$ from the segment, a ring is formed with an inner radius of $x$ and an outer radius of $\sqrt{x^{2}+d^{2}}$; the area of this ring is $\pi d^{2}$, i.e., it... | \pi^{3}\sqrt{3/24} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,132 |
3.27. Lines $A C$ and $B D$, the angle between which is $\alpha\left(\alpha<90^{\circ}\right)$, touch a sphere of radius $R$ at diametrically opposite points $A$ and $B$. Line $C D$ also touches the sphere, and the angle between $A B$ and $C D$ is $\varphi\left(\varphi<90^{\circ}\right)$. Find the volume of the tetrahe... | 3.27. Let $A C=x, B D=y ; D_{1}$ be the projection of $D$ onto the plane tangent to the sphere at point $A$. In triangle $C A D_{1}$, angle $A$ is equal to $\alpha$ or $180^{\circ}-\alpha$, so $x^{2}+y^{2} \mp 2 x y \cos \alpha=C D_{1}^{2}=$ $=4 R^{2} \operatorname{tg}^{2} \varphi$. It is also clear that $x+y=C D=2 R /... | \begin{cases}2R^{3}\operatorname{tg}(\alpha/2)/3&if\alpha\leqslant2\varphi<\pi-\alpha,\\2R^{3}\operatorname{ctg}(\alpha/2)/3&if\pi-\alpha\leqslant2\varphi< | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,133 |
3.28. Point $O$ lies on the segment connecting the vertex of a triangular pyramid with volume $V$ to the point of intersection of the medians of the base. Find the volume of the common part of the given pyramid and the pyramid symmetric to it with respect to point $O$, if point $O$ divides the specified segment in the ... | 3.28. In figures $30, a-d$, the common parts of the pyramids in all four cases are depicted.
a) The common part is a parallelepiped (Fig. 30, a). It is obtained from the original pyramid by cutting off three pyramids similar to it with a coefficient of $2 / 3$; at the same time, three pyramids similar to the original ... | \frac{} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,134 |
3.29. The sides of the spatial quadrilateral $K L M N$ are perpendicular to the faces of the tetrahedron $A B C D$, and their lengths are equal to the areas of the corresponding faces. Find the volume of the tetrahedron $K L M N$, if the volume of the tetrahedron $A B C D$ is $V$. | 3.29. The existence of such a spatial quadrilateral $K L M N$ for any tetrahedron $A B C D$ follows from the assertion of problem 7.19; there are several such quadrilaterals, but the volumes of all the tetrahedra they define are equal (problem 8.26).
Using the formula from problem 3.2, it is easy to prove that $V^{3}=... | \frac{3}{4}V^{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,135 |
3.30. The lateral edge of a regular prism $A B C A_{1} B_{1} C_{1}$ is equal to $a$; the height of the base of the prism is also equal to $a$. Through point $A$, planes are drawn perpendicular to the lines $A B_{1}$ and $A C_{1}$, and through point $A_{1}$ - planes perpendicular to $A_{1} B$ and $A_{1} C$. Find the vol... | 3.30. Let $M$ and $N$ be the midpoints of the edges $B_{1} C_{1}$ and $B C$. The planes of the considered spheres are symmetric relative to the plane $A A_{1} M N$. Take a point $K$ on the ray $M N$ such that $M K=2 M N$.
Since $A A_{1} M N$ is a square, $K A \perp A M$, and therefore, the line $A K$ is perpendicular ... | \frac{9^{3}\sqrt{3}}{4} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,136 |
3.31. Tetrahedra $A B C D$ and $A_{1} B_{1} C_{1} D_{1}$ are positioned such that all vertices of each lie in the corresponding planes of the faces of the other tetrahedron ( $A$ lies in the plane $B_{1} C_{1} D_{1}$, etc.). Moreover, $A_{1}$ coincides with the centroid of triangle $B C D$, and the lines $B D_{1}, C B_... | 3.31. Let $K, L$ and $M$ be the midpoints of segments $AB, AC$ and $AD$. First, we prove that $K$ is the midpoint of segment $DC_1$. Point $B$ lies in the plane $A_1C_1D_1$, therefore point $C_1$ lies in the plane $A_1LB$. We extend the tetrahedron $ABCD$ to a triangular prism by adding vertices $S$ and $T$, where $\ov... | \frac{3V}{8} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,137 |
3.32. Prove that the bisector plane of a dihedral angle at the edge of a tetrahedron divides the opposite edge into parts proportional to the areas of the faces enclosing this angle. | 3.32. The ratio of segments of the edge is equal to the ratio of the heights dropped from its ends onto the bisector plane, and the latter ratio is equal to the ratio of the volumes of the tetrahedra into which the plane divides the given tetrahedron. Since the heights dropped from any point on the bisector plane to th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,138 |
3.33. In the tetrahedron $A B C D$, 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 passing through the edge $A B$ and the center of the sphere inscribed in the tetrahedron. | 3.33. Let $a=AB, x$ be the area of the desired section. Using the formula from problem 3.3 for the volume of tetrahedron $ABCD$ and its parts, we get
$$
\frac{2}{3} \frac{p x \sin (\alpha / 2)}{a}+\frac{2}{3} \frac{q x \sin (\alpha / 2)}{a}=\frac{2}{3} \frac{p q \sin \alpha}{a}
$$
Therefore, $x=\frac{2 p q \cos (\alp... | \frac{2pq\cos(\alpha/2)}{p+q} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,139 |
3.34. 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
$$
\frac{x_{1}}{h_{1}}+\frac{x_{2}}{h_{2}}+\frac{x_{3}}{h_{3}}+\frac{x_{4}}{h_{4}}=1
$$ | 3.34. Let's cut the tetrahedron into 4 triangular pyramids, the bases of which are the faces of the tetrahedron, and the vertex is the given point. The specified sum of ratios is the sum of the ratios of the volumes of these pyramids to the volume of the tetrahedron. This sum is equal to 1, since the sum of the volumes... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,140 |
3.35. On the face $ABC$ of the tetrahedron $ABCD$, a point $O$ is taken, and through it, segments $OA_1$, $OB_1$, and $OC_1$ are drawn parallel to the edges $DA$, $DB$, and $DC$, respectively, until they intersect the faces of the tetrahedron. Prove that
$$
\frac{OA_1}{DA} + \frac{OB_1}{DB} + \frac{OC_1}{DC} = 1
$$ | 3.35. Parallel segments $A D$ and $O A_{1}$ form equal angles with the plane $B C D$, therefore the ratio of the lengths of the heights dropped from points $O$ and $A$ to this plane is equal to the ratio of the lengths of these segments. Consequently, $\frac{V_{O B C D}}{V_{A B C D}}=\frac{O A_{1}}{D A}$. Writing simil... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,141 |
3.36. Let $r$ be the radius of the inscribed sphere of a tetrahedron; $r_{a}, r_{b}, r_{c}$, and $r_{d}$ be the radii of the spheres, each of which touches one face and the extensions of the other three. Prove that
$$
\frac{1}{r_{a}}+\frac{1}{r_{b}}+\frac{1}{r_{c}}+\frac{1}{r_{d}}=\frac{2}{r}
$$ | 3.36. Let $S_{a}, S_{b}, S_{c}$ and $S_{d}$ be the areas of the faces $BCD, ACD$, $ABD$ and $ABC$; $V$ be the volume of the tetrahedron; $O$ be the center of the sphere that touches the face $BCD$ and the extensions of the other three faces. Then $3 V=r_{a}\left(-S_{a}+S_{b}+S_{c}+S_{d}\right), \quad$ hence, $\quad 1 /... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,142 |
3.37. Given a convex quadrilateral pyramid $M A B C D$ with vertex $M$. A plane intersects the edges $M A, M B, M C$ and $M D$ at points $A_{1}, B_{1}, C_{1}$ and $D_{1}$ respectively. Prove that
$$
S_{B C D} \frac{M A}{M A_{1}}+S_{A B D} \frac{M C}{M C_{1}}=S_{A B C} \frac{M D}{M D_{1}}+S_{A C D} \frac{M B}{M B_{1}}
... | 3.37. The pyramid $M A_{1} B_{1} C_{1} D_{1}$ can be cut into two tetrahedra by the plane $M A_{1} C_{1}$ or the plane $M B_{1} D_{1}$, 60
\[
V_{M B_{1} C_{1} D_{1}}+V_{M A_{1} B_{1} D_{1}}=V_{M A_{1} B_{1} C_{1}}+V_{M_{A_{1}} C_{1} D_{1}}
\]
Using the formula from problem 3.1, we get
\[
\begin{aligned}
& V_{M B_{1}... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,143 |
3.38. The lateral faces of a triangular pyramid are equal in area and form angles $\alpha, \beta$ and $\gamma$ with the base. 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 lateral faces. | 3.38. Let $r$ and $r^{\prime}$ be the radii of the inscribed and exscribed spheres, $S$ the area of the lateral face, $s$ the area of the base, and $V$ the volume of the pyramid. Then $V=(3 S+s) r / 3$. Similarly, it can be shown that $V=(3 S-s) r^{\prime} / 3$. Moreover, $s=(\cos \alpha + \cos \beta + \cos \gamma) S$ ... | \frac{3-\cos\alpha-\cos\beta-\cos\gamma}{3+\cos\alpha+\cos\beta+\cos\gamma} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,144 |
4.1. A plane touches two tangent spheres of radius $R$ and $r$ at points $A$ and $B$. Prove that $A B=$ $=2 \sqrt{\operatorname{Rr}}$. | 4.1. First, let's prove that the length of the common tangent to two touching circles with radii \( R \) and \( r \) is \( 2 \sqrt{R r} \). For this, consider a right triangle, the ends of the hypotenuse of which are the centers of the circles, and one of the legs is parallel to the common tangent. Applying the Pythago... | 2\sqrt{Rr} | Geometry | proof | Yes | Yes | olympiads | false | 24,145 |
4.2. Three spheres touch each other pairwise; a plane touches these spheres at points $A, B$, and $C$. Find the radii of the spheres if the sides of triangle $ABC$ are equal to $a, b$, and $c$. | 4.2. Let $x, y$ and $z$ be the radii of the spheres. According to problem 4.1, $a=2 \sqrt{x y}, b=2 \sqrt{y z}$ and $c=2 \sqrt{x z}$. Therefore, $a c / b=2 x$, hence $x=a c / 2 b$. Similarly, $y=a b / 2 c$ and $z=b c / 2 a$. | x=\frac{ac}{2b},y=\frac{}{2c},z=\frac{}{2a} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,146 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.