problem stringlengths 1 13.6k | solution stringlengths 0 18.5k ⌀ | answer stringlengths 0 575 ⌀ | problem_type stringclasses 8
values | question_type stringclasses 4
values | problem_is_valid stringclasses 1
value | solution_is_valid stringclasses 1
value | source stringclasses 8
values | synthetic bool 1
class | __index_level_0__ int64 0 742k |
|---|---|---|---|---|---|---|---|---|---|
202. Given a convex polygon into which no triangle of area 1 can be placed. Prove that this polygon can be placed in a triangle of area 4. | 202. Consider the set of vertices of a given polygon $P$ and choose three of them $-A, B, C$- such that the area of triangle $A B C$ is the largest. (It is clear that the area of triangle $A B C$ is not less than the area of any triangle that can be placed inside polygon $P$. We will use further specific cases of this ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,404 |
203. On the segment $0 \leqslant x \leqslant 1$, a function $f$ is given. It is known that this function is non-negative and $f(1)=1$. Moreover, for any two numbers $x_{1}$ and $x_{2}$ such that $x_{1} \geqslant 0, x_{2} \geqslant 0$ and $x_{1}+x_{2} \leqslant 1$, the inequality
$$
f\left(x_{1}+x_{2}\right) \geqslant ... | 203. a) The function $f$ is monotonically non-decreasing on the interval $[0,1]$, since from $1 \geqslant x \geqslant y \geqslant 0$ it follows that $f(x)=f((x-y)+y) \geqslant$ $\geqslant f(x-y)+f(y) \geqslant f(y)$. In addition, $f(2 x) \geqslant 2 f(x)$ for all $x_{1}$. Using this, we obtain:
$$
\begin{array}{ll}
\t... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 25,405 |
204. Given a triangle $ABC$ with area I. Let $A_{1}$, $B_{1}$, and $C_{1}$ be the midpoints of sides $BC$, $CA$, and $AB$ respectively. What is the minimum area that the common part of triangles $A_{1} B_{1} C_{1}$ and $KLM$ can have, if points $K$, $L$, and $M$ lie on segments $AB_{1}$, $CA_{1}$, and $BC_{1}$ respecti... | 204. Answer: 1/8. Since (notations are clear from Fig. 83)
$$
\frac{C_{1} M_{2}}{M_{2} M_{1}} \leqslant \frac{A K}{K C} \leqslant \frac{A B_{1}}{B_{1} C}=1
$$
then $C_{1} M_{2} \leqslant M_{2} M_{1}$, and therefore $S_{C_{1} M_{2} K_{1}} \leqslant S_{M_{2} M_{1} K_{1}}$. Similarly, it is proved that $S_{A L_{2} M_{1}... | \frac{1}{8} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,406 |
205. a) From triangle $A B C$, by rotating around the center of the circumscribed circle by some angle less than $180^{\circ}$, we obtained triangle $A_{1} B_{1} C_{1}$. Corresponding segments $A B$ and $A_{1} B_{1}$ intersect at point $C_{2}$, $B C$ and $B_{1} C_{1}$ at point $A_{2}$, and $C A$ and $C_{1} A_{1}$ at po... | 205. If the chord $Q_{1} Q_{2}$ of a circle with center $O$ is obtained by rotating the chord $P_{1} P_{2}$ by an angle $\alpha$ around the point $O$, then the intersection point $R$ of the lines $P_{1} P_{2}$ and $Q_{1} Q_{2}$ can be obtained by rotating the midpoint $M$ of the chord $P_{1} P_{2}$ around the center $O... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,407 |
206. Given a triangle $ABC$ with an area of 1. The first player chooses a point $X$ on side $AB$, the second player chooses a point $Y$ on side $BC$, and then the first player chooses a point $Z$ on side $AC$. The goal of the first player is to obtain a triangle $XYZ$ with the largest possible area, while the goal of t... | 206. Answer: 1/4. First of all, note that the second player can ensure that \( S_{X Y Z} \leqslant 1 / 4 \) regardless of the first player's move. For this, it is sufficient for the second player to choose \( Y \) such that \( X Y \| A C \) (Fig. 84). Then, for any point \( Z \) on the base \( A C \), the inequality
\... | \frac{1}{4} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,408 |
207. What is the smallest perimeter that a convex 32-gon can have, all of whose vertices lie at the nodes of a grid paper with a cell side of 1?
$208^{*}$. a) In a $7 \times 7$ square of cells, it is necessary to mark the centers of $k$ cells so that no four marked points form the vertices of a rectangle with sides pa... | 207. Answer: $4+4 \sqrt{2}+8 \sqrt{5}+8 \sqrt{10}+8 \sqrt{13}$.
The contour of the 32-sided polygon $A_{1} A_{2} \ldots A_{32}$ can be represented as the image of the sum of 32 vectors $\overrightarrow{A_{1} A_{2}}+\overrightarrow{A_{2} A_{3}}+\ldots+\overrightarrow{A_{32}} A_{1}=0$.
Since a convex 32-sided polygon c... | 4+4\sqrt{2}+8\sqrt{5}+8\sqrt{10}+8\sqrt{13} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 25,409 |
210. Prove that from the digits 1 and 2, one can form $2^{n+1}$ numbers, each of which is $2^{n}$-digit long, and any two of which differ in at least $2^{n-1}$ digits. | 210. For $n=1$, the four numbers $11,21,12,22$ satisfy the condition. We will prove the statement of the problem by induction.
Let $a^{\prime}$ denote the number obtained from $a$ by replacing the digit 1 with 2 and 2 with 1, and let $ab$ denote the number obtained by appending the number $b$ to $a$. Suppose we have c... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 25,411 |
211. In the plane, there is a finite set of polygons, each two of which have a common point. Prove that some line intersects all these polygons. | 211. Let's project all polygons onto some straight line. The projection of each polygon is a segment, and by the condition, any two segments have a common point. From this, it follows that all segments have a common point (to convince yourself of this, it is enough to consider the given line as a number line and take t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,412 |
212. Prove that for positive $a, b, c$ the inequality $a^{3}+b^{3}+c^{3}+3 a b c>$ $>a b(a+b)+b c(b+c)+a c(a+c)$ holds. | 212. Without loss of generality, we can assume that $a \geqslant b \geqslant c$. Then $c(a-c)(b-c) \geqslant 0$, from which $c^{3} + abc \geqslant ac^{2} + bc^{2}$. It is sufficient to prove that $a^{3} + b^{3} + 2abc \geqslant ab(a+b) + a^{2}c +$
, where $D E: B C=1$ $\vDash 2: 3$. Since one of the flies has been to all vertices, the center of gravity of the "triangle of flies" should belong to the three shaded triang... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,414 |
215. Given a horizontal strip on a plane, the edges of which are parallel lines, and $n$ lines intersecting this strip. Any two of these lines intersect inside the strip, and no three of them have a common point. Consider all paths starting on the lower edge of the strip, traveling along the given lines, and ending on ... | 215. Let $A_{1}, A_{2}, \ldots, A_{n}$ be the points of intersection of the lines with the lower edge of the strip, numbered in order (from left to right), and $B_{1}, B_{2}, \ldots, B_{n}$ be the points of intersection with the upper edge (also from left to right). We will number the paths starting from points $A_{1},... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 25,416 |
216. For which natural numbers $k$ can a cube of size $k \times k \times k$ be composed of white and black unit cubes $1 \times 1 \times 1$ such that for any unit cube, exactly two of its neighbors have the same color as itself? (Two unit cubes are considered neighbors if they share a face.) | 216. Answer: possible for even $k$, and impossible for odd $k$.
For even $k$, it is easy to construct an example: a cube can be made from alternating black and white blocks of $2 \times 2 \times 1$.
Now suppose we managed to construct the required cube for odd $k$. Connect the centers of adjacent white cubes with seg... | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 25,417 |
217. Given a polynomial $P(x)$ with
a) natural coefficients;
b) integer coefficients.
Denote by $a_{n}$ the sum of the digits in the decimal representation of the number $P(n)$. Prove that there is a number that occurs in the sequence $a_{1}, a_{2}, \ldots, a_{n}, \ldots$ infinitely many times. | 217. a) As such a number $S$, one can take the sum of all the digits in the representation of all the coefficients of the polynomial $P(x)$: then for any $n \geqslant n_{0}$, where $n_{0}$ is chosen so that $10^{n_{0}}$ is greater than all the coefficients, $a_{10}{ }^{n}=S$.
b) The result of a) can be applied to the ... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 25,418 |
218. In the World and European championships, 20 teams participate. Among them, there are $k$ European teams, the results of whose matches in the World Championship count towards the European Championship. The championship is played in a single round-robin format.
What is the largest $k$ for which it can happen that t... | 218. Let's consider the case of $n$ participating teams in the World Championship immediately.
c) Answer: $k=n-2$, if $n \geqslant 3$ (in particular, for $n=20$ we get $k=18$).
The total number of points played in the World Championship (2 points for a win, 1 point for a draw, 0 points for a loss) is $n(n-1)$, and in... | n-2 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 25,419 |
220. On the table, there are 50 correctly running clocks. Prove that at some moment, the sum of the distances from the center of the table to the ends of the minute hands will be greater than the sum of the distances from the center of the table to the centers of the clocks. | 220. The idea of the solution can be expressed in one phrase: the sum of the distances to the ends of the minute hands on average (over time) is greater than the sum of the distances to the centers of the clocks. The proof can be conducted as follows.
Consider the sums $s_{1}$ and $s_{2}$ of the distances from the cen... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,421 |
221. A row of 1000 numbers is written in a row. Below it, a second row of numbers is written according to the following rule: under each number $a$ in the first row, a natural number is written indicating how many times $a$ appears in the first row. From the second row, a third row is similarly constructed: under each ... | 221. The solution to part a) follows from these two considerations: (1) under each number $a$ in the $m$-th row ($m \geqslant 2$), there is a number not less than $a$; (2) each of these numbers does not exceed 1000. To solve part b), we also need to consider that if a number $a$ in the $m$-th row ($m \geqslant 2$) is s... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 25,422 |
222. On a plane, there are three circles of the same radius.
a) Prove that if all of them intersect at one point, as shown in Fig. $11, a$, then the sum of the marked arcs $A K, C K, E K$ is $180^{\circ}$.
 is a degenerate special case of b) and is solved similarly. Let's indicate the solution of task b).
$\delta$
Fig. 95
Let $O_{1}, O_{2}, O_{3}$ be the centers of the circles co... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,423 |
223. Natural numbers $x_{1}, x_{2}$ are less than 10000. From them, a sequence $x_{1}, x_{2}$, $x_{3}, \ldots, x_{n}$ is constructed, where the number $x_{3}$ is equal to $\left|x_{1}-x_{2}\right|$, the number $x_{4}$ is the smallest of the numbers $\left|x_{1}-x_{2}\right|,\left|x_{2}-x_{3}\right|$, $\left|x_{1}-x_{3}... | 223. If the first three numbers are rearranged in descending order, then for all members of the sequence, the condition $x_{k} \leqslant x_{k-r}-x_{k-1}$ will be satisfied, and the sequence will become decreasing: $x_{1} \geqslant x_{2} \geqslant x_{3} \geqslant \ldots \geqslant x_{21}$. If we assume that $x_{21} \geqs... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 25,424 |
225*. On the plane, vectors $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}, \boldsymbol{d}$ are given, the sum of which is 0. Prove the inequality
$$
|a|+|b|+|c|+|d| \geqslant|a+d|+|b+d|+|c+d|
$$ | 225. The right side of the inequality, like the left, is symmetric with respect to \(a, b, c\) and \(d\) (provided their sum is 0); since \(b+c=-(a+d)\) and so on, it is equal to half the sum of the lengths of all pairwise sums of the given vectors. Considering this, for a given quadruple of vectors with a sum of 0, we... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 25,426 |
226. In a regular 1976-gon, the midpoints of all sides and the midpoints of all diagonals are marked. What is the greatest number of marked points that can lie on one circle? | 226. Answer: 1976. All marked points, except for the center $O$ of the 1976-gon, lie 1976 each on 987 circles with center $O$. Any other circle $\gamma$ intersects each of these 987 circles at two points; besides these intersection points, there can be only one more marked point on $\gamma$: $O$. Therefore, there are n... | 1976 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,427 |
227. 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 all the rectangles are cut out, the number of pieces into which the remaining part of the sheet will split is no more than $n+1$. | 227. Let's mark 4 vertices (angles at these vertices are $90^{\circ}$ or $270^{\circ}$) on each of the $k$ pieces into which the remaining part of the sheet has split. Each of the $4 k$ marked points is a vertex of one of the $n$ cut-out rectangles or the original one, and if any point is marked twice, then two rectang... | k\leqn+1 | Geometry | proof | Yes | Yes | olympiads | false | 25,428 |
228. Three pedestrians are walking along three straight roads at constant speeds. At the initial moment, they were not on the same straight line. Prove that they can be on the same straight line no more than twice. | 228. Three points $\left(x_{0}, y_{0}\right),\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)$ lie on the same straight line if and only if
$$
\left(x_{1}-x_{0}\right)\left(y_{2}-y_{0}\right)-\left(x_{2}-x_{0}\right)\left(y_{1}-y_{0}\right)=0 .
$$
If $x_{i}$ and $y_{i}(i=0,1,2)$ are linear functions of time $t_{\t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,429 |
229. On a $99 \times 99$ chessboard, a figure (this figure will be different in parts a), b), and c)) is marked. In each cell of the figure $F$, there is a beetle. At some point, the beetles fly up and land again in the cells of the same figure $F$; in this process, several beetles could land in the same cell. After th... | 229. a) We can assume that the beetle from the central cell has moved to the right by $k \geqslant 2$ cells. Let's write in each of the 49 cells of the rightmost column how many cells the beetle has moved horizontally from the corresponding cell; a move to the right is considered positive, and a move to the left is neg... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 25,430 |
231. Given a natural number \( n \). A sequence of natural numbers \( a_{1}, a_{2}, \ldots, a_{k} (k \geqslant n) \) is called universal for the given \( n \) if any permutation of the numbers \( 1, 2, \ldots, n \) (i.e., any sequence of \( n \) numbers in which each of the numbers \( 1, 2, \ldots, n \) appears exactly... | 231. Example a) is obvious: it is sufficient to write the sequence "block" $123 \ldots n$ $n$ times under ${ }^{\prime}$. The $i$-th digit of any permutation can be taken from the $i$-th block. As an example for b), the sequence
$ and $(-1)$ are placed. In one step, it is allowed to change the sign of all numbers located at the vertices of any regular $k$-gon with center $O$ (we also allow 2-gons, understanding a 2-gon as a segment with midpoint at point $O$). Pro... | 233. Let's first note several facts relevant to the case of any natural $n$. There are a total of $2^n$ arrangements of the numbers +1 and -1 at the vertices of a regular $n$-gon. We will call two arrangements equivalent if one can be transformed into the other (and vice versa) using the operations specified in the pro... | K(n)=2^{\varphi(n)} | Combinatorics | proof | Yes | Yes | olympiads | false | 25,433 |
234*. On a sphere of radius 1, a great circle has been drawn, which we will call the equator. It will be convenient for us to use other geographical terms: pole, meridian, parallel.
a) Let's define a function $f$ on this sphere that assigns to each point of the sphere the square of the distance from this point to the ... | 234. We will only consider points of one "northern" hemisphere with pole $P$ (the values of the function $f$ at diametrically opposite points are the same). The pole $\boldsymbol{P}$ is considered the highest point of the sphere. Obviously, $f(P)=1,0f(Y)>f(N)$.
b) Similarly to the previous, for any pair of points $M$ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,434 |
235. On a plane, there is a non-self-intersecting closed broken line, no three vertices of which lie on the same straight line. We will call a pair of non-adjacent segments special if the extension of one of them intersects the other. Prove that the number of special pairs is even. | 235. A line passing through some segment $B C$ of a broken line intersects an even or odd number of other segments depending on whether the adjacent segments $A B$ and $C D$ lie on the same or different sides of the line $B C$ (in the latter case, we will call $B C$ a "zigzag"); indeed, the parity of the number of inte... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,435 |
236. On a plane, several points not lying on the same line are marked, and a number is written near each. It is known that if a line passes through two or more of the marked points, then the sum of all the numbers written near these points is zero. Prove that all the numbers are zero. | 236. Let the number $a_{x}$ be written at the marked point $X$, and let $n_{x}>1$ lines pass through it, containing other marked points. The sum of the numbers on each of them is 0 by condition. The total sum of the numbers on all $n_{x}$ lines (in which $a_{x}$ participates $n_{x}$ times) is $s+(n_{x}-1)a_{x}=0$, wher... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 25,436 |
238. Several black and white chips are arranged in a circle. Two players take turns performing the following operation: the first player removes all black chips that have a white neighbor (at least on one side), and the second player then removes all white chips that have a black neighbor. They continue doing this unti... | 238. Answers: a) yes, b) 8 moves (each player - per move).
Fig. 102 shows an example of arranging 41 chips; next to each chip is its rank-number, indicating how many moves before the end it will be removed. The construction of such an arrangement is convenient "from the end"; to the remaining last black chip of rank 0... | 8 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 25,438 |
239. Given an infinite numerical sequence $\left\{a_{n}\right\}$. It is known that $\lim _{n \rightarrow \infty}\left(a_{n+1}-\frac{a_{n}}{2}\right)=0$. Prove that $\lim _{n \rightarrow \infty} \alpha_{n}=0$.
240*. In a certain country, it is possible to travel from any city to any other city without passing through o... | 239. Let's use the definition of the limit of a sequence (P21). If for all $n \geqslant N$ and some $k$ the inequalities
$$
\left|a_{n}-a_{n+1} / 2\right|<\varepsilon \text { and }\left|a_{N}\right| / 2^{k}<\varepsilon,
$$
hold, then for $m \geqslant N+k$ we have
$$
\begin{aligned}
\left|a_{m}\right|<\frac{1}{2}\lef... | proof | Calculus | proof | Yes | Yes | olympiads | false | 25,439 |
241. At each vertex of a convex polyhedron $M$, three edges meet. It is known that each of its faces is a polygon around which a circle can be circumscribed. Prove that a sphere can be circumscribed around this polyhedron.
$242^{*}$. The polynomial $x^{10}+* x^{9}+* x^{8}+\ldots$ $\ldots+* x^{2}+* x+1$ is written. Two... | 241. Two circles, in which the faces adjacent to the edge $A B$ of the polyhedron are inscribed, uniquely determine a sphere on which these circles lie, and thus all the vertices of these two faces. If $B C$ and $B D$ are two other edges emanating from $B$, then the circle containing points $B, C, D$ (circumscribed aro... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,440 |
243. Seven dwarfs are sitting around a round table. In front of each of them is a mug. Some of these mugs contain milk. One of the dwarfs pours all his milk into the mugs of the others equally. Then his right neighbor does the same. Then the next right neighbor does the same, and so on. After the last, seventh dwarf ha... | 243. Answer: $\frac{6}{7}, \frac{5}{7}, \frac{4}{7}, \frac{3}{7}, \frac{2}{7}, \frac{1}{7}, 0$ liters.
To verify that the given numbers serve as the answer is not difficult: after the first gnome pours the milk (1/7 to each of the others), the distribution remains the same but shifted by one gnome, and the sum $(1+2+3... | \frac{6}{7},\frac{5}{7},\frac{4}{7},\frac{3}{7},\frac{2}{7},\frac{1}{7},0 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 25,441 |
244. We will call a $2n$-digit number special if it is a perfect square, and the numbers formed by its first $n$ digits and its last $n$ digits are also perfect squares (the second $n$-digit number may start with a 0, but it should not be zero, and the first one should not start with a 0).
a) Find all two-digit and fo... | 244. a) Answer: there is one two-digit number 49 and one four-digit number $1681=41^{2}$ that are special.
Let $(10 x+t)^{2}=100 x^{2}+20 x t+t^{2}$, where $20 x t+t^{2}$ is the square of a natural number less than 10, and $x$ and $t$ are integers from 1 to 9, and $x^{2}>10$. Then $x \geqslant 4$ and $x t \leqslant 4$... | notfound | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 25,442 |
246. There are a thousand tickets numbered $000, 001, \ldots, 999$ and a hundred boxes numbered $00, 01, \ldots, 99$. A ticket is allowed to be placed in a box if the number of the box can be obtained from the number of the ticket by erasing one of its digits. Prove that:
a) all tickets can be placed in 50 boxes;
b) ... | 246. a) Let's divide ten digits $0,1,2, \ldots, 9$ into two groups of 5 digits each (for example, from 0 to 4 - one group, from 5 to 9 - the other). It is sufficient to use boxes where both digits are taken from the same group, since such two digits are present in any three-digit number.
b) In addition to the 10 boxes... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 25,443 |
247. Given a square sheet of grid paper $100 \times 100$ cells. Several non-intersecting broken lines are drawn along the sides of the cells and have no common points. These broken lines strictly lie inside the square, and their ends exit onto its boundary. Prove that, besides the vertices of the square, there is a nod... | 247. Let's color all nodes in a checkerboard pattern in black and white. On the boundary, there are 4.99 nodes (excluding the vertices) - an equal number of black and white nodes. Let all of them serve as endpoints of broken lines. Then there is an equal number of broken lines with two white and two black ends. Therefo... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,444 |
249. On the plane, there are 1000 squares with sides parallel to the coordinate axes. Let $M$ be the set of centers of these squares. Prove that it is possible to mark some of the squares such that each point in the set $M$ is covered by no fewer than one and no more than four of the marked squares. | 249. Let's choose the largest of the given squares $K_{1}$, then the largest $K_{2}$ whose center does not lie in $K_{1}$, and then the largest of the remaining ones whose centers do not lie in the already marked squares $K_{1}$ and $K_{2}$, and so on.
Suppose that in this case, the center $C$ of some square falls int... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 25,445 |
250. On the table are pan scales and $n$ weights of different masses. The weights are placed on the pans of the scales one by one (at each step, any weight is taken from the table and added to one of the pans of the scales).
a) Prove that the weights can be placed in such an order that the left pan is heavier first, t... | 250. Let the masses of the weights be $m_{1}0, m_{3}>m_{2}, \ldots, m_{k}>m_{k-1}$.
This same lemma, but applied to the segment $m_{l}<m_{l+1}<\ldots<m_{k-1}<m_{k}$, consisting of $k-l+1$ numbers, is used in solving problem b).
6) It is convenient to describe the order of placing the weights corresponding to the give... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 25,446 |
252. Let $a_{n}$ denote the integer closest to $\sqrt{n}$. Find the sum
$$
\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{1980}}
$$ | 252. Answer: 88.
Each number $k=1,2,3, \ldots$ appears in the sequence $(a_{n})$ $2 k$ times, since the condition $a_{n}=k$ is equivalent to
$$
k-\frac{1}{2}<\sqrt{n}<k+\frac{1}{2}, \quad \text { or } \quad k^{2}-k<n \leqslant k^{2}+k
$$
Therefore, in the sum
$$
\begin{aligned}
&\left(\frac{1}{a_{1}}+\frac{1}{a_{2}... | 88 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 25,448 |
253. Inside the quadrilateral $A B C D$, a point $M$ is marked such that $A B M D$ is a parallelogram. Prove that if $\angle C B M=\angle C D M$, then $\angle A C D=$ $=\angle B C M$. | 253. Let $E$ be the vertex of triangle $A D E$, obtained by translating triangle $B M C$ (by vector $B A$, Fig. 103). Then $M D E C$ is a parallelogram, the angles $E A D$, $C B M$, $C D M$, and $E C D$ are equal, and the points $A$, $C$, $E$, and $D$ lie on the same circle. Therefore, the angles $A C D$, $A E D$, and ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,449 |
254. Prove that for no natural $m$ the number $1978^{m}-1$ is divisible by $1000^{m}-1$. | 254. If $1978^{m}-1$ were divisible by $1000^{m}-1=d$, then the number $1978^{m}-1000^{m}=2^{m} \quad(989^{m} - 500^{m})$ would also be divisible by $d$. But this is impossible, since $989^{m} - 500^{m} < d$ and $d$ is odd.
. To it are added all points that can be obtained by symmetric reflection of one point of this set relative to another. The resulting set is denoted as $K_{1}$. Similarly, from the set $K_{1}$, $K_{2}$ is obtained, from $K_{2}$ - $K_{3}$, and so on.
a) Let the... | 255. a) Answer: $n=7$.
The set $K_{n}$ consists of points on the line $A B$, removed from $A$ and $B$ by integer distances, and the extreme points of this set are removed from the midpoint of the segment $A B$ by a distance of $3^{n} / 2$ $(n=1,2,3, \ldots)$, i.e., from $A$ - by distances $\left(3^{n}-1\right) / 2$ an... | (3^{2n+1}-1)/2 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,451 |
257*. Prove that there exists an infinite bounded sequence $x_{n}$ such that for any distinct $m$ and $k$ the inequality
$$
\left|x_{m}-x_{k}\right| \geqslant \frac{1}{|m-k|}
$$
holds. | 257. The sequence $4\{n \sqrt{2}\}$, where $\{x\}=x-[x]$ is the fractional part of the number $x$ ($n=1,2,3, \ldots$), satisfies the condition of the problem. Indeed, if $p$ and $q$ are natural numbers, $p\frac{1}{4 q^{2}}
$$
Therefore, for $m>k \geqslant 1$
$$
|\{m \sqrt{2}\}-\{k \sqrt{2}\}|=|(m-k) \sqrt{2}-l|>\frac... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 25,452 |
258. Let $f(x)=x^{2}-x+1$. Prove that for any natural number $m>1$ the numbers $m, f(m)$, $f(f(m)), \ldots$ are pairwise coprime. | 258. Since $f(1)=f(0)=1$, the constant term $P_{n}(0)$ of the polynomial $P_{n}(x)=\underbrace{f(\ldots(f(x)) \ldots))}_{n}$ is equal to 1. Therefore, for any integer $m$, the remainder of the division of $P_{n}(m)$ by $m$ is 1.
Replacing $m$ with $m^{\prime}=P_{k}(m)$, we get that $P_{n+k}(m)$ and $m^{\prime}=P_{k}(m... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 25,453 |
259. Prove that there exists a number $A$ such that in the graph of the function $y=A \sin x$, one can inscribe no fewer than 1978 pairwise distinct squares. (A square is called inscribed if all its vertices belong to the graph.) | 259. Let $M$ be any point of intersection of the graph $y = A \sin x$ with its image $x = -A \sin y$ after a $90^{\circ}$ rotation around the origin (Fig. 105). Then the point $M$ and its images
$, outputs a new card $(a+1 ; b+1)$; the second, reading a card $(a ; b)$, outputs a card $(a / 2 ; b / 2)$ (it works only when $a$ and $b$ are even); the third machine, given two cards $... | 260. a) Answer: it is possible.
First, we get the card $(1 ; 8)$:
$(5 ; 19) \rightarrow(6 ; 20) \rightarrow(3 ; 10) \rightarrow \ldots \rightarrow(10 ; 17) \rightarrow(3 ; 17) \rightarrow$
$$
\rightarrow(4 ; 18) \rightarrow(2 ; 9) \rightarrow \ldots \rightarrow(9 ; 16) \rightarrow(2 ; 16) \rightarrow(1 ; 8)
$$
From... | proof | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 25,455 |
261. A regular $n$-sided polygon of area $S$ is inscribed in a circle of radius $R$. A point is marked on each side of the $n$-sided polygon. Prove that the perimeter of the $n$-sided polygon with vertices at the marked points is not less than $2 S / R$. | 261. Let $A_{i}$ be the vertices of an inscribed $n$-gon ( $i=1$, 2. $\ldots, n, n$); $b_{l}$ - a segment connecting two marked points on its sides emanating from $A_{i}$; $O$ - the center of the circle (Fig. 106):
$s_{i}$ and $s_{i}^{\prime}$ - the areas of triangles with base $b_{l}$, the vertices of which are $A_{l}... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,456 |
262. A chip is placed in the corner of an $n \times n$ chessboard. Each of the two players, in turn, moves it to an adjacent square (one that shares a side with the current square). It is not allowed to move to a square that the chip has already visited. The player who cannot make a move loses.
a) Prove that if $n$ is... | 262. a) If $n$ is even, the entire board can be divided into rectangles of size $1 \times 2$ cells («dominoes»). The first player will always have the opportunity to make a move (and thus win) if they follow this strategy: if the chip is on one of the cells of a certain domino, then they place it on the second cell of ... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 25,457 |
263. On a plane, several non-intersecting segments are given, no two of which lie on the same line. We want to draw several more segments connecting the ends of the given segments so that all segments together form one non-self-intersecting broken line. Is it always possible to do this? | 263. Answer: not always.
Fig. 107 shows an example of 6 segments - 3 short and 3 long, which cannot be included in a non-self-intersecting (even open) broken line in the required manner. Indeed, one of the short segments is not at the end of the broken line, but its ends can be connected by lines to the ends of the ne... | notalways | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,458 |
264. The numbers $x_{1}, x_{2}, \ldots, x_{n}$ belong to the interval $[a ; b]$, where $0<a<b$. Prove the inequality
$\left(x_{1}+x_{2}+\ldots+x_{n}\right)\left(\frac{1}{x_{1}}+\frac{1}{x_{2}}+\ldots+\frac{1}{x_{n}}\right) \leqslant \frac{(a+b)^{2}}{4 a b} n^{2}$. | 264. First solution. Using the inequality $a b \leqslant \leqslant(a+b)^{2} / 4$, for any number $c>0$ we have:
$$
\begin{aligned}
P=\left(x_{1}+x_{2}+\ldots+x_{n}\right)\left(\frac{1}{x_{1}}+\frac{1}{x_{2}}+\ldots+\frac{1}{x_{n}}\right)= \\
= & \left(\frac{x_{1}}{c}+\ldots+\frac{x_{n}}{c}\right)\left(\frac{c}{x_{1}}+... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 25,459 |
265*. Given a prime number $p>3$. Consider a set $M$ on the coordinate plane consisting of points with integer coordinates $(x, y)$ such that $0 \leqslant x < p, 0 \leqslant y < p$. Prove that it is possible to mark $p$ different points of the set $M$ such that no four of them lie at the vertices of a parallelogram and... | 265. As the required set, one can take the set consisting of points $A_{k}=(k ; r(k)), k=0,1,2, \ldots$
$\ldots, p-1$, where $r(k)$ is the remainder of the division of $k^{2}$ by $p$. For $p=7$, this set is shown in Fig. 109.
If some three points $A_{l}, A_{m}, A_{n}(l3$, and $l-m$ would also have to be divisible by $... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 25,460 |
266*. Prove that for any tetrahedron, there exist two planes such that the ratio of the areas of the projections of the tetrahedron onto these planes is not less than $\sqrt{2}$ | 266. Let's draw planes \( p \) and \( p' \) through two opposite edges \( a \) and \( b \) of a tetrahedron. Let \( h \) be the distance between them. We will only consider projections of the tetrahedron onto planes \( q \) perpendicular to \( p \), and among these, we will find those for which the ratio of the areas i... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,461 |
267. Consider $n$ numbers $a_{1}, a_{2}, \ldots, a_{n}$. Let
$$
\begin{aligned}
& b_{k}=\frac{a_{1}+a_{2}+\ldots+a_{k}}{k}(\text { for } k=1,2, \ldots, n) \\
& C=\left(a_{1}-b_{1}\right)^{2}+\left(a_{2}-b_{2}\right)^{2}+\ldots+\left(a_{n}-b_{n}\right)^{2} \\
& D=\left(a_{1}-b_{n}\right)^{2}+\left(a_{2}-b_{n}\right)^{2... | 267. Let
$$
f(x)=\left(x-a_{1}\right)^{2}+\ldots+\left(x-a_{n}\right)^{2}
$$
By completing the square in this quadratic trinomial, we get
$$
f(x)=n\left(x-b_{n}\right)^{2}+f\left(b_{n}\right)
$$
when verifying this equality, it is necessary to take into account that $n b_{n}=a_{1}+$ $+a_{2}+\ldots+a_{n}$.
For $n=1... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 25,462 |
268*. Consider the sequence of numbers $x_{n}=(1+\sqrt{2}+\sqrt{3})^{n}$. Each of them can be expressed in the form
$$
x_{n}=q_{n}+r_{n} \sqrt{2}+s_{n} \sqrt{3}+t_{n} \sqrt{6}
$$
where $q_{n}, r_{n}, s_{n}, t_{n}$ are integers. Find the limits
$$
\lim _{n \rightarrow \infty} \frac{r_{n}}{q_{n}}, \quad \lim _{n \righ... | 268. Let's consider together with the number $\lambda_{1}=1+\sqrt{2}+\sqrt{3}$ its "conjugates," which differ from it by the signs of the radicals: $\lambda_{2}=1-$ $-\sqrt{2}+\sqrt{3}, \lambda_{3}=1+\sqrt{2}-\sqrt{3}$ and $\lambda_{4}=1-\sqrt{2}-\sqrt{3}$. If
$$
\lambda_{1}^{n}=q_{n}+r_{n} \sqrt{2}+s_{n} \sqrt{3}+t_{... | \lim_{narrow\infty}\frac{r_{n}}{q_{n}}=\frac{1}{\sqrt{2}},\quad\lim_{narrow\infty}\frac{s_{n}}{q_{n}}=\frac{1}{\sqrt{3}},\quad\lim_{narrow\infty}\frac{t_{n}}{q_{n}}=\frac{} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 25,463 |
270. A kangaroo jumps in the corner $x \geqslant 0, y \geqslant 0$ of the coordinate plane $O x y$ as follows: from the point ( $x ; y$ ) the kangaroo can jump to the point ( $x-5$; $y+7)$ or to the point $(x+1 ; y-1)$, and jumping to points with a negative coordinate is not allowed. From which initial points $(x ; y)$... | 270. Answer: the set of points from which one cannot escape *to infinity* has an area of 15; this is the stepped figure $T$, shown in Fig. 112.
From any point outside $T$, one can reach the region $x \geq 5$ in several steps $(1; -1)$, and then make steps $(-5; 7) + 5(1; -1) = (0; 2)$
$\nabla$ Even more interesting s... | 15 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,465 |
271. In the parliament, each of its members has no more than three enemies. Prove that the parliament can be divided into two chambers so that each parliamentarian will have no more than one enemy in the same chamber. (It is assumed that if $B$ is an enemy of $A$, then $A$ is an enemy of $B$.) | 271. First, let's arbitrarily distribute the parliamentarians into two chambers. If in one of the chambers a member $A$ has at least two enemies, then in the other chamber $A$ has no more than one enemy. We will move $A$ to the other chamber; as a result, the total number $s$ of enemy pairs in both chambers decreases. ... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 25,466 |
272. In a notebook, several numbers are written. It is allowed to append to the already written numbers any number equal to the arithmetic mean of two or more of them, provided that it is different from all the already written ones. Prove that, starting with the two numbers 0 and 1, using such additions, one can obtain... | 272. All binary-rational fractions with denominators $2, 4, 8, 16$, etc., are easily obtained.
To obtain the fraction $1 / n$ from the pair $(0 ; 1)$, it is sufficient to choose $n$ different binary-rational numbers with a sum of 1 and take their arithmetic mean; for example, for $n=5:$
$$
\frac{1}{5}=\frac{\frac{3}{... | proof | Algebra | proof | Yes | Yes | olympiads | false | 25,467 |
273. A decreasing sequence $x_{n}$ of positive numbers is such that for any natural $n$
$$
x_{1}+\frac{x_{4}}{2}+\frac{x_{9}}{3}+\ldots+\frac{x_{n^{2}}}{n} \leqslant 1
$$
Prove that for any natural $n$
$$
x_{1}+\frac{x_{2}}{2}+\frac{x_{3}}{3}+\ldots+\frac{x_{n}}{n} \leqslant 3
$$ | 273. Consider segments of the number series $x_{m} / m, k^{2} \leqslant m \leqslant$ $\leqslant(k+1)^{2}-1 \quad(k=1,2,3, \ldots), \quad k$-th segment consists of $(k+1)^{2}-k^{2}=2 k+1$ numbers from $x_{k^{2}} / k^{2}$ to $x_{(k+1)^{2}-1} /\left((k+1)^{2}-1\right)$.
By replacing each term $x_{m} / m$ in the $k$-th se... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 25,468 |
274. On a plane, there are several points. For some pairs $A, B$ of these points, vectors $\overrightarrow{A B}$ are taken, such that in each point, the number of vectors starting from it is equal to the number of vectors ending at it. Prove that the sum of all selected vectors is 0. | 274. Let's choose any point $O$ and represent each of the vectors $\overrightarrow{A B}$ as $\overrightarrow{O B}-\overrightarrow{O A}$. In the sum of all vectors, each vector $\overrightarrow{O M}$, where $M$ is one of the given points, will appear as many times with a minus sign as with a plus sign, so the entire sum... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 25,469 |
275. What is the minimum number of chips that need to be placed on the fields of a chessboard of size
a) $8 \times 8$ cells,
b) $n \times n$ cells
so that on each line passing through the center of any field and parallel to any side or diagonal of the board, there is at least one chip? (Chips are placed in the cente... | 275. Answer: a) for $n=8-16$ chips; b) for $n$ even $\rightarrow$ $2 n$, for $n$ odd $-2 n+1$.
The arrangement of such a number of chips is clear from Fig. $113, a, 6$. The proof that a smaller number cannot suffice is simpler for even $n$; on each line parallel to one diagonal, there should be a chip, and on the diag... | )16chips;b)fornevenarrow2n,fornoddarrow2n-1 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 25,470 |
276. Find $x$ and $y$ from the system of equations
$$
\frac{x-y \sqrt{x^{2}-y^{2}}}{\sqrt{1-x^{2}+y^{2}}}=a, \quad \frac{y-x \sqrt{x^{2}-y^{2}}}{\sqrt{1-x^{2}+y^{2}}}=b
$$
($a$ and $b$ are given numbers). | 276. Answer: $x=\frac{a+b \sqrt{a^{2}-b^{2}}}{\sqrt{1-a^{2}+b^{2}}} ; y=\frac{b+a \sqrt{a^{2}-b^{2}}}{\sqrt{1-a^{2}+b^{2}}}$.
By adding and subtracting the given equations, and then multiplying the results, we obtain the following consequence of the system:
$$
x^{2}-y^{2}=a^{2}-b^{2}, \quad \text { i.e. } \sqrt{x^{2}... | \frac{+b\sqrt{^{2}-b^{2}}}{\sqrt{1-^{2}+b^{2}}};\frac{b+\sqrt{^{2}-b^{2}}}{\sqrt{1-^{2}+b^{2}}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 25,471 |
279. Natural numbers $p$ and $q$ are coprime. The segment $[0 ; 1]$ is divided into $p+q$ equal segments. Prove that in each of these segments, except for the two extreme ones, there lies exactly one of the $p+q-2$ numbers
$$
\frac{1}{p}, \frac{2}{p}, \ldots, \frac{p-1}{p}, \frac{1}{q}, \frac{2}{q}, \ldots, \frac{q-1}... | 279. Since $p$ and $q$ are coprime, each of them is coprime with $n=p+q$. Therefore, all numbers
$$
\frac{l}{p}, \frac{l}{q}, \frac{i+j}{n} \quad(i=1,2, \ldots, p-1 ; j=1,2, \ldots, q-1)
$$
are distinct. Note that $\frac{l+1}{p+q}$ always lies between $\frac{l}{p}$ and $\frac{l}{q}$; therefore, it is clear that all f... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 25,474 |
280. Through a point $O$ in space, 1979 lines $l_{1}, l_{2}, \ldots, l_{1979}$ are drawn, no two of which are mutually perpendicular. On the line $l_{1}$, a point $A_{1}$, different from $O$, is taken. Prove that it is possible to choose points $A_{i}$ on $l_{i}, i=2,3, \ldots, 1979$, such that the following 1979 pairs... | 280. Let $e_{l}$ be a unit vector directed along the line $l_{l}$. Put $e_{i} e_{l+1}=c_{l} (i=1,2, \ldots, 1978), e_{1979} e_{1}=c_{1979}$ ( $c_{l}$ - the cosine of the angle between $e_{l}$ and the next vector $\left.\boldsymbol{e}_{l+1}\right) ;$ let $\overrightarrow{O A}_{l}=a_{i} e_{l}$. The condition of perpendic... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,475 |
281*. The final sequence $a_{1}, a_{2}, \ldots, a_{n}$ of numbers 0 and 1 must satisfy the following condition: for any integer $k$ from 0 to $n-1$, the sum
$$
a_{1} a_{k+1}+a_{2} a_{k+2}+\ldots+a_{n-k} a_{n}
$$
is an odd number.
a) Come up with such a sequence for $n=25$.
b) Prove that such a sequence exists for s... | 281. The value specified in the problem condition
$$
p_{k}=a_{1} a_{k+1}+a_{2} a_{k+2}+\ldots+a_{n-k} a_{n}
$$
is conveniently calculated as follows: the sequence $A_{n}=\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ is signed under itself with a shift of $k$ digits; in this case, $p_{k}=p_{k}\left(A_{n}\right)$ is the nu... | 1201 | Combinatorics | proof | Yes | Yes | olympiads | false | 25,476 |
282. A convex quadrilateral $A B C D$ is divided by its diagonals into four triangles. Prove that if the radii of all four inscribed circles in these triangles are equal to each other, then the quadrilateral $A B C D$ is a rhombus. | 282. Suppose the diagonals of the quadrilateral $ABCD_{\text{s}}$ intersect at point $O$, and are not perpendicular to each other, for example, that angle $AOB$ is acute. Construct points $A'$ and $B'$, symmetric to $A$ and $B$ with respect to point $O$. The radii of the circles inscribed in triangles $A'OB$ and $B'OC$... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,477 |
283*. On a line, points $A_{1}, A_{2}, \ldots, A_{n}$ are arranged in order such that the lengths of the segments $A_{1} A_{2}, A_{2} A_{3}, \ldots, A_{n-1} A_{n}$ do not exceed 1. It is required to mark $k-1$ of the points $A_{2}, \ldots, A_{n-1}$ in red so that the lengths of any two of the $k$ parts into which the s... | 283. a) Let's consider possible arrangements $T$ of a pair of red points ($A_{l}, A_{r}$) that divide the segment $A_{0} A_{n}$ into three parts: $A_{0} A_{l}, A_{l} A_{r}, A_{r} A_{n}$. Let the largest of these parts be denoted by $M$, and the smallest by $m$. From all the arrangements $T$, we select those for which $... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 25,478 |
284. Two-digit numbers from 19 to 80 are written in sequence. Is the resulting number 192021 ,.. ... 7980 divisible by 1980? | 284. Answer: divisible. The given number is obviously divisible by 20. Divisibility by 99 follows from the fact that $100=99+1$, and the equality $19+20+\ldots+80=99 \cdot 31$. | divisible | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 25,479 |
285. The vertical side $AB$ of the square $ABCD$ is divided into $n$ segments such that the sum of the lengths of the segments with even numbers is equal to the sum of the lengths of the segments with odd numbers. Through the points of division, segments parallel to the side $AD$ are drawn, and then each of the resulti... | 285. If $S_{1}$ and $S_{2}$ are the sums of the areas of the left parts with even and odd numbers, and $S_{3}$ and $S_{4}$ are the analogous sums for the right parts, then
$$
S_{2}+S_{4}=S_{3}+S_{1}=S_{3}+S_{4}=S_{1}+S_{2}
$$ | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,480 |
287. Points $M$ and $P$ are the midpoints of sides $B C$ and $C D$ of the convex quadrilateral $A B C D$. It is known that $A M + A P = a$. Prove that the area of quadrilateral $A B C D$ is less than $a^{2} / 2$. | 287. $S_{A B C D}=2 S_{A M P C}$. Let $A M=x$, then
$$
S_{A M C D}<2 S_{A M P} \leqslant x(a-x) \leqslant \frac{a^{2}}{4}
$$ | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,482 |
288. Does the equation $x^{2}+y^{3}=z^{4}$ have solutions in prime numbers $x, y, z$? | 288. Answer: does not have. From $y^{3}=\left(z^{2}-x\right)\left(z^{2}+x\right)$ and the simplicity of $y$, we get either $z^{2}-x=1, z^{2}+x=y^{3}$, or $z^{2}-x=y$, $z^{2}+x=y^{2}$. Both systems do not have solutions in prime numbers | doesnothave | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 25,483 |
289. On the diameter $A C$ of a certain circle, a point $E$ is given. Draw a chord $B D$ through it so that the area of the quadrilateral $A B C D$ is maximized. | 289. Let $O$ be the center of the given circle, $R$ its radius, and $OE=a$. If $a \geqslant R / \sqrt{2}$, then the desired chord $BD$ subtends an arc of $90^{\circ}$ and is tangent to the circle of radius $R / \sqrt{2}$ centered at $O$. If $a<R / \sqrt{2}$, the desired chord is perpendicular to the diameter $AC$. It i... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,484 | |
290. On the shore of a large round lake, there are several settlements. Some of them are connected by ferry routes. It is known that two settlements are connected by a route if and only if the two next settlements in the counterclockwise direction are not connected by a route. Prove that from any settlement to any othe... | 290. Let $A, B$ and $C$ be three consecutive points on the shore of a lake. From the condition, it follows that $A$ and $B$ are connected if and only if $B$ and $C$ are not connected. Thus, all points are divided into pairs of adjacent points connected by a steamboat route. Moreover, any two such pairs are connected, i... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 25,485 |
291. A six-digit number, written with six different non-zero digits, is divisible by 37. Prove that by rearranging the digits of this number, you can obtain at least 23 other different six-digit numbers that are divisible by 37. | 291. Along with the number $N=\overline{a b c d e f}$, the numbers $\overline{a b c}+\overline{d e f}$, $\overline{b c a f d e}$, and $\overline{c a b d e f}$ are also divisible by 37. | proof | Number Theory | proof | Yes | Yes | olympiads | false | 25,486 |
304. Two identical chessboards $(8 \times 8$ cells) have a common center, with one of them rotated relative to the other by $45^{\circ}$ about the center. Find the total area of all intersections of the black cells of these two boards, if the area of one cell is 1. | 304. Answer: $32(\sqrt{2}-1)$. You need to use the fact that when one of the boards is rotated by $90^{\circ}$, the black cells are replaced by white ones and vice versa, and when both boards are rotated, the black cells are replaced by white ones on both boards. Therefore, the desired area is a quarter of the area of ... | 32(\sqrt{2}-1) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,497 |
305. On a circle, points $A, B, M$ and $N$ are given. From point $M$, chords $M A_{1}$ and $M B_{1}$ are drawn, perpendicular to the lines $N B$ and $N A$ respectively. Prove that the lines $A A_{1}$ and $B B_{1}$ are parallel. | 305. It is sufficient to prove that $\angle B_{1} M A=\angle A A_{1} B_{1}=$ $=\angle A_{1} B_{1} B$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,498 |
306. We will say that a number has the property $\mathrm{P}(k)$ if it can be factored into the product of $k$ consecutive natural numbers, all greater than 1.
a) Find $k$ such that some number $N$ simultaneously has the properties $\mathrm{P}(k)$ and $\mathrm{P}(k+2)$.
b) Prove that there are no numbers that simultane... | 306. a) $k=3 ; 720=2 \cdot 3 \cdot 4 \cdot 5 \cdot 6=8 \cdot 9 \cdot 10$.
b) If $m(m+1)=n(n+1)(n+2)(n+3)$, then $m^{2}+m+1=\left(n^{2}+3 n+1\right)^{2}$, which is impossible. | 3 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 25,499 |
307. The table has 4 rows. In the first row, arbitrary natural numbers are written, which may include identical ones. The second row is filled as follows: from left to right, the numbers in the first row are examined, and under the number $a$, the number $k$ is written if $a$ appears for the $k$-th time in the first ro... | 307. If in some column of the table the first three numbers, counting from the top, are $a, m$ and $n$, then each of the numbers $1,2, \ldots, m-1, m$ will appear in the second row to the left of the number $m$ at least $n$ times, and any number greater than $m$ will appear fewer than $n$ times. | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 25,500 |
308. Given a number $a$. Find the minimum value of the area of a rectangle with sides parallel to the coordinate axes $O x$ and $O y$ and containing the figure defined by the system of inequalities
$$
\left\{\begin{array}{l}
y \leqslant -x^{2} \\
y \geqslant x^{2}-2 x+a
\end{array}\right.
$$ | 308. Answer: $S=(1-a) \sqrt{1-a}$ for $a \leqslant 0$; $S=1-2a$ for $0<a<1/2$; $S=0$ for $a \geqslant 1/2$. | S=(1-)\sqrt{1-}for\leqslant0;S=1-2afor0<<1/2;S=0for\geqslant1/2 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 25,501 |
309. Equilateral triangles $A B C, C D E$ and $E H K$ (vertices are listed counterclockwise) are placed on a plane such that $\overrightarrow{A D}=\overrightarrow{D K}$. Prove that triangle $B H D$ is also equilateral. | 309. When rotated by $60^{\circ}$ around vertex $C$, triangle $C A D$ will transform into triangle $C B E$, and when rotated by $60^{\circ}$ around point $H$, triangle $H B E$ will transform into triangle $H D K$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,502 |
310. In a certain village, there are 1000 residents. Every day, each of them shares the news they learned yesterday with all their acquaintances. It is known that any news eventually becomes known to all residents of the village.
Prove that it is possible to choose 90 residents such that if they are all simultaneously... | 310. We will say that residents $X, A_{1}, A_{2}, \ldots, A_{k}, Y$ form a chain if $X$ is acquainted with $A_{1}, A_{2}$ is acquainted with $A_{3}, \ldots, A_{k}$ is acquainted with Y. From the condition, it follows that any two residents are connected by some chain. We will consider that there are no closed chains (i... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 25,503 |
311. For numbers $a$ and $b$, it is known that the inequality
$$
a \cos x + b \cos 3x > 1
$$
has no solutions. Prove that $|b| \leqslant 1$. | 311. We have $f(\pi n)=(-1)^{n} a+(-1)^{n} b ; \quad f(\pi / 3)=a / 2-b$; $(2 \pi / 3)=-a / 2+b$. Therefore $|a+b| \leqslant 1,|a-2 b| \leqslant 2$. | proof | Inequalities | proof | Yes | Yes | olympiads | false | 25,504 |
314*. Is it possible to color all cells of some rectangular table in white and black so that there are an equal number of white and black cells, and in each row and each column, more than $3 / 4$ of the cells are of one color? | 314. Let's call a row of an $m \times n$ table white if it contains more white cells, and black otherwise. Let $p$ and $q$ be the numbers of white and black rows, and $r$ and $s$ be the numbers of white and black columns ($p+q=m, r+s=n$). We can assume that $p \leqslant q$. Suppose that in each row (both rows and colum... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 25,507 |
315. Prove that if the quadrilaterals $A C P H$, $A M B E$, $A H B T$, $B K X M$, $C K X P$ are parallelograms, then the quadrilateral $A B T E$ is also a parallelogram (the vertices of all quadrilaterals are listed counterclockwise). | 315. It follows from the condition that $\overrightarrow{A E}=\overrightarrow{M B}=X \vec{K}=P \vec{C}=H \vec{A}=\overrightarrow{B T}$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,508 |
316. Solve the equation
$$
x^{3}-y^{3}=x y+61
$$
in natural numbers $x, y$. | 316. Answer: $x=5, y=6$. If $x=d+y$, where $d \geqslant 1$, then $(3 d-1) y^{2}+\left(3 d^{2}-d\right) y+d^{3}=61$, from which $d \leqslant 3$. | 5,6 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 25,509 |
317. In a football tournament, 18 teams played 8 rounds - each team played against eight different teams. Prove that there will be three teams that have not played against each other yet. | 317. For any team, there will be 9 teams that it has not yet played against, and among these nine, there will be two that have not played against each other. | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 25,510 |
318. Points $C_{1}, A_{1}, B_{1}$ are taken on the sides $A B, B C$, and $C A$ of triangle $A B C$ respectively, such that
$$
A C_{1}: C_{1} B=B A_{1}: A_{1} C=C B_{1}: B_{1} A=1 / 3
$$
Prove that the perimeter $P$ of triangle $A B C$ and the perimeter $p$ of triangle $A_{1} B_{1} C_{1}$ are related by the inequaliti... | 318. Let $A_{2}, B_{2}, C_{2}$ be points on the sides $B C, C A$, and $A B$ such that $A C_{2} / C_{2} B = B A_{1} / A_{1} C = C B_{2} / B_{2} A = 3$. The perimeter of the hexagon $A_{1} A_{2} B_{1} B_{2} C_{1} C_{2}$ is $3 / 4 P$ and is greater than the perimeter of the triangle $A_{1} B_{1} C_{1}$. Next, write the tr... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,511 |
320. A student needs to copy a convex polygon that fits inside a circle of radius 1. First, the student laid out the first side, then drew the second from its end, the third from the end of the second, and so on. Upon completing the construction, the student found that the polygon did not close, and the first and last ... | 320. Let $A$ and $B$ be the endpoints of the constructed broken line. Project the broken line onto the line $AB$. The relative error of each segment is $p$, so the sum of the absolute errors of all segments does not exceed $4 p$, hence $d \leqslant 4 p$. | \leqslant4p | Geometry | proof | Yes | Yes | olympiads | false | 25,513 |
322. Find at least one natural number $n$ such that each of the numbers $n, n+1, n+2, \ldots, n+20$ has a common divisor with the number $30030=2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13$ that is greater than one. | 322. Answer: for example, 9440. Let $m=2 \cdot 3 \cdot 5 \cdot 7 \cdot k$. By choosing $k$ such that $m-1$ is divisible by 11, and $m+1$ is divisible by 13, we get that the number $n=m-10$ satisfies the condition of the problem (P2). | 9440 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 25,515 |
325. a) Find the smallest possible value of the polynomial
$$
P(x, y)=4+x^{2} y^{4}+x^{4} y^{2}-3 x^{2} y^{2}
$$
b) ${ }^{*}$ Prove that this polynomial cannot be represented as a sum of squares of polynomials in the variables $x, y$. | 325. a) Answer: 3 at $x=y=1$.
From the inequality of the arithmetic mean, it follows that $1+x^{2} y^{4}+x^{4} y^{2} \geqslant 3 x^{2} y^{2}$.
b) Let $P(x, y)=g_{1}^{2}(x, y)+g_{2}^{2}(x, y)+\ldots+g_{n}^{2}(x, y)$, where $g_{i}(x, y), i=1,2, \ldots, n-$ are polynomials. Since $P(x, 0)=$ $=P(0, y)=4$, the polynomials... | 3 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 25,518 |
328. In the numerical sequences $\left(a_{n}\right)$ and $\left(b_{n}\right)$, each term, starting from the third, is equal to the sum of the two preceding ones, with $a_{1}=1, a_{2}=2$ and $b_{1}=2, b_{2}=1$. How many numbers exist that appear in both sequences? | 328. Answer: three numbers. By induction, it is proved that $a_{n-1}<b_{n}<a_{n}$ for $n \geqslant 4$. | three\ | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 25,521 |
329. a) Let $m$ and $n$ be natural numbers. Prove that if for some non-negative integers $k_{1}, k_{2}, \ldots, k_{n}$ the number $2^{k_{1}}+2^{k_{2}}+\ldots+2^{k_{n}}$ is divisible by $2^{m}-1$, then $n \geqslant m$.
b) Does there exist a natural number that is divisible by $\underbrace{111 \ldots 1}$ and has a sum o... | 329. a) From all numbers of the form $2^{k_{1}}+2^{k_{2}}+\ldots+2^{k_{n}}$ that are divisible by $2^{m}-1$, we select the numbers with the smallest $n$, and from the obtained numbers, we select the number with the smallest $k_{1}+k_{2}+\ldots+k_{n}$. All numbers in the set $(k_{1}, k_{2}, \ldots, k_{n})$ are distinct.... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 25,522 |
330. A non-negative real number is assigned to each vertex of a cube, with the sum of all these numbers being 1. Two players play the following game: the first player chooses any face of the cube, the second player chooses another face, and finally, the first player chooses a third face. It is not allowed to choose fac... | 330. Among eight numbers, there are three that do not exceed y/6, and two of them will be at the ends of the diagonal of one of the faces. The player should choose this face first. | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 25,523 |
332. In parallelogram $ABCD$, which is not a rhombus, the ratio of the lengths of the diagonals is given as $AC: BD = k$. Let ray $AM$ be symmetric to ray $AD$ with respect to line $AC$, and ray $BM$ be symmetric to ray $BC$ with respect to line $BD$, with $M$ being the common point of rays $AM$ and $BM$. Find the rati... | 332. Answer: $M A / M B=k^{2}$. Let $O$ be the center of the parallelogram. Triangles $A O M$ and $M O B$ are similar. | MA/MB=k^2 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,525 |
334. Inside a tetrahedron, a point $M$ is chosen. Prove that at least one edge of the tetrahedron is seen from point $M$ at an angle whose cosine is no greater than $-1 / 3$. | 334. Let $A M, B M, C M$ and $D M$ be segments of length 1, laid out on the rays connecting point $M$ with the vertices of the tetrahedron. The line $D M$ intersects the triangle $A B C$ at some point $P$. If the cosines of the angles $A M D, B M D$ and $C M D$ are greater than $-1 / 3$, then these angles are less than... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,526 |
335. The numbers $a, b, c$ lie in the interval $(0 ; \pi / 2)$ and satisfy the equations: $\cos a=a$, $\sin \cos b=b$, $\cos \sin c=c$. Arrange these numbers in ascending order. | 335. Answer: $b<a<c$.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly.
335. Answer: $b<a<c$. | b<< | Algebra | math-word-problem | Yes | Yes | olympiads | false | 25,527 |
336. A closed broken line $M$ has an odd number of vertices - sequentially $A_{1}, A_{2}, \ldots, A_{2 n+1}$. Denote by $S(M)$ a new closed broken line, the successive vertices $B_{1}, B_{2}, \ldots, B_{2 n+1}$ of which are the midpoints of the segments of the broken line $M: B_{1}$ - the midpoint of segment $A_{1} A_{... | 336. Let $\quad 0$-point, for which $\overrightarrow{O_{1}}+\overrightarrow{O A_{2}}+\ldots$ ... $+\overrightarrow{O A}_{2^{n+1}}=0$. Choose $k$ such that $2^{k}-1$ is divisible by
$2 n+1$. Then the broken line $M_{k}$ will be homothetic to $M$ with respect to $O$ with a coefficient of $2^{-k}$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,528 |
337. Natural numbers from 1 to 1982 are arranged in some order. A computer scans pairs of adjacent numbers (the first and second, the second and third, etc.) from left to right up to the last pair and swaps the numbers in the scanned pair if the larger number is to the left. Then it scans all pairs from right to left f... | 337. Answer: 100. Consider the number $a$-the largest of the first 99 numbers, and the number $b$-the smallest of the last 1882 numbers and ensure that $a<100<b$. | 100 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 25,529 |
338. The Cucumber River, flowing through the Flower City, has several islands in the harbor area, the total perimeter of which is 8 meters. Know-it-all claims that it is possible to set off in a boat from the harbor and cross to the other bank, traveling less than 3 meters. The banks of the river in the harbor area are... | 338. Answer: Know-it-all is right. The total length of the projections of the islands onto the shore is less than 4 m. Therefore, the distance from the pier to the nearest gap between the islands is less than $2 \mathrm{m}$. | 2\mathrm{} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,530 |
339. On the coordinate plane $O x y$, the graph of the function $y=x^{2}$ was drawn. Then the coordinate axes were erased - only the parabola remained. How can you restore the coordinate axes and the unit of length using a compass and a ruler? | 339. Sequentially perform: 1) two parallel lines, each intersecting the parabola at two points; 2) a line through the midpoints of the resulting segments; 3) a perpendicular to this line, intersecting the parabola at two points \(A\) and \(B\); 4) the perpendicular bisector of segment \(A, B\) - this is the \(O y\) axi... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,531 |
341. Prove that for all positive values of $x$ the inequality
$$
2^{12} \sqrt{x}+2^{\sqrt[4]{x}} \geqslant 2 \cdot 2^{\sqrt[6]{x}}
$$
holds. | 341. $2^{\sqrt[12]{\sqrt{x}}}+2^{\sqrt[4]{x}} \geqslant 2 \cdot 2^{\frac{\sqrt[4]{x}+\sqrt{x}}{2}} \geqslant 2 \cdot 2^{\sqrt[6]{x}}$. | proof | Inequalities | proof | Yes | Yes | olympiads | false | 25,533 |
345. In a square table of $n \times n$ cells, $n-1$ cells are marked. Prove that by permuting the rows among themselves and the columns among themselves, it is possible to achieve that all the marked cells lie below the diagonal of the table. | 345. A column without marked cells is swapped with the rightmost column, and the last row of the resulting table is swapped with some row containing marked cells. The problem is reduced to a $(n-1) \times(n-1)$ table. | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 25,536 |
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