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35. Four points not lying in the same plane are given by their coordinates: $A\left(x_{1}, y_{1}, z_{1}\right) ; B\left(x_{2}, y_{2}, z_{2}\right), C\left(x_{3}, y_{3}, z_{3}\right)$ and $D\left(x_{4}, y_{4}, z_{4}\right)$. Find the coordinates of a point $M$ such that the forces proportional to the vectors $\overright... | 35. We already know that the radius vector $\vec{r}_{0}$ of the center of mass of $n$ points $A_{1}, A_{2}, \ldots, A_{n}$ with masses $m_{1}, m_{2}, \ldots, m_{n}$, determined by the radius vectors $\vec{r}_{1}=\overrightarrow{O A}_{1}, \vec{r}_{2}=\overrightarrow{O A}_{2}, \ldots, \vec{r}_{n}=\overrightarrow{O A}_{n}... | x_{0}=\frac{\sum_{k=1}^{n}m_{k}x_{k}}{\sum_{k=1}^{n}m_{k}};\quady_{0}=\frac{\sum_{k=1}^{n}m_{k}y_{k}}{\sum_{k=1}^{n}m_{k}};\quadz_{0}=\frac{\sum_{k=1}^{n}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 44,358 |
36*. A wireframe model of a tetrahedron (without faces) is made from a uniform metal wire. The vertices of the tetrahedron $A, B$, $C, D$ have the following coordinates respectively: $\left(x_{1}, y_{1}, z_{1}\right),\left(x_{2}, y_{2}, z_{2}\right)$, $\left(x_{3}, y_{3}, z_{3}\right),\left(x_{4}, y_{4}, z_{4}\right)$.... | 36. Let's assume that the mass of an edge of the tetrahedron is proportional to the length of this edge and is concentrated at its midpoint.
The length of edge $AB$ by the distance formula is $\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}=r_{12}$. Similar formulas will b... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,359 |
37. Given points $M_{1}\left(x_{1}, y_{1}, z_{1}\right)$ and $M_{2}\left(x_{2}, y_{2}, z_{2}\right)$. Find the coordinates of the point lying on the $O x$ axis and equidistant from the given points. | 37. The coordinates of the desired point have the form ( $x, 0,0$ ). Using the formula for the distance between points, we get the equation:
$$
\left(x_{1}-x\right)^{2}+y_{1}^{2}+z_{1}^{2}=\left(x_{2}-x\right)^{2}+y_{2}^{2}+z_{2}^{2}
$$
After expanding the brackets and transformations, we get:
$$
x=\frac{x_{2}^{2}-x... | \frac{x_{2}^{2}-x_{1}^{2}+y_{2}^{2}-y_{1}^{2}+z_{2}^{2}-z_{1}^{2}}{2(x_{2}-x_{1})} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,360 |
38. Calculate the angle between the bisectors of the coordinate angles $x O y$ and $y O z$. | 38. The bisector of angle $x O y$ is defined by the vector $\vec{r}_{1}=\vec{i}+\vec{j}$; the bisector of angle $y O z$ is defined by the vector $\overrightarrow{r_{2}}=\vec{j}+\vec{k}$. Consider the scalar product: $\quad \vec{r}_{1} \cdot \vec{r}_{2}=(\vec{i}+\vec{j}) \cdot (\vec{j}+\vec{k})=1=$ $=r_{1} r_{2} \cos \v... | 60 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,361 |
39. The center of the lower base of the cube is connected by straight lines to the four vertices of the upper base of the cube. Calculate the angles between these lines. | 39. Let's place the origin of the coordinate system at the center of the base of the cube, direct the axes $O x$ and $O y$ along the diagonals of the base, and take the length of half the diagonal of the base as the unit. Then the coordinates of the four vertices will be: $(1,0, \sqrt{2}),(0,1, \sqrt{2}),(-1,0, \sqrt{2... | \varphi\approx4811^{\}23^{\\},\varphi^{\}\approx7031^{\}44^{\\} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,362 |
40. The line $l$ passes through the origin, its direction is determined by the unit vector $\vec{e}$, which forms angles $\alpha, \beta$, and $\gamma$ with the coordinate axes $O x, O y, O z$. Any point $M$ on this line defines the radius vector $\vec{r}=\overrightarrow{O M}$, which is collinear with the unit vector $\... | 40. Let $\cos \alpha, \cos \beta, \cos \gamma$ be the direction cosines of the unit vector $\vec{e}$. Since vectors $\vec{r}$ and $\vec{e}$ are collinear, from the equality $x \vec{i} +$ 132
$+y \vec{j}+z \vec{k}=t \vec{i} \cos \alpha+t \vec{j} \cos \beta+t \vec{k} \cos \gamma \quad$ we obtain: $\quad x=t \cos \alpha$,... | \frac{x}{\cos\alpha}=\frac{y}{\cos\beta}=\frac{z}{\cos\gamma} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 44,363 |
41. Prove that the system of equations of the form
$$
\frac{x}{m}=\frac{y}{n}=\frac{z}{p}
$$[^7]defines a line passing through the origin. How can these equations be transformed into the form (2) from the previous problem? For what values of $m, n, p$ will this line lie in one of the coordinate planes? coincide with o... | 41. The system of equations $\frac{x}{m}=\frac{y}{n}=\frac{z}{p}$ can be understood as the condition of collinearity of vectors $\vec{r}=(x, y, z)$ and $\vec{a}=(m, n, p)$, therefore such a system defines a line. Considering the unit vector $\vec{e}=\frac{\vec{a}}{a}$, where $a=|\vec{a}|=\sqrt{m^{2}+n^{2}+p^{2}}$, this... | proof | Algebra | proof | Yes | Yes | olympiads | false | 44,364 |
42. Point $M$ in space is defined by the radius vector $\vec{a}=\overrightarrow{O M}$ with coordinates ( $x_{1}, y_{1}, z_{1}$ ). A line passing through point $M$ is defined by the unit vector $\vec{e}$, which forms angles $\alpha, \beta, \gamma$ with the axes $O x, O y$, $O z$. Let $\vec{r}$ be the variable radius vec... | 42. The vector $\vec{r}-\vec{a}$ is collinear with the vector $\vec{e}$, hence: $(\vec{r}-\vec{a})=t \vec{e}$, where $t \in \boldsymbol{R}$. Transitioning to coordinates, we find:
$$
\left(x-x_{1}\right) \vec{i}+\left(y-y_{1}\right) \vec{j}+\left(z-z_{1}\right) \vec{k}=t \vec{i} \cos \alpha+t \vec{j} \cos \beta+t \vec... | \frac{x-x_{1}}{\cos\alpha}=\frac{y-y_{1}}{\cos\beta}=\frac{z-z_{1}}{\cos\gamma} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,365 |
43. Prove that the system of equations of the form $\frac{x-x_{i}}{m}=\frac{y-y_{\mathrm{t}}}{n}=$ $=\frac{z-z_{1}}{p}$ defines a line passing through the point with coordinates $\left(x_{1}, y_{1}, z_{1}\right)$.
How, using these equations, can we obtain the equations of a line passing through two points given by the... | 43. Just as in problem 41, using the coefficients $m, n$, $p$, we will compute the direction cosines and reduce the equations to the canonical form obtained in the previous problem. Thus, the equations
$$
\frac{x-x_{1}}{m}=\frac{y-y_{1}}{n}=\frac{z-z_{1}}{p}
$$
define a line passing through the point with coordinates... | \frac{x-x_{1}}{x_{2}-x_{1}}=\frac{y-y_{1}}{y_{2}-y_{1}}=\frac{z-z_{1}}{z_{2}-z_{1}} | Algebra | proof | Yes | Yes | olympiads | false | 44,366 |
44. Two lines are given by the systems of equations:
$$
\frac{x-x_{1}}{m_{1}}=\frac{y-y_{1}}{n_{1}}=\frac{z-z_{1}}{p_{1}} \quad \text { and } \quad \frac{x-x_{2}}{m_{2}}=\frac{y-y_{2}}{n_{2}}=\frac{z-z_{2}}{p_{2}}
$$
Write the condition for the parallelism of these lines. Write the condition for their perpendicularit... | 44. If the lines are parallel, then their direction vectors are collinear, the direction cosines are the same, and the coefficients determining these cosines must be proportional. Hence, the condition for parallelism is:
$$
\frac{m_{1}}{m_{2}}=\frac{n_{1}}{n_{2}}=\frac{p_{1}}{p_{2}}
$$
For perpendicular lines, the sc... | \frac{m_{1}}{m_{2}}=\frac{n_{1}}{n_{2}}=\frac{p_{1}}{p_{2}}\quad\quadm_{1}m_{2}+n_{1}n_{2}+p_{1}p_{2}=0 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,367 |
45. Write the system of equations of a line passing through the origin and forming equal angles with the three coordinate axes. Determine the magnitude of these angles. How many solutions does the problem have? | 45. According to the problem, $\cos \alpha=\cos \beta=\cos \gamma$ and the equations take the form: $x=y=z$. Further, we have:
$$
\begin{gathered}
\cos ^{2} \alpha+\cos ^{2} \alpha+\cos ^{2} \alpha=1 ; \cos ^{2} \alpha=\frac{1}{3} \\
\cos \alpha= \pm \frac{1}{\sqrt{3}} ; \alpha \approx 54^{\circ} 44^{\prime} 8^{\prime... | 4 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 44,368 |
46. Write the system of equations of the line passing through the point $(3, -2, 1)$ and perpendicular to the lines given by the systems:
$$
\frac{x-1}{2}=\frac{y+2}{3}=\frac{z-3}{-1} \quad \text{and} \quad \frac{x+3}{4}=\frac{y+1}{-1}=\frac{z+3}{3}
$$ | 46. Let $m, n, p$ be the coefficients of the desired line. The condition of perpendicularity gives two equations:
$$
\begin{aligned}
& 2 m+3 n-p=0 \\
& 4 m-n+3 p=0
\end{aligned}
$$
Solving them, we get: $m: n: p=4:(-5):(-7)$. From this, we obtain the equations of the line:
$$
\frac{x-3}{4}=\frac{y+2}{-5}=\frac{z-1}{... | \frac{x-3}{4}=\frac{y+2}{-5}=\frac{z-1}{-7} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 44,369 |
47. On a ray originating from the origin, there are two points with coordinates ( $x_{1}, y_{1}, z_{1}$ ) and ( $x_{2}, y_{2}, z_{2}$ ). The distances from these points to the origin are $r_{1}$ and $r_{2}$. Prove the equality:
$$
r_{1} r_{2}=x_{1} x_{2}+y_{1} y_{2}+z_{1} z_{2}
$$ | 47. In this case, the angle between the vectors $\vec{r}_{1}$ and $\overrightarrow{r_{2}}$ is 0 and $\cos \varphi=1$. The scalar product of the vectors is therefore $r_{1} r_{2}=x_{1} x_{2}+y_{1} y_{2}+z_{1} z_{2}$. | proof | Algebra | proof | Yes | Yes | olympiads | false | 44,370 |
48. A perpendicular [OP] is drawn from the origin to a given plane, defining the distance from this plane to the origin. Let's introduce the following notations: $\vec{p}=\overrightarrow{O P},|\vec{p}|=p-$ the mentioned distance; $\vec{e}$-the unit vector of $\vec{p}$ (then $\vec{p}=p \vec{e}$ ); $\alpha, \beta, \gamma... | 48. Let $\vec{r}$ be a variable radius vector. Since the vector $(\vec{r}-\vec{p})$ is perpendicular to the vector $\vec{p}$, we obtain the equation: $(\vec{r}-\vec{p}) \vec{p}=0 ; \quad \vec{r} \vec{e}=p . \quad$ Here $\quad \vec{r}=x \vec{i}+y \vec{j}+z \vec{k}, \quad \vec{p}=p \vec{e}$, $\vec{e}=\vec{i} \cos \alpha+... | x\cos\alpha+y\cos\beta+z\cos\gamma=p | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,371 |
49. Prove that any equation of the form $\vec{r} \vec{a}=m$, where $\vec{r}$ is a variable vector, $\vec{a}$ is a fixed non-zero vector, and $m$ is a real number, is the equation of a plane.
Prove also that any linear equation with three variables of the form $A x+B y+C z+D=0$, where $(A, B, C) \neq(0,0,0)$, is the eq... | 49. The equation $\vec{r} \vec{a}=m$ can be transformed into the form $\vec{r} \vec{e}=p$, where $\vec{e}=\frac{\vec{a}}{a}, p=\frac{m}{a}$. Further, by replacing $p$ with $p \vec{e} \cdot \vec{e}$, we will have: 134
$\vec{r} \vec{e}=p \vec{e} \cdot \vec{e} ;(\vec{r}-p \vec{e}) \vec{e}=0$. If $\vec{p} \vec{e}=\left(x_{... | proof | Algebra | proof | Yes | Yes | olympiads | false | 44,372 |
50. Given equations of two planes:
$$
A_{1} x+B_{1} y+C_{1} z+D_{1}=0 \text { and } A_{2} x+B_{2} y+C_{2} z+D_{2}=0
$$
Find the conditions for the parallelism and perpendicularity of these planes. | 50. The coefficients of the variables in the equations of planes are the coordinates of vectors perpendicular to these planes, i.e., normal vectors. Therefore, for the parallelism of planes, the necessary and sufficient condition is the proportionality of these coefficients to each other:
$$
\frac{A_{1}}{A_{2}}=\frac{... | \frac{A_{1}}{A_{2}}=\frac{B_{1}}{B_{2}}=\frac{C_{1}}{C_{2}}A_{1}A_{2}+B_{1}B_{2}+C_{1}C_{2}=0 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,373 |
51. Given the equations of a plane and a line:
$$
A x+B y+C z+D=0 \text { and } \frac{x-x_{1}}{m}=\frac{y-y_{1}}{n}=\frac{z-z_{1}}{p}
$$
Find the conditions for the parallelism and perpendicularity of the line and the plane | 51. For the parallelism of a line and a plane, it is necessary and sufficient that the direction vector of the line be perpendicular to the normal vector of the plane: a line perpendicular to the perpendicular of the plane will be parallel to this plane. Therefore, the condition for parallelism can be written as:
$$
A... | A+Bn+Cp=0 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,374 |
52. Find the equations of the line passing through the point with coordinates $(3,-2,4)$ and perpendicular to the plane given by the equation $5 x-3 y+3 z=10$.
53*. Prove that if the coordinates of any point are substituted into the left side of the normal equation of a plane, written in the form:
$$
x \cos \alpha + ... | 52. Using the conclusions from the previous problem, we conclude that the coefficients $m, n, p$ are proportional to the numbers: $5, -3, 3$. From this, we obtain the equation:
$$
\frac{x-3}{5}=\frac{y+2}{-3}=\frac{z-4}{3}
$$ | \frac{x-3}{5}=\frac{y+2}{-3}=\frac{z-4}{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,375 |
54*. Given the equations of two planes in normal form. How can we obtain the equation of the set of points that are at equal distances from both planes? (This set is the union of the bisector planes of the dihedral angles between the given planes.)
Given the equations of two planes in normal form. How can we obtain th... | 54. Let's denote the left side of the normal equation of a plane symbolically by the letter $P$ and write the equation itself in the form $P=0$. If $P_{1}$ and $P_{2}$ are the left sides of two normal equations of planes, then the equation $P_{1}=P_{2}$ or $P_{1}-P_{2}=0$ is the equation of the set of points equidistan... | P_{1}-P_{2}=0P_{1}+P_{2}=0 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,376 |
55. Points $A$ and $A^{\prime}$ are given by their coordinates: $(7,1,4)$ and $(3,5,2)$. Find the equation of the plane relative to which these points are symmetric. | 55. All points with coordinates ( $x, y, z$ ), equally distant from two given points, satisfy the conditions:
$$
(x-7)^{2}+(y-1)^{2}+(z-4)^{2}=(x-3)^{2}+(y-5)^{2}+\cdot(z-2)^{2} .
$$
Expanding the brackets, we get:
$$
\begin{gathered}
14 x-49+2 y-1+8 z-16=6 x-9+10 y-25+4 z-4 \\
8 x-8 y+4 z-28=0 ; 2 x-2 y+z-7=0
\end{... | \frac{2}{3}x-\frac{2}{3}y+\frac{z}{3}-\frac{7}{3}=0 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,377 |
56. Prove that the equation $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ defines a plane intersecting the coordinate axes at points with coordinates $a, b, c$. This equation is called the equation of a plane in intercepts. Under what conditions can the general equation $A x+B y+C z+D=$ $=0$ be transformed into the intercept... | 56. Substituting $y=0$ and $z=0$ into the equation, we get: $\frac{x}{a}=1 ; x=a$, which means the plane intersects the $O x$ axis at the point $x=a$. The points of intersection with the $O y$ and $O z$ axes are found in the same way. If all coefficients $A, B, C, D$ in the equation $A x+B y+$ †- $C z+D=0$ are non-zero... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,378 |
57*. Prove that the plane intersecting the coordinate axes at points with coordinates $a, b$, and $c$ along these axes is at a distance $p$ from the origin, satisfying the relation
$$
\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}=\frac{1}{p^{2}}
$$ | 57. To bring the equation in segments to a normal form, we divide it by $\sqrt{\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}}$ (or, equivalently, multiply by the normalizing factor: $\left.\frac{1}{\sqrt{\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}}}\right)$.
Then the distance $p$ from the origin to the plane wil... | \frac{1}{^{2}}+\frac{1}{b^{2}}+\frac{1}{^{2}}=\frac{1}{p^{2}} | Geometry | proof | Yes | Yes | olympiads | false | 44,379 |
58. Planes are given by the equations:
$$
\begin{aligned}
& 3 x+4 y-2 z=5 \\
& 2 x+3 y-z=3
\end{aligned}
$$
Write in standard form (see problems 42,43) the system of equations that defines the line of intersection of these planes. | 58. The sought line is simultaneously parallel to two planes. Based on the conclusions of problem 51, the coefficients of its canonical equation must satisfy the conditions:
$$
\left\{\begin{array}{l}
3 m+4 n-2 p=0 \\
2 m+3 n-p=0
\end{array}\right.
$$
Solving this system, we get:
$$
m: n: p=2:(-1): 1
$$
At the same... | \frac{x-3}{2}=\frac{y+1}{-1}=\frac{z}{1} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 44,380 |
60. The center of sphere $Q$ has coordinates ( $a, b, c$ ), and the radius of the sphere is $R$. Denoting the variable radius vector by $\vec{r}$, and the radius vector of point $Q$ by $\vec{q}=\overrightarrow{O Q}$, derive the vector equation of the sphere as the set of points located at a distance $R$ from point $Q$.... | 60. From the condition of the problem, we get: $(\vec{r}-\vec{q})^{2}=R^{2}$, therefore, $(x-a)^{2}+(y-b)^{2}+(z-c)^{2}=R^{2}$. | (x-)^{2}+(y-b)^{2}+(z-)^{2}=R^{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,382 |
61. Write the general equation of a sphere whose center coincides with the origin. Write the general equation of a sphere passing through the origin and with its center on the $O x$ axis. | 61. When $a=b=c=0$ we get: $x^{2}+y^{2}+z^{2}=R^{2}$. If the center of the sphere passing through the origin lies on the $O x$ axis, then the coordinates $(a, b, c)$ of the center are $a=R, b=0, c=0$ and the equation of the sphere takes the form:
$$
x^{2}-2 R x+y^{2}+z^{2}=0
$$ | x^{2}-2Rx+y^{2}+z^{2}=0 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,383 |
62*. Given two points: $A_{1}\left(x_{1}, y_{1}, z_{1}\right)$ and $A_{2}\left(x_{2}, y_{2}, z_{2}\right)$. Point $P$ defines two vectors: $\overrightarrow{P A}_{1}=\vec{q}_{1}$ and $\overrightarrow{P A}_{2}=\vec{q}_{2}$. What does the set of points $P$ represent, for which the condition $\vec{q}_{1} \vec{q}_{2}=0$ is ... | 62. According to the problem, we have: $\vec{q}_{1}=\vec{a}_{1}-\vec{r}, \vec{q}_{2}=\vec{a}_{2}-\vec{r}$. Here $\vec{r}=\overrightarrow{O P}, \vec{a}_{1}=\overrightarrow{O A}_{1}, \vec{a}_{2}=\overrightarrow{O A}_{2}$
$$
\vec{a}_{1} \vec{q}_{2}=\left(\overrightarrow{a_{1}}-\vec{r}\right)\left(\overrightarrow{a_{2}}-\... | (x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}-R^{2}=0 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,384 |
63. Prove that if the distance from the center of a sphere to a plane is equal to the radius of the sphere, then the sphere and the plane have exactly one common point. (In this case, the plane is called tangent to the sphere.) Prove the converse statement. | 63. Let's prove the converse statement. If a plane has only one common point $M$ with a sphere, then the other points of the plane lie outside the sphere (why?). Therefore, the distance from the other points of the plane to the center of the sphere $O$ is greater than the radius $|O M|$, i.e., the distance from the cen... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,385 |
65*. Based on the identity:
$$
\vec{a} \vec{b}=\frac{(\vec{a}+\vec{b})^{2}}{2}-\frac{(\vec{a}-\vec{b})^{2}}{2}
$$
prove that if given a sphere with center $Q$ and radius $R$ and an arbitrary point $M$ at a distance $m$ from the center, as well as a diameter of the sphere $[A B]$, then for the vectors $\vec{a}=\overri... | 65. Since $\frac{\vec{a}+\vec{b}}{2}=\frac{\overrightarrow{M A}+\overrightarrow{M B}}{2}=\overrightarrow{M Q}$, the length of this vector is $m$.
The vector $\frac{\vec{a}-\vec{b}}{2}$ is equal in length to half the length of the diameter, i.e., the radius of the sphere. Therefore, $\left(\frac{\vec{a}-\vec{b}}{2}\rig... | \vec{}\vec{b}=^{2}-R^{2} | Geometry | proof | Yes | Yes | olympiads | false | 44,387 |
66*. Prove: if the equation of a sphere is written in the form:
$$
\overrightarrow{(r}-\overrightarrow{q)^{2}}-R^{2}=0
$$
then, by substituting an arbitrary vector $\overrightarrow{r_{1}}=\overrightarrow{O M}$ into its left side, we obtain a number equal to the power of point $M$ relative to the sphere. | 66. If $M$ is an external point relative to the sphere, then $m>R$ and the power is positive; if $M$ is a point on the sphere, then $m=R$ and the power is zero; if $M$ is an internal point, then $m<R$ and the power is negative. Let $[A B]$ be the diameter of the sphere, and $(M B)$ intersects the sphere at point $P$; t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,388 |
67*. Prove that the set of points having the same power with respect to two non-concentric spheres is a plane perpendicular to the line of centers of these spheres. This plane is called the radical plane of the spheres. | 67. Let us consider two spheres defined by the equations:
$$
\left(\vec{r}-\vec{q}_{1}\right)^{2}-R_{1}^{2}=0 \text { and }\left(\vec{r}-\vec{q}_{2}\right)^{2}-R_{2}^{2}=0
$$
By equating the left-hand sides of these equations, we obtain the condition that points with the same power relative to both spheres must satis... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,389 |
68*. a) Prove that if the centers of three spheres do not lie on the same straight line, then the set of points having the same power with respect to all three spheres is a straight line—the radical axis of the three spheres. This line is perpendicular to the plane passing through the centers of the spheres.
b) Prove ... | 68. a) If the centers of three spheres do not lie on the same straight line, then their radical planes are not parallel. Therefore, the radical planes of the first and second spheres intersect with the radical plane of the second and third spheres. The line of their intersection contains all points that have the same p... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,390 |
69. Prove that there exists an infinite set of spheres having a common radical plane. They form what is called a pencil of spheres. The centers of all these spheres lie on a single line perpendicular to the common radical plane - on the line of centers of the pencil. | 69. Let $\sigma_{1}$ and $\sigma_{2}$ be non-concentric spheres, the radical plane of which is $\gamma$ intersecting the line of their centers $l$ at point $O$, which has a power $\lambda$ relative to both spheres. If the spheres intersect, $O$ is inside the spheres and $\lambda=-p^{2}$, if the spheres touch, $O$ is th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,391 |
70. a) Prove that there exists an infinite family of spheres having one common radical axis - so-called pencil of spheres. The centers of such spheres belong to the same plane of centers of the pencil, and this plane is perpendicular to the radical axis of the pencil.
b) Prove that there exists an infinite set of sphe... | 70. In the same way as in the previous problem, in the plane of the centers of three spheres, we take an arbitrary center of a new sphere and construct a sphere that has the same radical axis as the given spheres. As a result, we obtain a bundle of spheres. Depending on whether the axis intersects these spheres, touche... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,392 |
1. Prove that among all lines lying in the plane of one of the faces of an acute dihedral angle, the lines perpendicular to the edge of the dihedral angle form the greatest angle with the plane of the second face.
[If the second face of the given dihedral angle is horizontal, then the line drawn in the plane of the fi... | 1. Consider the dihedral angle $\alpha / \beta$, and let $A \in \alpha, A^{\prime}$ be the projection of $A$ on $\beta, P$ - the projection of $A$ on $l$ (in the plane $\alpha$). Triangle $A A^{\prime} P$ is a right triangle with a right angle at vertex $A^{\prime}$. If $Q \in l$, then triangle $A A^{\prime} Q$ is also... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,393 |
2. Two plane mirrors serve as the faces of a dihedral angle. A light ray, perpendicular to the edge of the angle and parallel to the first mirror, is reflected from the second mirror, then from the first, then again from the second, again from the first, and finally, reflecting for the fifth time from the second mirror... | 2. Since the light ray falls perpendicular to the edge of the dihedral angle, all reflections occur in one plane perpendicular to the edge (Fig. 81). Since after the third reflection the ray returns along the same path, at this reflection it must be directed perpendicular to the mirror plane. The exterior angle of the ... | 30 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,394 |
3. What does the set of points represent, the orthogonal projections of which onto two intersecting planes are two lines
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 3. The set of points under consideration is contained in each of two planes $\alpha$ and $\beta$, which are the union of the projecting lines. If $\alpha \neq \beta$, then this set is easily seen to be the line $l=\alpha \cap \beta$. The case $\alpha=\beta$ should be analyzed independently.
$ and $\left[O A^{\prime}\right)$ - the intersections of its faces with a given plane, $[O P)$ - the bisector of angle $A O A^{\prime}$. From point $P$, which lies on the bisector $[O P)$, we draw a perpendicular $[P M]$ to the edge $l$. The plane $O... | \operatorname{tg}\frac{\gamma}{2}=\frac{\operatorname{tg}\frac{\beta}{2}}{\sin\alpha} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,396 |
5. Given three pairwise intersecting lines, not parallel to one plane, and a point $P$ not belonging to any of them, construct a plane through this point so that it forms equal angles with the given lines.
保留源文本的换行和格式，这里直接输出翻译结果。 | 5. Draw through point $P$ the lines $A A^{\prime}, B B^{\prime}$, and $C C^{\prime}$, respectively parallel to the given lines $a, b$, and $c$. Point $P$ divides each of the lines $A A^{\prime}, B B^{\prime}, C C^{\prime}$ into two opposite rays $P A$ and $P A^{\prime}, P B$ and $P B^{\prime}, P C$ and $P C^{\prime}$. ... | 4 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,397 |
7*. On the two faces of a dihedral angle, points $A$ and $B$ are given. Find a point $M$ on the edge of the angle such that the angle $A M B$ is a right angle. | 7. Through the midpoint $O$ of segment $A B$ and the edge $l$ of a given angle, we draw a plane and in this plane we construct segment $OM$ of length $|OM|=\frac{1}{2}|AB|$, where $M \in l$. There can be two, one, or no such points $M$. By connecting point $M$ with $A$ and $B$, we obtain a triangle, the median of which... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,399 |
8*. What does the set of points represent, the sum of the distances from which to two intersecting planes is constant and equal to a given value $p$? | 8. Consider one of the dihedral angles formed by the given planes $\alpha$ and $\beta$. Construct a plane $\alpha^{\prime} \| \alpha$ and located at a given distance $p$ from $\alpha$. Let $\alpha^{\prime} \cap \beta=b$; then $b \| l$, where $l = \alpha \cap \beta$. We also construct a plane $\beta^{\prime} \| \beta$ a... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,400 |
9. Half-planes $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$ have a common edge $l$. What is the sum of the dihedral angles $\alpha_{1} \widehat{l \alpha_{2}}, \alpha_{2} l \alpha_{3}, \ldots, \alpha_{n-1} l \alpha_{n}$, $\widehat{\alpha_{n} l} l \alpha_{1}$, which together fill the entire space? | 9. By intersecting the entire figure with a plane perpendicular to the common edge $l$, we obtain the linear angles of these dihedral angles in the section. These planar angles have a common vertex and fill the entire plane, so the sum of their magnitudes is $2 \pi$ (or $360^{\circ}$). | 2\pi | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,401 |
10. A ray perpendicular to the face of a dihedral angle and directed into the half-space determined by the plane of this face and not containing the other face is called the external normal of this angle. Prove that the sum of the magnitude of the angle between two external normals of a dihedral angle and the magnitude... | 10. Let's take a point $O$ on the edge of a dihedral angle and draw a plane through it, perpendicular to the edge. In the intersection of the faces with this plane, we will get two sides of the linear angle of the given dihedral angle. In the same plane, there will also be the external normals of the angle, drawn from ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,402 |
11*. From a point $M$ on the edge of a dihedral angle in one of its faces, a ray is drawn. Draw from the same point $M$ in the other face a ray that forms an angle of a given magnitude with the first ray. | 11. Let's take an arbitrary point $A$ on this ray; let $A^{\prime}$ be the projection of this point onto the other face. Suppose the desired ray is drawn, and we construct a segment $M B$ on it such that $|M B|=|M A|$. Then, in the isosceles triangle $A M B$, the two sides and the angle between them are known, and it c... | 2 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,403 |
12*. Given perpendicular intersecting lines $a$ and $b$ and a point $P$. What does the set of points $M$ represent, for which the sum of the lengths of the projections of the segment $P M$ onto the lines $a$ and $b$ is constant? [The projection of a point $A$ onto a line $l$ is a point $A^{\prime} \in l$ such that $\le... | 12. Draw through $P$ lines $a^{\prime} \| a$ and $b^{\prime} \| b$. These lines lie in a plane $\alpha$, parallel to both lines. Let the projection of segment $P M$ onto plane $\alpha$ be segment $P M^{\prime}$ (Fig. 85). Since the projections of segment $P M$ onto lines $a$ and $b$ are parallel to plane $\alpha$, they... | |x|+|y|= | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,404 |
13. Through the given line, pass a plane that forms a dihedral angle of a given magnitude with another given plane. | 13. Let the given line $l$ intersect the given plane $\alpha$ at point $O$. Take an arbitrary point $P$ on $l$ and find its projection $P^{\prime}$ on $\alpha$. Let the given angle $\varphi<90^{\circ}$. If the desired plane $\beta$ is constructed, and $\angle P M P^{\prime}$ is the linear angle of the dihedral angle be... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,405 |
14. Prove that:
a) the sum of the magnitudes of the three plane angles of a trihedral angle is less than $2 \pi ;$
b) the sum of the magnitudes of two plane angles of a trihedral angle is greater than the magnitude of the third angle. | 14. On the edges of the trihedral angle $S a b c$, we lay off segments of equal length $|S A|=|S B|=|S C|$ and through points $A, B, C$ we draw a plane. Let $S^{\prime}$ be the projection of point $S$ onto this plane.
a) Using problem 22 from $\S 5$ and considering the cases when the projection $S^{\prime}$ lies insid... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,406 |
15. Four points $A, B, C, D$, not lying in the same plane, define a closed broken line $A B C D A$ (this figure is sometimes called a "skew" or spatial quadrilateral). Prove that $\widehat{A B} C+B \widehat{C} D+C \widehat{D} A+D \widehat{A B}<360^{\circ}$. | 15. Consider triangles \(ABD\) and \(CBD\) (Fig. 86). The angles at vertices \(B\) and \(D\) in the first triangle are denoted by \(\beta_{1}\) and \(\hat{\delta}_{1}\), and the angles at the same vertices in the second triangle are denoted by \(\beta_{2}\) and \(\delta_{2}\). Then we have: \(B \widehat{AD} + \beta_{1}... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,407 |
16. What does the set of points that are equidistant from the planes of all three faces of a trihedral angle and lying inside this angle represent? | 16. The bisector half-plane of one of the dihedral angles of the given trihedral angle is the set of points equidistant from the planes of the corresponding faces, for example, the first and second and lying inside this dihedral angle. The bisector half-plane of the second and third faces possesses an analogous propert... | SA | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,408 |
17. What does the set of points equidistant from three lines containing the edges of a trihedral angle and lying inside this angle represent? | 17. The plane of symmetry of edges $a$ and $b$ intersects the plane of symmetry of edges $b$ and $c$ along a line $S$, which is the set of points equidistant from the lines containing the three edges of the given trihedral angle. (Note that the plane of symmetry of edges $c$ and $a$ also passes through $S$.) | S | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,409 |
18. A family of parallel planes, intersecting all faces of a trihedral angle, forms a family of triangles. What does the set of centroids (points of intersection of medians) of these triangles represent? What does the set of orthocenters (points of intersection of altitudes) of these triangles represent? | 18. Let's take the vertex $S$ of the trihedral angle as the center of homothety and transform one of the section triangles into other triangles by changing the coefficient of homothety. In this case, the sets of all similar points, i.e., the set of centroids, orthocenters, etc., will represent rays emanating from point... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,410 |
19. Intersect the given trihedral angle with a plane that forms angles of equal magnitude with its edges. | 19. This problem is a particular case of problem 5. Let's provide another solution. On the edges of the angle from the vertex $S$, we lay off congruent segments $[S A] \cong[S B] \cong[S C]$ and draw the plane $A B C$. We describe a circle around triangle $A B C$ with center $O$. According to the conclusion of problem ... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,411 |
21. Prove that if in a trihedral angle two plane angles are congruent, then the dihedral angles opposite to them are also congruent. Prove the converse statement. | 21. Let in the dihedral angle $S A B C$ the plane angles $A S B$ and $A S C$ be congruent. Then the edge $[S A)$ forms congruent angles with the edges $[S B)$ and $[S C)$ and, therefore, belongs to the plane of symmetry of these edges. Let's draw this plane of symmetry. The same symmetry maps the dihedral angles at the... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,413 |
22. A trihedral angle in which all plane angles are right angles is called orthogonal (for example, the trihedral angle $0 x y z$ in a rectangular coordinate system¹). Prove that for an orthogonal trihedral angle: a) three points lying on its three edges and not coinciding with the vertex $O$ are the vertices of an acu... | 22. Let $O A B C$ be an orthogonal trihedral angle. Due to the perpendicularity of the edges, all its dihedral angles are right angles.
Fig. 87
a) The projection of triangle $A B C$ onto t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,414 |
23. The given orthogonal trihedral angle should be intersected by a plane in such a way that the intersection forms a triangle congruent to the given acute triangle. | 23. Let $ABC$ be a triangle congruent to the given one, which should result from the section. We will calculate the lengths of the legs of the right triangles $BCO$, $CAO$, and $ABO$. Imagine that we have rotated these triangles around the lines $BC$, $CA$, and $AB$ and placed them on the plane $ABC$, as shown in Figur... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,415 |
24. Let $O a b c$ be an orthogonal trihedral angle. Prove that for any points $A \in a, B \in b, C \in c$, the sum of the squares of the areas of triangles $B C O, C A O$, and $A B O$ is equal to the square of the area of triangle $A B C$. | 24. Let $|O A|=a,|O B|=b,|O C|=c$. We will prove that the square of the area $S$ of triangle $A B C$ is equal to the sum of the squares of the areas of triangles $B C O, A B O$, and $C A O$, which are, respectively, $\frac{1}{4} b^{2} c^{2}, \frac{1}{4} a^{2} b^{2}, \frac{1}{4} c^{2} a^{2}$. The volume of the tetrahedr... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,416 |
25. Let's draw external normals \(a', b', c'\) from the vertex \(S\) of the trihedral angle \(Sabc\) to the faces \(bSc, cSa,\) and \(aSb\) respectively: we obtain a new trihedral angle \(Sa'b'c'\), which is called the polar angle relative to the angle \(Sabc\). Prove: a) the trihedral angle polar to \(Sa'b'c'\) is the... | 25. a) Since $a^{\prime} \perp b, a^{\prime} \perp c, b^{\prime} \perp c, b^{\prime} \perp a, c^{\prime} \perp a, c^{\prime} \perp b$, then, consequently, $a \perp b^{\prime}, a \perp \bar{~}^{\prime}, b \perp c^{\prime}, b \perp a^{\prime}, c \perp a^{\prime}, c \perp b^{\prime}$. Therefore, the angle $S a b c$ is pol... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,417 |
26. Prove that in any trihedral angle, the sum of the measures of its dihedral angles is greater than $180^{\circ}$. | 26. The sums of the planar angles in the trihedral angle $S a^{\prime} b^{\prime} c^{\prime}$ satisfy the condition: $b^{\prime} \widehat{S c^{\prime}}+c^{\prime} \widehat{S a^{\prime}}+a^{\prime} \widehat{S b^{\prime}}<2 \pi ; \quad \pi-\alpha+\pi-$ $-\beta+\pi-\gamma<2 \pi$. Here $\alpha, \beta, \gamma$ are the magni... | \pi<\alpha+\beta+\gamma | Geometry | proof | Yes | Yes | olympiads | false | 44,418 |
28*. Let's define the magnitude of the trihedral angle $S A B C$ as $\widehat{A} + \widehat{B} + \widehat{C} - \pi$, where $\widehat{A}, \widehat{B}, \widehat{C}$ are the magnitudes of the dihedral angles at the edges $S A, S B, S C$. Prove that: a) the magnitude of the trihedral angle is always positive; b) if the giv... | 28. a) In the solution to problem 26, it is proven that the sum of the dihedral angles of any trihedral angle is greater than $\pi$. Therefore, the introduced value is always positive.
b) Let the plane $S A M$ divide the trihedral angle $S A B C$ into trihedral angles $S A M B$ and $S A M C$. Suppose the sum of the di... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,420 |
29. Consider an $n$-gon $A_{1} A_{2} \ldots A_{n}$ and a point $S$ outside its plane. The set of all points belonging to rays with origin at $S$ and passing through the points of the $n$-gon is an $n$-faceted angle. Prove that if the original $n$-gon is convex, then the corresponding $n$-faceted angle is the intersecti... | 29. If a plane is drawn through the vertex $S$ and one of the sides of the polygon $A_{1}, A_{2}, \ldots, A_{n}$, then in the case of the convexity of the polygon, this polygon, along with the corresponding polyhedral angle, will be in one half-space relative to this plane. Since this will be the case for any side of t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,421 |
30. Using the method of mathematical induction, prove that the sum of the magnitudes of the plane angles of a convex polyhedral angle (see problem 29) is less than $2 \pi$. | 30. Consider a convex $n$-hedral angle $S A_{1} A_{2} \ldots A_{n-1} A_{n}$ (Fig. 88 shows its projection onto the plane of the section). The statement has been proven for the case $n=3$ (see problem 14a). Assuming that the statement is true for any $(n-1)$-hedral angle, we will prove that it will also be true for an $... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,422 |
32. Prove that a convex polyhedral angle can be divided into trihedral angles by planes passing through its non-adjacent edges.
$33^{*}$. The magnitude of the polyhedral angle $S A_{1} A_{2} \ldots A_{n}$ is defined as the difference between the sum of the magnitudes of all its dihedral angles and the sum of the magni... | 32. The required division into trihedral angles will be obtained by drawing all diagonal sections of the polyhedral angle through one of its edges. | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,424 |
34*. What is the magnitude of an orthogonal trihedral angle? What is the sum of the magnitudes of polyhedral angles that share a common vertex, have no common internal points, and together cover the entire space? | 34. In an orthogonal trihedral angle, the three dihedral angles are right angles, and the sum of their measures is $\frac{3 \pi}{2}$. Subtracting $\pi$ from this, we get $\frac{\pi}{2}$ - the measure of the orthogonal trihedral angle.
Three mutually perpendicular planes divide space into 8 orthogonal trihedral angles.... | 4\pi | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,425 |
35. If rays are drawn from the center of a cube to its vertices, these rays can be considered as the edges of six tetrahedral angles into which the entire space is divided. Calculate the magnitudes of the planar and dihedral angles of these tetrahedral angles. | 35. Six congruent four-sided angles fill all space. Therefore, the magnitude of each of them is $\frac{4 \pi}{6}=\frac{2 \pi}{3}$. This magnitude is obtained by subtracting the number $2 \pi$ (the sum of the angles of a square) from the sum of the magnitudes of the dihedral angles. This sum is $\frac{2 \pi}{3}+2 \pi=\f... | 7031^{\}44^{\\} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,426 |
36*. How to construct a trihedral angle given the magnitudes of its dihedral angles?
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
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36*. How to construct a trihedral angle given the magnitudes of its dihedral angles? | 36. Let $\alpha, \beta, \gamma$ be the magnitudes of planar angles. From an arbitrary point $S$ on a plane, we draw 4 congruent segments $S A, S B$, $S C, S A^{\prime}$ such that $\widehat{A S} B=\alpha, \widehat{B S} C=\beta, C \widehat{S} A^{\prime}=\gamma$. By the segments $B C, C A^{\prime}$, and $A B$, we construc... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,427 |
37*. How to construct a trihedral angle given the magnitudes of its dihedral angles? | 37. If the dihedral angles are given, then the plane angles of the trihedral angle polar to the given one are determined. By constructing such a trihedral angle (see the previous problem), we draw external normals to its faces at the vertex - we obtain the desired trihedral angle. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,428 |
38. Let us recall that the symmetry of space with respect to the axis $s$ is a movement of space in which the points of the line $s$ remain fixed, i.e., map onto themselves, and each point $A \notin s$ is mapped to a point $A^{\prime}$ such that $A^{\prime} \neq A,\left(A A^{\prime}\right) \perp s,\left(A A^{\prime}\ri... | 38. a) Let the symmetry with axis $s$ map $A$ to $A^{\prime}$ and $B$ to $B^{\prime}$ (Fig. 89). We will prove that $|A B|=\left|A^{\prime} B^{\prime}\right|$. Let $\left(B B^{\prime}\right) \cap s=N$. Draw a line through $N$ parallel to ( $A A^{\prime}$ ), and lay off symmetric segments $N A_{1}$ and $N A_{1}^{\prime}... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,429 |
39. a) Prove that the composition of two axial symmetries with parallel axes is a parallel translation of space. How to determine the length and direction of this translation (vector)? b) Prove that any parallel translation of space can be represented as the composition of two axial symmetries. How to construct the axe... | 39. a) Let us have two axes of symmetry $s$ and $s'$, where $s \| s'$. Let the first symmetry map point $A$ to $A_{1}$, and the second map point $A_{1}$ to $A'$. Since the axes are parallel and the lines $A A_{1}$ and $A, A'$
150
are perpendicular to them, the three points $A, A_{1}, A'$ lie in a plane perpendicular t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,430 |
40. a) Prove that the composition of symmetries with respect to two mutually perpendicular planes is an axial symmetry. b) Prove that any axial symmetry of space can be represented as the composition of two symmetries with respect to mutually perpendicular planes. How to construct such planes? | 40. a) Let $\alpha \perp \beta$ and $\alpha \cap \beta=s$. Suppose the first symmetry maps point $A$ to $A_{1}$, and the second symmetry maps point $A_{1}$ to $A^{\prime}$. Since $\left(A A_{1}\right) \perp s$ and $\left(A_{1} A^{\prime}\right) \perp s$, the points $A, A_{1}$, and $A^{\prime}$ lie in the plane $\gamma$... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,431 |
41*. What does the composition of two axial symmetries with mutually perpendicular (intersecting or skew) axes represent? | 41. Suppose first that the axes \( s_{1} \) and \( s_{2} \) of two symmetries intersect and \( s_{1} \perp s_{2} \). Decompose the first symmetry into the composition of two symmetries with respect to planes \( \alpha_{1} \) and \( \beta_{1} \), such that \( \alpha_{1} \cap \beta_{1} = s_{1} \) and \( \alpha_{1} \perp ... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,432 |
42. A rotation of space about an axis $l$ is a movement of space in which the points of the line $l$ remain fixed, and each point $A$, not belonging to $l$, is mapped to a point $A^{\prime}$ such that points $A$ and $A^{\prime}$ belong to a plane $\gamma$ perpendicular to the line $l$, and the angle $A A_{0} A^{\prime}... | 42. If line \( a \) is perpendicular to the axis of rotation \( l \), then through it, we can draw a plane \( \alpha \) perpendicular to \( l \) and intersecting the axis of rotation at some point \( O \). In the plane \( \alpha \), we will obtain the figure shown in Figure 91. Let \( (O A) \perp a \), \( A \in a \). D... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,433 |
43. a) What kind of mapping is the composition of two symmetries with respect to two intersecting planes? b) Prove that any rotation of space about an axis can be represented as the composition of symmetries with respect to two planes. How to construct such planes? | 43. a) Let $\alpha$ and $\beta$ be given planes, $\alpha \cap \beta=l$. The first symmetry maps point $A$ to $A_{1}$, the second maps point $A_{1}$ to $A^{\prime}$. Since $\left(A A_{1}\right) \perp \alpha,\left(A_{1} A^{\prime}\right) \perp \beta$, the points $A, A_{1}, A^{\prime}$ belong to the plane $\gamma$, perpen... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,434 |
44*. a) Prove that the composition of two spatial axial symmetries with intersecting axes is a rotation about an axis. How to construct the axis of rotation? How to determine the angle of rotation? b) Prove that any rotation of space about an axis can be represented as the composition of two axial symmetries. How to co... | 44. a) Let us have two axial symmetries with axes $s_{1}$ and $s_{2}$, intersecting at point $O$ and forming an angle $\varphi=s_{1} \widehat{O s_{2}}$.
Decompose each of these symmetries into the composition of two symmetries with respect to planes $\alpha_{1} \perp \alpha_{2}$ and $\beta_{1} \perp \beta_{2}$, respec... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,435 |
45*. Prove that the composition of two rotations of space with intersecting axes is again a rotation. How to construct the axis and find the angle of the resulting rotation? | 45. Let given are rotations with axes $l_{1}$ and $i_{2}$, intersecting at point $O$. Decompose both rotations into compositions of symmetries with respect to planes $\alpha_{1}$ and $\alpha_{2}, \beta_{1}$ and $\beta_{2}$. Choose the plane $l_{1} \mathrm{Ol}_{2}$ as $\alpha_{2}$ and $\beta_{1}$. Then the composition o... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,436 |
46. In space, a line $l$ and points $A$ and $B$ are given, such that lines $A B$ and $l$ are skew. Find a point $M$ on line $l$ such that the sum of the distances $|A M|+|M B|$ is minimized. | 46. By rotating around the axis $l$, we will translate point $B$ into the plane passing through point $A$ and line $l$, and do so in such a way that points $A$ and $B^{\prime}$ - the image of $B$ - are on opposite sides of line $l$. The shortest path from $A$ to $B^{\prime}$ is the segment $A B^{\prime}$, intersecting ... | M | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,437 |
47*. Given two congruent triangles $A O B$ and $A^{\prime} O B^{\prime}$ (with a common vertex $O$), not lying in the same plane. Prove that there exists a rotation of space about an axis that maps the first triangle onto the second. | 47. Since by condition $|O A|=\left|O A^{\prime}\right|$ and $|O B|=\left|O B^{\prime}\right|$, the plane of symmetry $\alpha$ of points $A$ and $A^{\prime}$ passes through $O$. The corresponding symmetry maps $A$ to $A^{\prime}$ and $B$ to $B_{1}$, and $|O B|=$
 the trihedral angle from them. | 48. Let $SABC$ be the sought trihedral angle, $SA', SB', SC'$ - the bisectors of angles $BSC, CSA$, and $ASB$ respectively. The symmetry with axis $SA'$ maps $(SB)$ onto $(SC)$. The symmetry with axis $SB'$ maps $(SC)$ onto $(SA)$, and finally, the symmetry with axis $SC'$ maps $(SA)$ onto $(SB)$. The composition of th... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,439 |
49. Consider a rotation about an axis $l$ by an angle equal to $\frac{2 \pi}{n}$, where $n$ is a natural number, $n \geqslant 3$. From a point $S$ on the axis $l$, draw a ray forming an angle $\varphi < 90^{\circ}$ with $l$. Then $n$ rays, obtained from this ray by successive applications of the considered rotation, ca... | 49. Rotate this $n$-sided angle by an angle of $\frac{2 \pi}{n}$ about the axis $l$. In this process, each edge will coincide with the adjacent edge, and each face will coincide with the adjacent face. From this, it follows that all plane angles are congruent and all dihedral angles are congruent. | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,440 |
50. Prove the converse statement: if an n-sided angle is regular, then there exists an axis such that when rotated about this axis by an angle of $\frac{2 \pi}{n}$, the n-sided angle coincides with itself. This axis will be called the axis of symmetry of the regular n-sided angle. | 50. Let all planes and all dihedral angles of a certain $n$-sided angle be congruent to each other. Under this condition, each edge belongs to the plane of symmetry of two adjacent edges. This symmetry transforms the adjacent edges into each other, and due to the congruence of the dihedral angles at these edges, the an... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,441 |
53*. Given two skew lines $a$ and $a^{\prime}$. On the first one, a point $A$ is given, and on the second one, a point $A^{\prime}$. Find the rotation of space about an axis that maps $a$ to $a^{\prime}$ and $A$ to $A^{\prime}$ (construct the axis of such a rotation). | 53. Let's take an arbitrary point $M$ on the line $a$, and on the line $a^{\prime}$ from the given point $A^{\prime}$, lay off $\left[M^{\prime} A^{\prime}\right] \cong [M A]$. Let $\alpha$ be the plane of symmetry of points $A$ and $A^{\prime}$. Suppose that under symmetry relative to $\alpha$, point $M$ maps to point... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,444 |
54. Given intersecting lines $a$ and $b$, forming equal angles with some line $l$. Prove that there exists a rotation with axis $l$, mapping line $a$ to a line $a^{\prime}$, parallel to $b$. | 54. Let's take an arbitrary point $O$ on the line $l$ and draw two lines through it: $a_{1} \| a$ and $b_{1} \| b$. From the problem statement, it follows that the lines $a_{1}$ and $b_{1}$ form congruent angles with the line $l$, and there exists a rotation that maps $a_{1}$ to $b_{1}$. (To find the angle of rotation,... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,445 |
55*. Given four segments $A_{1} B_{1}, A_{2} B_{2}, A_{3} B_{3}, A_{4} B_{4}$, none of which lie in the same plane, and all of which are parallel to each other but pairwise non-congruent. How are the centers of the six homotheties, mapping $A_{i}$ to $A_{j}$ and $B_{i}$ to $B_{i}(i, j=1,2,3,4)$, arranged? | 55. Let the center of homothety defined by the segments $A_{i} B_{i}$ and $A_{i} B_{j}$ be denoted by $S_{i j}(i, j=1,2,3,4)$. In the solution to problem 26 from $\S 5$, it was proven that the centers of any two homotheties and the third homothety, which is their composition, lie on the same line. Considering the homot... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,446 |
56*. Prove that two similar but not congruent triangles in space can be mapped onto each other by a composition of a homothety and a rotation about an axis.
保留源文本的换行和格式，直接输出翻译结果。
(注：最后一句为说明，不需翻译，仅在输出时保持格式。) | 56. Let $A_{1} B_{1} C_{1}$ and $A_{2} B_{2} C_{2}$ be similar triangles, i.e., triangles with respectively congruent angles. We need to find a homothety that would map triangle $A_{2} B_{2} C_{2}$ to a triangle congruent to $\triangle A_{1} B_{1} C_{1}$, such that point $A_{2}$ is mapped to $A_{1}$. The center of such... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,447 |
57. Prove that any two spheres of different radii are homothetic to each other. How to determine the center and coefficient of the corresponding homothety? | 57. Let us have spheres with center $O_{1}$ of radius $R_{1}$ and with center $O_{2}$ of radius $R_{2}\left(R_{1}>R_{2}\right)$. Take an arbitrary point $A_{1}$ on the first sphere outside the line $O_{1} O_{2}$ and consider the vector $\overrightarrow{O_{2} A_{2}}$, collinear with the vector $\vec{O}_{1} \vec{A}_{1}$ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,448 |
58. Three spheres with different radii and centers not lying on the same straight line lie outside each other. What figure is formed by the lines of intersection of pairs of planes that are symmetric with respect to the plane of the centers of the spheres and simultaneously tangent to all three given spheres? | 58. The common tangent plane of two spheres passes through the center of their homothety, since the radii drawn to the points of tangency are parallel, and the line passing through the points of tangency also passes through the center of homothety (see the previous problem). The centers of homothety of three spheres be... | 4 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,449 |
1*. What are the necessary and sufficient conditions for a prism to be inscribed in a sphere? | 1. If the vertices of a prism lie on a sphere, then the vertices of each of its bases belong to two congruent circles obtained by the intersection of the sphere with the planes of the prism's bases. The line connecting the centers of these circles passes through the center of the sphere and is perpendicular to the plan... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,450 |
3. Does there exist a parallelepiped, different from a cube, all faces of which are congruent to each other? Does such a parallelepiped have an axis of symmetry, i.e., an axis of rotation about which the parallelepiped maps onto itself? | 3. Given the condition, all sides of the bases of the parallelepiped must be congruent to each other, which is derived from the congruence of the lateral faces (explain). Therefore, the bases of the parallelepiped, as well as its lateral faces, must be congruent rhombuses. Thus, all 12 edges of the parallelepiped are c... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,452 |
5. Calculate the volume of a triangular prism, knowing the area of its lateral face and the distance between the plane of this face and the line containing the opposite lateral edge.
保留源文本的换行和格式，这里已经是按照原文本的格式进行翻译的。 | 5. Each parallelepiped can be cut by a diagonal section into two congruent triangular prisms with equal volumes. Since the volume of a parallelepiped is equal to the product of the area of any of its faces by the distance from it to the opposite face, the volume of each of the prisms is equal to half the product of the... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,454 |
6. Prove that the sum of the squares of the lengths of the four diagonals of an arbitrary parallelepiped is equal to the sum of the squares of the lengths of its twelve edges. | 6. To prove this, it is sufficient to apply the formula for the sum of the squares of the sides of a parallelogram first to two diagonal sections of the parallelepiped, and then to its two bases. | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,455 |
7*. Prove that in a wooden model of a cube, a hole can be drilled of such a size that a cube congruent to the given one can be passed through it. | 7. In Figure 94, two projections of a cube are given: in one (a) the projecting lines are parallel to the diagonal of the cube, and in the other (b) they are parallel to its edge.
The second projection can be constructed from the first if we notice that on the first projection, the diagonals of the faces perpendicular... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,456 |
8*. Prove that the area of a triangle, which is the intersection of a tetrahedron and a plane, is less than the area of at least one of the tetrahedron's faces. | 8. Let's intersect the tetrahedron \(ABCD\) with a plane passing through the edge \(AB\) and a point \(M\) on the edge \(\lceil CD \mid\) (Fig. 95). The triangles \(ABD\), \(ABC\), and \(ABM\) share a common base, so to compare their areas, it is sufficient to compare the heights drawn to the side \(AB\). For this, we ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,457 |
9. In a regular tetrahedron $A B C D$, a line $M P$ is drawn through the midpoint $M$ of edge $A B$ and the centroid $P$ of face $B C D$. A line $N Q$ passes through the midpoint $N$ of edge $B C$ and the centroid $Q$ of face $A B D$. Calculate the angle between the lines $M P$ and $N Q$. | 9. Let $a$ be the length of the edges of the tetrahedron, then $|M P|=\frac{a}{2}$, since this is the median of the right triangle $A B P$ (Fig. 96). Similarly, $|N Q|=\frac{a}{2}$, and also $|M N|=\frac{a}{2}$. In triangle $M N D$, points $P$ and $Q$ divide the sides in the ratio $2: 3$, so $|P Q|=\frac{2}{3}|M N|=\fr... | \varphi\approx11253^{\}06^{\\} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,458 |
10*. Intersect the given (arbitrary) tetrahedron with a plane so that the intersection is a rhombus.
Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly. | 10. If a tetrahedron is intersected by a plane parallel to two of its opposite edges, the section will be a parallelogram (why?). Let's take an arbitrary point \( M \) on the edge \( AB \) of the tetrahedron \( ABCD \) and draw \( (MN) \parallel (AD) \) and \( (MP) \parallel (BC) \); \([MN]\) and \([MP]\) are the sides... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,459 |
11. Prove that the sum of the distances from any point inside a regular tetrahedron to all its faces is the same and equal to the height of the tetrahedron. | 11. Let $A B C D$ be a regular tetrahedron with height $h$. Let point $P$ inside it be at a distance $a$ from the plane $B C D$, at a distance $b$ from the plane $A C D$, at a distance $c$ from the plane $A B D$, and at a distance $d$ from the plane $A B C$. Denote by $S$ the area of a face of the tetrahedron. The volu... | +b++= | Geometry | proof | Yes | Yes | olympiads | false | 44,460 |
12. Prove: if the trihedral angles at vertices $O$ and $O^{\prime}$ of two tetrahedrons $O A B C$ and $O^{\prime} A^{\prime} B^{\prime} C^{\prime}$ are congruent, then the volumes of these tetrahedrons are in the ratio of the products of the lengths of the three edges emanating from vertices $O$ and $O^{\prime}$. | 12. Superimpose (by some movement of space) the second tetrahedron on the first so that the congruent trihedral angles \(O\) and \(O^{\prime}\) coincide. Let \(|O A|=a,|O B|=b,|O C|=c\) be the lengths of the edges of the first tetrahedron, and \(\left|O A^{\prime}\right|=a^{\prime},\left|O B^{\prime}\right|=b^{\prime},... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,461 |
13. On two intersecting lines, segments $A B$ and $C D$ are given. Prove that the volume of the tetrahedron $A B C D$ does not change if these segments, without changing their lengths, are moved along these lines. Calculate this volume if the lengths of the segments $m=|A B|, n=|C D|$, the distance $h$ between the line... | 13. Suppose first that segment $AB$ is fixed, while only segment $CD$ moves. In this case, the area of triangle $ACD$ remains constant, since the length $|CD|$ and the distance from point $A$ to the line $CD$ do not change. The height of the tetrahedron is also constant, as the distance from point $B$ to the plane dete... | \frac{1}{6}n\sin\varphi | Geometry | proof | Yes | Yes | olympiads | false | 44,462 |
14. From an arbitrary tetrahedron, four smaller tetrahedra are separated by four planes passing through the midpoints of the edges emanating from each vertex. Calculate the ratio of the volume of the remaining body to the volume of the original tetrahedron. | 14. Each of the separated tetrahedra is homothetic to the original with a coefficient of $\frac{1}{2}$, so its volume is $\frac{1}{8}$ of the volume of the tetrahedron. Since a total of 4 tetrahedra are removed, each pair of which does not share any internal points, it means that half of the volume has been removed. Th... | \frac{1}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,463 |
15*. Prove that any plane passing through the midpoints of two opposite edges of a tetrahedron divides it into two parts of equal volume. | 15. Let \(ABCD\) be the given tetrahedron (Fig. 97), \(M\) the midpoint of \([AB]\), \(N\) the midpoint of \([CD]\), \((MPNQ)\) the intersecting plane. One of the resulting parts of the tetrahedron consists of the quadrilateral pyramid \(AMPNQ\) and the tetrahedron \(DANQ\), the other part consists of the quadrilateral... | proof | Geometry | proof | Yes | Yes | olympiads | false | 44,464 |
16. If an internal point of some $n$-sided prism is connected to all its vertices, then we obtain $n$ quadrilateral pyramids with a common vertex at this point, the bases of which are the lateral faces of the prism. Find the ratio of the sum of the volumes of these pyramids to the volume of the given prism. | 16. Let $S$ be the area of the base of a prism, $h$ its height, and $P$ a given point. If all pyramids with vertex $P$ and lateral faces of the prism as bases are removed from the prism, two pyramids with vertex $P$ and bases coinciding with the bases of the prism will remain. The volume of one of them is $\frac{1}{3} ... | \frac{2}{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,465 |
17. Given three pairwise parallel lines $a, b$ and $c$, not lying in the same plane. On the first, $\left[A A^{\prime}\right]$ is given, on the second $\left[B B^{\prime}\right]$, and on the third $\left[C C^{\prime}\right]$. Calculate the volume of the polyhedron $A B C A^{\prime} B^{\prime} C^{\prime}$, bounded by th... | 17. Suppose we have denoted the segments such that $m<n<p$. From points $B$ and $C$, we lay off segments $\left|B B_{1}\right|=\left|C C_{1}\right|=m$ (Fig. 98). The desired volume is the sum of two volumes: an oblique prism with base $A B C$ and edge $\left|A A^{\prime}\right|=m$, and a quadrilateral pyramid with vert... | \frac{(+n+p)S}{3} | Geometry | proof | Yes | Yes | olympiads | false | 44,466 |
18. Two congruent cubes have a common diagonal, and one is obtained from the other by rotating around this diagonal by an angle of $60^{\circ}$. Find the volume of the intersection of these cubes if the edge length of the cube is $a$. | 18. Let a cube $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ be given. It is easy to see that the midpoints of the edges $C D, D D^{\prime}, D^{\prime} A^{\prime}, A^{\prime} B^{\prime}, B^{\prime} B$ and $B C$ are at equal distances from the points $A$ and $C^{\prime}$--the endpoints of the cube's diagonal, so... | \frac{3^{3}}{4} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 44,467 |
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