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29.6. a) Prove that there exists a unique affine transformation that maps a given point $O$ to a given point $O^{\prime}$, and a given basis of vectors $\boldsymbol{e}_{1}, \boldsymbol{e}_{2}$ to a given basis $\boldsymbol{e}_{1}^{\prime}, \boldsymbol{e}_{2}^{\prime}$.
b) Given two triangles $A B C$ and $A_{1} B_{1} C... | 29.6. a) Let's define the mapping $L$ as follows. Let $X$ be an arbitrary point. Since $e_{1}, e_{2}$ is a basis, there exist uniquely determined numbers $\boldsymbol{x}_{1}$ and $\boldsymbol{x}_{2}$ such that $\overrightarrow{O X}=x_{1} \boldsymbol{e}_{1}+x_{2} \boldsymbol{e}_{2}$. We will associate with the point $X$... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,178 |
29.7*. Each diagonal of a convex pentagon is parallel to one of its sides. Prove that the pentagon can be transformed into a regular pentagon by an affine transformation.
保留源文本的换行和格式，这里的翻译结果为：
29.7*. Each diagonal of a convex pentagon is parallel to one of its sides. Prove that the pentagon can be transformed into a ... | 29.7. Let \(ABCDE\) be a regular pentagon. According to problem 29.6, b), there exists an affine transformation that maps three consecutive vertices of this pentagon to points \(A, B, C\). Let \(D'\) and \(E'\) be the images of the remaining two vertices under this transformation. We will prove that they coincide with ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,179 |
29.8*. Prove that if under an affine (non-identity) transformation $L$ every point of some line $l$ is mapped to itself, then all lines of the form $M L(M)$, where $M$ is any point not lying on the line $l$, are parallel to each other. | 29.8. Let $M$ and $N$ be arbitrary points not lying on the line $l$. Denote by $M_{0}$ and $N_{0}$ their projections onto the line $l$, and by $M^{\prime}$ and $N^{\prime}$ the images of points $M$ and $N$ under the mapping $L$. The lines $M_{0} M$ and $N_{0} N$ are parallel, as they are both perpendicular to $l$, i.e.... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,180 |
29.10*. On a plane, there is a polygon $A_{1} A_{2} \ldots A_{n}$ and a point $O$ inside it. Prove that the equalities
$$
\begin{gathered}
\overrightarrow{O A_{1}}+\overrightarrow{O A_{3}}=2 \cos \frac{2 \pi}{n} \overrightarrow{O A_{2}} \\
\overrightarrow{O A_{2}}+\overrightarrow{O A_{4}}=2 \cos \frac{2 \pi}{n} \overr... | 29.10. First, let's prove that if \(A_{1} A_{2} \ldots A_{n}\) is a regular polygon inscribed in a unit circle, and \(O\) is its center, then the equalities given in the problem statement hold, i.e.,
\[
\overrightarrow{O A_{i-1}} + \overrightarrow{O A_{i+1}} = 2 k \overrightarrow{O A_{i}}, \quad i=1, \ldots, n
\]
(we... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,182 |
29.11*. Prove that any affine transformation can be represented as a composition of a scaling (compression) and an affine transformation that maps any triangle to a similar triangle.
保留源文本的换行和格式，但这里不需要额外的换行，因为原文本只有一行。如果需要严格保持格式，可以如下所示：
29.11*. Prove that any affine transformation can be represented as a composition o... | 29.11. Let $L$ be a given affine transformation, $O$ be an arbitrary point, $T$ be the translation by the vector $\overrightarrow{L(O) O}$, and let $L_{1}=T \circ L$. Then $O$ is a fixed point of the transformation $L_{1}$. Among all points of the unit circle centered at $O$, choose a point $A$ for which the length of ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,183 |
29.12*. Prove that if an affine transformation maps a certain circle to itself, then it is either a rotation or a reflection.
29.13 ${ }^{*}$. Prove that if $M^{\prime}$ and $N^{\prime}$ are the images of polygons $M$ and $N$ under an affine transformation, then the ratio of the areas of $M$ and $N$ is equal to the ra... | 29.12. First, let us prove that an affine transformation \( L \) that maps a given circle onto itself maps diametrically opposite points to diametrically opposite points. To do this, note that the tangent to the circle at point \( A \) is transformed into a line that, due to the one-to-one nature of the transformation ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,184 |
29.14. Through each vertex of a triangle, two lines are drawn that divide the opposite side of the triangle into three equal parts. Prove that the diagonals connecting the opposite vertices of the hexagon formed by these lines intersect at one point. | 29.14. Since any triangle is transformed into an equilateral one by an affine transformation (problem 29.6, b) and the ratios of lengths of parallel segments are preserved (problem 29.5), it is sufficient
 it follows that any parallelogram can be transformed into a square by an affine transformation. Since under this transformation, the ratios of the lengths of parallel segments are preserved (problem 29.5), it is sufficient to prove the statement of the problem in the case where $ABCD$ is a ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,186 |
29.16. Given a triangle $ABC$. Let $O$ be the point of intersection of its medians, and $M, N$, and $P$ be points on sides $AB, BC$, and $CA$, respectively, dividing these sides in the same ratio (i.e., $AM: MB = BN: NC = CP: PA = p: q$). Prove that:
a) $O$ is the point of intersection of the medians of triangle $MNP$... | 29.16. a) Consider an affine transformation that maps triangle $A B C$ to an equilateral triangle $A^{\prime} B^{\prime} C^{\prime}$. Let $O^{\prime}, M^{\prime}, N^{\prime}, P^{\prime}$ be the images of points $O$, $M, N, P$. When rotated by $120^{\circ}$ around point $O^{\prime}$, triangle $M^{\prime} N^{\prime} P^{\... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,187 |
29.17. In trapezoid $ABCD$ with bases $AD$ and $BC$, a line through point $B$ parallel to side $CD$ intersects diagonal $AC$ at point $P$, and a line through point $C$ parallel to side $AB$ intersects diagonal $BD$ at point $Q$. Prove that line $PQ$ is parallel to the bases of the trapezoid. | 29.17. Consider an affine transformation that maps $A B C D$ to an isosceles trapezoid $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$. As such a transformation, one can take an affine transformation that maps the triangle $A D E$ to an isosceles triangle (where $E$ is the intersection point of the lines $A B$ and $C D$)... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,188 |
29.18. In parallelogram $A B C D$, points $A_{1}, B_{1}, C_{1}, D_{1}$ lie on sides $A B, B C, C D, D A$ respectively. On the sides $A_{1} B_{1}, B_{1} C_{1}, C_{1} D_{1}$, $D_{1} A_{1}$ of quadrilateral $A_{1} B_{1} C_{1} D_{1}$, points $A_{2}, B_{2}$, $C_{2}, D_{2}$ are taken respectively. It is known that
$$
\frac{... | 29.18. Any parallelogram $A B C D$ can be transformed into a square by an affine transformation (for this, triangle $A B C$ needs to be transformed
Fig. 29.4
into an isosceles right triang... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,189 |
29.19. On the sides $AB$, $BC$, and $AC$ of triangle $ABC$, points $M$, $N$, and $P$ are given, respectively. Prove:
a) if points $M_{1}$, $N_{1}$, and $P_{1}$ are symmetric to points $M$, $N$, and $P$ with respect to the midpoints of the corresponding sides, then $S_{MNP} = S_{M_{1}N_{1}P_{1}}$.
b) if $M_{1}$, $N_{1... | 29.19. a) Since any triangle can be transformed into an equilateral one by an affine transformation, and in this process, midpoints of sides are transformed into midpoints of sides, centrally symmetric points into centrally symmetric points, and triangles of equal area into triangles of equal area (Problem 29.13), we c... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,190 |
29.20. Let $a, b, c, d$ be complex numbers, and the angles $a 0 b$ and $c 0 d$ are equal and oppositely oriented. Prove that then $\operatorname{Im} a b c d = 0$.
We will say that triangles $A B C$ and $A^{\prime} B^{\prime} C^{\prime}$ are properly similar if there exists a rotational homothety that maps $A$ to $A^{\... | 29.20. Let $A=a /|a|, B=$ $=b /|b|, C=c /|c|, D=d /|d|$. These points lie on the unit circle. There exists a rotation $R^{\alpha}$ such that $B=R^{\alpha}(A)$ and $D=R^{\alpha}(C)$. But $R^{\alpha}$ is multiplication by the complex number $w=\cos \alpha+i \sin \alpha$. Therefore, $B / A=C / D=w$. Let $k=$ $=|a b c d|$.... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,191 |
29.21. Prove that if triangles $a b c$ and $a^{\prime} b^{\prime} c^{\prime}$ on the complex plane are properly similar, then
$$
(b-a) /(c-a)=\left(b^{\prime}-a^{\prime}\right) /\left(c^{\prime}-a^{\prime}\right)
$$ | 29.21. Let's shift these triangles by vectors $-a$ and $-a^{\prime}$ respectively. As a result, we obtain similar triangles with vertices $0, b-a$, $c-a$ and $0, b^{\prime}-a^{\prime}, c^{\prime}-a^{\prime}$. In this case, the point $b-a$ is mapped to the point $c-a$ by the same rotational homothety that maps the point... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,192 |
29.23. Let $a$ and $b$ be complex numbers lying on a circle centered at the origin, and $u$ be the point of intersection of the tangents to this circle at points $a$ and $b$. Prove that $u=2 a b /(a+b)$. | 29.23. Let $v=(a+b) / 2$ be the midpoint of the segment $a b$. Then the right triangles $0 a u$ and $0 v b$ are similar, since they have equal angles at vertex 0. Therefore, according to problem $29.21 a / u=v / b$. Hence, $u=a b / v=2 a b /(a+$ $+b)$. | 2/(+b) | Geometry | proof | Yes | Yes | olympiads | false | 28,194 |
29.24. Let $a$ be a complex number lying on the unit circle $S$ centered at the origin, and $t$ be a real number (a point lying on the real axis). Let $b$ be the point of intersection of the line $a t$ with the circle $S$, distinct from $a$. Prove that $\bar{b}=(1-t a)(t-a)$. | 29.24. The right triangles formed by the points $0,(\bar{a}+\bar{b}) / 2, t$ and $0,(a+\bar{a}) / 2, t$ are similar. According to problem 29.21
$$
\frac{\bar{a}+\bar{b}}{2 t}=\frac{\frac{a+\bar{a}}{2 t}-a}{t-a}=\frac{\bar{a}-a}{2(t-a)}
$$
i.e., $\bar{a} t-1+\bar{b}(t-a)=t \bar{a}-t a$ (here we used the equality $a \b... | \bar{b}=(1-)/(-) | Algebra | proof | Yes | Yes | olympiads | false | 28,195 |
29.25. Given a triangle $A B C$ and a line $l$ passing through the center $O$ of the inscribed circle. Let $A_{1}$ (respectively $B_{1}, C_{1}$) be the foot of the perpendicular dropped from point $A$ (respectively $B, C$) to the line $l$, and let $A_{2}$ (respectively $B_{2}, C_{2}$) be the point on the inscribed circ... | 29.25. Let $A_{3}$ (respectively $B_{3}, C_{3}$) be the point of intersection of the line $A_{1} A_{2}$ (respectively $B_{1} B_{2}, C_{1} C_{2}$) with the inscribed circle, distinct from $A_{2}$ (respectively $B_{2}, C_{2}$). We need to prove that these three points coincide.
Place triangle $ABC$ on the complex plane ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,196 |
29.27. Let points $A^{*}, B^{*}, C^{*}, D^{*}$ be the images of points $A, B, C$, $D$ under inversion. Prove that:
a) $\frac{A C}{A D}: \frac{B C}{B D}=\frac{A^{*} C^{*}}{A^{*} D^{*}}: \frac{B^{*} C^{*}}{B^{*} D^{*}}$
b) $\angle(D A, A C)-\angle(D B, B C)=\angle\left(D^{*} B^{*}, B^{*} C^{*}\right)-\angle\left(D^{*} ... | 29.27. Establish a correspondence between points on the plane and complex numbers so that the center of inversion is at the origin. Then the image of the number $z$ under inversion with degree $R$ is the number $R / \bar{z}$. The double ratio of complex numbers $a, b, c, d$ is defined as the complex number
$$
(a b c d... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,197 |
29.28. Prove that the points corresponding to complex numbers $a$, $b$, $c$, lie on the same straight line if and only if the number $\frac{a-b}{a-c}$, called the simple ratio of three complex numbers, is real.
b) Prove that the points corresponding to complex numbers $a, b, c, d$, lie on the same circle (or on the sa... | 29.28. a) Let $A, B, C$ be points corresponding to the numbers $a, b, c$. The complex number $(a-b):(a-c)$ is real if and only if the vectors $\overrightarrow{A B}$ and $\overrightarrow{A C}$ are proportional.
b) Let $S$ be a circle (or a line) on which the points $b, c, d$ lie. By adding, if necessary, the same compl... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,198 |
29.29*. a) Prove that if $A, B, C$ and $D$ are arbitrary points on a plane, then $A B \cdot C D + B C \cdot A D \geqslant A C \cdot B D$ (Ptolemy's inequality).
b) Prove that if $A_{1}, A_{2}, \ldots A_{6}$ are arbitrary points on a plane, then
$$
\begin{aligned}
& A_{1} A_{4} \cdot A_{2} A_{5} \cdot A_{3} A_{6} \leq... | 29.29. a) The statement of the problem follows from the following properties of complex numbers: 1) $|z w|=|z| \cdot|w| ; 2)|z+w| \leqslant|z|+|w|$. Indeed, if $a, b, c, d$ are arbitrary complex numbers, then
$$
(a-b)(c-d)+(b-c)(a-d)=(a-c)(b-d) .
$$
Therefore
$$
|a-b| \cdot|c-d|+|b-c| \cdot|a-d| \geqslant|a-c| \cdot... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 28,199 |
29.30*. Prove that if $a, b, c$ and $d$ are the lengths of the consecutive sides of a convex quadrilateral $A B C D$, and $m$ and $n$ are the lengths of its diagonals, then $m^{2} n^{2}=a^{2} c^{2}+b^{2} d^{2}-2 a b c d \cos (A+C)$ (Bretschneider). | 29.30. First, let's prove that if \(u, v, w, z\) are complex numbers, and \(u + v + w + z = 0\), then
\[
|u w - v z|^{2} = |u + v|^{2} |v + w|^{2}
\]
Indeed,
\[
|u w - v z| = |u w + v(u + v + w)| = |u + v| \cdot |v + w|
\]
Let the complex numbers \(u, v, w, z\) correspond to vectors \(\overrightarrow{A B}, \overrig... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,200 |
29.31*. a) Given a point \( X \) and a triangle \( ABC \). Prove that
\[
\frac{XB}{b} \cdot \frac{XC}{c} + \frac{XC}{c} \cdot \frac{XA}{a} + \frac{XA}{a} \cdot \frac{XB}{b} \geqslant 1
\]
where \( a, b, c \) are the lengths of the sides of the triangle.
b) Points \( A_1, B_1, C_1 \) are taken on the sides \( BC, CA,... | 29.31. a) Let's place triangle $ABC$ on the complex plane so that point $X$ coincides with zero. Let $\alpha, \beta, \gamma$ be the complex numbers corresponding to the vertices of the triangle. The required inequality follows from the identity
$$
\frac{\beta}{\alpha-\gamma} \cdot \frac{\gamma}{\alpha-\beta}+\frac{\ga... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 28,201 |
29.32*. On the sides of an affinely regular polygon $A_{1} A_{2} \ldots A_{n}$ with center $O$, squares $A_{j+1} A_{j} B_{j} C_{j+1}$ are constructed externally ($j=$ $=1, \ldots, n$). Prove that the segments $B_{j} C_{j}$ and $O A_{j}$ are perpendicular, and their ratio is $2(1-\cos (2 \pi / n))$. | 29.32. Establish a correspondence between points
Fig. 29.6 planes and complex numbers such that point \( O \) coincides with zero. Then \( B_{j} - A_{j} = -i \left( A_{j+1} - A_{j} \right)... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,202 |
29.33*. On the sides of a convex $n$-gon, regular $n$-gons are constructed externally. Prove that their centers form a regular $n$-gon if and only if the original $n$-gon is affinely regular. | 29.33. Let $A_{1} \ldots A_{n}$ be the original $n$-gon, with its vertices numbered counterclockwise; $B_{j}$ is the center of the regular $n$-gon constructed outwardly on the side $A_{j} A_{j+1}$. We will identify points in the plane with complex numbers. Let $w$ be the complex number $\cos (2 \pi / n) + i \sin (2 \pi... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,203 |
29.34*. The vertices of a triangle correspond to complex numbers $a$, $b$, and $c$, lying on the unit circle centered at the origin. Prove that if points $z$ and $w$ are isogonal conjugates, then $z+w+a b c \bar{z} \bar{w}=a+b+c$ (Morley). | 29.34. According to problem 29.20
$$
\operatorname{Im}(a-z)(a-w)(\bar{a}-\bar{b})(\bar{a}-\bar{c})=0
$$
Let $(\bar{a}-\bar{b})(\bar{a}-\bar{c}), (\bar{b}-\bar{a})(\bar{b}-\bar{c})$, and $(\bar{c}-\bar{a})(\bar{c}-\bar{b})$ be denoted by $A, B$, and $C$ respectively. Then
$$
\operatorname{Im} a^{2} A - \operatorname{... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,204 |
29.35*. Points $Z$ and $W$ are isogonal conjugates with respect to an equilateral triangle. Under inversion with respect to the circumcircle, points $Z$ and $W$ map to $Z^{*}$ and $W^{*}$. Prove that the midpoint of segment $Z^{*} W^{*}$ lies on the incircle. | 29.35. Let us place the given equilateral triangle on the complex plane so that the center of its circumscribed circle is at the origin and the radius of the circumscribed circle is 1. Let \( z \) and \( w \) be the complex numbers corresponding to points \( Z \) and \( W \). According to problem 29.34, \( z + w + \bar... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,205 |
29.36*. Points $Z$ and $W$ are isogonal conjugates with respect to an equilateral triangle $ABC$ with center $O$; $M$ is the midpoint of segment $ZW$. Prove that $\angle A O Z + \angle A O W + \angle A O M = n \pi$ (angles are oriented).
## §4. Steiner Ellipses
According to problem 29.6, b) for a given triangle $ABC$... | 29.36. Let's place the given triangle so that the center of the circumscribed circle is at the origin, and point $A$ is at 1. Let $z$ and $w$ be the complex numbers corresponding to points $Z$ and $W$. Rotations of the plane around the origin by the specified angles correspond to multiplication by the complex numbers $... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,206 |
30.1*. Prove that there exists a projective mapping that transforms three given points on one line into three given points on another line.
Definition. The double ratio of a quadruple of points $A, B, C, D$, lying on the same line, is the number
$$
(A B C D)=\frac{c-a}{c-b}: \frac{d-a}{d-b}
$$
where $a, b, c, d$ are... | 30.1. Let us denote the given lines by $l_{0}$ and $l$, the given points on line $l_{0}$ by $A_{0}, B_{0}, C_{0}$, and the given points on line $l$ by $A, B, C$. Let $l_{1}$ be an arbitrary line not passing through point $A$. Take an arbitrary point $O_{0}$ not lying on lines $l_{0}$ and $l_{1}$. Denote by $P_{0}$ the ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,209 |
30.2*. a) Given lines $a, b, c, d$, passing through one point, and a line $l$, not passing through this point. Let $A, B, C, D$ be the points of intersection of line $l$ with lines $a, b, c, d$ respectively. Prove that $(a b c d) = (A B C D)$.
b) Prove that the cross-ratio of four points is preserved under projective ... | 30.2. a) Let the point of intersection of the four given lines be denoted by $O$; let $H$ be the projection of this point onto the line $l$ and $h=O H$. Then
\[
\begin{aligned}
& 2 S_{O A C}=O A \cdot O C \sin (a, c)=h \cdot A C \\
& 2 S_{O B C}=O B \cdot O C \sin (b, c)=h \cdot B C \\
& 2 S_{O A D}=O A \cdot O D \sin... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,210 |
30.3*. Prove that if $(A B C X)=(A B C Y)$, then $X=Y$ (all points are pairwise distinct, except possibly points $X$ and $Y$, and lie on the same line). | 30.3. Let $a, b, c, x, y$ be the coordinates of points $A, B, C, X, Y$. Then
$$
\frac{x-a}{x-b}: \frac{c-a}{c-b}=\frac{y-a}{y-b}: \frac{c-a}{c-b}
$$
Therefore, since all points are distinct, $(x-a)(y-b)=(x-b)(y-a)$. Expanding the brackets and combining like terms, we get $a x - b x = a y - b y$. Simplifying this equa... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,211 |
30.4*. Prove that a projective transformation of a line is uniquely determined by the images of three arbitrary points. | 30.4. Let the image of each of three given points under one projective transformation coincide with the image of this point under another projective transformation. We will prove that then the images of any other point under these transformations coincide. Denote the images of the given points by \(A, B, C\). Take an a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,212 |
30.6*. Given a mapping of line $a$ onto line $b$ that preserves the cross-ratio of any four points. Prove that this mapping is projective.
30.7*. Prove that a transformation $P$ of the real line is projective if and only if it can be represented in the form
$$
P(x)=\frac{a x+b}{c x+d}
$$
where $a, b, c, d$ are such ... | 30.6. We fix three distinct points on a line $a$. According to problem 30.1, there exists a projective mapping $P$ that maps these points in the same way as the given mapping. However, in the solution to problem 30.4, it was essentially proven that any mapping that preserves the cross-ratio is uniquely determined by th... | proof | Algebra | proof | Yes | Yes | olympiads | false | 28,214 |
30.8*. Points $A, B, C, D$ lie on the same line. Prove that if $(A B C D)=1$, then either $A=B$ or $C=D$. | 30.8. First solution. Let $a, b, c, d$ be the coordinates of the given points. Then, by the condition, $(c-a)(d-b)=(c-b)(d-a)$. Expanding the brackets and combining like terms, we get $cb + ad = ca + bd$. Moving everything to the left side and factoring, we obtain $(d-c)(b-a)=0$, i.e., either $a=b$ or $c=d$.
Second so... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,215 |
30.9*. Given a line $l$, a circle, and points $M, N$ lying on the circle and not on the line $l$. Consider the mapping $P$ of the line $l$ onto itself, which is the composition of the projection of the line $l$ onto the given circle from the point $M$ and the projection of the circle onto the line $l$ from the point $N... | 30.9. According to problem 30.6, it is sufficient to prove that the transformation $P$ preserves the cross-ratio of a quadruple of points. Let $A, B, C, D$ be any points on a line $l$. Denote by $A^{\prime}, B^{\prime}, C^{\prime}, D^{\prime}$ their images under the transformation $P$, and by $a, b, c, d$ and $a^{\prim... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,216 |
30.10*. Given a line $l$, a circle, and a point $M$ lying on the circle and not on the line $l$. Let $P_{M}$ be the projection of the line $l$ onto the given circle from the point $M$ (a point $X$ on the line is mapped to the other point of intersection of the line $X M$ with the circle), $R$ a motion of the plane that... | 30.10. Let $N=R^{-1}(M), m=R(l), P_{N}$ be the projection of line $l$ onto the circle from point $N, Q$ - the projection of line $m$ onto line $l$ from point $M$. Then $P_{M}^{-1} \circ R \circ P_{M}=Q \circ R \circ P_{N}^{-1} \circ P_{M}$. But according to the previous problem, the mapping $P_{N}^{-1} \circ P_{M}$ is ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,217 |
30.11*. Prove that if planes $\alpha_{1}$ and $\alpha_{2}$ intersect, then the central projection of $\alpha_{1}$ onto $\alpha_{2}$ with center $O$ defines a one-to-one mapping of the plane $\alpha_{1}$ with the removed line $l_{1}$ onto the plane $\alpha_{2}$ with the removed line $l_{2}$, where $l_{1}$ and $l_{2}$ ar... | 30.11. Lines passing through $O$ and parallel to the plane $\alpha_{1}$ (respectively $\alpha_{2}$) intersect the plane $\alpha_{2}$ (respectively $\alpha_{1}$) at points of the line $l_{2}$ (respectively $l_{1}$). Therefore, if a point lies on one of the planes $\alpha_{1}, \alpha_{2}$ and does not lie on the lines $l... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,218 |
30.12*. Prove that under central projection, a line that is not exceptional is projected into a line.
For central projection to be defined everywhere, it is convenient to assume that each line, in addition to ordinary points, has one more point called the point at infinity. In this case, if two lines are parallel, the... | 30.12. In central projection onto the plane $\alpha_{2}$ with center $O$, a line $l$ is projected to the intersection of the plane passing through $O$ and $l$ with the plane $\alpha_{2}$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,219 |
30.14*. a) Prove that a projective transformation $P$ of the plane, which maps the line at infinity to the line at infinity, is affine.
b) Prove that if points $A, B, C, D$ lie on a line parallel to the exceptional line of a projective transformation $P$ of the plane $\alpha$, then $P(A) P(B): P(C) P(D)=A B: C D$.
c)... | 30.14. a) From problem 30.13, c) it follows that if, along with ordinary points, we consider infinitely distant points, then the transformation $P$ is bijective. In this case, the infinitely distant line is mapped to the infinitely distant line. Therefore, the set of finite points is also bijectively mapped to the set ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,221 |
30.15. Given points $A, B, C, D$, no three of which lie on the same line, and points $A_{1}, B_{1}, C_{1}, D_{1}$, satisfying the same condition.
a) Prove that there exists a projective transformation that maps points $A, B, C, D$ to points $A_{1}, B_{1}, C_{1}, D_{1}$, respectively.
b) Prove that the transformation ... | 30.15. a) It is sufficient to show that points $A, B, C, D$ can be transformed by a projective transformation into the vertices of a square. Let $E$ and $F$ be the points (possibly at infinity) of intersection of line $AB$ with line $CD$ and line $BC$ with line $AD$, respectively. If line $EF$ is finite, then there exi... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,222 |
30.16*. a) Prove that there exists a projective transformation that maps a given circle to a circle and a given point inside the circle to the center of the image.
b) Prove that if a projective transformation maps a given circle to a circle and a point \( M \) to its center, then the exceptional line is perpendicular ... | 30.16. a) Consider on the coordinate plane $O x z$ the points $O(0,0), N(0,1)$, $E(1,0)$. For an arbitrary point $M$ lying on the arc $N E$ of the unit circle (see Fig. 30.2), denote by $P$ the midpoint of the segment $E M$, and by $M^{*}$ and $P^{*}$ the points of intersection of the lines $N M$ and $N P$ with the lin... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,223 |
30.17*. On a plane, there is a circle and a line that does not intersect it. Prove that there exists a projective transformation that translates the given circle into a circle, and the given line into the infinitely distant line. | 30.17. Consider on the coordinate plane $O x z$ the points $O(0,0)$, $N(0,1)$, $E(1,0)$. For an arbitrary point $M$ lying on the arc $N E$ of the unit circle, denote by $P$ the intersection of the segment $E M$ with the line $z=1$. Clearly, by moving the point $M$ along the arc $N E$, we can make the ratio $E M: M P$ e... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,224 |
30.18*. Prove that there exists a projective transformation that maps a given circle to a circle and a given chord to its diameter. | 30.18. Let $M$ be an arbitrary point on a given chord. According to problem 30.16, there exists a projective transformation that maps the given circle to a circle and the point $M$ to its center. Since a line is transformed into a line under a projective transformation, the given chord will be transformed into a diamet... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,225 |
30.19*. Given a circle $S$ and a point $O$ inside it. Consider all projective transformations that map $S$ to a circle and $O$ to its center. Prove that all such transformations map to infinity the same line.
This line is called the polar of the point $O$ with respect to the circle $S$. | 30.19. Draw two arbitrary chords $A C$ and $B D$ through $O$. Let $P$ and $Q$ be the points of intersection of the extensions of the opposite sides of the quadrilateral $A B C D$. Consider an arbitrary projective transformation that maps $S$ to a circle and $O$ to its center. It is clear that the quadrilateral $A B C D... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,226 |
30.20*. A projective transformation maps a certain circle to itself and leaves its center in place. Prove that this is a rotation or a reflection. | 30.20. A projective transformation maps a straight line to a straight line, and since the center remains in place, each diameter is transformed into a diameter. Therefore, each infinitely distant point, where the lines tangent to the circle at diametrically opposite points intersect, is transformed into an infinitely d... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,227 |
30.21*. Given two parallel lines $a, b$ and a point $O$. Then for each point $M$ the following construction can be performed. Draw through $M$ an arbitrary line $l$, not passing through $O$ and intersecting lines $a$ and $b$. Denote the points of intersection by $A$ and $B$, respectively, and let $M^{\prime}$ be the po... | 30.21. a) Point $M^{\prime}$ lies on the line $O M$, so its position is uniquely determined by the ratio $M O: O M^{\prime}$. However, since triangles $M B O$ and $M A M^{\prime}$ are similar, $M O: O M^{\prime}=M B: B A$, and the latter ratio does not depend on the choice of line $l$ by Thales' theorem.
b) First solu... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,228 |
30.22*. Prove that the transformation of the coordinate plane which maps each point with coordinates $(x, y)$ to a point with coordinates $\left(\frac{1}{x}, \frac{y}{x}\right)$ is a projective transformation. | 30.22. First solution. Denote this transformation by $P$. Extend its definition to the points of the line $x=0$ and to the infinitely distant points, setting $P(0, k)=M_{k}, P\left(M_{k}\right)=(0, k)$, where $M_{k}$ is the infinitely distant point on the line $y=k x$. It is easy to see that the thus extended mapping $... | proof | Algebra | proof | Yes | Yes | olympiads | false | 28,229 |
30.24*. Prove that the geometric locus of the points of intersection of the diagonals of quadrilaterals $ABCD$, for which sides $AB$ and $CD$ lie on two given lines $l_{1}$ and $l_{2}$, and sides $BC$ and $AD$ intersect at a given point $P$, is a line passing through the point $Q$ of intersection of the lines $l_{1}$ a... | 30.24. Consider a projective transformation for which the line $P Q$ is exceptional. The images $l_{1}^{\prime}$ and $l_{2}^{\prime}$ of the lines $l_{1}$ and $l_{2}$ under this transformation are parallel, and the images of the considered quadrilaterals are parallelograms, with two sides lying on the lines $l_{1}^{\pr... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,231 |
30.25*. Let $O$ be the point of intersection of the diagonals of quadrilateral $ABCD$, and $E, F$ be the points of intersection of the extensions of sides $AB$ and $CD$, $BC$ and $AD$ respectively. The line $EO$ intersects sides $AD$ and $BC$ at points $K$ and $L$, and the line $FO$ intersects sides $AB$ and $CD$ at po... | 30.25. Let's perform a projective transformation with the exceptional line $E F$. Then the quadrilateral $A B C D$ will transform into a parallelogram, and the lines $K L$ and $M N$ will transform into lines parallel to its sides and passing through the intersection of the diagonals, i.e., into the midlines. Therefore,... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,232 |
30.26*. Lines $a, b, c$ intersect at one point $O$. In triangles $A_{1} B_{1} C_{1}$ and $A_{2} B_{2} C_{2}$, vertices $A_{1}$ and $A_{2}$ lie on line $a$; $B_{1}$ and $B_{2}$ lie on line $b$; $C_{1}$ and $C_{2}$ lie on line $c$. $A, B, C$ are the points of intersection of lines $B_{1} C_{1}$ and $B_{2} C_{2}$, $C_{1} ... | 30.26. Let's perform a projective transformation with the exceptional line $A B$. The images of the points under this transformation will be denoted by letters with a prime. Consider a homothety with center at point $O^{\prime}$ (or a parallel translation if $O^{\prime}$ is an infinitely distant point), which maps poin... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,233 |
30.27*. Points $A, B, C$ lie on line $l$, and points $A_{1}, B_{1}, C_{1}$ lie on line $l_{1}$. Prove that the points of intersection of the lines $A B_{1}$ and $B A_{1}, B C_{1}$ and $C B_{1}, C A_{1}$ and $A C_{1}$ lie on one line (Pappus). | 30.27. Consider a projective transformation, the exceptional line of which passes through the points of intersection of the lines $A B_{1}$ and $B A_{1}, B C_{1}$ and $C B_{1}$, and denote by $A^{\prime}, B^{\prime}, \ldots$ the images of the points $A, B, \ldots$ Then $A^{\prime} B_{1}^{\prime} \parallel B^{\prime} A_... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,234 |
30.29*. Given two triangles $A B C$ and $A_{1} B_{1} C_{1}$. It is known that the lines $A A_{1}, B B_{1}$ and $C C_{1}$ intersect at one point $O$, and the lines $A B_{1}, B C_{1}$ and $C A_{1}$ intersect at one point $O_{1}$. Prove that the lines $A C_{1}, B A_{1}$ and $C B_{1}$ also intersect at one point $O_{2}$ (t... | 30.29. This task is a rephrasing of the previous one. Indeed, suppose that the pair of lines $O O_{1}$ and $O B$ separates the pair of lines $O A$ and $O C$, and the pair of lines $O_{1} O$ and $O_{1} B$ separates the pair of lines $O_{1} A$ and $O_{1} C$ (consider other possible arrangements of these lines similarly o... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,236 |
30.30*. Given two triangles $A B C$ and $A_{1} B_{1} C_{1}$. It is known that the lines $A A_{1}, B B_{1}$ and $C C_{1}$ intersect at one point $O$, the lines $A A_{1}, B C_{1}$ and $C B_{1}$ intersect at one point $O_{1}$, and the lines $A C_{1}, B B_{1}$ and $C A_{1}$ intersect at one point $O_{2}$. Prove that the li... | 30.30. Consider a projective transformation with the exceptional line $O_{1} O_{2}$ and denote the images of points $A, B, \ldots$ by $A^{\prime}, B^{\prime}, \ldots$ Then $A^{\prime} C_{1}^{\prime}\left\|C^{\prime} A_{1}^{\prime}\right\| B^{\prime} B_{1}^{\prime}, B^{\prime} C_{1}^{\prime}\left\|C^{\prime} B_{1}^{\pri... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,237 |
30.31*. Given a quadrilateral $A B C D$ and a line $l$. Denote by $P$, $Q, R$ the points of intersection of the lines $A B$ and $C D, A C$ and $B D, B C$ and $A D$, and by $P_{1}, Q_{1}, R_{1}$ the midpoints of the segments that these pairs of lines intercept on the line $l$. Prove that the lines $P P_{1}, Q Q_{1}$ and... | 30.31. By performing a projective transformation with the exceptional line parallel to $l$ and passing through the intersection point of the lines $P P_{1}$ and $Q Q_{1}$, and then an affine transformation that makes the images of the lines $l$ and $P P_{1}$ perpendicular, we can assume that the lines $P P_{1}$ and $Q ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,238 |
30.32*. Given a triangle $A B C$ and a line $l$. Denote by $A_{1}, B_{1}, C_{1}$ the midpoints of the segments cut off on the line $l$ by the angles $A, B, C$, and by $A_{2}$, $B_{2}, C_{2}$ the points of intersection of the lines $A A_{1}$ and $B C, B B_{1}$ and $A C, C C_{1}$ and $A B$. Prove that the points $A_{2}, ... | 30.32. By making a projective transformation with the exceptional line parallel to $l$ and passing through point $A$, we can consider point $A$ to be infinitely distant, i.e., lines $A B$ and $A C$ are parallel. According to problem 30.14, b), points $A_{1}, B_{1}, C_{1}$ will still be the midpoints of the correspondin... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,239 |
30.33*. Given four points $A, B, C, D$. Let $P, Q, R$ be the points of intersection of the lines $A B$ and $C D, A D$ and $B C, A C$ and $B D$ respectively; $K$ and $L$ be the points of intersection of the line $Q R$ with the lines $A B$ and $C D$ respectively. Prove that $(Q R K L)=-1$ (the theorem of the complete qua... | 30.33. Let's perform a projective transformation, the exceptional line of which is the line $P Q$. Through $A^{\prime}, B^{\prime}, \ldots$ we denote the images of points $A, B, \ldots$ Then $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ is a parallelogram, $R^{\prime}$ is the point of intersection of its diagonals, $Q^... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,240 |
30.34*. A circle intersects the lines $B C, C A, A B$ at points $A_{1}$ and $A_{2}, B_{1}$ and $B_{2}, C_{1}$ and $C_{2}$. Let $l_{a}$ be the line connecting the points of intersection of the lines $B B_{1}$ and $C C_{2}, B B_{2}$ and $C C_{1}$; lines $l_{b}$ and $l_{c}$ are defined similarly. Prove that the lines $l_{... | 30.34. According to Pascal's theorem, the points of intersection of the lines $A_{1} B_{2}$ and $C_{1} C_{2}$, $B_{1} C_{2}$ and $A_{1} A_{2}$, $C_{1} A_{1}$ and $B_{1} B_{2}$ lie on the same line. Translate this line to infinity. After this, we can use the result of problem 14.15. | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,241 |
30.35*. Prove that for any odd $n \geqslant 3$, on the plane it is possible to indicate $2 n$ distinct points, not lying on the same line, and to pair them such that any line passing through two points from different pairs would also pass through one of these $2 n$ points.
## §4. Application of projective transformati... | 30.35. Let $A_{1} \ldots A_{n}$ be a regular $n$-gon, $l_{i}$ be the line containing its side opposite to vertex $A_{i}$, and $B_{i}$ be the point of intersection of line $l_{i}$ with the line at infinity. We will partition the points $A_{1}, \ldots, A_{n}, B_{1}, \ldots, B_{n}$ into pairs $\left(A_{i}, B_{i}\right)$. ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,242 |
30.39*. Given a circle $S$, a line $l$, a point $M$ lying on $S$ and not on $l$, and a point $O$ not lying on $S$. Consider the transformation $P$ of the line $l$, which is the composition of the projection of $l$ onto $S$ from $M$, $S$ onto itself from $O$, and $S$ onto $l$ from $M$, i.e., $P(A)$ is the intersection o... | 30.39. Let $m$ be the line that is the desired locus of points in problem 30.38, b), and let $N$ be the point of intersection of $S$ with the line $O M$, distinct from $M$. Let $Q$ be the composition of the projections of $l$ onto $S$ from $M$ and of $S$ onto $m$ from $N$. According to problem 30.9, this mapping is pro... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,246 |
30.40*. Given a circle $S$, a point $P$ located outside $S$, and a line $l$ passing through $P$ and intersecting the circle at points $A$ and $B$. The point of intersection of the tangents to the circle at points $A$ and $B$ is denoted by $K$.
a) Consider all possible lines passing through $P$ and intersecting $A K$ a... | 30.40. Both tasks become obvious after a projective transformation that maps the circle $S$ to a circle and the line $K P$ to the line at infinity (see problem 30.17).
a) The required locus of points lies on a line equidistant from the images of the lines $A K$ and $B K$.
b) The required point is the center of the im... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,247 |
30.41*. The excircle of triangle $ABC$ touches side $BC$ at point $D$, and the extensions of sides $AB$ and $AC$ at points $E$ and $F$. Let $T$ be the point of intersection of lines $BF$ and $CE$. Prove that points $A$, $D$, and $T$ lie on the same line. | 30.41. Let $A^{\prime}, B^{\prime}, \ldots$ be the images of points $A, B, \ldots$ under a projective transformation that maps the excircle of triangle $ABC$ to a circle and the chord $EF$ to a diameter (see problem 30.18). Then $A^{\prime}$ is the point at infinity of the lines perpendicular to the diameter $E^{\prime... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,248 |
30.43*. In a circle $S$, a hexagon $A B C D E F$ is inscribed. Prove that the points of intersection of the lines $A B$ and $D E, B C$ and $E F, C D$ and $F A$ lie on one straight line (Pascal). | 30.43. Consider a projective transformation that maps the circle $S$ to a circle, and the intersection points of the lines $A B$ and $D E$, $B C$ and $E F$ - to the points at infinity (see problem 29.19). Our task has been reduced to problem 2.11. | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,250 |
30.44*. Let $O$ be the midpoint of the chord $AB$ of the circle $S, MN$ and $PQ$ be arbitrary chords passing through $O$, such that points $P$ and $N$ lie on the same side of $AB, E$ and $F$ are the points of intersection of the chord $AB$ with the chords $MP$ and $NQ$ respectively. Prove that $O$ is the midpoint of th... | 30.44. Consider a projective transformation that maps a circle $S$ to another circle and a point $O$ to its center $O^{\prime}$ (see problem 30.16, a)). Let $A^{\prime}$, $B^{\prime}, \ldots$ be the images of points $A, B, \ldots$ Then $A^{\prime} B^{\prime}$, $M^{\prime} N^{\prime}$, and $P^{\prime} Q^{\prime}$ are di... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,251 |
30.45*. Points $A, B, C$ and $D$ lie on a circle, $S A$ and $S D$ are tangents to this circle, $P$ and $Q$ are the points of intersection of the lines $A B$ and $C D$, $A C$ and $B D$ respectively. Prove that the points $P, Q$ and $S$ lie on the same line.
## §5. Application of projective transformations of a line in ... | 30.45. Consider a projective transformation that maps a given circle to a circle and the segment $A D$ to its diameter (see problem 30.18). Let $A^{\prime}$, $B^{\prime}, \ldots$ be the images of points $A, B, \ldots$ Then $S$ is mapped to the infinitely distant point $S^{\prime}$ of the lines perpendicular to the line... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,252 |
30.46*. On the side $AB$ of the quadrilateral $ABCD$, a point $M_{1}$ is taken. Let $M_{2}$ be the projection of $M_{1}$ onto the line $BC$ from $D$, $M_{3}$ be the projection of $M_{2}$ onto $CD$ from $A$, $M_{4}$ be the projection of $M_{3}$ onto $DA$ from $B$, $M_{5}$ be the projection of $M_{4}$ onto $AB$ from $C$,... | 30.46. According to problem 30.15, it is sufficient to consider only the case when $A B C D$ is a square. We need to prove that the composition of the projections described in the condition is an identity transformation. According to problem 30.4, a projective transformation is identical if it has three distinct fixed ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,253 |
30.49*. Using projective transformations of the line, solve the butterfly problem (problem 30.44 ).
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | 30.49. Let $F^{\prime}$ be the point symmetric to $F$ with respect to $O$. We need to prove that $F^{\prime}=E$. According to problem 30.9, the composition of the projection of the line $A B$ onto the circle $S$ from point $M$, and then from $S$ back onto $A B$ from $Q$ is a projective transformation of the line $A B$.... | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 28,255 |
30.50*. Points $A, B, C, D, E, F$ lie on the same circle. Prove that the points of intersection of the lines $A B$ and $D E, B C$ and $E F, C D$ and $F A$ lie on the same line (Pascal).
## §6. Application of projective transformations of a line in construction problems | 30.50. Let the points of intersection of the lines $A B$ and $D E$, $B C$ and $E F$, $C D$ and $F A$ be denoted by $P, Q, R$ respectively, and the point of intersection of the lines $P Q$ and $C D$ - by $R^{\prime}$. We need to prove that the points $R$ and $R^{\prime}$ coincide. Let $G$ be the point of intersection of... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,256 |
30.51*. Given a circle, a line, and points $A, A^{\prime}, B, B^{\prime}, C, C^{\prime}, M$ lying on this line. According to problems 30.1 and 30.3, there exists a unique projective transformation of this line onto itself that maps points $A, B, C$ to $A^{\prime}, B^{\prime}, C^{\prime}$, respectively. Denote this tran... | 30.51. Let's denote the given line and circle as $l$ and $S$ respectively. Let $O$ be an arbitrary point on the circle, and let $A_{1}, A_{1}^{\prime}, B_{1}, B_{1}^{\prime}, C_{1}, C_{1}^{\prime}$ be the images of points $A, A^{\prime}, B, B^{\prime}, C, C^{\prime}$ when projecting the line $l$ onto the circle $S$ fro... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,257 |
30.52*. Given two lines $l_{1}$ and $l_{2}$ and two points $A$ and $B$, not lying on these lines. Using a compass and a straightedge, construct a point $X$ on the line $l_{1}$ such that the lines $A X$ and $B X$ intercept a segment on the line $l_{2}$, a) having a given length $a ;$ b) bisected at a given point $E$ on ... | 30.52. a) The desired point $X$ is the fixed point of the composition of the projection from $l_{1}$ to $l_{2}$ from point $A$, a translation along the line $l_{2}$ by a distance $a$, and the projection from $l_{2}$ to $l_{1}$ from point $B$. The fixed point of the projective transformation is constructed in problem 30... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,258 |
30.56*. a) Given a line $l$ and a point $P$ outside it. Using a compass and a straightedge, construct a segment $X Y$ of a given length on $l$ that is seen from $P$ at a given angle $\alpha$.
b) Given two lines $l_{1}$ and $l_{2}$ and points $P$ and $Q$ not lying on these lines. Using a compass and a straightedge, con... | 30.56. a) Draw an arbitrary circle $S$ through the point $P$. According to problem 30.10, the composition of the projection of $l$ onto $S$ from $P$, a rotation around the center of the circle $S$ by an angle of $2 \alpha$, and the projection of $S$ onto $l$ from $P$ is a projective transformation of the line $l$. Then... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,262 |
1. Coin game. Two players take turns placing five-kopeck coins on a rectangular table. Coins can only be placed on free spaces (i.e., so that they do not overlap each other even partially). Once placed, coins cannot be moved. It is assumed that each player has a sufficient number of coins. The player who places the las... | 1. The player starting the game must place a coin at the center of the table. Thereafter, he places his coin each time symmetrically (with respect to the center of the table) to the coin placed by the second player (Fig. 48).
, the sides of which are respectively equal to $1,3 / 2,9 / 4$ and $3 / 2,9 / 4,27 / 8$. These triangles have equal angles (due to the proportionality of the sides, they are similar) and two corresponding equal sides,
Note. In general, the condi... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,268 |
5. What is the greatest number of acute angles that can occur in a convex polygon? | 5. The maximum number of acute angles in a convex polygon is three. Indeed, the sum of all exterior angles of an $n$-sided polygon is always $4 d$. If any $n$-sided polygon ($n \geqslant 4$) had four acute interior angles, then the exterior angles supplementary to these interior angles would be obtuse, and their sum wo... | 3 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,269 |
8. a) Prove that any (not necessarily convex!) polygon can be divided into triangles by non-intersecting diagonals.
b) Prove that the sum of the interior angles of any (not necessarily convex) $n$-gon is $2 d(n-2)$.
Note. For the case of a convex polygon, the statements of problems 8 a) and b) are well-known. | 8. a) First of all, let's prove that for any polygon, we can find a diagonal that divides it into two polygons with a smaller number of sides. Consider the rightmost (or one of the rightmost) vertex $A$ of the polygon (Fig. $59, a$). Both vertices $B$ and $C$ adjacent to it lie no further to the right. Draw the diagona... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,272 |
9. a) Let $A B C D$ and $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ be two convex quadrilaterals with equal corresponding sides ($A B=A^{\prime} B^{\prime}$, $B C=B^{\prime} C^{\prime}$, etc.). Prove that if $\angle A>\angle A^{\prime}$, then $\angle B>\angle B^{\prime}$, $\angle C>\angle C^{\prime}$, and $\angle D<\... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,273 | |
10. A circle with a radius equal to the height of a certain equilateral triangle rolls along the side of this triangle. Prove that the arc, cut off by the sides of the triangle from the circle, is always equal to $60^{\circ}$. Also prove that the "lens" formed by reflecting this arc relative to the chord that spans it ... | 10. Let $ABC$ be a given equilateral triangle, and $AB'C'$ be a triangle symmetric to it with respect to point $A$ (Fig. 64, a). A circle rolls along the side $BC$ of the triangle; $O$ is its center at some position, and $K$ is the point of tangency of side $BC$ with the circle. The sides $AB$ and $AC$ intercept arc $M... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,274 |
11. A circle has the property that an inscribed equilateral triangle can be moved inside it so that each vertex of the triangle describes this circle. Find a plane closed non-self-intersecting curve, different from a circle, inside which an equilateral triangle can also be moved so that each of its vertices describes t... | 11. The simplest curve, similar to a circle, which has the required property, is shown in Fig. 65. It consists of two equal arcs of a circle, each containing $120^{\circ}$; the radius of the arcs is equal to the side of the triangle. The triangle $A B C$, shown in Fig. 65, can rotate around vertex $A$, so that vertices... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,275 |
12. A circle is inscribed in a triangle, and a square is circumscribed around it (Fig. 4). Prove that more than half of the perimeter of the square is inside the triangle.
Fig. 4. | 12. A triangle has three points of tangency with its inscribed circle, and a square has four. Therefore, at least between one pair of tangency points of the circle with the triangle, there are two tangency points of the circle with the square. Consequently, at least one "corner" of the square ${ }^{1}$ lies entirely wi... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,276 |
15. Given two triangles $A B C$ and $D E F$ and a point $O$. Any point $X$ inside triangle $A B C$ and any point $Y$ inside triangle $D E F$ are taken; triangle $O X Y$ is completed to form a parallelogram $O X Z Y$ (Fig. 5).
.
... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,279 |
17. a) On a plane, there are five points $A, B, C, D, E$, none of which are collinear. Prove that among them, one can choose four points that are the vertices of a convex quadrilateral.
b) Inside a square $A_{1} A_{2} A_{3} A_{4}$, there is a convex quadrilateral $A_{5} A_{6} A_{7} A_{8}$; inside $A_{5} A_{6} A_{7} A_... | 17. a) Let $A$ be the leftmost (or one of the two leftmost) of our five points (Fig. $70, a$). We draw a vertical line through $A$ and rotate it until it passes through another of our points, point $B$. We then continue to rotate the line $AB$ in the same direction around $B$ until it passes through a third point $C$; ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,281 |
19. In triangle $ABC$, a triangle $PQR$ is inscribed. Prove that the area of at least one of the triangles $BPQ$, $APR$, $CRQ$ (Fig. 7) does not exceed the area of triangle $PQR$.
Fig. 7. | 19. Let $A_{1} B_{1}, B_{1} C_{1}, C_{1} A_{1}$ be the midlines of triangle $A B C$. We will call corresponding sides of triangles $A_{1} B_{1} C_{1}$ and $P Q R$ those whose endpoints lie on the sides of the same angle of triangle $A B C$. Then two cases may arise: either there will be a pair of non-intersecting corre... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,283 |
22. a) Two points $A$ and $B$ in a plane, the distance between which is 1, are connected by a convex broken line $A P_{1} \ldots P_{n} B$ (i.e., such a broken line that the polygon $A P_{1} P_{2} \ldots P_{n} B \xrightarrow{n}$ is convex). Prove that if the sum of the exterior angles of the broken line at points $P_{1}... | 22. a) First of all, note that the theorem is valid if the broken line consists of only two segments $A P$ and $P B$ (Fig. $88, a$). Indeed, extend
Fig. 88.
in this case, the line $A P$ be... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,285 |
23. Prove that no triangle inscribed in a convex polygon \( M \) can have a larger area than the largest (by area) of all triangles whose vertices coincide with any three vertices of \( M \).
Note. The theorem of problem 23 indicates the solution to the following problem: to inscribe a triangle of the largest possible... | 23. Let $X Y Z$ be some triangle inscribed in a given polygon $M$ (Fig. $90, a$). If $P P_{1}$ is a side of $M$ on which the vertex $X$ of the triangle lies, then either all triangles $P Y Z, \quad X Y Z$ and $P_{1} Y Z$ have the same area (which will be the case if $P P_{1} \| Y Z$), or at least one of the triangles $... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,286 |
24. In space, two triangles $A B C$ and $A^{\prime} B^{\prime} C^{\prime}$ are given. Prove that the greatest distance between points of triangle $A B C$ and points of triangle $A^{\prime} B^{\prime} C^{\prime}$ will be the greatest of the nine segments $A A^{\prime}, A B^{\prime}, A C^{\prime}, B A^{\prime}, B B^{\pri... | 24. Let the greatest of the nine distances between the vertices of our two triangles be equal to $a$ and let $M$ be an arbitrary point of triangle $ABC$ and $M'$ be an arbitrary point of triangle $A'B'C'$ (Fig. 92). We will prove that $MM' \leqslant a$. Draw an arbitrary line through $M$ lying in the plane of triangle ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,287 |
25. Let $ABC$ be some triangle with $BC=a$, $CA=b$, $AB=c$, $\angle BAC=\alpha$, $\angle ABC=\beta$, $\angle ACB=\gamma$ (angles are measured in radians). Prove that
$$
\frac{\pi}{3} \leqslant \frac{a \alpha + b \beta + c \gamma}{a + b + c} < \frac{\pi}{2}
$$
Note. For a tetrahedron (i.e., an arbitrary triangular pyr... | 25. 26) Since opposite the larger side of a triangle lies the larger angle, the product $(\alpha-\beta)(a-b) \geqslant 0$ (both factors have the same sign) and equals zero only if $\alpha=\beta, a=b$, i.e., if the triangle $A B C$ is isosceles. From this, it follows that
$$
(\alpha-\beta)(a-b)+(\beta-\gamma)(b-c)+(\ga... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 28,288 |
28. On the side $A_{1} A_{2}$ of a regular $n$-gon $A_{1} A_{2} A_{3} \ldots A_{n}$, a point $M_{1}$ is taken. This point is projected from vertex $A_{n}$ to point $M_{2}$ on side $A_{2} A_{3}$ or its extension (i.e., $M_{2}$ is the point of intersection of line $A_{n} M_{1}$ with line $A_{2} A_{3}$); then point $M_{2}... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,290 | |
29. A segment of length 1 is completely covered by some number of smaller segments lying on it. Prove that among these segments, one can find non-intersecting segments whose total length is greater than or equal to $1 / 2$. | 29. First, we will exclude from consideration all those segments of the covering that are entirely covered by one or several of these segments. After this, we will number all the remaining segments in a certain order as follows. We will assume that our original segment \( O \) of length one is located horizontally. We ... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 28,291 |
30. The plane is completely covered by some number of half-planes. Prove that among these half-planes, three can be chosen which together already cover the entire plane. | 30. We will prove the theorem by mathematical induction. For the case when the total number of half-planes is three, the theorem is obvious. Suppose now that we have already proved the theorem in the case when the total number of half-planes is $n$; we will prove that in this case the theorem remains valid for $n+1$ ha... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,292 |
31. a) There is a certain number of half-planes, any three of which have a common point. Prove that there exists a point that belongs simultaneously to all half-planes.
b) On a plane, a certain number of convex polygons are given, any three of which have a common point. Prove that all polygons have a common point.
Not... | 31. a) We will prove the statement by mathematical induction. Suppose we have already proven that $k$ of our half-planes have a common point; we need to show that in this case, the $(k+1)$-th half-plane $\pi$ intersects the common part $\Phi$ of the first $k$ half-planes, in other words, that $k+1$ half-planes have a c... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,293 |
32. a) Inside a square $A B C D$ with side length 1, there is a convex polygon $M$ with an area greater than $1 / 2$. Prove that there exists a line $l$, parallel to any arbitrarily chosen side of the square, that intersects the polygon $M$ along a segment of length greater than $1 / 2$.
b)* Inside a square $A B C D$ ... | 32. a) Draw lines through all vertices of the polygon \( M \) parallel to any side \( AB \) of the square. Then the polygon will be divided into a series of triangles and trapezoids (Fig. 102). The area of each of these triangles and trapezoids is equal to the length of the midline multiplied by the height. If all midl... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,294 |
33. There exist figures having an infinite number of centers of symmetry (for example, the strip between two parallel lines). Can a figure have more than one, but a finite number of centers of symmetry? | 33. Suppose a figure has two centers of symmetry $S$ and $O$. Reflect point $S$ symmetrically with respect to $O$ (Fig. 104).
We will prove that the resulting point $P$ is also a center of symmetry. Let $A$ be an arbitrary point of the figure. Then the point $A_{1}$, symmetric to $A$ with respect to $O$, also belongs ... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,295 |
34. Prove that if a polygon has several axes of symmetry, then all of them intersect at one point. | 34. First of all, it is clear that any two axes of symmetry intersect
Fig. 105. inside the polygon. Indeed, suppose this is not the case. Let \( AB \) and \( CD \) be two axes of symmetry ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,296 |
37. a) A plane is covered with a grid of squares. Is it possible to construct an equilateral triangle with vertices at the grid points?
b) In space, a regular grid of cubes is given. Is it possible to construct a regular tetrahedron with vertices coinciding with the grid points? | 37. a) Let \(ABC\) be any triangle, the vertices of which lie at the nodes of a grid.
Fig. 109. Consider a rectangle composed of grid squares, the sides of which pass through the vertices o... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,299 |
41. The diameter of a closed (possibly self-intersecting) $n$-segment broken line, all segments of which have a length of 1, of course, cannot be less than 1. For which $n$ can the diameter of the broken line be equal to 1? | 41. It is quite obvious that a broken line with any odd number of segments satisfying the conditions of the problem can be constructed: for $n=3$ it will be an equilateral triangle (Fig. $119, \quad$) for $n>3$ - a regular $n$-pointed star
. No... | \frac{\sqrt{3}}{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,304 |
43. It is obvious that any figure with a diameter of 1 can be enclosed within a square with a side of 2: for this, it is sufficient for the center of the square to coincide with any point of the figure. What is the side of the smallest square that can enclose any figure with a diameter of 1?
Note. One can also pose a ... | 43. It is obvious that a circle with a diameter of 1 cannot be enclosed in any square whose side is less than 1. On the other hand, it is almost equally clear that any figure with a diameter of 1 can be enclosed in a square with a side of 1. Indeed, any square that encloses some figure $\Phi$ can always be reduced so t... | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,305 |
44. It is obvious that any flat closed broken line with a perimeter of 1 can be enclosed in a circle of radius $1 / 2$: for this, it is sufficient for the center $O$ of the circle to coincide with any point of the broken line (in this case, for any point $A$ of the broken line, the length of one of the two segments of ... | 44. We will prove that any closed broken line of perimeter 1 can be enclosed within a circle of radius $1 / 4$. Let $A$ be an arbitrary point on our broken line, and $B$ be a point on the broken line such that both parts of the broken line connecting points $A$ and $B$ have the same length $1 / 2$. Let $O$ be the midpo... | \frac{1}{4} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,306 |
47. What is the radius of the smallest circle in which $n$ points ( $n=2,3,4, \ldots, 10,11$ ) can be placed, one of which coincides with the center of the circle, so that the distance between any two points is at least 1? | 47. Let's denote the radius of the smallest "circle in which $n$ points can be placed, one of which coincides with the center of the circle and the distances between each pair are at least 1" by $R_{n}$. Our task is to determine the values of $R_{n}$ for the first few values of $n$.
$1^{\circ}$ It is completely obviou... | R_{2}=R_{3}=R_{4}=R_{5}=R_{6}=R_{7}=1,\quadR_{8}=\frac{1}{2\sin\frac{180}{7}}=1.15\ldots,\quadR_{9}=\frac{1}{2\sin\frac{180}{8}}=1.30\ldots | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,309 |
49. How many circles of radius 1 can be applied ${ }^{1}$ ) to a given unit circle $\mathcal{S}$ so that no two of these circles intersect? So that no one of these circles contains the center of another circle inside itself? | 49. Since a unit circle tangent to $S$ is seen from the center $O$ of circle $S$ at an angle of $60^{\circ}$ (Fig. $132, a$), no more than $6\left(=\frac{360}{60^{\circ}}\right)$ non-overlapping unit circles can be applied to $S$. Six circles can obviously be applied (Fig. 132, b).
Further, if $O_{1}$ and $O_{2}$ are ... | 12 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,311 |
50*. What is the greatest number of circles of radius 1 that can be placed on a plane so that they all intersect a certain fixed unit circle $S$ and no one of them contains the center of $S$ or the center of another circle inside it? | 50. Fig. 134 shows that 18 circles satisfying the condition of the problem can be placed (six circles in Fig. 134 have centers at the vertices of a regular hexagon inscribed in $S$, the other 12 - at the vertices of squares constructed on the sides of the inscribed hexagon outside it; compare with Fig. 130). The fact t... | 18 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,312 |
51. What is the greatest number of squares with side 1 that can be placed ${ }^{1}$ ) next to a given unit square $K$ so that no two of them intersect?
$^{1}$) The superscript "1" is kept as is, since it might refer to a footnote or additional information in the original text. | 51. First solution. Let $O$ be the center of the main square $K$, and $O_{1}$ and $O_{2}$ be the centers of the non-overlapping squares $K_{1}$ and $K_{2}$ attached to it (Fig. $135, a$). Since the smallest distance from the center of a unit square to its boundary is $\frac{1}{2}$, the segments $O O_{1}$, $O O_{2}$, an... | 8 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,313 |
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