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23.39*. The triangulation of a polygon is its division into triangles with the property that these triangles either share a side, share a vertex, or have no common points (i.e., a vertex of one triangle cannot lie on the side of another). Prove that the triangles of the triangulation can be colored with three colors su... | 23.39. We will prove this statement by induction on the number of triangles in the triangulation. For a single triangle, the required coloring exists. Now suppose that any triangulation consisting of fewer than \( n \) triangles can be colored in the required manner, and we will prove that any triangulation consisting ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,027 |
23.41*. Several circles of the same radius were placed on a table so that no two overlap. Prove that the circles can be colored with four colors so that any two touching circles are of different colors. | 23.41. We will prove the statement by induction on the number of circles $n$. For $n=1$, the statement is obvious. Let $M$ be any point, and $O$ be the center of the circle farthest from it. Then the circle centered at $O$ touches no more than three other given circles. Remove it and color the remaining circles accordi... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 28,029 |
24.1*. Does there exist an equilateral triangle with vertices at the nodes of an integer lattice
翻译完成，保留了原文的换行和格式。 | 24.1. Suppose the vertices of an equilateral triangle $A B C$ are located at the nodes of an integer lattice. Then the tangents of all angles formed by the sides $A B$ and $A C$ with the grid lines are rational. For any position of the triangle $A B C$, the sum or difference of some two such angles $\alpha$ and $\beta$... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,030 |
24.2*. Prove that for $n \neq 4$ a regular $n$-gon cannot be placed such that its vertices lie on the nodes of an integer lattice. | 24.2. For $n=3$ and $n=6$, the statement follows from the previous problem, so we will henceforth assume that $n \neq 3,4,6$. Suppose there exist regular $n$-gons with vertices at the nodes of an integer lattice $(n \neq 3,4,6)$. Among all such $n$-gons, we can choose one with the smallest side length. (To prove this, ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,031 |
24.3*. Can a right-angled triangle with integer sides be placed so that its vertices lie on the nodes of an integer lattice, but none of its sides lie along the grid lines? | 24.3. It is easy to verify that the triangle with vertices at points with coordinates $(0,0),(12,16)$, and $(-12,9)$ has the required properties.
Fig. 24.1
 are located at the nodes of an integer lattice. Inside it, there are $n$ lattice nodes, and on its boundary, there are $m$ nodes. Prove that its area is equal to $n + m / 2 - 1$ (Pick's formula). | 24.5. To each polygon \( N \) with vertices at the nodes of an integer lattice, we assign the number \( f(N) = n + m/2 - 1 \). Suppose a polygon \( M \) is cut into polygons \( M_1 \) and \( M_2 \) with vertices at the nodes of the lattice. We need to prove that if Pick's formula is true for two of the polygons \( M, M... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,034 |
24.6*. The vertices of triangle $A B C$ are located at the nodes of an integer lattice, and there are no other nodes on its sides, while there is exactly one node $O$ inside it. Prove that $O$ is the point of intersection of the medians of triangle $A B C$.
See also problem 23.32.
## §2. Various Problems | 24.6. According to Pick's formula $S_{A O B}=S_{B O C}=S_{C O A}=1 / 2$, which means that $O-$ is the point of intersection of the medians of triangle $A B C$ (see problem 4.2). | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,035 |
24.7*. On an infinite sheet of graph paper, $N$ cells are colored black. Prove that from this sheet, a finite number of squares can be cut out such that two conditions are met: 1) all black cells lie within the cut-out squares; 2) in any cut-out square $K$, the area of black cells will be at least 0.2 and at most 0.8 o... | 24.7. Let's take a sufficiently large square with side $2^{n}$ such that all black cells lie inside it and constitute less than 0.2 of its area. We will cut this square into four equal squares. Each of them is painted in less than 0.8. Those that are painted more than 0.2, we leave, and the rest we cut further in the s... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 28,036 |
24.8*. The origin is the center of symmetry of a convex figure with an area greater than 4. Prove that this figure contains at least one point with integer coordinates, different from the origin (Minkowski). | 24.8. Consider all convex figures obtained from a given one by translations by vectors, both coordinates of which are even. We will prove that at least two of these figures intersect. The original figure can be enclosed in a circle of radius $R$ centered at the origin, and $R$ can be chosen as an integer. Let's take th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,037 |
24.9*. In all nodes of the integer lattice, except one where the hunter is located, trees grow, the trunks of which have a radius $r$. Prove that the hunter will not be able to see a hare located at a distance greater than $1 / r$ from him. | 24.9. Let the hunter be at point $O$, and the hare at point $A$; $A_{1}$ is the point symmetric to $A$ with respect to $O$. Consider the figure $\Phi$, containing all points,
Fig. 24.5 the ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,038 |
24.10*. Inside a convex figure with area $S$ and semiperimeter $p$, there are $n$ lattice nodes. Prove that $n > S - p$. | 24.10. Consider an integer lattice defined by the equations $x=k+$ $+1 / 2$ and $y=l+1 / 2$, where $k$ and $l$ are integers. We will prove that each square of this lattice gives a non-negative contribution to the quantity $n-S+p$. Consider two cases.
1. The figure contains the center of the square. Then $n^{\prime}=1$... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,039 |
24.11*. Prove that for any $n$ there exists a circle inside which there lie exactly $n$ integer points. | 24.11. First, let's prove that on the circle with center $A=(\sqrt{2}, 1 / 3)$, there cannot be more than one integer point. If $m$ and $n$ are integers, then $(m-\sqrt{2})^{2}+(n-(1 / 3))^{2}=q-2 m \sqrt{2}$, where $q$ is a rational number. Therefore, from the equality
$$
\left(m_{1}-\sqrt{2}\right)^{2}+\left(n_{1}-1... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,040 |
24.12*. Prove that for any $n$ there exists a circle on which there lie exactly $n$ integer points. | 24.12. First, let's prove that the equation \(x^{2}+y^{2}=5^{k}\) has exactly \(4(k+1)\) integer solutions. For \(k=0\) and \(k=1\), this statement is obvious. We will prove that the equation \(x^{2}+y^{2}=5^{k}\) has exactly eight solutions \((x, y)\) such that \(x\) and \(y\) are not divisible by 5; together with \(4... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 28,041 |
25.2. Cut an arbitrary triangle into pieces from which a triangle symmetrical to the original one with respect to some line can be formed (pieces cannot be flipped). | 25.2. Let $A$ be the largest angle of the triangle. Cut the triangle $ABC$ into isosceles triangles and rearrange them as shown in Fig. 25.5.
Fig. 25.5
 \pi \leqslant k \pi$ or $n-2 \leqslant k$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,057 |
25.18*. On a square sheet of paper, $n$ rectangles with sides parallel to the sides of the sheet are drawn. No two of these rectangles have any common interior points. Prove that if these rectangles are cut out, the number of pieces into which the remaining part of the sheet is divided does not exceed $n+1$. | 25.18. The sum of the exterior angles of a polygon, adjacent to the interior angles less than $\pi$, is not less than $2 \pi$ (see problem 22.20). The exterior angles of the figures into which the remaining part of the sheet splits are either exterior angles of the square or interior angles of the cut-out rectangles. T... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,058 |
25.19*. Prove that if a convex quadrilateral $A B C D$ can be cut into two similar quadrilaterals, then $A B C D$ is a trapezoid or a parallelogram. | 25.19. Let the segment $M N$, where points $M$ and $N$ lie on sides $A B$ and $C D$, divide the quadrilateral $A B C D$ into two similar quadrilaterals. Then the angle $A M N$ of the quadrilateral $A M N D$ is equal to one of the angles of the quadrilateral $N M B C$. On the other hand, $\angle N M B=180^{\circ}-\angle... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,059 |
25.20*. In a square with side 1, a finite number of segments parallel to its sides are drawn, and these segments can intersect each other. The sum of the lengths of the segments is 18. Prove that the area of one of the parts into which the square is divided is not less than 0.01. | 25.20. The sum of the lengths of the boundaries of all figures into which the square is divided is $2 \cdot 18 + 4 = 40$. Indeed, the drawn segments contribute twice to this sum, while the sides of the square contribute once. Let the sum of the lengths of the horizontal parts of the boundary of the $i$-th figure be $2 ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,060 |
25.21*. A triangle, all angles of which do not exceed $120^{\circ}$, is cut into several triangles. Prove that at least one of the resulting triangles has all angles not exceeding $120^{\circ}$.
See also problems $22.25,22.26$.
## §4. Dissections into Parallelograms | 25.21. Consider all points, different from the vertices of the original triangle and being vertices of the obtained triangles. Let $m$ of these points lie inside the original triangle and $n$ on its boundary. The sum of all angles of the obtained triangles is $\pi+\pi n+2 \pi m$, i.e., the number of these triangles is ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,061 |
25.22*. Prove that the following properties of a convex polygon $F$ are equivalent: 1) $F$ has a center of symmetry; 2) $F$ can be cut into parallelograms. | 25.22. Consider a convex polygon \(A_{1} \ldots A_{n}\). We will prove that each of properties 1 and 2 is equivalent to property 3: "For any vector \(\overrightarrow{A_{i} A_{i+1}}\), there exists a vector \(\overrightarrow{A_{j} A_{j+1}} = -\overrightarrow{A_{i} A_{i+1}}\)."
It is clear that property 1 implies proper... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,062 |
25.23*. Prove that if a convex polygon can be cut into centrally symmetric polygons, then it has a center of symmetry. | 25.23. Let's use the result of the previous problem. If a convex polygon $M$ is cut into convex centrally symmetric polygons, then they can be cut into parallelograms. Therefore, $M$ can be cut into parallelograms, i.e., $M$ has a center of symmetry. | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,063 |
25.25*. A regular octagon with side 1 is cut into parallelograms. Prove that among them there are at least two rectangles, and the sum of the areas of all rectangles is 2.
## §5. Plane, cut by lines
Let $n$ pairwise non-parallel lines be drawn on the plane, with no three intersecting at the same point. Problems 25.26... | 25.25. Let us consider two mutually perpendicular pairs of opposite sides in a regular octagon and, as in problem 25.1, chains of parallelograms connecting opposite sides. At the intersections of these chains, there are rectangles. By considering two other pairs of opposite sides, we will obtain at least one more recta... | 2 | Geometry | proof | Yes | Yes | olympiads | false | 28,065 |
25.28. a) Prove that when $n=2 k$ among the obtained figures there are no more than $2 k-1$ angles.
b) Can it be that when $n=100$ among the obtained figures there are only three angles? | 25.28. a) All intersection points of the given lines can be enclosed in some circle. The lines divide this circle into $4 k$ arcs. It is clear that two adjacent arcs cannot simultaneously belong to angles, so the number of angles does not exceed $2 k$, and equality can only be achieved if the arcs belonging to angles a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,068 |
25.33*. The parts into which the plane is divided by lines are painted red and blue so that adjacent parts are of different colors (see problem 27.1). Let $a$ be the number of red parts, $b$ be the number of blue parts. Prove that
$$
a \leqslant 2 b-2-\sum(\lambda(P)-2),
$$
with equality holding if and only if the re... | 25.33. Let $a_{k}^{\prime}$ be the number of red $k$ -
Fig. 25.25 polygons, $a^{\prime}$ - the number of bounded red regions, the number of segments into which these lines are divided by t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,073 |
25.35*. Prove that any convex $n$-gon, where $n \geqslant 6$, can be cut into convex pentagons. | 25.35. We will prove by induction that any convex $n$-gon, where $n \geqslant 5$,
Fig. 25.26
can be cut into pentagons. For $n=5$ this is obvious, and how to do it for $n=6$ and 7 is shown ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,075 |
25.36*. Prove that for any natural number $n$, where $n \geqslant 6$, a square can be cut into $n$ squares. | 25.36. Let a square be cut into $m$ smaller squares. If we cut one of these smaller squares into 4 smaller squares, the original square will be divided into $m+3$ squares. It remains to note that a square can be cut into 6, 7, and 8 squares (Fig. 25.27).
$-gon cannot be cut into $k$ pentagons along its diagonals. For $k=1$, this statement is obvious. Now suppose it has been proven for all $(3k+1)$-gons, and we will prove it for a $(3k+4)$-gon. Suppose a $(3k+4)$-gon is cut along its diagonals into $k+1$ pentagons. If each... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,077 |
25.38*. Can a regular triangle be cut into 1000000 convex polygons such that any straight line intersects no more than 40 of them? | 25.38. If cuts are made close to the vertices of a convex $n$-gon, then one can cut off $n$ triangles from it and obtain a convex $2n$-gon. It is easy to verify that in this case any line intersects no more than two of the cut-off triangles.
Cut off 3 triangles from an equilateral triangle, then 6 triangles from the r... | 1,000,000 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,078 |
25.40*. A rectangle is cut into rectangles, the length of one of the sides of each of which is an integer. Prove that the length of one of the sides of the original rectangle is an integer.
See also problems 23.17 and 23.18.
## §7. Partitioning Figures into Segments | 25.40. Let's introduce a coordinate system with the origin at one of the vertices of the original rectangle and the axes directed along its sides. We will cut the coordinate plane with the lines $x=n / 2$ and $y=m / 2$, where $m$ and $n$ are integers, and color the resulting parts in a checkerboard pattern. If the side... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,080 |
25.42*. Prove that a triangle can be divided into segments.
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.
Note: The instruction part is for your reference, you don't need to include it in the output. Here is the translatio... | 25.42. Let's take a point $P$ on side $B C$ of triangle $A B C$, and points $Q$ and $R$ on side $A C$. The "degenerate quadrilateral" $Q R C P$ and quadrilateral $A Q P B$ can be divided into segments as was done in the previous problem (Fig. 25.29). | Number Theory | proof | Yes | Yes | olympiads | false | 28,082 | |
25.43*. Prove that a circle can be divided into segments.
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.
Note: The sentence "Translate the text above into English, please retain the original text's line breaks and format, a... | 25.43. Divide the diameter $A B$ into 5 equal parts by points $C, D, E$ and $F$. Let points $M_{t}$ and $P_{t}$ lie on segments $C D$ and $E F$, such that $C M_{t}: M_{t} D=F P_{t}: P_{t} E=t:(1-t)$, where $0<t<1$, and points $Q_{t}$ and $N_{t}$ lie on the different arcs of the circle defined by points $A$ and $B$, and... | Number Theory | proof | Yes | Yes | olympiads | false | 28,083 | |
25.46*. A segment of length 1 is covered by several segments lying on it. Prove that among them, one can choose several pairwise non-intersecting segments, the sum of the lengths of which is not less than 0.5. | 25.46. We will sequentially discard segments that are covered by one or several remaining segments until it is no longer possible. Let's direct the coordinate axis along the given segment and denote the coordinates of the ends of the remaining segments as $a_{k}$ and $b_{k}\left(a_{k}<b_{k}\right)$.
 A square with side 1 is covered by several smaller squares with sides parallel to its sides. Prove that among them, one can choose non-overlapping squares whose total area is not less than $1 / 9$.
b) The area of the union of several circles is 1. Prove that from them, one can choose several pairwise non-ov... | 25.48. a) Consider the largest square $K$ in the covering and remove all squares that intersect with it. They lie within a square whose side is three times the side of $K$, so the area they occupy is no more than 81, where $s$ is the area of $K$. We include the square $K$ in the selected ones and do not consider it fur... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 28,088 |
25.50*. The length of the projection of the figure $\Phi$ onto any line does not exceed 1. Is it true that $\Phi$ can be covered by a circle of diameter: a) 1; b) $1.5 ?$ | 25.50. a) Not correct. Let $\Phi$ be an equilateral triangle with side length 1. It is easy to verify that the length of the projection of $\Phi$ onto any line does not exceed 1. On the other hand, since triangle $\Phi$ is acute-angled, it cannot be covered by a circle with a radius smaller than the radius of the circu... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,090 |
25.52*. On a round table of radius $R$, $n$ round coins of radius $r$ are placed without overlapping, and it is impossible to place any more coins. Prove that $R / r \leqslant 2 \sqrt{n}+1$.
See also problems $20.12,20.16,20.31,22.4,22.9,22.14,22.15$.
## §9. Tiling with Dominoes and Tiles | 25.52. Inflate all the coins by a factor of 2, i.e., for each of them, consider a circle of radius $2r$ with the same center. If the center of one coin does not belong to the inflation of the second coin, then the distance between their centers is greater than $2r$, which means these coins do not intersect. Moreover, i... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,092 |
25.59*. a) Can a $6 \times 6$ square be tiled with $1 \times 2$ dominoes in such a way that there is no "seam," i.e., a straight line that does not cut through any dominoes?
b) Prove that any rectangle $m \times n$, where $m$ and $n$ are greater than 6 and $mn$ is even, can be tiled with $1 \times 2$ dominoes in such ... | 25.59. a) It is impossible. Suppose that a $6 \times 6$ square is tiled with $1 \times 2$ dominoes such that there is no "seam." Consider the 10 segments that divide the square into 36 cells (we do not consider the sides of the square itself). Each of these segments cuts through at least two dominoes. Indeed, if a segm... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 28,096 |
25.60*. There is an unlimited number of tiles in the form of a polygon $M$. We will say that a parquet can be made from these tiles if they can cover a circle of arbitrarily large radius without any gaps or overlaps.
a) Prove that if $M$ is a convex $n$-gon, where $n \geqslant 7$, then it is impossible to make a parqu... | 25.60. a) Suppose that a circle $K$ of radius $R$ is covered by a parquet floor consisting of identical convex $n$-gons. Consider all points lying inside the circle $K$ and being vertices of the $n$-gons. These points can be of two types: points of the 1st type lie on the sides of other $n$-gons and the sum of the angl... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,097 |
26.1. a) An architect wants to arrange four skyscrapers so that while walking around the city, one can see their spires in any order (i.e., for any set of building numbers $i, j, k, l$, one can stand at some point and by turning in the direction "clockwise" or "counterclockwise," see the spire of building $i$ first, th... | 26.1. a) It is easy to verify that by constructing the fourth building inside the triangle formed by the other three buildings, we obtain the required arrangement.
b) It is impossible to arrange five buildings in the required manner. Indeed, if we see buildings $A_{1}, A_{2}, \ldots, A_{n}$ in sequence, then $A_{1} A_... | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 28,098 |
26.2. On a plane, there are $n$ points, and from any quadruple of these points, one point can be removed so that the remaining points lie on one line. Prove that from the given points, one point can be removed so that all the remaining points lie on one line. | 26.2. We can assume that $n \geqslant 4$ and not all points lie on the same line. Then we can choose four points $A, B, C$ and $D$ that do not lie on the same line. By the condition, three of them lie on the same line. Let's assume, for definiteness, that points $A$, $B$, and $C$ lie on the line $l$, and $D$ does not l... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 28,099 |
26.3*. On a plane, there are 400 points. Prove that the number of different distances between them is not less than 15. | 26.3. Let the number of distinct distances between points be $k$. Fix two points. Then all other points are intersection points of two families of concentric circles, each containing $k$ circles. Therefore, the total number of points does not exceed $2 k^{2}+2$. It remains to note that $2 \cdot 14^{2}+2=394<400$. | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 28,100 |
26.4*. On the plane, there are $n \geqslant 3$ points. Let $d$ be the greatest distance between pairs of these points. Prove that there are no more than $n$ pairs of points, the distance between which is equal to $d$. | 26.4. Let's call a diameter a segment of length $d$ connecting a pair of given points. The ends of all diameters emanating from point $A$ lie on a circle centered at $A$ with radius $d$. Since the distance between any two points does not exceed $d$, the ends of all diameters emanating from $A$ lie on an arc whose angul... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,101 |
26.5*. On a plane, there are 4000 points, no three of which lie on the same line. Prove that there exist 1000 non-intersecting quadrilaterals (possibly non-convex) with vertices at these points. | 26.5. Let's draw all the lines connecting pairs of the given points, and choose a line $l$ that is not parallel to any of them. By lines parallel to $l$, we can divide the given points into quartets. The quadrilaterals with vertices at these quartets of points are the ones we are looking for (Fig. 26.2). | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 28,102 |
26.6*. On a plane, there are 22 points, and no three of them lie on the same line. Prove that they can be paired in such a way that the segments defined by the pairs intersect in at least five points. | 26.6. Let's divide the given points arbitrarily into six groups: four groups of four points each, one group of five points, and one group of a single point. Consider the group of five points. From these, we can select four points that are the vertices of some convex quadrilateral $ABCD$ (see problem 22.2). We will pair... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 28,103 |
26.7*. Prove that for any natural $N$ there exist $N$ points, no three of which lie on the same line and all pairwise distances between which are integers.
See also problems $20.13-20.15,22.7$.
## §2. Systems of segments, lines, and circles | 26.7. Since ${\frac{2 n}{n^{2}+1}}^{2}+{\frac{n^{2}-1}{n^{2}+1}}^{2}=1$, there exists an angle $\varphi$ with the property that $\sin \varphi=2 n /\left(n^{2}+1\right)$ and $\cos \varphi=\left(n^{2}-1\right) /\left(n^{2}+1\right)$, and $0<2 N \varphi<\pi / 2$ for sufficiently large $n$. Consider a circle of radius $R$ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,104 |
26.10*. A point $O$ lying inside a convex polygon $A_{1} \ldots A_{n}$ has the property that any line $O A_{i}$ contains another vertex $A_{j}$. Prove that no other point except $O$ has this property. | 26.10. It follows from the condition that all vertices of the polygon are divided into pairs, defining diagonals $A_{i} A_{j}$ that pass through the point $O$. Therefore, the number of vertices is even, and on either side of each such diagonal $A_{i} A_{j}$, there is an equal number of vertices. Hence, $j=i+m$, where $... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,107 |
26.11*. On a circle, $4 n$ points are marked and colored alternately in red and blue. The points of each color are paired, and the points of each pair are connected by segments of the same color. Prove that if no three segments intersect at the same point, then there will be at least $n$ intersection points of red segm... | 26.11. If $A C$ and $B D$ are intersecting red segments, then the number of intersection points of any line with segments $A B$ and $C D$ does not exceed the number of intersection points of this line with segments $A C$ and $B D$. Therefore, by replacing the red segments $A C$ and $B D$ with segments $A B$ and $C D$, ... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 28,108 |
26.12*. On a plane, there are $n \geqslant 5$ circles such that any three of them have a common point. Prove that then all the circles have a common point.
## §3. Examples and Counterexamples
There are many false statements that seem true at first glance. To refute such statements, one needs to construct a correspond... | 26.12. Let $A$ be a common point of the first three circles $S_{1}, S_{2}$, and $S_{3}$. Denote the points of intersection of circles $S_{1}$ and $S_{2}, S_{2}$ and $S_{3}, S_{3}$ and $S_{1}$ by $B, C$, and $D$ respectively. Suppose there exists a circle $S$ that does not pass through point $A$. Then the circle $S$ pas... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,109 |
26.13. Does there exist a triangle in which all altitudes are less than 1 cm, and the area is greater than $1 \mathrm{m}^{2}$? | 26.13. Consider a rectangle $A B C D$ with sides $A B=1$ cm and $B C=500$ m. Let $O$ be the point of intersection of its diagonals. It is easy to verify that the area of triangle $A O D$ is greater than $1 \mathrm{~m}^{2}$, and all its altitudes are less than 1 cm. | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,110 |
26.15*. The list of ordered side lengths and diagonals of one convex quadrilateral in ascending order coincides with the same list for another quadrilateral. Must these quadrilaterals be equal? | 26.15. Not necessarily. It is easy to check that the list of side lengths and diagonals for an isosceles trapezoid with height 1 and bases 2 and 4 matches the same list for a quadrilateral with perpendicular diagonals of lengths 2 and 4, divided by the point of intersection into segments of lengths 1 and 1, and 3 (Fig.... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,112 |
26.16*. Let $n \geqslant 3$. Do there exist $n$ points, not lying on the same line, such that the pairwise distances between them are irrational, and the areas of all triangles with vertices at these points are rational? | 26.16. Yes, they do exist. Consider the points $P_{i}=\left(i, i^{2}\right)$, where $i=1, \ldots, n$. The areas of all triangles with vertices at nodes of the integer lattice are rational (see problem 24.5), while the numbers $P_{i} P_{j}=|i-j| \sqrt{1+(i+j)^{2}}$ are irrational. | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,113 |
26.17*. Do there exist three points $A, B$, and $C$ in the plane such that for any point $X$ the length of at least one of the segments $X A, X B$, and $X C$ is irrational? | 26.17. Yes, they exist. Let $C$ be the midpoint of segment $A B$. Then $X C^{2}=$ $=\left(2 X A^{2}+2 X B^{2}-A B^{2}\right) / 2$. If the number $A B^{2}$ is irrational, then the numbers $X A, X B$ and $X C$ cannot all be rational simultaneously. | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,114 |
26.18*. In an acute-angled triangle $A B C$, the median $A M$, the bisector $B K$, and the altitude $C H$ are drawn. Can the area of the triangle formed by the intersection points of these segments be greater than $0.499 S_{A B C}$? | 26.18. Let's consider a right-angled triangle \( A B C_{1} \) with legs \( A B = 1 \) and \( B C_{1} = 2n \). In this triangle, we draw the median \( A M_{1} \), the angle bisector \( B K_{1} \), and the altitude \( C_{1} H_{1} \). The area of the triangle formed by these segments is greater than \( S_{A B M_{1}} - S_{... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,115 |
26.20*. Can a finite set of points contain for each of its points exactly 100 points at a distance of $1$? | 26.20. Yes, it can. We will prove this statement by induction, replacing 100 with $n$. For $n=1$, we can take the endpoints of a segment of length 1. Suppose the statement is proven for $n$ and $A_{1}, \ldots, A_{k}$ is the required set of points. Let $A_{1}^{\prime}, \ldots, A_{k}^{\prime}$ be the images of the points... | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 28,117 |
26.23*. The circus arena is illuminated by $n$ different spotlights. Each spotlight illuminates a convex shape. It is known that if any one spotlight is turned off, the arena will still be fully illuminated, but if any two spotlights are turned off, the arena will not be fully illuminated. For which $n$ is this possibl... | 26.23. This is possible for any $n \geqslant 2$. Inscribing a regular $k$-gon in the arena, where $k$ is the number of different pairs that can be formed from $n$ spotlights, i.e., $k=n(n-1) / 2$. Then, a one-to-one correspondence can be established between the segments cut off by the sides of the $k$-gon and the pairs... | n\geqslant2 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,120 |
27.1. Prove that if a plane is divided into parts by straight lines and circles, then the resulting map can be colored with two colors in such a way that parts sharing a segment or arc will be of different colors. | 27.1. We will prove this by induction on the total number of lines and circles. For one line or circle, the statement is obvious. Now suppose that any map defined by $n$ lines and circles can be colored in the required manner, and we will show how to then color a map defined by $n+1$ lines and circles. Remove one of th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,121 |
27.2*. Prove that in a convex $n$-gon, it is impossible to choose more than $n$ diagonals such that any two of them have a common point. | 27.2. We will prove by induction on $n$ that in a convex $n$-gon, it is impossible to choose more than $n$ sides or diagonals such that any two of them have a common point. For $n=3$, this is obvious. Suppose the statement is true for any convex $n$-gon, and we will prove it for an $(n+1)$-gon. If from each vertex of t... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 28,122 |
27.3*. Let $E$ be the point of intersection of the non-parallel sides $A D$ and $B C$ of trapezoid $A B C D, B_{n+1}$ be the point of intersection of the lines $A_{n} C$ and $B D\left(A_{0}=A\right)$, $A_{n+1}$ be the point of intersection of the lines $E B_{n+1}$ and $A B$. Prove that $A_{n} B=$ $=A B /(n+1)$. | 27.3. It is clear that $A_{0} B=A B$. Let $C_{n}$ be the point of intersection of the lines $E A_{n}$ and $D C, D C: A B=k, A B=a, A_{n} B=a_{n}$ and $A_{n+1} B=x$. Since $C C_{n+1}: A_{n} A_{n+1}=$ $=D C_{n+1}: B A_{n+1}$, then $k x:\left(a_{n}-x\right)=(k a-k x): x$, i.e., $x=a a_{n} /\left(a+a_{n}\right)$. If $a_{n}... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,123 |
27.4*. On a line, points $A_{1}, \ldots, A_{n}$ and $B_{1}, \ldots, B_{n-1}$ are given. Prove that
$$
\sum_{i=1}^{n} \frac{\prod_{k=1}^{n-1} \overline{A_{i} B_{k}}}{\prod_{j \neq 1} \overline{A_{i} A_{j}}}=1
$$ | 27.4. Let's first prove the required statement for $n=2$. Since $\overrightarrow{A_{1} B_{1}} + \overrightarrow{B_{1} A_{2}} + \overrightarrow{A_{2} A_{1}} = \overrightarrow{0}$, it follows that $\left(\overline{A_{1} B_{1}} / \overline{A_{1} A_{2}}\right) + \left(\overline{A_{2} B_{1}} / \overline{A_{2} A_{1}}\right) ... | proof | Algebra | proof | Yes | Yes | olympiads | false | 28,124 |
27.5*. Prove that if $n$ points do not lie on the same line, then among the lines connecting them, there are at least $n$ different ones.
See also problems $2.12,5.109,22.7,22.10-22.13,22.21$ b), $22.23,22.24$, $22.30,23.39-23.41,26.20$.
## §2. Combinatorics | 27.5. We will prove this by induction on $n$. For $n=3$, the statement is obvious. Suppose we have proven it for $n-1$ points, and we will prove it for $n$ points. If on every line passing through two given points, there lies another given point, then all the given points lie on one line (see problem 20.13). Therefore,... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 28,125 |
27.6. Several points are marked on a circle, $A$ is one of them. Which are there more of: convex polygons with vertices at these points that contain point $A$ or those that do not contain it? | 27.6. Any polygon that does not contain point $A$ can be associated with a polygon that does contain point $A$ by adding it to its vertices. However, the reverse operation, i.e., removing vertex $A$, can only be performed for $n$-gons with $n \geqslant 4$. Therefore, there are more polygons containing point $A$ than po... | (n-1)(n-2)/2 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 28,126 |
27.7. Ten points are marked on a circle. How many non-closed, non-self-intersecting nine-segment broken lines with vertices at these points exist? | 27.7. The first point can be chosen in ten ways. Each of the following eight points can be chosen in two ways, as it must be adjacent to one of the previously chosen points (otherwise, a self-intersecting broken line would result). Since the beginning and end are not distinguished in this calculation, the result must b... | 1280 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 28,127 |
27.8*. In a convex $n$-gon $(n \geqslant 4)$, all diagonals are drawn, and no three of them intersect at the same point. Find the number of intersection points of the diagonals. | 27.8. Any intersection point of diagonals determines two diagonals, the intersection of which it is, and the ends of these diagonals determine a convex quadrilateral. Conversely, any four vertices of a polygon determine one point of intersection of diagonals. Therefore, the number of intersection points of diagonals is... | \frac{n(n-1)(n-2)(n-3)}{24} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 28,128 |
27.9*. In a convex $n$-gon $(n \geqslant 4)$, all diagonals are drawn. Into how many parts do they divide the $n$-gon, if no three of them intersect at the same point? | 27.9. B. Let's draw the diagonals one by one. When we draw the next diagonal, the number of parts into which the previously drawn diagonals divide the polygon increases by \(m+1\), where \(m\) is the number of intersection points of the new diagonal with the previously drawn ones, i.e., each new diagonal and each new i... | \frac{n(n-3)}{2}+\frac{n(n-1)(n-2)(n-3)}{24}+ | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 28,129 |
27.10*. On a plane, there are $n$ points, no three of which lie on the same line. Prove that there exist at least $C_{n}^{5} /(n-4)$ different convex quadrilaterals with vertices at these points (the definition of the number $C_{n}^{k}$ see on p. 400$)$. | 27.10. If we choose any five points, then there exists a convex quadrilateral with vertices among them (problem 22.2). It remains to note that a set of four points can be supplemented to a set of five in $n-4$ different ways. | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 28,130 |
27.11*. Prove that the number of non-congruent triangles with vertices at the vertices of a regular $n$-gon is the nearest integer to $n^{2} / 12$.
See also problem 25.6. | 27.11. Let there be a total of $N$ unequal triangles with vertices at the vertices of a regular $n$-gon, of which $N_{1}$ are equilateral, $N_{2}$ are isosceles but not equilateral, and $N_{3}$ are scalene. Each equilateral triangle is equal to one triangle with a fixed vertex $A$, an isosceles but not equilateral tria... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 28,131 |
28.1. Let during inversion with center $O$ point $A$ goes to $A^{*}$, and point $B$ goes to $B^{*}$. Prove that triangles $O A B$ and $O B^{*} A^{*}$ are similar. | 28.1. Let $R^{2}$ be the degree of inversion. Then $O A \cdot O A^{*}=O B \cdot O B^{*}=R^{2}$, from which $O A: O B=O B^{*}: O A^{*}$ and $\triangle O A B \sim \triangle O B^{*} A^{*}$, since $\angle A O B=\angle B^{*} O A^{*}$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,132 |
28.2. Prove that under inversion with center $O$, a line $l$ that does not pass through $O$ transforms into a circle passing through $O$. | 28.2. Drop a perpendicular $OC$ from point $O$ to line $l$ and take an arbitrary point $M$ on $l$. From the similarity of triangles $OCM$ and $OM^{*}C^{*}$ (problem 28.1), it follows that $\angle OM^{*}C^{*} = \angle OCM = 90^{\circ}$, i.e., point $M^{*}$ lies on the circle $S$ with diameter $OC^{*}$. If $X$ is any poi... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,133 |
28.3. Prove that under inversion with center $O$, a circle passing through $O$ transforms into a line, and a circle not passing through $O$ transforms into a circle. | 28.3. The case when the circle $S$ passes through $O$ was actually discussed in the previous problem (and formally follows from it, since $\left(M^{*}\right)^{*}=M$). Suppose now that the point $O$ does not belong to $S$. Let $A$ and $B$ be the points of intersection of the circle $S$ with the line passing through $O$ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,134 |
28.4. Prove that touching circles (a circle and a line) transform under inversion into touching circles or into a circle and a line, or into a pair of parallel lines.
Definition. Let two circles intersect at point A. The angle between the circles is defined as the angle between the tangents to the circles at point A. ... | 28.4. If the point of tangency does not coincide with the center of inversion, then after inversion these circles (circle and line) will still have one common point, i.e., the tangency will be preserved.
If circles with centers $A$ and $B$ touch at point $O$, then under inversion with center $O$ they will transform in... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,135 |
28.5*. Prove that under inversion, the angle between circles (between a circle and a line, between lines) is preserved.
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | 28.5. Let's draw tangents $l_{1}$ and $l_{2}$ through the point of intersection of the circles. Since touching circles and lines transform into touching ones under inversion (see problem 28.4), the angle between the images of the circles is equal to the angle between the images of the tangents to them. Under inversion ... | Number Theory | proof | Yes | Yes | olympiads | false | 28,136 | |
28.6*. Prove that two non-intersecting circles $S_{1}$ and $S_{2}$ (or a circle and a line) can be transformed into a pair of concentric circles using inversion. | 28.6. First solution. Draw a coordinate axis through the centers of the circles. Let $a_{1}$ and $a_{2}$ be the coordinates of the points of intersection of the axis with $S_{1}$, and $b_{1}$ and $b_{2}$ with $S_{2}$. Let $O$ be a point on the axis with coordinate $x$. Then, under inversion with center $O$ and power $k... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,137 |
28.7*. Through point $A$, a line $l$ is drawn, intersecting circle $S$ with center $O$ at points $M$ and $N$ and not passing through $O$. Let $M^{\prime}$ and $N^{\prime}$ be the points symmetric to $M$ and $N$ with respect to $OA$, and let $A^{\prime}$ be the point of intersection of the lines $MN^{\prime}$ and $M^{\p... | 28.7. Let point $A$ lie outside $S$, then $A^{\prime}$ lies inside $S$ and $\angle M A^{\prime} N = \left(\smile M N + \smile M^{\prime} N^{\prime}\right) / 2 = \smile M N = \angle M O N$, i.e., the quadrilateral $M N O A^{\prime}$ is cyclic. But under inversion with respect to $S$, the line $M N$ will transform into a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,138 |
28.9. Construct a circle passing through two given points and tangent to a given circle (or line). | 28.9. If both given points $A$ and $B$ lie on the given circle $S$ (or line), the problem has no solution. Now let point $A$ not lie on $S$. Under inversion with center $A$, the desired circle will transform into a line passing through $B^{*}$ and tangent to $S^{*}$. This leads to the following construction. Perform an... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,140 |
28.11*. Construct a circle that is tangent to three given circles (Apollonius's problem). | 28.11. Let's reduce this problem to problem 28.10. Suppose circle $S$ of radius $r$ touches circles $S_{1}, S_{2}, S_{3}$ of radii $r_{1}, r_{2}, r_{3}$ respectively. The contact of circle $S$ with each of $S_{i} (i=1,2,3)$ can be either external or internal, so there are eight different cases of contact in total. Let,... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,142 |
28.13*. Construct a circle that is tangent to a given circle $S$ and orthogonal to two given circles $S_{1}$ and $S_{2}$. | 28.13. Let's perform an inversion that transforms circles $S_{1}$ and $S_{2}$ into a pair of lines (if they have a common point) or into a pair of concentric circles (see problem 28.6) with a common center $A$. In the latter case, the circle perpendicular to both of them will transform into a line passing through $A$ (... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,144 |
28.15. a) Construct a segment that is twice as long as the given segment.
b) Construct a segment that is $n$ times as long as the given segment. | 28.15. a) Let $AB$ be the given segment. Draw a circle with center $B$ and radius $AB$. By laying off chords $AX, XY$, and $YZ$ equal in length to $AB$ on this circle, we obtain equilateral triangles $ABX, XBY$, and $YBZ$. Therefore, $\angle ABZ=180^{\circ}$ and $AZ=2AB$.
b) The solution to part a) describes how to la... | nAB | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,146 |
28.16. Construct a point symmetric to point $A$ with respect to the line passing through the given points $B$ and $C$. | 28.16. Let's draw circles centered at $B$ and $C$, passing through $A$. Then the point of intersection of these circles, distinct from $A$, will be the desired one. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,147 |
28.17*. Construct the image of point $A$ under inversion with respect to a given circle $S$ with a given center $O$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 28.17. Suppose first that point $A$ lies outside the circle $S$. Let $B$ and $C$ be the points of intersection of $S$ and the circle of radius $AO$ centered at $A$. Draw circles with centers at $B$ and $C$ and radii $BO = CO$; let $O$ and $A'$ be their points of intersection. We need to prove that $A'$ is the desired p... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 28,148 |
28.18*. Construct the midpoint of the segment with the given endpoints.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
---
28.18*. Construct the midpoint of the segment with the given endpoints. | 28.18. Let $A$ and $B$ be given points. If point $C$ lies on ray $AB$ and $AC = 2AB$, then under inversion with respect to a circle of radius $AB$ centered at $A$, point $C$ will map to the midpoint of segment $AB$. The construction is reduced to problems 28.15, a) and 28.17 | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,149 |
28.20*. Construct a circle passing through three given points. | 28.20. Let $A, B, C$ be given points. Construct (Problem 28.17) the images of points $B$ and $C$ under the inversion with center $A$ and an arbitrary power. Then the circle passing through $A, B$, and $C$ will be the image of the line $B^{*} C^{*}$ under this inversion, and its center is constructed according to the pr... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,151 |
28.24. Prove that the inversion with center at vertex $A$ of isosceles triangle $ABC (AB = AC)$ and power $AB^2$ maps the base $BC$ of the triangle to the arc $BC$ of the circumscribed circle. | 28.24. This inversion transforms the line $B C$ into a circle passing through points $A, B$, and $C$, and the image of segment $B C$ must remain inside angle $B A C$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,155 |
28.26*. No three of the four points $A, B, C, D$ lie on the same line. Prove that the angle between the circumcircles of triangles $A B C$ and $A B D$ is equal to the angle between the circumcircles of triangles $A C D$ and $B C D$. | 28.26. Let's perform an inversion with center $A$. The angles of interest to us will then be equal (see problem 28.5) to the angle between the lines $B^{*} C^{*}$ and $B^{*} D^{*}$ and the angle between the line $C^{*} D^{*}$ and the circumcircle of triangle $B^{*} C^{*} D^{*}$. Both of these angles are equal to half t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,157 |
28.28*. Two circles intersecting at point $A$ touch the circle (or line) $S_{1}$ at points $B_{1}$ and $C_{1}$, and the circle (or line) $S_{2}$ at points $B_{2}$ and $C_{2}$ (the tangency at $B_{2}$ and $C_{2}$ is the same as at $B_{1}$ and $C_{1}$). Prove that the circles circumscribed around triangles $A B_{1} C_{1}... | 28.28. From the condition on the types of tangency, it follows that after inversion with center $A$, we obtain two circles inscribed in the same angle or in a pair of vertical angles. In any case, the circles $S_{1}^{*}$ and $S_{2}^{*}$ are transformed into each other by a homothety with center $A$. This homothety tran... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,159 |
28.30*. a) Prove that the circle passing through the midpoints of the sides of a triangle is tangent to its inscribed and three excircles (Feuerbach).
b) On the sides $AB$ and $AC$ of triangle $ABC$, points $C_{1}$ and $B_{1}$ are taken such that $AC_{1} = B_{1}C_{1}$ and the inscribed circle $S$ of triangle $ABC$ is ... | 28.30. a) Let $A_{1}, B_{1}$ and $C_{1}$ be the midpoints of sides $BC, CA$ and $AB$. We will prove, for example, that the circumcircle of triangle $A_{1} B_{1} C_{1}$ is tangent to the incircle $S$ and the excircle $S_{a}$, which is tangent to side $BC$. Let points $B'$ and $C'$ be symmetric to $B$ and $C$ with respec... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,161 |
28.31. Given four circles, where circles $S_{1}$ and $S_{3}$ intersect both circles $S_{2}$ and $S_{4}$. Prove that if the points of intersection of $S_{1}$ with $S_{2}$ and $S_{3}$ with $S_{4}$ lie on one circle or line, then the points of intersection of $S_{1}$ with $S_{4}$ and $S_{2}$ with $S_{3}$ also lie on one c... | 28.31. After the inversion with the center at the intersection point of $S_{1}$ and $S_{2}$, we obtain lines $l_{1}$, $l_{2}$, and $l$, intersecting at one point. Line $l_{1}$ intersects the circle $S_{4}^{*}$ at points $A$ and $B$, line $l_{2}$ intersects $S_{3}^{*}$ at points $C$ and $D$, and line $l$ passes through ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,162 |
28.32*. Given four circles $S_{1}, S_{2}, S_{3}, S_{4}$. Let $S_{1}$ and $S_{2}$ intersect at points $A_{1}$ and $A_{2}, S_{2}$ and $S_{3}$ at points $B_{1}$ and $B_{2}, S_{3}$ and $S_{4}$ at points $C_{1}$ and $C_{2}, S_{4}$ and $S_{1}$ at points $D_{1}$ and $D_{2}$ (Fig. 28.3). Prove that if points $A_{1}, B_{1}, C_{... | 28.32. Let's perform an inversion with the center at point $A_{1}$. Then the circles $S_{1}, S_{2}$, and $S_{4}$ will transform into the lines $A_{2}^{*} D_{1}^{*}, B_{1}^{*} A_{2}^{*}$, and $D_{1}^{*} B_{1}^{*}$, while the circles $S_{3}$ and $S_{4}$ will transform into the circles $S_{3}^{*}$ and $S_{4}^{*}$, circums... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,163 |
28.34*. On a plane, six points $A_{1}, A_{2}, A_{3}, B_{1}, B_{2}$, $B_{3}$ are taken. Prove that if the circumcircles of triangles $A_{1} A_{2} B_{3}, A_{1} B_{2} A_{3}$ and $B_{1} A_{2} A_{3}$ pass through one point, then the circumcircles of triangles $B_{1} B_{2} A_{3}$, $B_{1} A_{2} B_{3}$ and $A_{1} B_{2} B_{3}$ ... | 28.34. After inversion with the center at the intersection point of the circumcircles of triangles $A_{1} A_{2} B_{3}, A_{1} B_{2} A_{3}$ and $B_{1} A_{2} A_{3}$, these circles will transform into lines, and the statement of the problem will reduce to proving that the circumcircles of triangles $B_{1}^{*} B_{2}^{*} A_{... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,165 |
28.35*. On a plane, six points $A_{1}, A_{2}, B_{1}, B_{2}, C_{1}, C_{2}$ are taken. Prove that if the circumcircles of triangles $A_{1} B_{1} C_{1}, A_{1} B_{2} C_{2}, A_{2} B_{1} C_{2}, A_{2} B_{2} C_{1}$ pass through one point, then the circumcircles of triangles $A_{2} B_{2} C_{2}, \quad A_{2} B_{1} C_{1}$, $A_{1} ... | 28.35. After the inversion with the center at the intersection point of the circumcircles of triangles $A_{1} B_{1} C_{1}, A_{1} B_{2} C_{2}, A_{2} B_{1} C_{2}$ and $A_{2} B_{2} C_{1}$, we obtain four lines and four circumcircles of the triangles formed by these lines. According to problem 2.85, a) these circles pass t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,166 |
28.36*. In this problem, we will consider sets of $n$ lines in general position, i.e., sets in which no two lines are parallel and no three lines pass through the same point.
To a set of two lines in general position, we will correspond a point - their intersection point, and to a set of three lines in general positio... | 28.36. a) Let $M_{ij}$ denote the point of intersection of the lines $l_i$ and $l_j$, and $S_{ij}$ the circle corresponding to the remaining three lines. Then the point $A_1$ is the point of intersection of the circles $S_{15}$ and $S_{12}$, distinct from the point $M_{34}$.
Repeating this reasoning for all points $A_... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,167 |
28.37*. Let points $M_{1}$ and $M_{2}$ be chosen on two intersecting lines $l_{1}$ and $l_{2}$, not coinciding with the intersection point $M$ of these lines. We will associate with them a circle passing through $M_{1}, M_{2}$, and $M$.
If $\left(l_{1}, M_{1}\right),\left(l_{2}, M_{2}\right),\left(l_{3}, M_{3}\right)$... | 28.37. a) Let $M_{ij}$ denote the point of intersection of lines $l_i$ and $l_j$. Then the point $A_1$, corresponding to the triplet $l_2, l_3, l_4$, is the point of intersection of the circumcircles of triangles $M_2 M_3 M_{23}$ and $M_3 M_4 M_{34}$. Reasoning similarly for points $A_2, A_3$, and $A_4$, we obtain that... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,168 |
28.38*. Circles $S_{1}, S_{2}, \ldots, S_{n}$ touch two circles $R_{1}$ and $R_{2}$ and, moreover, $S_{1}$ touches $S_{2}$ at point $A_{1}, S_{2}$ touches $S_{3}$ at point $A_{2} \ldots, S_{n-1}$ touches $S_{n}$ at point $A_{n-1}$. Prove that the points $A_{1}, A_{2}, \ldots, A_{n-1}$ lie on one circle. | 28.38. If circles \( R_{1} \) and \( R_{2} \) intersect or touch, then the inversion with the center at their point of intersection will transform circles \( S_{1}, S_{2}, \ldots, S_{n} \) into circles touching a pair of lines and each other at points \( A_{1}^{*}, A_{2}^{*}, \ldots, A_{n-1}^{*} \), lying on the bisect... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,169 |
28.39*. Prove that if there exists a chain of circles $S_{1}, S_{2}, \ldots, S_{n}$, each of which touches two adjacent ones ( $S_{n}$ touches $S_{n-1}$ and $S_{1}$ ) and two given non-intersecting circles $R_{1}$ and $R_{2}$, then there are infinitely many such chains. Specifically, for any circle $T_{1}$, touching $R... | 28.39. Let's perform an inversion that transforms $R_{1}$ and $R_{2}$ into a pair of concentric circles. Then the circles $S_{1}^{*}, S_{2}^{*}, \ldots, S_{n}^{*}$ and $T_{1}^{*}$ are equal to each other (Fig. 28.12). By rotating the chain $S_{1}^{*}, \ldots, S_{n}^{*}$ around the center of the circle $R_{1}^{*}$ so th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,170 |
28.41*. Each of the six circles touches four of the remaining five (Fig. 28.6). Prove that for any pair of non-touching circles (from these six) their radii and the distance between their centers are related by the equation $d^{2}=r_{1}^{2}+r_{2}^{2} \pm 6 r_{1} r_{2}$ (with “plus” if the circles do not lie one inside ... | 28.41. Let $R_{1}$ and $R_{2}$ be any pair of non-intersecting circles. The remaining four circles form a chain, so by the previous problem, the circles $S^{\prime}$ and $S^{\prime \prime}$, which touch $R_{1}$ and $R_{2}$ at the points of their intersection with the line of centers, intersect at right angles (Fig. 28.... | ^{2}=r_{1}^{2}+r_{2}^{2}\6r_{1}r_{2} | Geometry | proof | Yes | Yes | olympiads | false | 28,172 |
29.1. Prove that the stretching of a plane is an affine transformation. | 29.1. We need to prove that if \(A^{\prime}, B^{\prime}, C^{\prime}\) are the images of points \(A, B, C\) under a stretch relative to a line \(l\) with coefficient \(k\) and point \(C\) lies on line \(A B\), then point \(C^{\prime}\) lies on line \(A^{\prime} B^{\prime}\). Let \(\overrightarrow{A C}=t \overrightarrow{... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,173 |
29.2. Prove that under an affine transformation, parallel lines are transformed into parallel lines. | 29.2. By definition, the images of lines are lines, and from the mutual uniqueness of the affine transformation, it follows that the images of non-intersecting lines do not intersect. | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,174 |
29.3. Let $A_{1}, B_{1}, C_{1}, D_{1}$ be the images of points $A, B, C, D$ under an affine transformation. Prove that if $\overrightarrow{A B}=\overrightarrow{C D}$, then $\overrightarrow{A_{1} B_{1}}=\overrightarrow{C_{1} D_{1}}$.
From the previous problem, it follows that we can define the image of the vector $\ove... | 29.3. Let $\overrightarrow{A B}=\overrightarrow{C D}$. First, consider the case when points $A, B, C$, $D$ do not lie on the same line. Then $A B C D$ is a parallelogram. From the previous problem, it follows that $A_{1} B_{1} C_{1} D_{1}$ is also a parallelogram, so $\overrightarrow{A_{1} B_{1}}=\overrightarrow{C_{1} ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,175 |
29.5. Let $A^{\prime}, B^{\prime}, C^{\prime}$ be the images of points $A, B, C$ under the affine transformation $L$. Prove that if $C$ divides the segment $A B$ in the ratio $A C: C B = p: q$, then $C^{\prime}$ divides the segment $A^{\prime} B^{\prime}$ in the same ratio. | 29.5. By task 29.4, v) from the condition $q \overrightarrow{A C}=p \overrightarrow{C B}$ it follows that $q \overrightarrow{A^{\prime} C^{\prime}}=$ $=q L(\overrightarrow{A C})=L(q \overrightarrow{A C})=L(p \overrightarrow{C B})=p L(\overrightarrow{C B})=p \overrightarrow{C^{\prime} B^{\prime}}$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,177 |
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