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8.5. Given a point $A$ and a circle $S$. Draw a line through point $A$ such that the chord cut off by the circle $S$ on this line has a given length $d$. | 8.5. Let $R$ be the radius of the circle $S$, and $O$ its center. If the circle $S$ intersects a line passing through point $A$ at chord $P Q$ and $M$ is the midpoint of $P Q$, then $O M^{2}=O Q^{2}-M Q^{2}=R^{2}-d^{2} / 4$. Therefore, the desired line is tangent to the circle of radius $\sqrt{R^{2}-d^{2} / 4}$ centere... | R^{2}-^{2}/4 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,226 |
8.7. Construct a triangle given $a, m_{c}$ and angle $A$. | 8.7. Suppose that triangle $A B C$ is constructed. Let $A_{1}$ and $C_{1}$ be the midpoints of sides $C B$ and $A B$. Since $C_{1} A_{1} \| A C$, then $\angle A_{1} C_{1} B=\angle A$. From this, the following construction follows. First, construct segment $C B$ of length $a$ and its midpoint $A_{1}$. Point $C_{1}$ is t... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,228 |
8.9. The extensions of sides $A B$ and $C D$ of rectangle $A B C D$ intersect a certain line at points $M$ and $N$, and the extensions of sides $A D$ and $B C$ intersect the same line at points $P$ and $Q$. Construct the rectangle $A B C D$ if points $M, N, P, Q$ and the length $a$ of side $A B$ are given. | 8.9. Suppose that rectangle $A B C D$ is constructed. Drop a perpendicular $P R$ from point $P$ to line $B C$. Point $R$ can be constructed since it lies on the circle with diameter $P Q$ and $P R=A B=a$. By constructing point $R$, we construct lines $B C$ and $A D$ and drop perpendiculars from points $M$ and $N$ to th... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,230 |
8.10*. Construct a triangle given the bisector, median, and altitude drawn from the same vertex.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 8.10. Suppose that triangle $A B C$ is constructed, $A H$ is the altitude, $A D$ is the angle bisector, and $A M$ is the median. According to problem 2.68, point $D$ lies between $M$ and $H$. Point $E$, the intersection of line $A D$ and the perpendicular from point $M$ to side $B C$, lies on the circumcircle of triang... | Other | math-word-problem | Yes | Yes | olympiads | false | 27,231 | |
8.11*. Construct triangle $A B C$ given side $a$, angle $A$, and the radius of the inscribed circle $r$.
## §3. Similar Triangles and Homothety | 8.11. Suppose that triangle $A B C$ is constructed and $O$ is the center of its inscribed circle. Then $\angle B O C=90^{\circ}+\angle A / 2$ (problem 5.3). From point $O$, segment $B C$ is seen at an angle of $90^{\circ}+\angle A / 2$, and it is removed from line $B C$ by a distance $r$, so it can be constructed. Then... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,232 |
8.12. Construct a triangle given two angles $A, B$ and the perimeter $P$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 8.12. Construct an arbitrary triangle with angles $A$ and $B$ and find its perimeter $P_{1}$. The desired triangle is similar to the constructed triangle with a coefficient of $P / P_{1}$. | notfound | Other | math-word-problem | Yes | Yes | olympiads | false | 27,233 |
8.13. Construct triangle $A B C$ given $m_{a}, m_{b}$ and $m_{c}$. | 8.13. Suppose that triangle $A B C$ is constructed. Let $A A_{1}, B B_{1}$ and $C C_{1}$ be its medians, $M$ be the point of their intersection, and $M^{\prime}$ be the point symmetric to $M$ with respect to point $A_{1}$. Then $M M^{\prime}=2 m_{a} / 3, M C=2 m_{c} / 3$ and $M^{\prime} C=2 m_{b} / 3$, so triangle $M M... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,234 |
8.14. Construct triangle $A B C$ given $h_{a}, h_{b}$ and $h_{c}$. | 8.14. It is clear that $B C: A C: A B=\frac{S}{h_{a}}: \frac{S}{h_{b}}: \frac{S}{h_{c}}=\frac{1}{h_{a}}: \frac{1}{h_{b}}: \frac{1}{h_{c}}$. Let's take an arbitrary segment $B^{\prime} C^{\prime}$ and construct triangle $A^{\prime} B^{\prime} C^{\prime}$ such that $B^{\prime} C^{\prime}: A^{\prime} C^{\prime}=$ $=h_{b}:... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,235 |
8.15. Inscribed in the acute-angled triangle $A B C$ is a square $K L M N$ such that vertices $K$ and $N$ lie on sides $A B$ and $A C$, and vertices $L$ and $M$ lie on side $B C$. | 8.15. Let's take an arbitrary point \( K' \) on side \( AB \), drop a perpendicular \( K' L' \) to side \( BC \), and then construct a square \( K' L' M' N' \) lying inside angle \( ABC \). Let the line \( BN' \) intersect side \( AC \) at point \( N \). It is clear that the desired square is the image of square \( K' ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,236 |
8.16*. Construct triangle $A B C$ given $h_{a}, b-c$ and $r$.
See also problems $19.16-19.21,19.40,19.41$.
## §4. Construction of Triangles from Various Elements
In the problems of this paragraph, it is required to construct a triangle according to the elements specified in the condition. | 8.16. Suppose the desired triangle \(ABC\) is constructed. Let \(Q\) be the point of tangency of the incircle with side \(BC\), \(PQ\) be the diameter of this circle, and \(R\) be the point of tangency of the excircle with side \(BC\). Clearly, \(BR = \frac{a+b+c}{2} - c = \frac{a+b-c}{2}\) and \(BQ = \frac{a+c-b}{2}\)... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,237 |
8.30. Construct triangle $ABC$, given three points $A', B', C'$, which are symmetric to the center $O$ of the circumcircle of this triangle with respect to the sides $BC, CA, AB$. | 8.30. Let the midpoints of the sides $B C, C A, A B$ of the triangle be denoted by $A_{1}, B_{1}, C_{1}$ respectively. Since $B C\left\|B_{1} C_{1}\right\| B^{\prime} C^{\prime}$ and $O A_{1} \perp B C$, then $O A^{\prime} \perp B^{\prime} C^{\prime}$. Similarly, $O B^{\prime} \perp A^{\prime} C^{\prime}$ and $O C^{\pr... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,250 |
8.33. Construct triangle $A B C$, given the positions of three points $A_{1}, B_{1}, C_{1}$, which are the centers of the excircles of triangle $A B C$. | 8.33. According to problem 5.2, points $A, B$ and $C$ are the feet of the altitudes of triangle $A_{1} B_{1} C_{1}$. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,253 |
8.34*. Construct triangle $A B C$ given the center of the circumscribed circle $O$, the centroid $M$, and the foot of the altitude $H$ from $C$ to $AB$. | 8.34. Let $H_{1}$ be the point of intersection of the altitudes of triangle $ABC$. According to problem $5.116$, $O M: M H_{1}=1: 2$ and point $M$ lies on the segment $O H_{1}$. Therefore, we can construct point $H_{1}$. Then we draw the line $H_{1} H$ and erect a perpendicular $l$ to this line at point $H$. Dropping a... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,254 |
8.35*. Construct triangle $A B C$ given the centers of the inscribed, circumscribed, and one of the excircles.
## §6. Triangle | 8.35. Let $O$ and $I$ be the centers of the circumcircle and incircle, and $I_{c}$ be the center of the excircle tangent to side $AB$. The circumcircle of triangle $ABC$ bisects the segment $I I_{c}$ (Problem 5.120, b), and the segment $I I_{c}$ bisects the arc $AB$. It is also clear that points $A$ and $B$ lie on the ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,255 |
8.36. Construct points $X$ and $Y$ on sides $A B$ and $B C$ of triangle $A B C$ such that $A X = B Y$ and $X Y \parallel A C$. | 8.36. Suppose we have constructed points $X$ and $Y$ on sides $A B$ and $B C$ of triangle $A B C$ such that $A X = B Y$ and $X Y \parallel A C$. We draw $Y Y_{1} \parallel A B$ and $Y_{1} C_{1} \parallel B C$ (points $Y_{1}$ and $C_{1}$ lie on sides $A C$ and $A B$). Then $Y_{1} Y = A X = B Y$, i.e., $B Y Y_{1} C$ is a... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,256 |
8.37. Construct a triangle given sides $a$ and $b$, if it is known that the angle opposite one of them is three times the angle opposite the other.
Construct a triangle given sides $a$ and $b$, if it is known that the angle opposite one of them is three times the angle opposite the other. | 8.37. Let $a<b$ for definiteness. Suppose triangle $ABC$ is constructed. Take a point $D$ on side $AC$ such that $\angle ABD = \angle BAC$. Then $\angle BDC = 2 \angle BAC$ and $\angle CBD = 3 \angle BAC - \angle BAC = 2 \angle BAC$, i.e., $CD = CB = a$. In triangle $BCD$, all sides are known: $CD = CB = a$ and $DB = A... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,257 |
8.39. Draw a line through the given point $M$ so that it cuts off from the given angle with vertex $A$ a triangle $A B C$ of a given perimeter $2 p$.
保留源文本的换行和格式，直接输出翻译结果。 | 8.39. Suppose that triangle $A B C$ is constructed. Let $K$ and $L$ be the points where the excircle, tangent to side $B C$, touches the extensions of sides $A B$ and $A C$ respectively. Since $A K=A L=p$, this excircle can be constructed; it remains to draw a tangent from the given point $M$ to the constructed circle. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,259 |
8.40. Construct triangle $A B C$ given the median $m_{c}$ and the angle bisector $l_{c}$, if $\angle C=90^{\circ}$. | 8.40. Let the extension of the bisector $C D$ intersect the circumcircle of triangle $A B C$ (with right angle $C$) at point $P$, $P Q$ be the diameter of the circumcircle, and $O$ be its center. Then $P D: P O=P Q: P C$, i.e., $P D \cdot P C=2 R^{2}=m_{c}^{2}$. Therefore, by drawing a tangent of length $\sqrt{2} m_{c}... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,260 |
8.41*. Given a triangle $A B C$, where $A B < B C$. Construct a point $D$ on the side $A C$ such that the perimeter of triangle $A B D$ is equal to the length of side $B C$. | 8.41. Let's construct point $K$ on side $A C$ such that $A K = B C - A B$. Let point $D$ lie on segment $A C$. The equality $A D + B D + A B = B C$ is equivalent to the equality $A D + B D = A K$. For point $D$ lying on segment $A K$, the latter equality can be rewritten as $A D + B D = A D + D K$, and for point $D$ no... | DistheintersectionoftheperpendicularbisectorofsegmentBKsegmentAC | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,261 |
8.42*. Construct triangle $ABC$ given the radius of the circumscribed circle and the angle bisector of angle $A$, if it is known that the difference between angles $B$ and $C$ is $90^{\circ}$. | 8.42. Suppose that triangle $A B C$
Fig. 8.4 is constructed. Draw the diameter $C D$ of the circumscribed circle. Let $O$ be the center of the circumscribed circle, and $L$ be the point of... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,262 |
8.43*. On the side $A B$ of triangle $A B C$, a point $P$ is given. Draw a line through point $P$ (different from $A B$) intersecting the rays $C A$ and $C B$ at points $M$ and $N$ such that $A M = B N$. | 8.43. Let's take points \(A_1\) and \(B_1\) on sides \(BC\) and \(AC\) such that \(PA_1 \parallel AC\) and \(PB_1 \parallel BC\). Then, we will lay off segments \(A_1 B_2 = AB_1\) and \(B_1 A_2 = BA_1\) on the rays \(A_1 B\) and \(B_1 A\). We will prove that the line \(A_2 B_2\) is the desired one. Indeed, let \(k = \f... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,263 |
8.44*. Construct triangle $ABC$ given the radius of the inscribed circle $r$ and (non-zero) lengths of segments $AO$ and $AH$, where $O$ is the center of the inscribed circle, and $H$ is the orthocenter.
See also problems 15.13 b), 17.12-17.15, 18.11, 18.31.
## §7. Quadrilaterals | 8.44. Suppose that triangle $A B C$ is constructed. Let $B_{1}$ be the point of tangency of the inscribed circle with side $A C$. In the right triangle $A O B_{1}$, the leg $O B_{1}=r$ and the hypotenuse $A O$ are known, so we can construct the angle $O A B_{1}$, and hence the angle $B A C$. Let $O_{1}$ be the center o... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,264 |
8.45. Construct a rhombus, two sides of which lie on two given parallel lines, and the other two pass through two given points. | 8.45. Let the distance between the given parallel lines be $a$. We need to draw parallel lines through points $A$ and $B$ such that the distance between them is $a$. For this, we construct a circle with segment $AB$ as its diameter and find the points $C_{1}$ and $C_{2}$ where this circle intersects with the circle of ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,265 |
8.46. Construct a quadrilateral $A B C D$ given four sides and the angle between $A B$ and $C D$. | 8.46. Suppose that quadrilateral $A B C D$ is constructed. Denote the midpoints of sides $A B, B C, C D$ and $D A$ by $P, Q, R$ and $S$ respectively, and the midpoints of diagonals $A C$ and $B D$ by $K$ and $L$. In triangle $K S L$, it is known that $K S = C D / 2$, $L S = A B / 2$, and angle $K S L$, which is equal t... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,266 |
8.50*. Given vertices $A$ and $C$ of an isosceles circumscribed trapezoid $A B C D(A D \| B C)$; the directions of its bases are also known. Construct vertices $B$ and $D$.
保留源文本的换行和格式，直接输出翻译结果。 | 8.50. Let $A B C D$ be an inscribed isosceles trapezoid with bases $A D$ and $B C$, where $A D > B C$; $C_{1}$ is the projection of point $C$ onto the line $A D$. We will prove that $A B = A C_{1}$. Indeed, if $P$ and $Q$ are the points of tangency of the sides $A B$ and $A D$ with the inscribed circle, then $A B = A P... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,270 |
8.52*. Construct a convex quadrilateral if the lengths of all its sides and one of its midlines are given (a midline of a quadrilateral is a segment connecting the midpoints of opposite sides). | 8.52. Suppose we have constructed a quadrilateral $A B C D$ with given side lengths and a given midline $K P$ (where $K$ and $P$ are the midpoints of sides $A B$ and $C D$). Let $A_{1}$ and $B_{1}$ be the points symmetric to points $A$ and $B$ with respect to point $P$. The triangle $A_{1} B C$ can be constructed becau... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,272 |
8.53*. Construct a cyclic quadrilateral from four sides (Brahmagupta).
See also problems $15.11,15.14,16.17,17.4,17.5$.
## §8. Circles
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 8.53. Using the formulas from problems 6.37 and 6.38, it is easy to express the diagonals of a cyclic quadrilateral in terms of its sides. The obtained formulas can be used to construct the diagonals (for convenience, an arbitrary segment $e$ can be introduced as a unit length segment, and segments of length $p q, p / ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,273 |
8.54. Inside an angle, two points $A$ and $B$ are given. Construct a circle passing through these points and intercepting equal segments on the sides of the angle. | 8.54. A circle cuts off equal segments on the sides of an angle if and only if its center lies on the bisector of the angle. Therefore, the center of the desired circle is the point of intersection of the perpendicular bisector of the segment $A B$ and the bisector of the given angle. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,274 |
8.55. Given a circle $S$, a point $A$ on it, and a line $l$. Construct a circle that is tangent to the given circle at point $A$ and to the given line. | 8.55. Suppose we have constructed a circle $S^{\prime}$ that is tangent to a given circle $S$ at point $A$ and to a given line $l$ at some point $B$. Let $O$ and $O^{\prime}$ be the centers of circles $S$ and $S^{\prime}$, respectively (Fig. 8.6). It is clear that points $O, O^{\prime}$, and $A$ lie on the same line an... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,275 |
8.56. a) Given two points $A, B$ and a line $l$. Construct a circle passing through points $A, B$ and tangent to the line $l$.
b) Given two points $A$ and $B$ and a circle $S$. Construct a circle passing through points $A$ and $B$ and tangent to the circle $S$. | 8.56. a) Let $l_{1}$ be the perpendicular bisector of the segment $A B$, $C$ be the point of intersection of the lines $l_{1}$ and $l$, and $l^{\prime}$ be the line symmetric to $l$ with respect to the line $l_{1}$. The problem reduces to constructing a circle passing through point $A$ and tangent to the lines $l$ and ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,276 |
8.58*. Construct a circle such that the tangents drawn from three given points \(A, B\) and \(C\) have lengths \(a, b\) and \(c\) respectively.
See also problems 15.9, 15.10, 15.12, 15.13 a), 16.13, 16.14, 16.18-16.20, 18.26.
## §9. Apollonian Circle | 8.58. Suppose we have constructed a circle $S$, and the tangents $A A_{1}, B B_{1}$, and $C C_{1}$ to it have lengths $a, b$, and $c$ respectively (where $A_{1}, B_{1}$, and $C_{1}$ are the points of tangency). We construct circles $S_{a}, S_{b}$, and $S_{c}$ with centers $A, B$, and $C$ and radii $a, b$, and $c$ respe... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,278 |
8.59. Construct a triangle given $a, h_{a}$ and $b / c$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 8.59. First, construct the segment $B C$ of length $a$. Then construct the locus of points $X$ such that $C X: B X=b: c$ (see problem 7.14). As vertex $A$, we can take any of the points of intersection of this locus with the line that is at a distance $h_{a}$ from the line $B C$. | Other | math-word-problem | Yes | Yes | olympiads | false | 27,279 | |
8.60. Construct triangle $ABC$, if the length of the bisector $CD$ and the lengths of the segments $AD$ and $BD$, into which it divides side $AB$, are known. | 8.60. Given the lengths of segments $A D^{\prime}$ and $B D$, one can construct segment $A B$ and point $D$ on this segment. Point $C$ is the intersection of the circle of radius $C D$ centered at $D$ and the locus of points $X$ such that $A X: B X = A D: B D$. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,280 |
8.61*. On a line, four points $A, B, C, D$ are given in the specified order. Construct a point $M$ from which the segments $A B, B C, C D$ are seen at equal angles. | 8.61. Let $X$ be a point not lying on the line $A B$. It is clear that $\angle A X B = \angle B C X$ if and only if $A X: C X = A B: C B$. Therefore, the point $M$ is the intersection of the locus of points $X$ for which $A X: C X = A B: C B$, and the locus of points $Y$ for which $B Y: D Y = B C: D C$ (these loci may ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,281 |
8.64. a) On parallel lines $a$ and $b$, points $A$ and $B$ are given. Draw a line $l$ through the given point $C$ that intersects lines $a$ and $b$ at points $A_{1}$ and $B_{1}$ such that $A A_{1}=B B_{1}$.
b) Draw a line through point $C$ that is equidistant from the given points $A$ and $B$. | 8.64. a) If the line $l$ does not intersect the segment $A B$, then $A B B_{1} A_{1}$ is a parallelogram and $l \| A B$. If the line $l$ intersects the segment $A B$, then $A A_{1} B B_{1}$ is a parallelogram and $l$ passes through the midpoint of the segment $A B$.
b) One of the desired lines is parallel to the line ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,284 |
8.67*. Given the diameter $A B$ of a circle and a point $C$ on it. Construct points $X$ and $Y$ on this circle, symmetric with respect to the line $A B$, such that the lines $A X$ and $Y C$ are perpendicular.
See also problems $15.8,16.15,16.16,16.21,17.9-17.11,17.27-17.29,18.43$.
## §11. Unusual Constructions | 8.67. Suppose that points $X$ and $Y$, possessing the required properties, are constructed. Denote the intersection point of lines $A X$ and $Y C$ as $M$, and the intersection point of lines $A B$ and $X Y$ as $K$. Right triangles $A X K$ and $Y X M$ have a common acute angle $X$, so $\angle X A K = \angle X Y M$. Angl... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,286 |
8.68. Using a compass and a ruler, divide an angle of $19^{\circ}$ into 19 equal parts. | 8.68. If there is an angle of magnitude $\alpha$, then angles of magnitude $2 \alpha, 3 \alpha$, etc., can be constructed. Since $19 \cdot 19^{\circ}=361$, an angle of $361^{\circ}$, which coincides with an angle of $1^{\circ}$, can be constructed. | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,287 |
8.69. Prove that an angle of $n^{\circ}$, where $n$ is an integer not divisible by 3, can be divided into $n$ equal parts using a compass and a straightedge. | 8.69. First, let's construct an angle of $36^{\circ}$ (see problem 8.65). Then we can construct the angle $\left(36^{\circ}-30^{\circ}\right) / 2=3^{\circ}$. If $n$ is not divisible by 3, then having angles $n^{\circ}$ and $3^{\circ}$, we can construct an angle of $1^{\circ}$. Indeed, if $n=3 k+1$, then $1^{\circ}=n^{\... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,288 |
8.71*. Using a two-sided ruler, construct the center of the given circle, the diameter of which is larger than the width of the ruler.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 8.71. Using a two-sided ruler, construct two parallel chords $AB$ and $CD$. Let $P$ and $Q$ be the points of intersection of the lines $AC$ and $BD$, and $AD$ and $BC$. Then the line $PQ$ passes through the center of the given circle. By constructing another such line in the same way, we can find the center of the circ... | Other | math-word-problem | Yes | Yes | olympiads | false | 27,290 | |
8.75*. Given two parallel lines and a segment lying on one of them. Double this segment.
保留源文本的换行和格式，所以翻译结果如下：
8.75*. Given two parallel lines and a segment lying on one of them. Double this segment. | 8.75. Let $AB$ be a given segment, and $C$ and $D$ be arbitrary points on a second given line. According to the previous problem, we can construct the midpoint $M$ of segment $CD$. Let $P$ be the intersection point of lines $AM$ and $BD$, and $E$ be the intersection point of lines $PC$ and $AB$. We will prove that $EB$... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,294 |
8.79*. Prove that if on a plane there is a circle $S$ and its center $O$, then with the help of a single ruler one can:
a) from any point draw a line parallel to a given line and drop a perpendicular to the given line;
b) on a given line from a given point lay off a segment equal to a given segment;
c) construct a s... | 8.79. a) Let $A$ be a given point, and $l$ a given line. First, consider the case where point $O$ does not lie on line $l$. Draw two arbitrary lines through point $O$ that intersect line $l$ at points $B$ and $C$. According to problem 8.78, in triangle $OBC$ we can drop perpendiculars to sides $OB$ and $OC$. Let $H$ be... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,297 |
8.80. a) Construct the bisector of the given angle $A O B$.
b) Given an acute angle $A O B$. Construct angle $B O C$, for which the ray $O A$ is the bisector. | 8.80. a) Draw lines parallel to lines $O A$ and $O B$, at a distance $a$ from them, and intersecting the sides of the angle. The point of intersection of these lines lies on the required bisector.
b) Draw a line parallel to $O B$, at a distance $a$ from it, and intersecting the ray $O A$ at some point $M$. Through poi... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,298 |
8.81. Construct a perpendicular to the given line $l$ at the given point $A$. | 8.81. Draw an arbitrary line through point $A$, and then draw lines $l_{1}$ and $l_{2}$, parallel to it and at a distance $a$ from it; these lines intersect line $l$ at points $M_{1}$ and $M_{2}$. Draw another pair of parallel lines $l_{a}$ and $l_{m}$ through points $A$ and $M_{1}$, with the distance between them bein... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,299 |
8.84. Given a segment $A B$, a straight line $l$ not parallel to it, and a point $M$ on it. Construct the points of intersection of the line $l$ with the circle of radius $A B$ centered at $M$. | 8.84. Complete the triangle $A B M$ to a parallelogram $A B M N$. Draw lines through point $N$ parallel to the bisectors of the angles between lines $l$ and $M N$. The points of intersection of these lines with line $l$ are the desired points. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,302 |
8.85*. A line $l$ and a segment $O A$, parallel to $l$, are given. Construct the points of intersection of the line $l$ with the circle of radius $O A$ centered at $O$.
$\mathbf{8 . 8 6}{ }^{*}$. Segments $O_{1} A_{1}$ and $O_{2} A_{2}$ are given. Construct the radical axis of the circles of radii $O_{1} A_{1}$ and $O... | 8.85. Draw a line $l_{1}$ parallel to the line $OA$ and at a distance $a$ from it. Take an arbitrary point $B$ on the line $l$. Let $B_{1}$ be the point of intersection of the lines $OB$ and $l_{1}$. Draw a line through the point $B_{1}$ parallel to $AB$; this line intersects the line $OA$ at the point $A_{1}$. Now dra... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,303 |
8.87. Draw a line through the given point $A$ parallel to the given line $l$. | 8.87. First, let's construct an arbitrary line $l_{1}$ perpendicular to line $l$, and then through point $A$ we will draw a line perpendicular to line $l_{1}$. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,304 |
8.88. Given a segment $A B$. Construct:
a) the midpoint of segment $A B$
b) segment $A C$, the midpoint of which is point $B$. | 8.88. a) Draw lines $A P$ and $B Q$ through points $A$ and $B$, perpendicular to line $A B$, and then draw an arbitrary perpendicular to line $A P$. As a result, we obtain a rectangle. It remains to drop a perpendicular from the intersection of its diagonals onto line $A B$.
b) Construct a perpendicular $l$ from point... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,305 |
8.89. Given angle $A O B$. Construct:
a) an angle twice as large as angle $A O B$;
b) an angle half as large as angle $A O B$. | 8.89. a) Drop a perpendicular $A P$ from point $A$ to line $O B$ and construct segment $A C$, with point $P$ being its midpoint. Then angle $A O C$ is the required one.
b) Take points $B$ and $B_{1}$ on line $O B$ such that $O B=O B_{1}$. Place a right angle so that its sides pass through points $B$ and $B_{1}$, and i... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,306 |
8.90*. Given angle $A O B$ and line $l$. Draw line $l_{1}$ such that the angle between lines $l$ and $l_{1}$ is equal to angle $A O B$. | 8.90. Draw a line $l^{\prime}$ through point $O$, parallel to line $l$. Drop perpendiculars $B P$ and $B Q$ from point $B$ to lines $l^{\prime}$ and $O A$, and then drop a perpendicular $O X$ from point $O$ to line $P Q$. Then line $X O$ is the desired one (see problem 2.3); if point $Y$ is symmetric to point $X$ with ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,307 |
8.91*. Given a segment $A B$, a line $l$ and a point $O$ on it. Construct a point $X$ on the line $l$ such that $O X=A B$.
Construct the point $X$ on the line $l$ so that the distance $O X$ is equal to the length of the segment $A B$. | 8.91. Complete the triangle $O A B$ to form a parallelogram $O A B C$, and then construct the segment $C C_{1}$, with point $O$ being the midpoint. Place a right angle so that its sides pass through points $C$ and $C_{1}$, and its vertex lies on the line $l$. The vertex of the right angle then coincides with the desire... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,308 |
9.1. Prove that $(a+b-c) / 2 < m_{c} < (a+b) / 2$. | 9.1. Let $C_{1}$ be the midpoint of side $A B$. Then $C C_{1} + C_{1} A > C A$ and $B C_{1} + C_{1} C > B C$. Therefore, $2 C C_{1} + B A > C A + B C$, i.e., $m_{c} > (a + b - c) / 2$.
Let point $C^{\prime}$ be symmetric to $C$ with respect to point $C_{1}$. Then $C C_{1} = C_{1} C^{\prime}$ and $B C^{\prime} = C A$. ... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 27,310 |
9.2. Prove that in any triangle the sum of the medians is greater than $3 / 4$ of the perimeter, but less than the perimeter. | 9.2. From the previous problem, we have $m_{a}B A, A O+O C>A C$ and $C O+O B>C B$. Adding these inequalities and considering that $A O=2 m_{a} / 3, B O=2 m_{b} / 3, C O=2 m_{c} / 3$, we get $m_{a}+m_{b}+$ $+m_{c}>3(a+b+c) / 4$. | m_{}+m_{b}+m_{}>3(+b+)/4 | Inequalities | proof | Yes | Yes | olympiads | false | 27,311 |
9.3. Given $n$ points $A_{1}, \ldots, A_{n}$ and a circle of radius 1. Prove that there exists a point $M$ on the circle such that $M A_{1}+\ldots+M A_{n} \geqslant n$.
Given $n$ points $A_{1}, \ldots, A_{n}$ and a circle of radius 1. Prove that there exists a point $M$ on the circle such that $M A_{1}+\ldots+M A_{n} ... | 9.3. Let $M_{1}$ and $M_{2}$ be diametrically opposite points on a circle. Then $M_{1} A_{k} + M_{2} A_{k} \geqslant M_{1} M_{2} = 2$. Adding these inequalities for $k=1, \ldots, n$, we get $\left(M_{1} A_{1} + \ldots + M_{1} A_{n}\right) + \left(M_{2} A_{1} + \ldots + M_{2} A_{n}\right) \geqslant 2 n$. Therefore, eith... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,312 |
9.4. Points $A_{1}, \ldots, A_{n}$ do not lie on the same line. Let two different points $P$ and $Q$ have the property that $A_{1} P+\ldots+A_{n} P=A_{1} Q+\ldots+$ $+A_{n} Q=s$. Prove that then $A_{1} K+\ldots+A_{n} K<s$ for some point $K$. | 9.4. We can take $K$ to be the midpoint of segment $P Q$. Indeed, in this case, $A_{i} K \leqslant\left(A_{i} P+A_{i} Q\right) / 2$ (see problem 9.1), and at least one of the inequalities is strict, since the points $A_{i}$ cannot all lie on the line $P Q$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,313 |
9.5*. On the table, there are 50 correctly running clocks. Prove that at some moment, the sum of the distances from the center of the table to the ends of the minute hands will be greater than the sum of the distances from the center of the table to the centers of the clocks.
## §2. Algebraic Problems Involving the Tr... | 9.5. Let $A_{i}$ and $B_{i}$ be the positions of the minute hand ends of the $i$-th clock at moments $t$ and $t+30$ min, $O_{i}$ be the center of the $i$-th clock, and $O$ be the center of the table. Then $O O_{i} \leqslant\left(O A_{i}+O B_{i}\right) / 2$ for any $i$ (see problem 9.1). It is clear that at some moment,... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,314 |
9.8. For any natural number $n$, the numbers $a^{n}, b^{n}$ and $c^{n}$ can form a triangle. Prove that among the numbers $a, b$ and $c$ there are two equal. | 9.8. We can assume that $a \geqslant b \geqslant c$. Let's prove that $a=b$. Indeed, if $b<a$, then $b \leqslant \lambda a$ and $c \leqslant \lambda a$, where $\lambda<1$. Therefore, $b^{n}+c^{n} \leqslant 2 \lambda^{n} a^{n}$. For sufficiently large $n$, we have $2 \lambda^{n}<1$ and we obtain a contradiction with the... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 27,317 |
9.9. Prove that
$$
a(b-c)^{2}+b(c-a)^{2}+c(a-b)^{2}+4 a b c>a^{3}+b^{3}+c^{3}
$$ | 9.9. Since $c(a-b)^{2}+4 a b c=c(a+b)^{2}$, then $a(b-c)^{2}+b(c-a)^{2}+c(a-b)^{2}+$ $+4 a b c-a^{3}-b^{3}-c^{3}=a\left((b-c)^{2}-a^{2}\right)+b\left((c-a)^{2}-b^{2}\right)+c\left((a+b)^{2}-c^{2}\right)=(a+b-c)(a-b+$ $+c)(-a+b+c)$. (The last equality is verified by simple calculation.) All three factors are positive du... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 27,318 |
9.10. Let \( p = \frac{a}{b} + \frac{b}{c} + \frac{c}{a} \) and \( q = \frac{a}{c} + \frac{c}{b} + \frac{b}{a} \). Prove that \( |p - q| < 1 \). | 9.10. It is easy to check that $a b c|p-q|=|(b-c)(c-a)(a-b)|$. Since $|b-c|<a,|c-a|<b$ and $|a-b|<c$, then $|(b-c)(c-a)(a-b)|<a b c$. | proof | Algebra | proof | Yes | Yes | olympiads | false | 27,319 |
9.11*. Five segments are such that any three of them can form a triangle. Prove that at least one of these triangles is acute. | 9.11. Let's denote the lengths of the segments such that $a_{1} \leqslant a_{2} \leqslant a_{3} \leqslant a_{4} \leqslant a_{5}$. If all triangles that can be formed from these segments are not acute-angled, then $a_{3}^{2} \geqslant a_{1}^{2}+a_{2}^{2}, a_{4}^{2} \geqslant a_{2}^{2}+a_{3}^{2}$ and $a_{5}^{2} \geqslant... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,320 |
9.12*. Prove that
$$
(a+b-c)(a-b+c)(-a+b+c) \leqslant a b c
$$ | 9.12. First solution. Introduce new variables $x=-a+b+c, y=$ $=a-b+c, z=a+b-c$. Then $a=(y+z) / 2, b=(x+z) / 2, c=(x+y) / 2$, i.e., we need to prove the inequality $x y z \leqslant(x+y)(y+z)(x+z) / 8$ or $6 x y z \leqslant x\left(y^{2}+z^{2}\right)+$ $+y\left(x^{2}+z^{2}\right)+z\left(x^{2}+y^{2}\right)$. The latter in... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 27,321 |
9.13*. Prove that
$$
a^{2} b(a-b)+b^{2} c(b-c)+c^{2} a(c-a) \geqslant 0
$$
## §3. The sum of the lengths of the diagonals of a quadrilateral
## Translation of the text into English, preserving the original text's line breaks and format, is provided as requested. | 9.13. Let's introduce new variables $x=(-a+b+c) / 2, y=(a-b+c) / 2$ and $z=$ $=(a+b-c) / 2$. Then the numbers $x, y, z$ are positive and $a=y+z, b=x+z, c=x+y$. Simple but somewhat cumbersome calculations show that $a^{2} b(a-b)+$ $+b^{2} c(b-c)+c^{2} a(c-a)=2\left(x^{3} z+y^{3} x+z^{3} y-x y z(x+y+z)\right)=2 x y z \fr... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 27,322 |
9.14. Let $A B C D$ be a convex quadrilateral. Prove that $A B + C D < A C + B D$. | 9.14. Let $O$ be the point of intersection of the diagonals of quadrilateral $ABCD$. Then $AC + BD = (AO + OC) + (BO + OD) = (AO + OB) + (OC + OD) > AB + CD$. | proof | Inequalities | proof | Yes | Yes | olympiads | false | 27,323 |
9.15. Let $A B C D$ be a convex quadrilateral such that $A B + B D \leqslant A C + C D$. Prove that $A B < A C$. | 9.15. According to the previous problem, $A B+C D<A C+B D$. Adding this inequality to the inequality $A B+B D \leqslant A C+C D$, we get $2 A B<2 A C$. | AB<AC | Geometry | proof | Yes | Yes | olympiads | false | 27,324 |
9.16*. Inside a convex quadrilateral with the sum of the lengths of the diagonals $d$, there is a convex quadrilateral with the sum of the lengths of the diagonals $d^{\prime}$. Prove that $d^{\prime}<2 d$. | 9.16. Let's first prove that if $P$ is the perimeter of a convex quadrilateral $ABCD$, and $d_{1}$ and $d_{2}$ are the lengths of its diagonals, then $P > d_{1} + d_{2} > P / 2$. It is clear that $A CP / 2$, and since $P' < P$ (Problem 9.27, b), then $d' < P' < P < 2d$. | '<2d | Geometry | proof | Yes | Yes | olympiads | false | 27,325 |
9.17*. Given a closed broken line, and any other closed broken line with the same vertices has a greater length. Prove that this broken line is non-self-intersecting. | 9.17. Let the broken line be the shortest self-intersecting one. Consider two intersecting segments. The vertices of these segments can be connected in one of three ways (see Fig. 9.2). Consider a new broken line where the two intersecting segments are replaced by dashed segments (see Fig. 9.2). In this case, a closed ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,326 |
9.18*. How many sides can a convex polygon have if all its diagonals have the same length? | 9.18. We will prove that the number of sides of such a polygon does not exceed 5. Suppose that all diagonals of the polygon $A_{1} \ldots A_{n}$ have the same length and $n \geqslant 6$. Then the segments $A_{1} A_{4}, A_{1} A_{5}, A_{2} A_{4}$, and $A_{2} A_{5}$ have the same length, as they are diagonals of this poly... | 5 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,327 |
9.19*. On a plane, there are $n$ red and $n$ blue points, no three of which lie on the same line. Prove that it is possible to draw $n$ segments with endpoints of different colors, having no common points. | 9.19. Consider all partitions of these points into
Fig. 9.3 pairs of points of different colors. The number of these partitions is finite, so there exists a partition for which the sum of ... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 27,328 |
9.20*. Prove that the arithmetic mean of the lengths of the sides of an arbitrary convex polygon is less than the arithmetic mean of the lengths of all its diagonals. | 9.20. Let $A_{p} A_{p+1}$ and $A_{q} A_{q+1}$ be non-adjacent sides of an $n$-gon $A_{1} \ldots A_{n}$ (i.e., $|p-q| \geqslant 2$). Then $A_{p} A_{p+1} + A_{q} A_{q+1} < A_{p} A_{q} + A_{p+1} A_{q+1}$. Write down all such inequalities and add them. For each side, there are exactly $n-3$ non-adjacent sides, so each side... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 27,329 |
9.21*. Let a convex $(2 n+1)$-gon $A_{1} A_{3} A_{5} \ldots A_{2 n+1} A_{2} \ldots A_{2 n}$ be given. Prove that among all closed broken lines with vertices at its vertices, the one with the greatest length is the broken line $A_{1} A_{2} A_{3} \ldots A_{2 n+1} A_{1}$.
## §4. Various Problems on the Triangle Inequalit... | 9.21. Consider an arbitrary closed broken line with vertices at the vertices of a given polygon. If it has two non-intersecting segments, then by replacing these segments with the diagonals of the quadrilateral they define, we increase the sum of the lengths of the segments; however, one closed broken line may split in... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,330 |
9.22. In a triangle, the lengths of two sides are 3.14 and 0.67. Find the length of the third side, given that it is an integer. | 9.22. Let the length of the third side be $n$. By the triangle inequality, $3.14-0.67<n<3.14+$ $+0.67$. Since $n$ is an integer, then $n=3$. | 3 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,331 |
9.23. Prove that the sum of the lengths of the diagonals of a convex pentagon $A B C D E$ is greater than the perimeter but less than twice the perimeter. | 9.23. It is clear that \(AB + BC > AC\), \(BC + CD > BD\), \(CD + DE > CE\), \(DE + EA > DA\), \(EA + AB > EB\). Adding all these inequalities, we get that the sum of the lengths of the diagonals of the pentagon is less than twice the perimeter.
^{2} \leqslant 4 c^{2}$. Therefore, $a^{2}+b^{2}<5 c^{2}$. A contradiction is obtained. | proof | Inequalities | proof | Yes | Yes | olympiads | false | 27,333 |
9.25. Two altitudes of a triangle are 12 and 20. Prove that the third altitude is less than 30. | 9.25. Since $c>|b-a|$ and $a=2 S / h_{a}, b=2 S / h_{b}, c=2 S / h_{c}$, then $\frac{1}{h_{c}}>\frac{1}{h_{a}}-\frac{1}{h_{b}}$. Therefore, in our case $h_{c}<20 \cdot 12 / 8=30$. | h_{}<30 | Geometry | proof | Yes | Yes | olympiads | false | 27,334 |
9.26*. Points $C_{1}, A_{1}, B_{1}$ are taken on the sides $A B, B C, C A$ of triangle $A B C$ such that $B A_{1}=\lambda \cdot B C, C B_{1}=\lambda \cdot C A, A C_{1}=\lambda \cdot A B$, where $1 / 2<\lambda<1$. Prove that the perimeter $P$ of triangle $A B C$ and the perimeter $P_{1}$ of triangle $A_{1} B_{1} C_{1}$ ... | 9.26. Let's take points \(C_{2}, A_{2}, B_{2}\) on the sides \(A B, B C, C A\) such that \(A_{1} B_{2} \| A B\), \(B_{1} C_{2} \| B C\), \(C_{1} A_{2} \| C A\) (Fig. 9.6). Then \(A_{1} B_{1} B_{1} C\), i.e., \(A_{1} B_{1} + (1-\lambda) B C > \lambda \cdot C A\). Similarly, \(B_{1} C_{1} + (1-\lambda) C A > \lambda \cdo... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,335 |
9.27*. a) Prove that when transitioning from a non-convex polygon to its convex hull, the perimeter decreases. (The convex hull of a polygon is the smallest convex polygon containing it.)
b) Inside a convex polygon, there lies another convex polygon. Prove that the perimeter of the outer polygon is not less than the p... | 9.27. a) When transitioning from a non-convex polygon to its convex hull, some broken lines formed by the sides are replaced by straight-line segments (Fig. 9.7). It remains to note that the length of the broken line is greater than the length of
(A D+$ $+B C) / 4$ | 9.31. Since $E H$ is the midline of triangle $A B D$, then $S_{A E H}=S_{A B D} / 4$. Similarly, $S_{C F G}=S_{C B D} / 4$. Therefore, $S_{A E H}+S_{C F G}=S_{A B C D} / 4$. Similarly, $S_{B F E}+S_{D G H}=S_{A B C D} / 4$. Hence, $S_{A B C D}=2 S_{E F G H}=E G \cdot H F \sin \alpha$, where $\alpha$ is the angle betwee... | S_{ABCD}\leqslantEG\cdotHF\leqslant(AB+CD)(BC+AD)/4 | Geometry | proof | Yes | Yes | olympiads | false | 27,340 |
9.32. The perimeter of a convex quadrilateral is 4. Prove that its area does not exceed 1. | 9.32. According to problem $9.31 S_{A B C D} \leqslant(A B+C D)(B C+A D) / 4$. Since $a b \leqslant(a+$ $+b)^{2} / 4$, then $S_{A B C D} \leqslant(A B+C D+A D+B C)^{2} / 16=1$. | 1 | Geometry | proof | Yes | Yes | olympiads | false | 27,341 |
9.33. Inside triangle $ABC$, a point $M$ is taken. Prove that $4 S \leqslant AM \cdot BC + BM \cdot AC + CM \cdot AB$, where $S$ is the area of triangle $ABC$. | 9.33. Drop perpendiculars $B B_{1}$ and $C C_{1}$ from points $B$ and $C$ to the line $A M$. Then $2 S_{A M B}+2 S_{A M C}=A M \cdot B B_{1}+A M \cdot C C_{1} \leqslant A M \cdot B C$, since $B B_{1}+C C_{1} \leqslant B C$. Similarly, $2 S_{B M C}+2 S_{B M A} \leqslant B M \cdot A C$ and $2 S_{C M A}+2 S_{C M B} \leqsl... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 27,342 |
9.34*. A polygon of area $S$, containing the center of a circle of radius $R$, is inscribed in the circle, and a point is chosen on each of its sides. Prove that the perimeter of the convex polygon with vertices at the chosen points is not less than $2 S / R$.
保留源文本的换行和格式，直接输出翻译结果。 | 9.34. Let points $B_{1}, \ldots, B_{n}$ be chosen on the sides $A_{1} A_{2}, A_{2} A_{3}, \ldots, A_{n} A_{1}$; $O$ is the center of the circle. Let further $S_{k}=S_{O B_{k} A_{k+1} B_{k+1}}=$ $=\left(O A_{k+1} \cdot B_{k} B_{k+1} \sin \varphi\right) / 2$, where $\varphi$ is the angle between $O A_{k+1}$ and $B_{k} B_... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,343 |
9.35*. Inside a convex quadrilateral $ABCD$ with area $S$, a point $O$ is taken such that $AO^2 + BO^2 + CO^2 + DO^2 = 2S$. Prove that then $ABCD$ is a square and $O$ is its center.
## §6. Inequalities with Areas | 9.35. We have $2 S_{A O B} \leqslant A O \cdot O B \leqslant\left(A O^{2}+B O^{2}\right) / 2$, with equality possible only if $\angle A O B=90^{\circ}$ and $A O=B O$. Similarly, $2 S_{B O C} \leqslant\left(B O^{2}+\right.$ $\left.+C O^{2}\right) / 2, 2 S_{C O D} \leqslant\left(C O^{2}+D O^{2}\right) / 2$ and $2 S_{D O ... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 27,344 |
9.36. Points $M$ and $N$ lie on sides $AB$ and $AC$ of triangle $ABC$, respectively, such that $AM = CN$ and $AN = BM$. Prove that the area of quadrilateral $BMNC$ is at least three times the area of triangle $AMN$.
保留源文本的换行和格式，直接输出翻译结果。 | 9.36. It is necessary to prove that $S_{A B C} / S_{A M N} \geqslant 4$. Since $A B=A M+M B=$ $=A M+A N=A N+N C=A C$, then
$$
\frac{S_{A B C}}{S_{A M N}}=\frac{A B \cdot A C}{A M \cdot A N}=\frac{(A M+A N)^{2}}{A M \cdot A N} \geqslant 4
$$ | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,345 |
9.37. The areas of triangles $A B C, A_{1} B_{1} C_{1}, A_{2} B_{2} C_{2}$ are $S, S_{1}, S_{2}$ respectively, and $A B=A_{1} B_{1}+A_{2} B_{2}, A C=A_{1} C_{1}+A_{2} C_{2}, B C=$ $=B_{1} C_{1}+B_{2} C_{2}$. Prove that $S \leqslant 4 \sqrt{S_{1} S_{2}}$. | 9.37. Let's use Heron's formula: $S^{2}=p(p-a)(p-b)(p-c)$. Since $p-a=$ $=\left(p_{1}-a_{1}\right)+\left(p_{2}-a_{2}\right)$, and $(x+y)^{2} \geqslant 4 x y$, then $(p-a)^{2} \geqslant 4\left(p_{1}-a_{1}\right)\left(p_{2}-a_{2}\right)$. Similarly, $(p-b)^{2} \geqslant 4\left(p_{1}-b_{1}\right)\left(p_{2}-b_{2}\right),(... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 27,346 |
9.38. $A B C D$ is a convex quadrilateral with area $S$. The angle between the lines $A B$ and $C D$ is $\alpha$, and the angle between $A D$ and $B C$ is $\beta$. Prove that
$$
A B \cdot C D \sin \alpha + A D \cdot B C \sin \beta \leqslant 2 S \leqslant A B \cdot C D + A D \cdot B C
$$ | 9.38. For definiteness, we can assume that the rays $BA$ and $CD$, $BC$ and $AD$ intersect (Fig. 9.9). Then, if we complete the triangle $ADC$ to a parallelogram $ADCK$, the point $K$ will be inside the quadrilateral $ABCD$. Therefore, $2S \geqslant 2S_{ABK} + 2S_{BCK} = AB \cdot AK \sin \alpha + BC \cdot CK \sin \beta... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 27,347 |
9.39. Through a point lying inside a triangle, three lines parallel to its sides are drawn. Let the areas of the parts into which these lines divide the triangle be denoted as shown in Fig. 9.1. Prove that $a / \alpha+b / \beta+c / \gamma \geqslant 3 / 2$. | 9.39. According to the inequality between the geometric mean and the arithmetic mean $\frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma} \geqslant 3 \sqrt[3]{a b c /(\alpha \beta \gamma)}=3 / 2$, since $\alpha=2 \sqrt{b c}, \beta=2 \sqrt{c a}$ and $\gamma=2 \sqrt{a b}$ (see problem 1.34 ). | \frac{}{\alpha}+\frac{b}{\beta}+\frac{}{\gamma}\geqslant\frac{3}{2} | Inequalities | proof | Yes | Yes | olympiads | false | 27,348 |
9.40. The areas of triangles $A B C$ and $A_{1} B_{1} C_{1}$ are $S$ and $S_{1}$, respectively, and triangle $A B C$ is not obtuse. The greatest of the ratios $a_{1} / a, b_{1} / b$ and $c_{1} / c$ is $k$. Prove,
. In addition to the original polygon, some quadrilaterals will remain, from which a polygon circumscribed around a circle of radius $h$ can be formed. The sum of the areas of these quadrilaterals is greater th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,351 |
9.44*. Prove that the sum of the areas of five triangles formed by pairs of adjacent sides and the corresponding diagonals of a convex pentagon is greater than the area of the entire pentagon. | 9.44. Let, for definiteness, $A B C$ be the triangle of the smallest area. Denote the intersection point of diagonals $A D$ and $E C$ by $F$. Then $S_{A B C D E} < S_{A E D} + S_{E D C} + S_{A B C F}$. Since point $F$ lies on segment $E C$ and $S_{E A B} \geqslant S_{C A B}$, then $S_{E A B} \geqslant S_{F A B}$. Simil... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,352 |
9.45*. a) Prove that in any convex hexagon with area $S$, there exists a diagonal that cuts off a triangle with area no more than $S / 6$.
b) Prove that in any convex octagon with area $S$, there exists a diagonal that cuts off a triangle with area no more than $S / 8$.
See also problem 17.19.
## §7. Area. One figur... | 9.45. a) Let the points of intersection of the diagonals $A D$ and $C F$, $C F$ and $B E$, $B E$ and $A D$ be denoted by $P, Q, R$ respectively (Fig. 9.11). The quadrilaterals $A B C P$ and $C D E Q$ do not have common internal points, since the sides $C P$ and $Q C$ lie on the line $C F$, and the segments $A B$ and $D... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,353 |
9.46. A convex polygon with an area greater than 0.5 is placed inside a square with a side length of 1. Prove that a segment of length 0.5, parallel to the side of the square, can be placed inside the polygon. | 9.46. Let's draw lines through all the vertices of the polygon, parallel to one pair of sides of the square, and thereby divide the square into strips. Each such strip cuts off a trapezoid or a triangle from the polygon. It is sufficient to prove that the length of one of the bases of these trapezoids is greater than 0... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,354 |
9.47*. Inside a square with side 1, there are $n$ points. Prove that:
a) the area of one of the triangles with vertices at these points or at the vertices of the square does not exceed $1 /(2(n+1))$;
b) the area of one of the triangles with vertices at these points does not exceed $1 /(n-2)$. | 9.47. a) Let $P_{1}, \ldots, P_{n}$ be given points. Connect point $P_{1}$ with the vertices of the square. This will result in four triangles. Then for $k=2, \ldots, n$, perform the following operation. If point $P_{k}$ lies strictly inside one of the previously obtained triangles, connect it with the vertices of this... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,355 |
9.48*. a) In a circle of area $S$, a regular $n$-sided polygon of area $S_{1}$ is inscribed, and a regular $n$-sided polygon of area $S_{2}$ is circumscribed around this circle. Prove that $S^{2}>S_{1} S_{2}$.
b) In a circle with circumference $L$, a regular $n$-sided polygon with perimeter $P_{1}$ is inscribed, and a... | 9.48. a) It can be assumed that the described $n$-gon $A_{1} \ldots A_{n}$ and the inscribed $n$-gon $B_{1} \ldots B_{n}$ are arranged such that the lines $A_{i} B_{i}$ intersect at the center $O$ of the given circle. Let $C_{i}$ and $D_{i}$ be the midpoints of the sides $A_{i} A_{i+1}$ and $B_{i} B_{i+1}$. Then $S_{O ... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 27,356 |
9.49*. A polygon of area $B$ is inscribed in a circle of area $A$ and circumscribed around a circle of area $C$. Prove that $2 B \leqslant A+C$. | 9.49. Let $O$ be the center of homothety that maps the inscribed circle to the circumscribed circle. Divide the plane by rays emanating from point $O$ and passing through the vertices of the polygon and the points of tangency of its sides with the inscribed circle (Fig. 9.13). It is sufficient to prove the required ine... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,357 |
9.52*. Prove that the area of a parallelogram lying inside a triangle does not exceed half the area of the triangle. | 9.52. Let's first consider the case when two sides of the parallelogram lie on the lines $A B$ and $A C$, and the fourth vertex $X$ lies on the side $B C$. If $B X: C X=x:(1-x)$, then the ratio of the area of the parallelogram to the area of the triangle is $2 x(1-x) \leqslant 1 / 2$.
In the general case, we draw para... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,359 |
9.53*. Prove that the area of a triangle, whose vertices lie on the sides of a parallelogram, does not exceed half the area of the parallelogram.
$$
* * *
$$ | 9.53. Let's first consider the case where two vertices $A$ and $B$ of triangle $ABC$ lie on one side $PQ$ of the parallelogram. Then $AB \leqslant PQ$ and the height dropped to side $AB$ is no greater than the height of the parallelogram. Therefore, the area of triangle $ABC$ is no more than half the area of the parall... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,360 |
9.54*. Prove that any acute triangle of area 1 can be placed in a right triangle of area $\sqrt{3}$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 9.54. Let $M$ be the midpoint of the largest side $BC$ of the given acute-angled triangle $ABC$. The circle with radius $MA$ and center $M$ intersects the rays $MB$ and $MC$ at points $B_1$ and $C_1$. Since $\angle BAC < 90^\circ$, we have $MB < MB_1$. Let us assume for definiteness that $\angle AMB \leqslant \angle AM... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,361 | |
9.55*. a) Prove that a convex polygon of area $S$ can be placed in some rectangle of area no more than $2 S$.
b) Prove that a parallelogram of area no less than $S / 2$ can be inscribed in a convex polygon of area $S$.
9.56 ${ }^{*}$. Prove that in any convex polygon of area 1, a triangle can be placed whose area is ... | 9.55. a) Let $AB$ be the largest diagonal or side of the given polygon $M$. The polygon $M$ is contained within a strip formed by the perpendiculars to the segment $AB$ passing through points $A$ and $B$. Draw two supporting lines to the polygon $M$ parallel to $AB$; let these lines intersect the polygon $M$ at points ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,362 |
9.57*. A convex $n$-gon is placed inside a square with side 1. Prove that there exist three vertices $A, B$, and $C$ of this $n$-gon such that the area of triangle $ABC$ does not exceed: a) $8 / n^{2}$; b) $16 \pi / n^{3}$.
See also problem 15.7.
## §8. Broken lines inside a square
Translate the above text into Engl... | 9.57. We will prove that there will even be three consecutive vertices that satisfy the required condition. Let $\alpha_{i}$ be the angle between the $i$-th and $(i+1)$-th sides, $\beta_{i}=\pi-\alpha_{i}$, and $a_{i}$ be the length of the $i$-th side.
a) The area of the triangle formed by the $i$-th and $(i+1)$-th si... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,363 |
9.58*. Inside a square with side 1, there is a non-self-intersecting broken line of length 1000. Prove that there is a straight line, parallel to one of the sides of the square, intersecting this broken line at least at 500 points. | 9.58. Let $l_{i}$ be the length of the $i$-th segment of the broken line, and $a_{i}$ and $b_{i}$ be the lengths of its projections on the sides of the square. Then $l_{i} \leqslant a_{i}+b_{i}$. Therefore, $1000=l_{1}+\ldots+l_{n} \leqslant\left(a_{1}+\right.$ $\left.+\ldots+a_{n}\right)+\left(b_{1}+\ldots+b_{n}\right... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,364 |
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