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152. Let a random quadrilateral $ABCD$ be given, a) inscribed in a circle $S$;
b) circumscribed around a circle $S$.
Prove that the perpendicular dropped from the center of $S$ to the line connecting the points of intersection of the opposite sides of the quadrilateral passes through the point of intersection of its ... | 152. a) Let $P$ and $Q$ be the points of intersection of the opposite sides of a quadrilateral inscribed in a circle $S$, and let $R$ be the point of intersection of the diagonals; then, by Theorem 1 (p. 84), $R$ is the midpoint of $PQ$ (see Fig. 66a on p. 84). From this, the statement of the problem follows (see p. 86... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,645 |
153. Given a circle $S$ and a point $A$ outside it. Draw tangents from point $A$ to the circle using only one ruler.
Compare with problem 147 from $\S 3$ (p. 83). | 153. The points of tangency of circle $S$ and the tangents drawn to it from point $A$ coincide with the points of intersection of $\mathcal{S}$ with the polar $a$ of point $A$ (see p. 86). But the polar $a$ can easily be constructed using a single ruler (see Fig. 66, b in the text). After this, it only remains to conne... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,646 |
156. Prove that the lines connecting the vertices of triangle $A B C$ with the poles $A^{\prime}, B^{\prime}, C^{\prime}$ of the opposite sides of the triangle relative to a certain circle $S$ intersect at one point.
The theorem of problem 156 can also be formulated in another way. Two triangles $A B C$ and $A^{\prime... | 156. Let $a$ be the polar of point $A$ with respect to circle $S$. If we transform circle $S$ into a new circle $\bar{S}$ and point $A$ into point $\bar{A}$ by central projection, then line $a$ will transform into the polar $\bar{a}$ of point $\bar{A}$ with respect to $\bar{S}$; similarly, the pole $B$ of any line $b$ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,649 |
157. Given two triangles $A B C$ and $A_{1} B_{1} C_{1}$ and a circle $S$. Prove that if the lines connecting the corresponding vertices of these triangles intersect at one point, then the lines connecting the poles of the sides of triangle $A B C$ (relative to $S$) with the poles of the corresponding sides of triangle... | 157. Let $A^{\prime}, B^{\prime}$, $C^{\prime}$ be the poles of the sides of triangle $A B C$ and $A_{1}^{\prime}, \quad B_{1}^{\prime}, \quad C_{1}^{\prime}$ be the poles of the sides of triangle $A_{1} B_{1} C_{1}$ (Fig. 323). According to the theorem, the polar of point $A$ is the line $B^{\prime} C^{\prime}$, and t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,650 |
159. What theorems are obtained by means of polar transformation from the theorems of problems 118a), б); 122a), б); 123; $125 ; 126 ; 127 ; 129$ a), б) | 159. Theorems of problems 118a) and b) transform into each other under polar transformation (therefore, it would be sufficient to prove only one of these theorems).
Theorems of problems 122a) and b) also transform into each other (and again, it would be sufficient to prove only one of these theorems).
The direct and ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,652 |
164. Given a circle $S$ and three lines $l, l_{1}$ and $l_{2}$. Describe a quadrilateral $A B C D$ around $S$ such that vertices $A$ and $C$ lie on line $l$, vertex $B$ lies on line $l_{1}$, and vertex $D$ lies on line $l_{2}$.
A generalization of this problem is problem 183b) from § 5 (p. 120). | 164. This problem is dual to problem 143 of the previous paragraph (problems 143a and b correspond to the cases when the line $l$ does not intersect the circle $S$ or intersects it). Thus, this problem can be solved as follows: by performing a polar transformation with respect to the given circle, we will arrive at pro... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,657 |
167. Let $l$ be an arbitrary tangent to the inscribed circle $S$ of triangle $ABC$; $M, N, P$ are the points of intersection of $l$ with the sides of the triangle (see Fig. 83). Draw perpendiculars from the center $O$ of the circle $S$ to the lines $OM, ON$, $OP$; let $M_{1}, N_{1}$, and $P_{1}$ be the points of inters... | 167. Perform a polar transformation with respect to the circle $S$. The sides $B C, C A, A B$ of the triangle $A B C$ will transform into points $A^{\prime}, B^{\prime}, C^{\prime}$ on the circle $S^{\prime}$, such that the circumcircle of triangle $A B C$ will transform into the inscribed triangle $A^{\prime} B^{\prim... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,660 |
169. What theorem does the Simson line theorem transform into under polar transformation (Problem 85 from § 1, Chapter II of the second part of the book)? | 169. Under polar transformation with respect to the circle $S$, the inscribed triangle $A B C$ transforms into the circumscribed triangle $A^{\prime} B^{\prime} C^{\prime}$. An arbitrary point $L$ on the circle $S$ transforms into an arbitrary tangent $l$ to the same circle, and the foot $P$ of the perpendicular droppe... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,662 |
170. Into which theorem does the theorem "the medians of a triangle intersect at one point" transform under polar transformation, if the circle of polar transformation is taken to be the circumcircle $S$ of the triangle? | 170. In the polar transformation with respect to the circumscribed circle $S$ of triangle $A B C$, it transforms into the triangle $A^{\prime} B^{\prime} C^{\prime}$ inscribed around $S$
t... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,663 |
172. Which theorems do the following theorems transform into under polar transformation:
a) the altitudes of a triangle intersect at one point?
b) the angle bisectors of a triangle intersect at one point? | 172. a) Polar transformation translates triangle $ABC$ into a new triangle $A'B'C'$, and the heights $AP, BQ$, and $CR$ of triangle $ABC$ into points $P', Q'$, and $R'$ on the sides of triangle $A'B'C'$, such that $\angle A'OP' = \angle B'OQ' = \angle C'OR' = 90^\circ$, where $O$ is the center of the circle $S$ relativ... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,665 |
175. Use the properties of projective transformation of a line to prove theorem 136a) from § 8, p. $65-66$. | 175. Let the intersection points of the opposite sides of the quadrilateral $ABCD$ be denoted by $S_{1}$ and $S_{2}$ (Fig. 345). The projection of the line $AB$ onto the line $BC$ from the center $D$ translates the points $S_{1}, A, B$ on the line $AB$ to the points $C, S_{2}, B$ on the line $BC$; the projection of the... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,667 |
177. Through the midpoint $O$ of the chord $AB$ of circle $S$, two arbitrary chords $MN$ and $PQ$ are drawn. Prove that the segment $EF$, which is cut on $AB$ by the chords $MP$ and $NQ$, is bisected by the point $O$ (figure 93). | 177. If we project the circle $S$ onto the line $A B$ from the point $M$, the quadruple of points $A, B, N, P$ on the circle will transform into the points $A, B, O, E$ on the line $A B$; if we project $S$ onto $A B$ from the point $Q$, the same quadruple of points will transform into the points $A, B, F, O$. Therefore... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,669 |
182. a) A straight line $l$ and a point $P$ are given. On the line $l$, find a segment $X Y$ of a given length $a$, which is seen from $P$ at a given angle $\alpha$.
b) Two straight lines $l_{1}$ and $l_{2}$, and two points $P$ and $Q$, not lying on these lines, are given. Find a point $X$ on the line $l_{1}$ and a po... | 182. a) Draw an arbitrary circle $S$ passing through point $P$; let the required lines $P X$ and $P Y$ intersect it at points $X_{1}$ and $Y_{1}$ (Fig. $352, a$). Consider the following projective transformation of the circle $S$: the circle is projected from point $P$ onto a line $l$, then the line is shifted parallel... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,674 |
186. Given two lines $l_{1}$ and $l_{2}$, point $A$ on line $l_{1}$, point $B$ on line $l_{2}$, and point $P$, not belonging to either $l_{1}$ or $l_{2}$. Draw a line through $P$ intersecting $l_{1}$ and $l_{2}$ at points $X$ and $Y$ such that
a) $A X: B Y=m: n$, where $m: n$ is given;
b) $A X \cdot B Y=k^{2}, \quad$ ... | 186. a) Let $X$ and $Y$ be the points of intersection of a given line with the lines $l_{1}$ and $l_{2}$ (Fig. 354), a). We project the line $l_{1}$ onto the line $l_{2}$ from the point $P$. We then align the line $l_{2}$ with the line $l_{1}$ in such a way that the point $B$ coincides with the point $A$, and then subj... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,678 |
187. On a plane, three lines $l_{1}, l_{2}$ and $l_{3}$ intersect at point $P$. Draw a line through $P$ such that the given three lines cut off equal segments on it. | 187. Let $A, B, C$ be the pairwise intersection points of the lines $l_{1}, l_{2} \| l_{3} ; X, Y, Z$ be the points of intersection of a transversal line $l$ with the lines $l_{1}, l_{2}$ and $l_{3}$; by the given condition, $X Z=Z Y$ (Fig. $3 \overline{\text { che }}$ ). Let $T$ be the point of intersection of the lin... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,679 |
190. a) Draw a line through the given point $P$ parallel to the given line $l$.
b) From the given point $M$, lay off a segment $M N$ equal to and parallel to the given segment $A B$.
c) Drop a perpendicular from the given point $P$ to the given line $l$. | 190. a) It can be considered that the position of the sought line $m$ (see Fig. 358, a) is determined by two points: point $P$ and the infinitely distant point of line $l$. Consequently, point $M$, which is the pole of $m$ relative to the given circle $S$, is the intersection point of the polar $p$ of point $P$ and the... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,682 |
200. a) Prove that any two lines $l_{1}$ and $l_{2}$ in Lobachevsky's non-Euclidean geometry have an axis of symmetry $l$ (symmetry with respect to a line is defined in non-Euclidean geometry of Lobachevsky in exactly the same way as in Euclidean geometry; see the beginning of § 1 of Chapter II of the first part). In t... | 200. a) Let $l_{1}$ and $l_{2}$ be two intersecting lines in non-Euclidean geometry of Lobachevsky. By a non-Euclidean motion, align their point of intersection with the center of the circle
... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,692 |
201. A quadrilateral in Lobachevsky's non-Euclidean geometry, where the diagonals intersect and are divided proportionally at the point of intersection, is called a non-Euclidean parallelogram. Prove that:
a) opposite sides of a non-Euclidean parallelogram are equal;
b) opposite angles of a non-Euclidean parallelogra... | 201. Let $A B C D$ be a non-Euclidean parallelogram. By a non-Euclidean motion, the point of intersection of its diagonals coincides with the center $O$ of the circle $K$. From the fact that the non-Euclidean lengths of the segments $O A$ and $O C$ (respectively, $O B$ and $O D$) are equal, it follows that these segmen... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,693 |
203. Prove that in Lobachevsky's non-Euclidean geometry, the altitudes of an acute triangle intersect at one point. Does this theorem remain true for obtuse triangles? | 203. Let $H$ be the intersection point of the altitudes $A K$ and $B L$ of the acute-angled triangle $A B C$. [These altitudes intersect because both lie inside the triangle: if the altitude dropped from vertex $A$ intersected the extension of side $B C$ beyond point $C$, then angle $C$ of the triangle would be obtuse ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,695 |
204. Prove that in non-Euclidean Lobachevsky geometry, the medians of a triangle intersect at one point. | 204. First of all, let's prove that the line $DE$, connecting the midpoints of sides $AB$ and $AC$ of triangle $ABC$ in Lobachevsky geometry, and the perpendicular raised from the midpoint of side $BC$, are mutually perpendicular. Drop perpendiculars $AK$, $BL$, and $CM$ from the vertices of the triangle to the midline... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,696 |
205. Does the theorem that the medians of a triangle are divided in the ratio $2: 1$, counting from the vertices, hold in non-Euclidean Lobachevsky geometry? | 205. No. For proof, it is sufficient to consider an equilateral triangle.
Fig. 375. Triangle \(ABC\) with the center at the midpoint \(O\) of the circle \(K\) (Fig. 375). The medians \(AD\)... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,697 |
206. Prove that in Lobachevsky's non-Euclidean geometry, the three perpendiculars erected at the midpoints of the sides of a triangle belong to one pencil (i.e., either all intersect at one point, or all are parallel to each other, or all three are perpendicular to one line; see p. 152). | 206. If the perpendiculars erected to the two sides of a triangle $ABC$ at their midpoints intersect at one point $O$, then the third perpendicular also passes through $O$; the proof of this fact does not differ from the usual one (compare with the proof that in non-Euclidean Lobachevsky geometry, the three bisectors o... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,698 |
207. a) Prove that the sum of the angles of a triangle in non-Euclidean Lobachevsky geometry is always less than $180^{\circ}{ }^{1}$ ).
b) Prove that the sum of the angles of an $n$-sided polygon in non-Euclidean Lobachevsky geometry is always less than $180^{\circ}(n-2)$. | 207. a) Move the triangle $ABC$ so that the point of intersection of its bisectors (see figure, p. 144) coincides with the center $O$ of the circle $\mathbb{K}$ (Fig. 377). In this case, each of the angles $OAB$, $OAC$, $OBA$, $OBC$, $OCA$, and $OCB$ will be acute (simultaneously in the Euclidean and non-Euclidean sens... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,699 |
212. Let circle $S$ touch two circles $S_{1}$ and $S_{2}$ simultaneously. Prove that the line connecting the points of tangency passes through the center of similarity of circles $S_{1}$ and $S_{2}$.
In another connection, this problem is given in the second part of the book (see problem 55 from § 1 of chapter I). | 212. Let circle $S$ touch $S_{1}$ and $S_{2}$ at points $A$ and $B$; $O_{1}, O_{2}$, and $\bar{O}$ be the centers of $S_{1}, S_{2}$, and $S$, respectively; $O$ be the intersection point of $A B$ and $O_{1} O_{2}$ (Fig. 384). We perform an inversion with center $O$ and power $k=O A \cdot O B$. In this case, point $A$ is... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,703 |
215. On the segments $A M$, $M B$ and $A B$ of one straight line, as diameters, semicircles $S_{1}$, $S_{2}$ and $S$ are constructed (Fig. 140). A perpendicular $M D$ is erected to the line $A B$ and circles $\Sigma_{1}$ and $\Sigma_{2}$ are inscribed in the curvilinear triangles $A D M$ and $B D M$. Prove that:
a) th... | 215. a) Let $r_{1}, r_{2}$, and $r$ be the radii of the circles $S_{1}, S_{2}$, and $S$; $r=r_{1}+r_{2}$. Under inversion with center $M$ and (negative!) power $k=M A \cdot M B$, the circle $S$ and the line $M D$ map to themselves, the circles $S_{1}$ and $S_{2}$ map to tangents to $S$ at points $B$ and $A$, and the ci... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,706 |
216. Prove that if each of the four circles $S_{1}, S_{2}, S_{3}$ and $S_{4}$ touches two adjacent ones (with $S_{1}$ being adjacent to $S_{2}$ and $S_{4}$; see Fig. 141), then the four points of tangency lie on one circle $\Sigma$. | 216. Inversion with the center at point $A$ of tangency of $S_{2}$ and $S_{1}$ transforms figure 141 into figure 389; it is, of course, sufficient to show that the points $B^{\prime}, C^{\prime}$ and $D^{\prime}$ of the latter figure lie on the same straight line $\mathrm{\Sigma}^{\prime}$. Let $M N$ be the common tang... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,707 |
217. a) On the plane, there are six points $A_{1}, A_{2}, A_{3}$; $B_{1}, B_{2}, B_{3}$. Prove that if the circumcircles of triangles $A_{1} A_{2} B_{3}, A_{1} A_{3} B_{2}$, and $A_{2} A_{3} B_{1}$ intersect at one point, then the circumcircles of triangles $B_{1} B_{2} A_{3}, B_{1} B_{3} A_{2}$, and $B_{2} B_{3} A_{1}... | 217. a) Let's perform an inversion with the center at point $P$ of intersection of the circumcircles of triangles $A_{1} A_{2} B_{3}$,
Fig. 389. $A_{1} A_{3} B_{2}$, $A_{2} A_{3} B_{1}$. In... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,708 |
221. Prove that the radius $r$ of the inscribed circle of a triangle cannot exceed half the radius $R$ of the circumscribed circle; moreover, $r=\frac{1}{2} R$ if and only if the triangle is equilateral.
), it follows that
$$
d^{2}=R^{2}-2 R r=R(R-2 r)
$$
and, therefore,
$$
R-2 r \geqslant 0, r \leqslant \frac{R}{2}
$$
which was to be proved.
If $r=\frac{R}{2}$, then $d=0$, i.e., the circumscribed and inscribed circles of the ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,712 |
224. Let $\Sigma_{1}$ and $\Sigma_{2}$ be two circles touching each other internally. In the figure formed by them (Fig. 153), circles $S_{0}, S_{1}, S_{2}, \ldots$ are inscribed sequentially; the center of $S_{0}$ lies on the line of centers $A B$ of the circles $\Sigma_{1}$ and $\Sigma_{2}$, and $S_{n}$ touches $S_{n... | 224. a) Let us perform an inversion that transforms the circles $\Sigma_{1}$ and $\Sigma_{2}$ into two parallel lines $\Sigma_{1}^{\prime}$ and $\Sigma_{2}^{\prime}$ (see Theorem 2 on p. 194); the center of this inversion will be the point A of tangency of $\Sigma_{1}$ and $\Sigma_{2}$. In this case, the circles $S_{0}... | r_{n}=\frac{4R_{1}R_{2}(R_{1}-R_{2})}{(R_{1}+R_{2})^{2}+(4n^{2}-1)(R_{1}-R_{2})^{2}} | Geometry | proof | Yes | Yes | olympiads | false | 28,715 |
226. Let $S_{0}, S_{1}, S_{2}, S_{3}, \ldots$ be circles that touch the inside of the semicircle $\Sigma$ and its diameter $AB$; $S_{v}$ passes through the center 2, and $S_{n}$ touches $S_{n-1} (n=1,2,3, \therefore)$
 Let's perform an inversion that transforms the circle $\Sigma$ into two intersecting lines $\Sigma'$ and $AB$ (the center of this inversion will be the endpoint $A$ of the diameter); by the property of inversion, the line $\Sigma'$ is perpendicular to $AB$. We set the degree of inversion to $2R$; in this case, ... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,717 |
228. Given an angle MAN and a point $O$ not lying on the side of the angle. Draw a line through $O$ intersecting the sides of the angle at points $X$ and $Y$, such that the product $O X \cdot O Y$ has a given value $k$. | 228. Suppose the problem is solved. From OX. $O Y=k$, it follows that $X$ is obtained from point $Y$ by the inversion with center $O$ and power $k$; therefore, $X$ lies on the circle $S$, which is obtained from the line $A N$ by the inversion with center $O$ and power $k$ (i.e., $k$), i.e., $X$ is the intersection poin... | 4 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,719 |
230. Given three points $A, B$ and $C$. Draw a line $l$ through point $A$ such that:
a) the product of the distances from $B$ and $C$ to the line $l$ has a given value;
b) the difference of the squares of the distances from $B$ and $C$ to the line $l$ has a given value. | 230. a) Suppose a line $l$ is drawn, and let $X$ and $Y$ be the feet of the perpendiculars dropped from points $B$ and $C$ to this line; $B X \cdot C Y = k$ (Fig. 404). Translate the triangle $A Y C$ parallel to the segment $C B$ by the length of $C B$ to the position $A' Y' B$; in this case, point $A'$ can be easily d... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,721 |
232. Construct a circle,
a) passing through two given points $A$ and $B$ and touching a given circle (or line) $S$;
b) passing through a given point $A$ and touching two given circles (or two lines, or a circle and a line) $S_{1}$ and $S_{2}$.
See also problems 49a), b) from § 1 of part I and 247a) from § 3 of this ... | 232. a) Let's perform an inversion with center $A$. In this case, point $B$ will transition to another point $B^{\prime}$, the circle (or line) $S$ - to a circle (or line) $S^{\prime}$, and some circle $\Sigma$ - to a line $\Sigma^{\prime}$, which passes through point $b^{\prime \prime}$ and is tangent to the circle $S... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,723 |
233. Construct a circle passing through two given points $A$ and $B$ and
a) perpendicular to a given circle (or line) $S$;
b) intersecting the given circle $S$ at diametrically opposite points.
A generalization of problem 233a) is problem 235z). | 233. a) First solution. According to the definition of symmetry with respect to a circle (see above, pp. 172-173), the sought circle (or line) $\Sigma$ must also pass through the point $A^{\prime}$, which is symmetric to $A$ with respect to the circle (or line) $S$. The problem has a unique solution if $A^{\prime}$ doe... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,724 |
234. Construct a circle passing through a given point $A$ and
a) perpendicular to two given circles (or two lines, or a circle and a line) $S_{1}$ and $S_{2}$;
b) perpendicular to a given circle (or line) $S_{1}$ and intersecting another given circle $S_{8}$ at diametrically opposite points;
c) intersecting two give... | 234. a) First solution. According to the definition of symmetry with respect to a circle, the sought circle (or line) $\mathrm{\Sigma}$ must pass through the points $A^{\prime}$ and $A^{\prime \prime}$, which are symmetric to $A$ with respect to the circles (or lines) $S_{1}$ and $S_{2}$: The problem generally has a un... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,725 |
235. Construct a circle,
a) passing through two given points $A$ and $B$, intersecting a given circle (or line) $S$ at a known angle $a$;
b) passing through a given point $A$, intersecting two given circles (or two lines, or a line and a circle) $S_{1}$ and $S_{2}$ at known angles $\alpha$ and $\beta$. | 235. a) Inversion with center $A$ translates the point $B$ into another point $B^{\prime}$, and the circle (or line) $S$ - into a circle
Fig. 405. (or line) $S^{\prime}$; the desired circle... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,726 |
239. On the plane, a segment $AB$ is given. Using only one compass, double this segment (i.e., find a point $C$ on the extension of $AB$ beyond point $B$ such that $AC = 2AB$).
From the solution to problem 239, it follows that with the help of one compass, any given segment $AB$ can be increased by any integer number ... | 239. Let's draw a circle with the center at point $B$ and radius $BA$, and make marks $M$, $N$, $C$ on this circle such that $AM = MN = NC = BA$ (Fig. 414). Clearly, $AM$, $MN$, and $NC$ are the sides of an inscribed equilateral triangle; therefore, $C$ and $A$ are diametrically opposite points on the circle, i.e., $C$... | AC=2AB | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,730 |
240. On the plane, there is a circle $\Sigma$ with a known center $O$ and a point $A$. Using only one compass, find the point $A^{\prime}$, symmetric to $A$ with respect to the circle $\Sigma$. | 240. Let point $A$ be outside the circle $\Sigma$ (Fig. $415, a$). Draw a circle with center at $A$ and radius $A O$ (where $O$ is the center of $\Sigma$): let $M$ and $N$ be the points of intersection of this circle with the circle $\Sigma$. From points $M$ and $N$ as centers, draw two circles with radii $M O$ and $N ... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,731 |
241. On a plane, a circle S is given. Using only one compass, find the center of this circle. | 241. Draw an arbitrary circle $\Sigma$ with center at point $A$ of the given circle $\mathcal{S}$, intersecting circle $S$ at two points $P$ and $Q$; let $K$ be the point symmetric to $A$ with respect to the line $PQ$ (Fig. 417; constructing point $K$ using only a compass is obvious). From the similarity of isosceles t... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,732 |
245. Prove that the common chord of two intersecting circles bisects the segment of their common external tangent, enclosed between the points of tangency. | 245. Since the common chord is the radical axis of two circles, the tangents drawn to the circles from any point on it must be equal. | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,735 |
246. Given a circle $S$ and a point $M$ outside it. Through the point $M$ a variable circle $\Sigma$ is drawn, intersecting $S$ at points $A, B$. Find the geometric locus of the points of intersection of the line $A B$ with the tangent to $\Sigma$ at the point $M$. | 246. If $P$ is a point of the required geometric locus, then $P A \cdot P B = P M^2$ (by the property of the tangent and secant of a circle). Therefore, the power $P A \cdot P B$ of point $P$ with respect to $S$ is equal to the power $P M^2$ of this point with respect to point $M$ (see above, p. 222). It follows that $... | 0 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,736 |
247. Construct a circle,
a) passing through two given points $A$ and $B$ and tangent to a given circle (or line) $S$;
b) perpendicular to two given circles $S_{1}$ and $S_{2}$ and tangent to a given circle (or line) $S$.
See also problem 496) from § 1 of Chapter I of the second part of the book and 232a) and 2366) f... | 247. a) If $S$ is a line parallel to $A B$, then the desired circle $\Sigma$ touches $S$ at the midpoint of the segment $A_{1} B_{1}$, where $A_{1}$ and $B_{1}$ are the projections of $A$ and $B$ onto $S$ (Fig. 421, a). If $S$ is a line intersecting $A B$ at point $M$, then the power of point $M$ relative to the desire... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,737 |
248. Construct a circle passing through a given point $M$
a) perpendicular to two given circles $S_{1}$ and $S_{2}$;
b) perpendicular to a given circle $S_{1}$ and intersecting another given circle $S_{2}$ at diametrically opposite points;
c) intersecting two given circles $S_{1}$ and $S_{1}$ at diametrically opposi... | 248. a) First solution. Circles perpendicular to $S_{1}$ and $S_{2}$ form a bundle; it is required to find the circle of this bundle passing through $M$. If $S_{1}$ and $S_{2}$ do not intersect, then all circles of the bundle pass through two specific points $A$ and $B$ (compare with the solution of problem 247 b)); th... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,738 |
249. Given three circles $S_{1}, S_{2}$ and $S_{3}$. Construct a circle $S$,
a) perpendicular to $\dot{S_{1}}, S_{2}$ and $S_{3}$;
b) intersecting $S_{1}, S_{2}$ and $S_{3}$ at diametrically opposite points
c) such that $S_{1}, S_{2}$ and $S_{3}$ intersect $S$ at diametrically opposite points.
See also problem 236a... | 249. a) The center $O$ of the desired circle $\Sigma$ coincides with the radical center of $S_{1}, S_{2}$, and $S_{3}$ (see above, p. 226); the radius is equal to the length of the tangent drawn from $O$ to $S_{1}$. The problem has one solution if $O$ lies outside $S_{1}, S_{2}, S_{3}$, and no solutions in the opposite... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,739 |
250. Given two circles $S_{1}$ and $S_{2}$. Find the geometric locus of points $M$ such that
a) the difference of the squares of the lengths of the tangents drawn from $M$ to $S_{1}$ and to $S_{2}$ is a given value $a$;
b) the ratio of the lengths of the tangents drawn from $M$ to $S_{1}$ and to $S_{2}$ is a given va... | 250. a) Let $M$ be a variable point of a certain geometric locus, $Q$ - the projection of $M$ onto the line of centers $O_{1} O_{2}$, of the given circles $S_{1}$ and $S_{2}$ with radii $r_{1}$ and $r_{2}$ (fig. $424, a$). From the reasoning, analogous to that given on pages $224-225$, we obtain:
$$
2 O_{1} O_{2} \cdo... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,740 | |
251. Let there be two circles $S_{1}$ and $S_{2}$ and a line $l$ intersecting $S_{1}$ at points $A_{1}$ and $B_{1}$ and $S_{2}$ at points $A_{2}$ and $B_{2}$. Prove that:
a) if $l$ passes through the center of similarity of $S_{1}$ and $S_{2}$, then the tangents to $S_{1}$ at points $A_{1}$ and $B_{1}$ intersect with t... | 251. a) Let $l$ pass through the center of similarity $O$ of circles $S_{1}$ and $S_{2}$ (see diagram $178, a$ in the text). In this case, the central similarity transformation with center $O$ maps $S_{1}$ to $S_{2}$ and $l$ to itself; therefore, $l$ forms the same angles $\alpha$ with $S_{1}$ and $S_{2}$. From this, i... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,741 |
253. a) Let $d_{1}, d_{2}, d_{8}, \ldots, d_{n}$ be the distances from a point $M$, located on the arc $A_{2} A_{n}$ of the circle circumscribed around a regular $n$-gon $A_{1} A_{2} \ldots A_{n}$, to the vertices $A_{2}, A_{2}, A_{2}, \ldots, A_{n}$ of this $n$-gon. Prove that
$$
\frac{1}{d_{1} d_{2}}+\frac{1}{d_{2} ... | 253. a) Let's perform inversion with the center at point $M$ and the radius of inversion 1. In this case, points $A_{1}, A_{2}, A_{3}, \ldots, A_{n}$ transition to points $A_{1}^{\prime}, A_{2}^{\prime}, A_{3}^{\prime}, \ldots, A_{n}^{\prime}$, located on one straight line (Fig. 426). Denote the length of the side of t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,743 |
254. a) Let $p_{1}, p_{2}, \ldots, p_{2 n-1}, p_{2 n}$ be the distances from an arbitrary point $M$ on the circle $S$ to the sides $A_{1} A_{2}$, $A_{2} A_{3}, \ldots, A_{2 n-1} A_{2 n}, A_{2 n} A_{2}$ of the inscribed $2 n$-gon $A_{1} A_{2} A_{3} \ldots A_{3 n}$. Prove that
$$
p_{1} p_{8} p_{5} \ldots p_{2 n-1}=p_{2}... | 254. a) Let us perform an inversion with the center at point \( M \). In this case, the vertices \( A_{1}, A_{2}, A_{3}, \ldots, A_{2 n} \) of the inscribed \( 2 n \)-gon will be mapped to \( 2 n \) points \( A_{1}^{\prime}, A_{2}^{\prime}, A_{3}^{\prime}, \ldots, A_{2 n}^{\prime} \), located on a single straight line ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,744 |
255. Let $a_{1}, a_{2}, \ldots, a_{n-1}, a_{0}$ be the lengths of the sides $A_{1} A_{2}$, $A_{2} A_{8}$, $A_{8} A_{4}$, $\ldots$, $A_{n} A_{1}$ of the n-sided polygon $A_{1} A_{8} A_{3} \ldots A_{n}$ inscribed in the circle $S$, and $p_{2}, p_{2}, \ldots, p_{n-1}, p_{0}$ be the distances from an arbitrary point $M$ on... | 255. Let's perform an inversion with the center at point $M$. The vertices $A_{1}, A_{2}, A_{3}, \ldots, A_{n}$ of the $n$-sided polygon will transition to $n$ points $A_{1}^{\prime}, A_{2}^{\prime}, A_{3}^{\prime}, \ldots, A_{n}^{\prime}$, located on the same straight line (see Fig. 427); in this case,
$$
A_{1}^{\pri... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,745 |
257. Find the geometric locus of points, the ratio of whose distances from two given points is constant. | 257. Obviously, the sought geometric locus is characterized by the fact that the double ratio $\frac{A M_{1}}{B M_{1}}: \frac{A M_{2}}{B M_{2}}$, where $M_{1}$, $M_{2}$ are any two points of this geometric locus, equals one. By the property $\Gamma$, it follows that if some inversion maps points $A, B$ to points $A^{\p... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,747 |
258. Prove Ptolemy's theorem: if a quadrilateral can be inscribed in a circle, then the sum of the products of the opposite sides of this quadrilateral is equal to the product of its diagonals.
See also problem 269 (p. 246), and problem 86 b) from § 1 of chapter II of the second part of the book ${ }^{1}$ ).[^44] | 258. The ratio of the sum of the products of the opposite sides of an arbitrary quadrilateral $ABCD$ to the product of its diagonals
$$
\frac{AB \cdot CD + AD \cdot BC}{AC \cdot BD}
$$
can be represented as
$$
\begin{aligned}
& \frac{AB \cdot CD}{AC \cdot BD} + \frac{AD \cdot BC}{AC \cdot BD} = \\
= & \frac{AB}{DB} ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,748 |
260. Find the geometric locus of points $M$ such that the ratio of the lengths of the tangents drawn from $M$ to two given circles $S_{1}$ and $S_{2}$ has a constant value.
See also problem 88 from 1 of Chapter II of the second part of the book and problem 2506) from § 3 of this chapter (p. 231). If we consider $S_{1}... | 260. This problem is very close to problem 257. First of all, it is obvious that if the ratio of the lengths of the tangents drawn from some point $M_{1}$ to two circles $S_{2}$ and $S_{1}$ is equal to the ratio of the lengths of the tangents drawn from another point $M_{2}$ to the same circles, then the double ratio $... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,750 |
261. Prove the following $o b o$ consequence of Ptolemy's theorem: if circles $S_{1}, S_{2}, S_{3}$ and $S_{4}$ touch the same fifth circle (or line) $\Sigma$ (Fig. 184), then the following relation holds:
$$
t_{12} t_{34} + t_{24} t_{13} = t_{14} t_{23}
$$
where $t_{12}$ is the segment of the common tangent of circl... | 261. The current problem is very close to problem 2.58. First of all, it is not difficult to see that if $S_{1}, S_{2}, S_{\mathrm{z}}$ and $S_{1}$ are four circles, then the ratio
$$
\frac{t_{12} t_{31}+t_{21} t_{23}}{t_{13} t_{24}}
$$
can be rewritten in the form
 $\Sigma$, this formula is a direct ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,752 |
268. Into which theorem does the proposed statement turn under the niversin: a circle is the geometric locus of points equidistant from one point (definition of a circle)?
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 268. When inverting with respect to a circle, the center $O$ of which does not lie on the given circle $S$, $S$ transforms into some other circle $S'$. Let $Z'$ be the point into which the center $Z$ of the circle $S$ transforms, and $M'$ be the point into which some point $M$ of the circle $S$ transforms (Fig. 438). A... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 28,756 |
269. a) Into which theorem does the theorem "each side of a triangle is less than the sum of the other two sides" transform under inversion?
b) Prove the theorem converse to Ptolemy's theorem (see the problem 258, p. 236): if the sum of the products of the opposite sides of a certain quadrilateral is equal to the prod... | 269. Perform an inversion with the center at point $O$; let $A^{\prime}, B^{\prime}$, and $C^{\prime}$ be the three points into which the vertices $A, B, C$ of triangle $ABC$ (Fig. 439, a) are transformed by this inversion.
Using formula (*) on page 233, we have:
$A B=A^{\prime} B^{\prime} \frac{k}{O A^{\prime} \cdot... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,757 |
276. a) Apply axial inversion to restate the Apollonius problem (see problem 237a) from § 2, p. 208) in the case where the axis of similarity of the given circles $S_{1}, S_{2}$ and $S_{3}$ does not intersect any of them.
b) Apply axial inversion to prove the theorem in problem 261 (p. 243) in the case where the axis ... | 276. a) We will consider the circles $S_{1}, S_{2}$ and $S_{3}$ as directed and transform them using a specially chosen axial inversion into points $S_{1}^{\prime}, S_{2}^{\prime}$ and $S_{3}^{\prime}$ (see above, p. 295).
In this case, the circle $\Sigma$, which is tangent to $S_{1}, S_{2}$, and $S_{3}$, will transfo... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,763 |
291. Prove that in Lobachevsky's non-Euclidean geometry, the altitudes of an acute triangle intersect at one point. Does this theorem also hold for obtuse triangles? | 291. Let's translate the intersection point of the heights $A K$ and $B L$ of an acute-angled triangle $A B C$ by a non-Euclidean motion to the center $O$ of the circle $\mathfrak{K}$ (the heights of an acute-angled triangle necessarily intersect at an interior point; see the beginning of the solution to problem 203); ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,778 |
292. Prove that the sum of the angles of any triangle in non-Euclidean Lobachevsky geometry is less than $180^{\circ}$. | 292. Translate the vertex $A$ of an arbitrary triangle $A B C$ by a parallel motion to the center $O$ of the circle $U$; then the triangle $A B C$ will transform into the triangle $O B^{\prime} C^{\prime}$, as shown in Fig. 466. It is evident that the sum of the angles of the triangle $O B^{\prime} C^{\prime}$ (which i... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,779 |
298. The set of all cycles in Lobachevsky's non-Euclidean geometry that are perpendicular to two given cycles \( S_{1} \) and \( S_{2} \) is called a bundle of cycles. List all possible types of bundles of cycles in Lobachevsky's non-Euclidean geometry. Prove that for each bundle II, there are infinitely many cycles th... | 298. Obviously, the pencil of cycles of Lobachevsky's hyperbolic geometry does not differ from the pencil of circles of ordinary (Euclidean) geometry (see § 3 of this chapter); only in accordance with the definition of points in hyperbolic geometry, here one should consider not the entire circles but only arcs of them,... | 19 | Geometry | proof | Yes | Yes | olympiads | false | 28,784 |
1.1. (England, 68). Let $a_{1}, a_{2}, \ldots, a_{7}$ be integers, and $b_{1}, b_{2}, \ldots, b_{7} \ldots$ the same numbers taken in a different order. Prove that the number
$$
\left(a_{1}-b_{1}\right)\left(a_{2}-b_{2}\right) \ldots\left(a_{7}-b_{7}\right)
$$
is even. | 1.1. The product of the numbers $c_{i}=a_{i}-b_{i}, i=1,2, \ldots, 7$, is even, since at least one of them is even (if each of the numbers $c_{i}$ were odd, then their sum would also be odd, equal to
$$
\begin{aligned}
c_{1}+c_{2}+\ldots+c_{7}= & \left(a_{1}-b_{1}\right)+\left(a_{2}-b_{2}\right)+\ldots+\left(a_{7}-b_{... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 28,787 |
1.2. (New York, 76). Let $a, a_{0}, a_{1}, \ldots, a_{n}$ be arbitrary integers. Is it true that the integer
$$
\sum_{k=0}^{n}\left(a^{2}+1\right)^{3 k} a_{k}
$$
is divisible by $a^{2}+a+1$ (or by $a^{2}-a+1$) if and only if the number
$$
\sum_{k=0}^{n}(-1)^{k} a_{k}
$$
is divisible by $a^{2}+a+1$ (or respectively ... | 1.2. Let $b_{\varepsilon}=a^{2}+\varepsilon a+1$, where $\varepsilon^{2}=1$. Then from the equalities
$$
\begin{gathered}
\left(a^{2}+1\right)^{3}=\left(b_{\mathrm{e}}-\varepsilon a\right)^{3}=-\varepsilon a^{3}\left(\bmod b_{\varepsilon}\right), \\
-\varepsilon a^{3}=-\varepsilon a b_{\varepsilon}+a^{2}+\varepsilon a... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 28,788 |
1.3. (Czechoslovakia, 52; England, 65). In an infinite "triangular" table
\[
\begin{aligned}
& a_{1,0} \Leftrightarrow \&, \\
& a_{2,-1} a_{2,0} a_{2, i} \\
& a_{3,-2} a_{3,-1} a_{3,0} a_{3,1} \quad a_{3,2} \\
& a_{4,-3} a_{4,-2} a_{4,-1} a_{4,0} a_{4,1} a_{4,2} a_{4,3}
\end{aligned}
\]
$a_{i, 0}=1$, and each number ... | 1.3. Consider the function defined on the set of integers:
$$
f(m)=\left\{\begin{array}{l}
0, \text { if } m \text { is even, } \\
1, \text { if } m \text { is odd, }
\end{array}\right.
$$
and construct a table of numbers defined by the formula $b_{n, k}=f\left(a_{n, k}\right)$. Then for $n>1$ we have
$$
\begin{alig... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 28,789 |
1.4. (CSSR, 71). Prove that for any prime number $p>2$ the numerator $m$ of the fraction
$$
\frac{m}{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{p-1} \quad(m, n \in \mathbf{N})
$$
is divisible by $p$. | 1.4. Note that the number $p-1$ is even, and transform the fraction $m / n$ as follows:
$$
\begin{aligned}
& \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{p-1}= \\
& =\left(\frac{1}{1}+\frac{1}{p-1}\right)+\left(\frac{1}{2}+\frac{1}{p-2}\right)+\left(\frac{1}{3}+\frac{1}{p-3}\right)+\ldots \\
& \cdots+\left(\fra... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 28,790 |
1.5. (New York, 75). Prove that for every integer value $n>1$ the number $n^{n}-n^{2}+n-1$ is divisible by $(n-1)^{2}$.
| 1.5. Let $n>2$, then by theorem 4 the equalities $n^{n}-n^{2}+n-1=\left(n^{n-2}-1\right) n^{2}+(n-1)=$ $=(n-1)\left(n^{n-3}+\ldots+1\right) n^{2}+(n-1) n^{0}=(n-1)\left(n^{n-1}+\ldots+n^{2}+n^{0}\right)$ hold.
Since $n=1(\bmod (n-1))$, for each value of $k=0,2, \ldots, \ldots, n-1$ we have
$$
n^{k}=1(\bmod (n-1)),
$$... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 28,791 |
1.6. (NPR, 65). Prove that only one triplet of natural numbers, greater than one, has the property that the product of any two of these numbers, increased by 1, is divisible by the third.
| 1.6. Let the triple of numbers $a, b, c \in \mathbf{N}$ satisfy the conditions
$$
(a b+1) \div c, \quad(a c+1) \div b, \quad \text { and } \quad(b c+1) \div a
$$
Note that the numbers $a, b, c$ are pairwise coprime (if, for example, $(a, b)>1$, then $(a c, b)=d>1$ and the number $a c+1$ does not divide $d$, let alone... | 2,3,7 | Number Theory | proof | Yes | Yes | olympiads | false | 28,792 |
1.7. (England, 76). Prove that for any value of $n \in \mathbf{Z}^{+}$ the number $19 \cdot 8^{n}+17$ is composite. | 1.7. If $n=2 k$ (here and throughout $k \in \mathbf{Z}^{+}$), then
$$
19 \cdot 8^{2 k}+17=18 \cdot 8^{2 k}+1 \cdot(1+63)^{k}+(18-1) \Rightarrow 0(\bmod 3) .
$$
If $n=4 k+1$, then
$$
\begin{aligned}
19 \cdot 8^{4 k+1}+17 & =13 \cdot 8^{4 k+1}+6 \cdot 8 \cdot 64^{2 k}+17= \\
& =13 \cdot 8^{4 k+1}+39 \cdot 64^{2 k}+9 \... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 28,793 |
1.8. (Canada, 83). Prove that for any prime number $p$ there are infinitely many numbers of the form $2^{n}-n$ (where $n \in \mathbf{N}$), divisible by $p$. | 1.8. If $p=2$, then each of the numbers $2^{(2 k)}-(2 k)$, where $k \in \mathrm{N}$, is divisible by $p$. Let $p>2$, then, considering Fermat's Little Theorem (Theorem 25), we get
$$
2^{p-1}=1(\bmod p), 2^{m(p-1)}=1(\bmod p), \quad m \in \mathbf{N}
$$
If $m=-1(\bmod p)$, then we have
$$
2^{m(p-1)}-m(p-1)=2^{m(p-1)}+... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 28,794 |
1.9. (CSSR, 73). Prove that there are infinitely many values of $n \in \mathbf{N}$, for which any number of the form $m^{4}+n(m \in \mathbf{N})$ is composite. | 1.9. Let $n=4 k^{4}$, where $k=2,3, \ldots$ Then for any value of $m \in \mathbf{N}$, the number
$$
\begin{aligned}
m^{4}+n & =m^{4}+4 k^{4}=\left(m^{4}+4 m^{2} k^{2}+4 k^{4}\right)-4 m^{2} k^{2}= \\
& =\left(m^{2}+2 k^{2}\right)^{2}-(2 m k)^{2}=\left(m^{2}+2 m k+2 k^{2}\right)\left(m^{2}-2 m k+2 k^{2}\right)= \\
& =\... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 28,795 |
1.10. (CSSR, 79). Find all natural numbers $n>2$, not exceeding the number 10000000 and possessing the following property: any number $m$, coprime with $n$ and satisfying the inequalities $1<m<n$, is prime. | 1.10. Let the number $n$ have the property required by the problem. If it is not divisible by 2, then $np_{k+1}^{2}
$$
which contradicts the necessary condition
$$
p_{1} p_{2} . . p_{k} \leqslant n < p_{k+1}^{2}
$$
The obtained contradiction shows that the considered case is impossible. Thus, it is proven that all p... | {3;4;6;8;12;18;24;30} | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 28,796 |
1.11. (FRG, 77). Let $a>1$ be a natural number. Find all numbers that are divisors of at least one of the numbers
$$
a_{n}=\sum_{k=0}^{n} a^{k}, \quad n \in \mathbf{N}
$$ | 1.11. We will prove that the desired set $M$ consists of all numbers $m \in \mathbf{N}$ that are coprime with the number $a$. If some number $m \in \mathbf{N}$ has a common divisor $d>1$ with the number $a$, then $m \notin M$. Indeed, for any $n \in \mathbf{N}$ we have
$$
\left(a_{n}, a\right)=\left(\sum_{k=0}^{n} a^{... | proof | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 28,797 |
1.12. (New York, 74). For the given pair of natural numbers $m<n$, determine whether any set of $n$ consecutive integers contains two distinct numbers whose product is divisible by $\mathrm{mn}$.
保留源文本的换行和格式，直接输出翻译结果。 | 1.12. We will prove that the answer to the question of the problem is positive. Let $n$ consecutive integers $a_{1}, a_{2}, \ldots, a_{n}$ be given. Then from the inequalities $ma_{n}, a_{i}-dn$, but $n \vdots d$, so $d=n>m$, which contradicts the condition $m \vdots d$. Therefore, $a_{i}$ and $a_{i}+d$ (or respectivel... | proof | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 28,798 |
1.13. (New York, 76). Let $f(n) \in \mathbf{N}$ be the smallest number for which the sum $\sum_{k=1}^{(n)} k$ is divisible by $n$. Prove that the equality $f(n)=2 n-1$ holds for numbers of the form $n=2^{m}\left(m \in \mathbf{Z}^{+}\right)$ and only for them. | 1.13. First, let's prove that if \( n=2^{m} \), where \( m \in \mathbb{Z}^{+} \), then \( f(n) = 2n - 1 \). Indeed, on one hand, the sum
\[
\sum_{k=1}^{2n-1} k = \frac{(2n-1) \cdot 2n}{2} = (2^{m+1} - 1) \cdot 2^{m}
\]
is divisible by \( 2^{m} = n \). On the other hand, if \( l \leq 2n - 2 \), then the sum
\[
\sum_{... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 28,799 |
1.14. (Jury, FRG, 79; GDR, 81). Prove that if the number $1+2^{n}+4^{n}$ is prime for some value of $n \in \mathbf{N}$, then $n=3^{k}$, where $k \in \mathbf{Z}^{+}$. | 1.14. Let $n=3^{k} r_{r}$, where $k \in \mathrm{Z}^{+}$, and the number $r \in \mathrm{N}$ is not divisible by 3. We will prove that the number
$$
\begin{aligned}
& p=1+2^{n}+4^{n} \\
& q=1+2^{3 k}+4^{3^{k}}
\end{aligned}
$$
is divisible by the number
Let's consider two possible cases.
1st case: $r=3 s+1, s \in \ma... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 28,800 |
1.15. (CRR, 78). Let numbers $m, n \in \mathbf{N}$ be such that for any value $k \in \mathbf{N}$ the greatest common divisors of the pairs of numbers $11 k-1, m$ and $11 k-1, n$ coincide. Prove that for some value $l \in \mathbf{Z}$ the equality $m=11^{\prime} n$ holds. | 1.15. Let $m=11^{i} p, n=11^{j} q$, where $i, j \in \mathrm{Z}^{+}$, and the numbers $p, q \in \mathbf{N}$ are not divisible by 11. We will prove that $p=q$ (from which it will follow that $m=11^{i-j_{n}}$). Suppose this is not the case, and $p>q$ (the case $p<q$ is similar). Then there exists a number $a>0$ satisfying... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 28,801 |
1.16. (New York, 75). Let the greatest common divisor of numbers $a, b, c, d \in \mathbf{Z}$ be 1. Is it true that any prime divisor of the number $a d - b c$ is a divisor of the numbers $a$ and $c$ if and only if for each value of $n \in \mathbf{Z}$, the numbers $a n + b$ and $c n + d$ are coprime? | 1.16. We will prove that the statement is true. Let each prime divisor of the number $a d-b c$ be a divisor of the numbers $a$ and $c$, however, contrary to the statement, for some value $n \in \mathbf{Z}$ the numbers $a n+b$ and $c n+d$ are divisible by a prime number $p$. Then the number
$$
a d-b c=a(c n+d)-c(a n+b)... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 28,802 |
1.17*. (USA, 82). Prove that there exists a number $k \in \mathbf{N}$, such that for any value of $n \in \mathbf{N}$, the number $k \cdot 2^{n}+1$ is composite. | 1.17. Note that the numbers $a_{m}=2^{2^{m}}+1$ for $m=0,1,2,3,4$ are prime, and the number
$$
\begin{aligned}
& 2^{33}+1=\left(2^{32}-1\right)+2= \\
& =\left(2^{16}+1\right)\left(2^{3}+1\right)\left(2^{4}+1\right)\left(2^{2}+1\right)(2+1)(2-1)+2= \\
& =a_{0} a_{1} a_{2} a_{3} a_{4}+2=2\left(\bmod a_{m}\right)
\end{al... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 28,803 |
1.18*. (CRR, 78). Prove that for any value of $a \in \mathbf{N}$, greater than 2, there exist infinitely many numbers $n \in \mathbf{N}$ such that the number $a^{n}-1$ is divisible by $n$. Is the analogous statement true for $\alpha=2$? | 1.18. Let a natural number $a \supseteq 3$ be given. We will prove by induction on $k \in \mathbf{N}$ that the sequence $\left\{n_{k}\right\}$, defined by the relations
$$
n_{1}=1, n_{k+1}=a^{n_{k}}-1, k \in \mathbf{N}
$$
satisfies the condition $a^{n_{k}}-1: n_{k}$. For $k=1$, we have $a-1 \vdots 1$. Suppose for som... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 28,804 |
1.19. (Jury, Belgium, 83). Prove that there are infinitely many numbers $n \in \mathbf{N}$, satisfying for all values $k=1,2, \ldots, n-1$ the inequalities
$$
\frac{\sigma(n)}{n}>\frac{\sigma(k)}{k}
$$
where $\sigma(n)$ denotes the sum of all divisors of the number $n$. | 1.19. Suppose, contrary to the statement of the problem, that only a finite set of values $n \in N$ satisfies the inequalities $a_{n}>a_{k}$ for $k=1,2, \ldots, n-1$, where $a_{n}=\sigma(n) / n$. Let $N$ be the largest of such values of $n$. Then the sequence of numbers
$$
A_{n}=\max _{1 \leqslant i \leqslant n}\left\... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 28,805 |
1.20*. (BPR, 82). For a given natural number $k>1$, $Q(n), n \in \mathbf{N}$, denotes the least common multiple of the numbers $n, n+1, \ldots, n+k$. Prove that there are infinitely many values of $n \in \mathbf{N}$, satisfying the inequality $Q(n)>Q(n+1)$. | 1.20. We will prove that the inequality $Q(n)>Q(n+1)$ is satisfied by any number
$$
\begin{gathered}
n=r \cdot k!-1, \text { where } r \in \mathbb{N}, r \geq 3 \\
m=[n+1, \ldots, n+k]
\end{gathered}
$$
For $f=1, \ldots, k$ we have $n=-1(\bmod j)$, hence
$$
(n, j)=1 \text { and }(n, n+j)=1
$$
Therefore,
$$
(n, m)=1... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 28,806 |
1.21*. (Austria, 73). Prove that for any value of $n \in \mathbf{N}$ the inequalities
$$
0<\sum_{k=1}^{n} \frac{g(k)}{k}-\frac{2 n}{3}<\frac{2}{3}
$$
hold, where $g(k)$ denotes the greatest odd divisor of the number $k$. | 1.21. Let $m(k)$ denote the exponent of 2 in the prime factorization of $k \in \mathrm{N}$, then $k=2^{m(k)} g(k)$ and
$$
S=\sum_{k=1}^{n} \frac{g(k)}{k}=\sum_{k=1}^{n} \frac{1}{2^{m(k)}}
$$
Notice that among the numbers $1,2, \ldots, n$ there are exactly $[n / 2]$ even numbers, $[n / 2^{2}]$ numbers divisible by fou... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 28,807 |
1.22*. (Jury, SFRY, 79). Let $h(n)$ denote the largest prime divisor of the number $n \in \mathbb{N} (n \geqslant 2)$. Is the set of values of $n$ satisfying the condition
$$
h(n)<h(n+1)<h(n+2)
$$
infinite? | 1.22. Let an odd prime number $p$ be chosen. We will prove that any two numbers from the increasing sequence of even numbers $a_{m}=p 2^{m}+1\left(m \in \mathbf{Z}^{+}\right)$ do not have common divisors greater than two. Indeed, if $m>l \geqslant 0$, then the number
$$
\begin{aligned}
p^{p 2^{m}}-1=\left(p 2^{m-1}+1\... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 28,808 |
2.1. (New York, 77). Solve the equation $2^{x}+1=y^{2}$ in natural numbers. | 2.1. Let's rewrite the equation as
$$
2^{x}=(y-1)(y+1)
$$
then for the sought values $x, y \in \mathbf{N}$, we have that the numbers $y-1, y+1 \in \mathbf{Z}^{+}$ are divisors of the number $2^{x}$, i.e., $y-1=2 p$ and $y+1=2^{q}$, where $p, q \in \mathbf{Z}^{+}, p<q$. Since the difference between $y-1$ and $y+1$ is ... | 3,3 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 28,810 |
2.2. (England, 72). Prove that for any values of $a, b, c, d \in \mathbf{Z}, a \neq b$, the equation
$$
(x+a y+c)(x+b y+d)=2
$$
has no more than four solutions in integers. Determine for which values of $a, b, c, d$ there are exactly four distinct solutions. | 2.2. Since $x, y, a, b, c, d \in \mathbb{Z}$, the equation
$$
(x+a y+c)(x+b y+d)=2
$$
is equivalent to the set of all systems of the form
$$
\left\{\begin{array}{l}
x+a y+c=p \\
x+b y+d=q
\end{array}\right.
$$
where the numbers $p, q \in \mathbb{Z}$ satisfy the equation $p q=2$. Each such system cannot have more th... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 28,811 |
2.3. (GDR, 73). Solve the equation
$$
x(x+1)(x+7)(x+8)=y^{2}
$$
in integers. | 2.3. Let the numbers $x, y \in \mathbf{Z}$ satisfy the equation, then
$$
y^{2}=(x(x+8))((x+1)(x+7))=\left(x^{2}+8 x\right)\left(x^{2}+8 x+7\right)=z^{2}+7 z
$$
where we denote $z=x^{2}+8 x$. If $z>9$, then
$$
(z+3)^{2}=z^{2}+6 z+9<z^{2}+7 z=y^{2}<z^{2}+8 z+16=(z+4)^{2}
$$
which means that the number $y^{2}$ is trap... | (-9;12),(-9;-12),(-8;0),(-7;0),(-4;12),(-4;-12),(-1;0),(0;0),(1;12),(1;-12) | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 28,812 |
2.4. (SRP, 81). Solve the equation
$$
x^{6}+3 x^{3}+1=y^{4}
$$
in integers. | 2.4. Let the equation be satisfied by some pair of numbers $x, y \in Z$. Suppose that $x>0$. Then we have
$$
\left(x^{3}+1\right)^{2}=x^{6}+2 x^{3}+1<x^{6}+3 x^{3}+1=y^{4}<x^{8}+4 x^{3}+4=\left(x^{3}+2\right)^{2}
$$
from which we obtain that the number $\boldsymbol{y}^{2}$ cannot be an integer, since
$$
x^{3}+1<y^{2... | 0,\1 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 28,813 |
2.5. (Yugoslavia, 74). Solve the equation
$$
x^{2}+x y+y^{2}=x^{2} y^{2}
$$
in integers. | 2.5. Let the pair of numbers $x, y \in \mathbf{Z}$ satisfy the equation, then $x^{2}+2 x y+y^{2}=x^{2} y^{2}+x y$, i.e., $(x+y)^{2}=x y(x y+1)$.
If $x y>0$, then
$$
x y+1>\sqrt{x y(x y+1)}>x y
$$
$$
|x+y|=\sqrt{x y(x y+1)}
$$
lies between two consecutive integers $x y$ and $x y+1$, and thus cannot be an integer. Si... | (0,0),(1,-1),(-1,1) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 28,814 |
2.6. (GDR, 74). Solve the equation
$$
(x+2)^{4}-x^{4}=y^{3}
$$
in integers. | 2.6. Let the pair of numbers $x, y \in \mathbf{Z}$ satisfy the equation. Suppose that $x \geqslant 0$, then
$$
y^{3}=8 x^{3}+24 x^{2}+32 x+16=8\left(x^{3}+3 x^{2}+4 x+2\right)
$$
Therefore, $y=2 z(2 \in \mathbb{Z})$ and
$$
2^{3}=x^{3}+3 x^{2}+4 x+2
$$
Notice that
$$
(x+1)^{3}=x^{3}+3 x^{2}+3 x+1<z^{3}<x^{3}+6 x^{2... | -1,0 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 28,815 |
2.7. (USA, 79). Solve the equation
$$
x_{1}^{4}+x_{2}^{4}+\ldots+x_{14}^{4}=1599
$$
in integers. | 2.7. Note that if the number $n$ is even, then $n=2 k$ and
$$
n^{4}=16 k^{4} \equiv 0(\bmod 16) ;
$$
if $n$ is odd, then the number
$$
n^{4}-1=(n-1)(n+1)\left(n^{2}+1\right)
$$
is divisible by 16 (since each of the numbers $n-1, n+1, n^{2}+1$ is even, and one of the numbers $n-1$ or $n+1$ is divisible by 4), i.e.
... | proof | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 28,816 |
2.8. (GDR, 70; PRC, 80). Prove that for any odd values of $a, b, c \in \mathbb{Z}$ the equation $a x^{2}+b x+c=0$ has no solutions in rational numbers. | 2.8. Let at least one of the possible solutions
$$
\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}
$$
of the original equation be a rational number. Then the number $\sqrt{b^{2}-4 a c}$ is rational, and hence (see Theorem 61), the number $b^{2}-4 a c$ is the square of some number $d \in \mathbf{Z}$. Since the numbers $a, b, c$... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 28,817 |
2.9. (England, 70). Solve the equation
$$
\sqrt{2 \sqrt{3}-3}=\sqrt{x \sqrt{3}}-\sqrt{y \sqrt{3}}
$$
in rational numbers. | 2.9. Let the numbers $x, y \in Q$ satisfy the equation. Then the following equalities hold
$$
\begin{gathered}
2 \sqrt{3}-3=x \sqrt{3}+y \sqrt{3}-2 \sqrt{3 x y} \\
(x+y-2) \sqrt{ } \overline{3}=2 \sqrt{3 x y}-3
\end{gathered}
$$
Since
$$
(x+y-2)^{2} \cdot 3=9+12 x y-12 \sqrt{3 x y}
$$
the number $\sqrt{3 x y}$ is r... | 3/2,1/2 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 28,818 |
2.10. (Brazil, 83). Prove that the equation
$$
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{1983}
$$
has only a finite set of solutions in natural numbers. | 2.10. Note that the given equation has a solution: for example, $x=y=z=3 \cdot 1983$. We will now prove that there is only a finite set of number triples $x, y, z \in N$ that satisfy the original equation and the inequalities $x \leqslant y \leqslant 2$. Indeed, for any such triple, the following relations hold:
$$
0<... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 28,819 |
2.11. (SFRY, 81). Prove that for any values of $a, b \in \mathbf{Z}$, satisfying the inequalities $5 a \geqslant 7 b \geqslant 0$, the system
$$
\left\{\begin{aligned}
x+2 y+3 z+7 u & =a \\
y+2 z+5 u & =b
\end{aligned}\right.
$$
has a solution in non-negative integers. | 2.11. Let the numbers $a, b \in \mathbf{Z}$ satisfy the inequalities $5 a \geqslant 7 b \geqslant 0$. Let $u=[b, 5]$, then the value $v=b-5 u$ can only take values from the set
$$
\{0 ; 1 ; 2 ; 3 ; 4\} .
$$
Using the equalities
$$
7 b=7(5 u+v)=35 u+7 v
$$
we obtain the relations
$$
a-7 u \geqslant \frac{7 b}{5}-7 ... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 28,820 |
2.12. (GDR, 77). How many pairs of values $p, q \in \mathbf{N}$, not exceeding 100, exist for which the equation
$$
x^{5}+p x+q=0
$$
has solutions in rational numbers? | 2.12. Let for some values
$$
p, q \in\{1 ; 2 ; \ldots ; 100\}
$$
the number $x \in \mathbf{Q}$ satisfies the equation
$$
x^{5}+p x+q=0
$$
Since all coefficients of the polynomial on the left side of the equation are integers, and the coefficient of the leading term is 1, by Theorem 60, any rational root of this pol... | 133 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 28,821 |
2.13. (CSSR, 76). Solve the equation $x^{2}+y^{2}=3 z^{2}$ in integers. | 2.13. The set of numbers $x=y=z=0$ is a solution to the equation. Suppose the equation has other solutions. Among them, we choose the set of numbers $x, y, z$ for which the quantity
$$
\alpha=|x|+|y|+|z|
$$
takes the smallest (natural) value. Note that for any value $n \in \mathbf{Z}$, the following holds: either $n=... | proof | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 28,822 |
2.14. (VNR, 83). Prove that the equation
$$
x^{3}+3 y^{3}+9 z^{3}-9 x y z=0
$$
has the unique solution $x=$ $=y=z=0$ in rational numbers. | 2.14. Note that if the set of numbers $x, y, z$ is a solution to the original equation, then any set of numbers $t x, t y, t z$, where $t \in \mathbb{Q}$, is also a solution. Therefore, if some non-zero set (i.e., not coinciding with the set $x=y=z=0$) of rational numbers
$$
x=m / n, y=l / k, z=p / q
$$
where $m, l, ... | proof | Algebra | proof | Yes | Yes | olympiads | false | 28,823 |
2.15. (USA, 76). Solve the equation
$$
x^{2}+y^{2}+z^{2}=x^{2} y^{2}
$$
in integers. | 2.15. The set of numbers $x_{0}=y_{0}=z_{0}=0$ is a solution to the equation. Suppose the equation has another solution $(x ; y ; z)$. Then the right-hand side of the equation $x^{2} y^{2}$ is divisible by 4 (otherwise, $x$ and $y$ are odd numbers, from which
$$
x^{2}=1(\bmod 4), \quad y^{2} \equiv 1(\bmod 4), \quad x... | proof | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 28,824 |
2.16. (England, 70). For each value of $n \in \mathbf{N}$, denote by $a_{n} \in \mathbf{Z}^{+}$ the number of solutions to the equation
$$
n^{2}+x^{2}=y^{2}
$$
in natural numbers greater than $n$.
a) Prove that for any number $M$, the inequality $a_{n}>M$ holds for at least one value of $n \in \mathbf{N}$. b) Is it ... | 2.16. For each pair of numbers $x, y \in N$ satisfying the equation $n^{2}=y^{2}-x^{2}$, there corresponds a pair of numbers $p=y+x, q=y-x$, the product of which equals $n^{2}$. Therefore, the number $a_{n}$ of these pairs does not exceed the number of distinct natural divisors of the number $n^{2}$. Consequently, for ... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 28,825 |
2.18. (GDR, 80). Prove that the equation
$$
(2 x)^{2 x}-1=y^{z+1}
$$
has no solutions in natural numbers. | 2.18. Suppose that the triple of numbers $x, y, z \in \mathbf{N}$ satisfies the equation. Then we have the factorization
$$
\left((2 x)^{x}+1\right)\left((2 x)^{x}-1\right)=y^{x+7}
$$
where the numbers
$$
k=(2 x)^{x}+1 \text { and } m=(2 x)^{x}-1
$$
are coprime, since they are odd and
$$
(k, m)=(k, k-m)=(k, 2)=1
$... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 28,827 |
2.19. (GDR, 81). Prove that for any value of $n \in \mathbf{N}$ the equation
$$
x_{1}^{2}+\ldots+x_{n}^{2}=y^{2}
$$
has a solution in natural numbers. | 2.19. We will prove by induction on $n$ a stronger statement: for any value $n \in N$ there exist numbers
$$
x_{1}, \ldots, x_{n} \in \mathbf{N}
$$
and an odd number $y_{n}>1$, satisfying the equation
$$
x_{1}^{2}+\ldots+x_{n}^{2}=y_{n}^{2}
$$
For $n=1$, the statement is true (it is sufficient to set $\boldsymbol{x... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 28,828 |
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