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[ Quadratic equations. Vieta's theorem ] Invariants
On the board, there was an equation of the form $x^{2}+p x+q=0$ with integer non-zero coefficients $p$ and $q$. From time to time, different schoolchildren approached the board, erased the equation, and then composed and wrote down an equation of the same form, the r... | According to Vieta's theorem, the coefficients of the new quadratic polynomial are $-(p+q)$ and $pq$ respectively. Note that the second coefficient (of $x$) of none of the written quadratic polynomials can be zero. Indeed, in this case, the constant term of all subsequent quadratic polynomials would be zero, which mean... | x^{2}+x-2 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 54,532 |
Solve the system
$$
\begin{aligned}
& y^{2}=4 x^{3}+x-4 \\
& z^{2}=4 y^{3}+y-4 \\
& x^{2}=4 z^{3}+z-4
\end{aligned}
$$ | The function $4 x^{3}+x-4$ is negative for $x \leq 0$, so all unknowns are positive. The function $4 x^{3}-x^{2}+x-4$ is increasing over the entire number line and equals zero at $x=1$. Therefore, if $x>1$, then $y^{2}=4 x^{3}+x-4>x^{2}$
> 1. Then, similarly, $z^{2}>x^{2}$ and
$x^{2}>z^{2}>y^{2}>x^{2}$, which is impo... | (1,1,1) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 54,533 |
The castle is surrounded by a circular wall with nine towers, where knights are on duty. After each hour, they all move to adjacent towers, with each knight moving either always clockwise or counterclockwise. During the night, each knight manages to take their turn on each tower. It is known that there was an hour when... | Let's imagine that knights stand on 18 platforms, located on two circular platforms that rotate in opposite directions by $40^{\circ}$ every hour. According to the condition, at some moment, five platforms were empty, while the corresponding platforms (in the same towers) had one knight each.
Let's consider two cases.... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,534 |
Berolov S.L.
In a convex $n$-gon, several diagonals are drawn. A drawn diagonal is called good if it intersects (at interior points) with exactly one of the other drawn diagonals. Find the maximum possible number of good diagonals. | We will prove by induction on $n$ that the number of good diagonals does not exceed $n-2$ if $n$ is even, and $n-3$ if $n$ is odd. We will consider a segment as a 2-gon without diagonals. The base cases ($n=2,3$) are obvious.
Inductive step. Let $n \geq 4$; denote our polygon by $P=A_{1} A_{1} \ldots A_{n}$. If no two... | n-2forevenn,n-3foroddn | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 54,536 |
In the Republic of Mathematicians, a number $\alpha>2$ was chosen and coins of denominations 1 ruble, as well as $\alpha^{k}$ rubles for each natural number $k$, were issued. In this case, $\alpha$ was chosen such that the denominations of all coins, except the smallest one, are irrational. Could it be that any amount ... | Let's show that mathematicians could choose the number $\alpha=\frac{\sqrt{29}-1}{2}-$ as a root of the equation $\alpha^{2}+\alpha=7$. It is clear that $\alpha > 2$. It is not difficult to see that for natural $m(2 \alpha)^{m}=a_{m}+b_{m} \sqrt{29}$, where $a_{m}$ and $b_{m}$ are integers, and $a_{m} > 0 > b_{m}$ for ... | proof | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 54,537 |
Authors: Bogdanov I.I. and Ivanov G.
In a safe, there are $n$ cells numbered from 1 to $n$. Initially, each cell contained a card with its number. Vasya rearranged the cards in some order so that the $i$-th cell now contains a card with the number $a_{i}$. Petya can swap any two cards with numbers $x$ and $y$, paying ... | Let $\left(b_{1}, \ldots, b_{n}\right)$ be an arbitrary arrangement of cards (here $b_{i}$ is the number on the card in the $i$-th cell). We call its cost the number
$\left|b_{1}-1\right|+\left|b_{2}-2\right|+\ldots+\left|b_{n}-n\right|$
Lemma. For any arrangement $(b_{1}, \ldots, b_{n})$, in which not all cards are ... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,538 |
Consider $\Gamma$. Petya calculated the number of all possible $m$-letter words, in which only four letters T, O, W, and N can be used, and in each word, the letters T and O are equal in number. Vasya calculated the number of all possible $2m$-letter words, in which only two letters T and O can be used, and in each wor... | Let's establish a one-to-one correspondence between Petya's and Vasya's words. We will divide Vasya's word, consisting of $2 m$ letters, into blocks of two letters. We will replace each TT block with the letter T, the OO block with the letter O, the TO block with the letter W, and the OT block with the letter N. This w... | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 54,539 |
In a bus with $n$ seats, all tickets are sold to $n$ passengers. The first to enter the bus is the Absent-Minded Scholar, who, without looking at the ticket, takes the first available seat. Subsequently, passengers enter one by one. If an entering passenger sees that their seat is free, they take their seat. If their s... | Let's number all passengers starting with the Scientist in the order in which they entered the bus. The last passenger has the number $n$. For simplicity, let's number the seats in the same way. Suppose all but the last passenger have already entered and taken their seats. There is one free seat left. If this were the ... | \frac{1}{2} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 54,540 |
Tompongo A.K.
Two players are involved. The first player has 1000 even cards (2, 4, ..., 2000), and the second player has 1001 odd cards (1, 3, ..., 2001). They take turns, with the first player starting. A turn consists of the following: the player whose turn it is lays down one of their cards, and the other player, ... | Let's call a move critical if the card 2001 is used. We will prove that the first player can score points on all of the second player's moves, except, perhaps, the critical one.
At any moment, we will denote the first player's cards as $a_{1}b_{k}$. We will prove that the first player can maintain this situation until... | First\player\-\499\points,\\player\-\501\points | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 54,541 |
Goovanovo $A . C$.
A natural number $N$ is represented as $N=a_{1}-a_{2}=b_{1}-b_{2}=c_{1}-c_{2}=d_{1}-d_{2}$, where $a_{1}$ and $a_{2}$ are squares, $b_{1}$ and $b_{2}$ are cubes, $c_{1}$ and $c_{2}$ are fifth powers, and $d_{1}$ and $d_{2}$ are seventh powers of natural numbers. Is it necessary that among the number... | Let's provide an example of a number $N$ for which all the indicated numbers will be distinct. Let $N=\left(3^{2}-2^{2}\right)^{105}\left(3^{3}-\right.$ $\left.2^{3}\right)^{70}\left(3^{5}-2^{5}\right)^{126}\left(3^{7}-2^{7}\right)^{120} \cdot$ Then
$N=A^{2}\left(3^{2}-2^{2}\right)=B^{3}\left(3^{3}-2^{3}\right)=C^{5}\... | Notnecessarily | Number Theory | proof | Yes | Yes | olympiads | false | 54,544 |
M. V. Murraikin
The city is a rectangular grid, with a five-story building in each cell. The renovation law allows choosing two adjacent cells (by side) with buildings and demolishing the building with fewer floors (or the same number). At the same time, the number of floors from the demolished building is added to th... | a) A square $20 \times 20$ can be divided into 25 squares $4 \times 4$, in each of which one house can be left. Indeed, in a $2 \times 2$ square, it is easy to gather all the houses in one cell. We will gather them in the marked cells (left image). Similarly, all the marked cells are gathered.
 Let's cut the original piece, for example, in the ratio $3:2$.
Suppose we already have several pieces of weights $2d > 2c > \ldots > b > a$, satisfying the condition of the problem, that is, $a > d$ (possibly, there are only two or three pieces, then some weights are written twice). We will show that we can make on... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 54,547 |
Poguin G.A.
At Ivan's birthday, $\$ 3 n$ guests arrived. Ivan has $\$ 3 n$ cylinders with letters A, B, and V written on top, with \$n\$ of each type. Ivan wants to organize a ball: put the cylinders on the guests and arrange them in circles (one or more) such that the length of each circle is divisible by \$3\$, and ... | First solution. We will divide all guests into ordered triples; we will put a cylinder with the letter A on the first person in the triple, B on the second, and C on the third. For this, we will line up the guests (which can be done in \$(3n)!\$ ways), group the first three into one triple, the next three into another,... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,549 |
Diding M.
In a country of knights (who always tell the truth) and liars (who always lie), 10 people are sitting at the vertices of a regular decagon around a round table, and among them, there are liars. A traveler can stand somewhere and ask the seated people: "What is the distance from me to the nearest liar among y... | Example. The first question is asked from an arbitrary point. If all answers to it are the same, then everyone sitting at the table is a liar, since a knight and a liar give different answers.
Otherwise, there will be neighbors who answered differently. We will stand in the middle of the arc between them. Since at lea... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 54,551 |
Rorbalako $A$.
In the cells of a square table $n \times n$, where $n > 1$, it is required to place different integers from 1 to $n^2$ such that every two consecutive numbers are in adjacent cells, and every two numbers that give the same remainder when divided by $n$ are in different rows and different columns. For wh... | Let's number the columns and rows from 1 to $n$ respectively from left to right and from top to bottom, and color the board in a checkerboard pattern such that the corner cell in the first column and first row is black.
Let $n$ be even. Fill the table with numbers from 1 to $n^2$ as follows: place them one after anoth... | Forallevenn | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 54,552 |
Petya has a deck of 36 cards (4 suits with 9 cards each). He chooses half of the cards (whichever he wants) and gives them to Vasya, while keeping the other half for himself. Then, each turn, the players take turns laying one card on the table (of their choice, face up); Petya starts. If, in response to Petya's move, V... | If Petya takes all the hearts, all the aces, kings, and queens, then Vasya will not be able to score points on the ace, king, and queen of hearts, i.e., he will score no more than 15 points.
Let's rephrase the problem. Consider a $4 \times 9$ board. Petya colors 18 cells black. We need to prove that Vasya can select a... | 15 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 54,554 |
Rai Gorodskii A.M.
Andrey Mikhailovich wrote on the board all possible sequences of length \$2022\$, consisting of 1011 zeros and 1011 ones. We will call two sequences compatible if they coincide in exactly 4 positions. Prove that Andrey Mikhailovich can divide all the sequences into 20 groups so that no two compatibl... | It is clear that compatible sequences coincide in units in two positions and in zeros in two positions. Let's consider the first five positions. There are \$C_5^3=10\$ ways to place three units in these five positions. For each of these ten ways, Andrey Mikhailovich selects a group of sequences written on the board, th... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,558 |
Authors: Binnov A.D., Binkov Yu.A., Gorskaya E.S.
A casino offers a game with the following rules. A player bets any whole number of dollars (but no more than they currently have) on either heads or tails. Then a coin is flipped. If the player guesses correctly how it will land, they get back their bet and an equal am... | The main observation in this problem is this: as soon as a player guesses correctly how the coin will land, they can guess incorrectly in all remaining games. Therefore, all bets after a win should be $1.
To begin, let's show that the player will not be able to leave having lost $1 or less. Let's see what they would h... | 98 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 54,559 |
Bakayev E.V.
Does there exist a natural number that can be represented as the product of two palindromes in more than 100 ways? (A palindrome is a natural number that reads the same backward as forward.) | Exists. Consider a palindrome consisting of \$n \$ ones \$ 1 \_n=1 \ldots 1 \$.
Method 1. If \$n\$ is divisible by \$k\$, then \$1_k\$ divides \$1_n\$. For example, \$111111\$ is divisible by \$111\$ and \$11\$. It remains to choose a number \$n\$ that has more than 100 proper divisors. For example, \$ 2 \wedge{101} \... | 128 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 54,560 |
[ Algorithm theory (miscellaneous) ]
Petya bought a microcalculator at the "Turing Machines and Other Computing Devices" store, which can compute $x y+x+y+1$ for any real numbers $x$ and $y$ and has no other operations. Petya wants to write a "program" to compute the polynomial $1+x+x^{2}+\ldots+x^{1982}$. By "program... | a) Let $\varphi(x, y)=x y+x+y+1=(x+1)(y+1)$.
The polynomial $x-1$ is computed by the program $P(x-1): x,-\frac{3}{2},-3, \varphi(-3 / 2, x)=-1 / 2(x+1), \varphi(-3,-1 / 2(x+1))=$ $-2\left(\frac{1}{2}-x / 2\right)=x-1$.
From this, it follows that if there is a program for computing the polynomial $g(x)$, then there is... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 54,563 |
5- }
$$
At King Arthur's court, $2 n$ knights have gathered, and each of them has no more than $n-1$ enemies among those present.
Prove that Merlin, Arthur's advisor, can seat the knights around a round table in such a way that no one sits next to their enemy. | Let's agree to call "friends" any two knights who are not enemies. Let's seat all the knights at a round table in any order. Suppose somewhere at the table sit knights $A$ and his enemy $B$; for definiteness, let's assume that $B$ is sitting to the right of $A$.
We claim that at the table, there will be a place where ... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,564 |
All possible $n$-digit numbers composed of the digits 1, 2, and 3 are considered. At the end of each of these numbers, a digit 1, 2, or 3 is appended such that to two numbers, which have different digits in all positions, different digits are appended. Prove that there exists an $n$-digit number, in which only one digi... | Let's prove that there are two numbers that differ by only one digit, to which different digits are appended. Indeed, suppose that to any two numbers that differ by only one digit, the same digit is appended. Then, by induction, it can be proven that to any two numbers that differ by $k$ digits, the same digit is appen... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,565 |
10,11
[ Pigeonhole Principle (other) ]
Author: Lakitanov E.
Two players are playing a card game. They have a deck of p pairwise distinct cards. For any two cards in the deck, it is known which one beats the other (however, if $A$ beats $B$, and $B$ beats $C$, it can happen that $C$ beats $A$). The deck is distributed b... | Let's list all possible situations that can occur in the game (that is, all possible pairs of decks of the participants). We will call a situation final if all the cards are with one player; critical if one of the players has exactly one card; and regular if both players have at least two cards. We will draw an arrow f... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,569 |
Franklin 5.
A convex $n$-gon $P$, where $n>3$, is cut into equal triangles by diagonals that do not intersect inside it.
What are the possible values of $n$, if the $n$-gon is cyclic? | Lemma. Let a convex $n$-gon be cut into equal triangles by diagonals that do not intersect inside it. Then, for each of the triangles in the partition, at least one side is a side (not a diagonal) of the $n$-gon.
Proof. Let a triangle in the partition have angles $\alpha \leq \beta \leq \gamma$ with vertices $A, B, C$... | 4 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 54,570 |
Mitranov I.V.
The King decided to reward a group of $n$ sages. They will be placed in a row, all facing the same direction, and a black or white hat will be placed on each of them. Each sage will see the hats of all those in front of them. The sages will take turns, from the last to the first, naming a color (white or... | Evaluation. $ $\$$ Madmen, obviously, may not guess correctly. The first speaking wise man can also not guess correctly, as he has no information about the color of his hat. Therefore, more than $\$ n-\mathrm{k}-1$ \$ correct guesses cannot be guaranteed.
Algorithm. Let all the wise men uniformly encode the coloring o... | n-k-1 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 54,572 |
Each cell of a chessboard is colored in one of two colors - blue or red. Prove that the cells of one of the colors have the property that a chess queen can visit them (the queen can visit the cells of this color more than once, she does not place herself on cells of the other color, but can jump over them).
# | If on the board there is a monochromatic row or column, the statement is obvious. Otherwise, we will act as follows: we will traverse all the red cells of the first row, move to the second, and there also traverse all the red cells, and so on. Let's say we couldn't move from the $i$-th row to the $(i+1)$-th. Then we wi... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,574 |
[Level $B$. .
Several boxes are arranged in a circle. Each of them can contain one or several balls (or it can be empty). In one move, it is allowed to take all the balls from any box and distribute them, moving clockwise, starting from the next box, placing one ball in each box.
a) Prove that if on each subsequent m... | a) The current state of the system described in the problem is determined by the number of balls in each box and by indicating the box from which the balls should be distributed next. Therefore, the number of possible states of the system is finite. From each state, by distributing the balls, one can transition to anot... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,576 |
Proizvolov V.V.
Consider an arbitrary polygon (not necessarily convex).
a) Is there always a chord of the polygon that divides it into two equal-area parts?
b) Prove that any polygon can be divided by some chord into parts, the area of each of which is not less than $1 / 3$ of the area of the polygon. (A chord of a ... | a) An example is a polygon consisting of three identical squares (halls) connected by thin curved corridors (see figure).
Let \( S \) be the area of the polygon, \( 0.3S \) be the area of o... | proof | Geometry | proof | Yes | Yes | olympiads | false | 54,577 |
Gooovanov A.S.
Prove that for any natural number $a_{1}>1$ there exists an increasing sequence of natural numbers $a_{1}, a_{2}, a_{3}, \ldots$ such that $a_{1}^{2}+a_{2}^{2}+\ldots+a_{k}^{2}$ is divisible by $a_{1}+a_{2}+\ldots+a_{k}$ for all $k \geq 1$. | Let's prove that for any numbers $a_{1}, \ldots, a_{n}$ satisfying the condition of the problem, we can find a number $a_{n+1}$ such that $S_{n+1}=a_{1}^{2}+\ldots+a_{n+1}^{2}$
is divisible by $s_{n+1}=a_{1}+\ldots+a_{n}+a_{n+1}$. From the equality $S_{n+1}=S_{n}+\left(a_{n+1}-s_{n}\right)\left(a_{n+1}+s_{n}\right)+s_... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 54,578 |
Authors: Dolnikov V.L., Karasev R.
On the plane, a family $S$ of equilateral triangles is drawn, which are obtained from each other by parallel translations, and any two triangles intersect. Prove that there are three points such that any triangle in the family $S$ contains at least one of them. | Let $ABC$ be one of the triangles in the family $S$. Let its height be taken as a unit. Since the triangles in $S$ intersect pairwise, they lie within a certain strip of width 2, parallel to side $AB$. Similarly, by considering strips parallel to $BC$ and $CA$, we examine their intersection - this will be a hexagon $H$... | proof | Geometry | proof | Yes | Yes | olympiads | false | 54,580 |
Pastor A.
In the city, there are several squares. Some pairs of squares are connected by one-way streets in such a way that from each square, you can exit onto exactly two streets. Prove that the city can be divided into 1014 districts such that the streets connect only squares from different districts, and for any tw... | First, let's prove that the squares can be painted in 13 colors such that it is impossible to reach a square of the same color by traveling fewer than three streets. For this, consider an auxiliary directed graph: its vertices will be the squares, and directed edges will connect pairs of squares between which there is ... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,581 |
Bogdanov I.I.
On the reverse sides of 2005 cards, different numbers are written (one on each). In one question, it is allowed to point to any three cards and find out the set of numbers written on them. What is the minimum number of questions needed to find out what numbers are written on each card?
# | Let there be $N$ questions asked. It is clear that each card participates in at least one question; otherwise, we would not be able to determine the number on it.
Suppose there are $k$ cards that participate in exactly one question.
Then, two such cards cannot appear in the same question. Indeed, if two such cards we... | 1003 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 54,582 |
Shapovalov A.V.
Natural numbers are painted in $N$ colors. There are infinitely many numbers of each color. It is known that the color of the half-sum of two different numbers of the same parity depends only on the colors of the addends.
a) Prove that the half-sum of numbers of the same parity and the same color is a... | a) Consider some color, for example, red. Find two red numbers whose difference is divisible by 8 (such numbers exist because the number of remainders when divided by 8 is finite, and by taking two red numbers with the same remainder, we get the desired pair). Denote these numbers by \(a\) and \(b\), and the color of t... | N | Number Theory | proof | Yes | Yes | olympiads | false | 54,583 |
Zaslavsky A.A.
In the infinite sequence $a_{1}, a_{2}, a_{3}, \ldots$ the number $a_{1}$ is equal to 1, and each subsequent number $a_{n}$ is constructed from the previous $a_{n-1}$ according to the rule: if the largest odd divisor of the number $n$ has a remainder of 1 when divided by 4, then $a_{n}=a_{n-1}+1$, if th... | a) Let $a_{n}=a_{n-1}+b_{n}$. Then $a_{n}=1+b_{2}+\ldots+b_{n}$. Note that $b_{2 n}=b_{n}$. Therefore, $a_{4 n}=1+b_{2}+\ldots+b_{4 n}=b_{2}+b_{4}+\ldots+b_{4 n}+\left(1+b_{3}\right)+\left(b_{5}+b_{7}\right)+\ldots+\left(b_{4 n-3}+b_{4 n-1}\right)=1+b_{2}+b_{3}+\ldots+b_{2 n}=a_{2 n}$
(expressions in parentheses are ze... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 54,584 |
10,11 [ Properties of polynomial coefficients $\quad$]
Author: Shapovalov A.B.
The polynomial $P(x)$ with real coefficients is such that the equation $P(m)+P(n)=0$ has infinitely many solutions in integers $m$ and $n$.
Prove that the graph of $y=P(x)$ has a center of symmetry. | Consider the polynomial $P_{a}(x)=P(a+x)+P(a-x)$. The sign of the coefficient of this polynomial at $x^{k}$ coincides with the sign of the $k$-th derivative of the function $P_{a}$ at $x=0$. For even $k$, this derivative is equal to $2 P^{(k)}(a)$, and for odd $k$, it is zero. In turn, the sign of $P^{(k)}(a)$ for larg... | proof | Algebra | proof | Yes | Yes | olympiads | false | 54,585 |
Petrov $\Phi$.
For a natural number $n$, denote $S_{n}=1!+2!+\ldots+n!$. Prove that for some $n$, the number $S_{n}$ has a prime divisor greater than $10^{2012}$. | For a prime $p$ and a natural number $n$, denote by $v_{p}(n)$ the exponent to which $p$ is raised in the prime factorization of $n$. Note that if
$v_{p}(n) \neq v_{p}(k)$, then $v_{p}(n+k)=\min \left\{v_{p}(n), v_{p}(k)\right\}$. From this, it immediately follows that if $v_{p}\left(S_{n}\right)n$.
Suppose that all ... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 54,588 |
10 |
| Task | $\left[\begin{array}{l}{[\text { Pigeonhole Principle (angles and lengths). }} \\ {[\underline{\text { Pairing and grouping; bijections }}]}\end{array}\right.$ | Classes: 10, |
Prove that it is impossible to arrange three great circle arcs, each $300^{\circ}$ long, on a sphere such that no two of them sh... | Assume that the required arrangement is possible. Draw the great circles of the sphere containing the given arcs. Let $A, B, C, A^{\prime}, B^{\prime}, C^{\prime}$ be their pairwise points of intersection (if some of them coincide, the solution is similar), with the pairs of points $A$ and $A^{\prime}, B$ and $B^{\prim... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,589 |
Given a convex figure and a point A inside it. Prove that there exists a chord (i.e., a segment connecting two boundary points of the convex figure) passing through point A and bisected by point A. | Rotate the chord continuously. Prove that during this rotation, we will achieve the desired position.
## Solution
Draw any chord through point A. Let a and b denote the lengths of the segments into which the chord is divided by point A. We start rotating the chord counterclockwise (thereby continuously changing the d... | proof | Geometry | proof | Yes | Yes | olympiads | false | 54,590 |
Bogdanov I.I.
On each of 2013 cards, a number is written, and all these 2013 numbers are distinct. The cards are face down. In one move, it is allowed to point to ten cards, and in response, one of the numbers written on them will be reported (it is unknown which one).
For what largest $t$ can it be guaranteed to fin... | 1) Let's first show that it is impossible to guess $1987=2013-26$ cards. We will number the cards $A_{1}, \ldots, A_{2013}$, place the numbers from 1 to 2013 on them in the same order, and indicate how to answer so that none of the numbers on the cards $A_{1}, \ldots, A_{27}$ can be determined.
For each $i=1, \ldots, ... | 1986 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 54,592 |
Berolov S.l.
Does there exist an infinite increasing sequence $a_{1}, a_{2}, a_{3}, \ldots$ of natural numbers such that the sum of any two distinct terms of the sequence is coprime with the sum of any three distinct terms of the sequence? | Let $a_{1}=1, a_{n+1}=\left(3 a_{n}\right)!+1$. Note that all these numbers are odd. To show that this sequence meets the requirements, we will need to strengthen these requirements somewhat. We will say that a pair (triple) of numbers is good if all its elements, except for one, are distinct (and one can appear severa... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 54,596 |
[ Counting in two ways ]
Numbers $x_{1}, x_{2}, \ldots, x_{n}$, each equal to 1 or -1, are written around a circle, and the sum of the products of adjacent numbers is zero, and generally for each $k=1,2, \ldots, n-1$ the sum of $n$ products of numbers that are $k$ places apart is zero
(that is, $x_{1} x_{2}+x_{2} x_{... | a) $\left(x_{1}+\ldots+x_{n}\right)^{2}=\left(x_{1}^{2}+\ldots+x_{n}^{2}\right)+\left(x_{1} x_{2}+x_{2} x_{3}+\ldots+x_{n} x_{1}\right)+$ $+\left(x_{1} x_{3}+x_{2} x_{4}+\ldots+x_{n} x_{2}\right)+\ldots+\left(x_{1} x_{n}+x_{2} x_{1}+\ldots+x_{n} x_{n-1}\right)$.
In the right-hand side of this equation, the first term ... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 54,598 |
In the middle between two parallel streets, identical houses with a side length of $a$ are lined up in a row. The distance between the streets is $3 a$, and the distance between two adjacent houses is $2 a$ (see figure).
. Connect the point where the gangster is located with segments to all points where the police are located. Number these segments with integers from $-\infty$ to $+\i... | 2vorv/2 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 54,600 |
Bernstein I. D:
a) Prove that the sum of the digits of the number $K$ does not exceed the sum of the digits of the number $8K$ by more than 8 times.
b) For which natural numbers $k$ does there exist a positive number $c_{k}$ such that $\frac{S(k N)}{S(N)} \geq c_{k}$ for all natural numbers $N$? Find the largest suit... | Let $S(N)$ be the sum of the digits of the number $N$. We will need the following properties of the function $S(N)$:
1) $S(A+B) \leq S(A)+S(B)$
2) $S\left(A_{1}+A_{2}+\ldots+A_{n}\right) \leq S\left(A_{1}\right)+S\left(A_{2}\right)+\ldots+S\left(A_{n}\right)$
3) $S(n A) \leq n S(A)$
4) $S(A B) \leq S(A) S(B)$
To veri... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 54,601 |
Ilychev V. On one side of an infinite corridor, there is an infinite number of rooms, numbered from minus infinity to plus infinity. In the rooms, 9 pianists live (several pianists can live in one room), and there is a piano in each room. Every day, some two pianists living in adjacent rooms (the $k$-th and ($k+1$)-th)... | Consider any three consecutive rooms (with numbers $n, n+1, n+2$). If a pianist ever ends up in one of them, this triplet of rooms will never become empty again: to leave this triplet, the pianist must move from the $n$-th room to the $(n-1)$-th room (or from the $(n+2)$-th to the $(n+3)$-th, which is symmetric), but t... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,603 |
Author: Shapovalov A.V., Kuukakov A.
There is a set of weights, the masses of which in grams are: $1, 2, 4, \ldots, 512$ (consecutive powers of two) - one weight of each mass. It is allowed to weigh a load using this set, placing the weights on both pans of the scales.
a) Prove that no load can be weighed using these... | Let $K_{n}(P)$ be the number of ways to weigh a weight $P$ using weights of $1,2, \ldots, 2^{n}$, and $K_{n}=\max _{P} K_{n}(P)$ (the maximum number of ways to weigh any weight using these weights). Obviously, $K_{0}=1, K_{1}=2$.
a) Our task is to prove that $K_{9} \leq 89$. We will prove that $K_{n+1} \leq K_{n}+K_{n... | 171 | Combinatorics | proof | Yes | Yes | olympiads | false | 54,604 |
Perrin $A$.
Players $A$ and $B$ take turns moving a knight on a $1994 \times 1994$ chessboard. Player $A$ can only make horizontal moves, that is, moves where the knight moves to an adjacent row. Player $B$ is allowed only vertical moves, where the knight moves to an adjacent column. Player $A$ places the knight on th... | Since the game ends in no more than $1994^{2}$ moves, one of the two players must have a winning strategy. If player $A$ does not have a winning strategy, then player $B$, playing correctly, wins regardless of $A$'s first move. We will prove that this is impossible. To do this, we will organize two games on two boards ... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,608 |
All integers from 1 to 100 are written in a string in an unknown order. With one question about any 50 numbers, you can find out the order of these 50 numbers relative to each other. What is the minimum number of questions needed to definitely find out the order of all 100 numbers?
# | To find the desired order $a_{1}, a_{2}, \ldots, a_{100}$ of numbers in a row, it is necessary that each pair $\left(a_{i}, a_{i+1}\right), i=1,2, \ldots, 99$, appears in at least one of the sets about which questions are asked; otherwise, for two sequences $a_{1}, \ldots, a_{i}, a_{i+1}, \ldots, a_{100}$ and $a_{1}, \... | 5 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 54,609 |
100 natural numbers, coprime in total, are arranged in a circle. It is allowed to add to any number the greatest common divisor of its neighbors. Prove that using such operations, it is possible to make all the numbers pairwise coprime.
# | Let for convenience $a_{n+100}=a_{n}$ when $n=1,2, \ldots, 100$. Notice that under the described procedure, the numbers remain pairwise coprime.
Lemma. Let $a_{1}, a_{2}, \ldots, a_{n}$ and $d$ be natural numbers. Then there exists a natural number $k$ such that the GCD $\left(a_{1}+k d, a_{i}\right) \leq d$ for any $... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 54,610 |
Zaslavsky A.A.
Given an acute triangle ABC. For an arbitrary line $l$, denote by $l_{a}, l_{b}, l_{c}$ the lines symmetric to $l$ with respect to the sides of the triangle, and by $I_{l}$ the incenter of the triangle formed by these lines. Find the geometric locus of points $I_{l}$. | Let's draw a line $m$ through the orthocenter $H$ of triangle $ABC$, parallel to $l$. The lines $m_{a}, m_{b}$, and $m_{c}$, which are symmetric to $m$ with respect to the corresponding sides, intersect at a point $I_{m}$, which lies on the circumcircle $\Omega$ of triangle $ABC$ (see problem 55657). Let the distance b... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 54,617 |
Ionin Yu.i.
In each cell of an infinite sheet of graph paper, a number is written such that the sum of the numbers in any square, the sides of which lie along the grid lines, does not exceed one in absolute value.
a) Prove the existence of a number $c$ such that the sum of the numbers in any rectangle, the sides of w... | Suppose in a certain rectangle with sides $a$ and $b (a < b)$. We construct four squares, each of which has three sides along some three sides of this rectangle $a \times b$; then the lines on which the fourth sides of these squares lie form a new rectangle with sides $2b - a$ and $|2a - b|$ (see Fig. 1 and 2; the case... | 3 | Combinatorics | proof | Yes | Yes | olympiads | false | 54,618 |
Р Рубанов и.с.
Znayka writes 10 numbers on the board, then Neznayka adds another 10 numbers, with all 20 numbers being positive and distinct. Could Znayka have written such numbers to ensure that he could later form 10 quadratic trinomials of the form $x^{2}+p x+q$, where the coefficients $p$ and $q$ include all the w... | Lemma 1. 1) If $p>4$ and $p>q$, then the quadratic polynomial $x^{2}+p x+q$ has two distinct roots.
2) If $00$, then at least one of the polynomials $x^{2}+p x+q, x^{2}+q x+p$ has no roots.
Proof. 1) The discriminant $p^{2}-4 q>4 q-4 q=0$.
2) If $q>p$, then the discriminant of the first polynomial $p^{2}-4 qr$ $=-\l... | Can | Algebra | math-word-problem | Yes | Yes | olympiads | false | 54,620 |
| Problem 109870 Topics | [ Reconfigurations |
| :---: | :---: |
| Problem $1098 / 0$ themes | $\left[\begin{array}{l}\text { Colorings } \\ {[\text { Invariants }}\end{array}\right.$ |
The streets of the city of Duzhinsk - simple broken lines, not intersecting each other at internal points. Each street connects two i... | Let's replace each white street in the city with two - a blue one and a red one, connecting the ends of the blue streets adjacent to the white one with a blue color, and the ends of the adjacent red streets with a red color (see the figure). According to these figures, we will call the white streets types a, b, v, and ... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,621 |
Ivanov I.
In the country, there are 100 cities, some pairs of which are connected by roads. For any four cities, there are at least two roads between them. It is known that there is no route passing through each city exactly once. Prove that it is possible to choose two cities such that each of the remaining cities is... | Consider a graph where the vertices correspond to cities and the edges to roads. Let's choose the longest path S in this graph, and let vertices $A$ and $B$ be the endpoints of this path. From the condition, it follows that the path $S$ contains no more than 99 vertices. Note that the endpoints of the path $S$ - vertic... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,624 |
Solyoviov I.
In the cells of an infinite sheet of graph paper, real numbers are written. Two figures are considered, each consisting of a finite number of cells. It is allowed to move the figures parallel to the grid lines by an integer number of cells. It is known that for any position of the first figure, the sum of... | Let's choose an arbitrary cell and denote its center by $O$. We introduce a Cartesian coordinate system with the origin at $O$, axes parallel to the grid lines, and a unit segment equal to the side of the cell. Fix some positions of the given figures I and II and denote the centers of the cells they cover by $A_{1}, A_... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,627 |
k vertices of a regular n-gon are colored. The coloring is called almost uniform if for any natural number $m$ the following condition holds: if $M_{1}$ is a set of $m$ consecutive vertices and $M_{2}$ is another such set, then the number of colored vertices in $M_{1}$ differs from the number of colored vertices in $M_... | 1) Induction on $n$. The base case $(n=2)$ is obvious.
Induction step. Suppose the statement is proven for regular polygons with fewer than $n$ vertices.
For $k=1$, the statement is obvious. The filled (red) vertices can be swapped with the unfilled ones. Therefore, we will assume from now on that
$1n$ ). Consequent... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,631 |
Klimov A.V.
A rectangular sheet of paper measuring $a \times b$ cm is cut into rectangular strips, each of which has a side of 1 cm. The cut lines are parallel to the sides of the original sheet. Prove that at least one of the numbers $a$ or $b$ is an integer. | Let's call a rectangular strip, which is an element of the given partition, a block. Denote one of the sides of the original rectangle of length $b$ as $B$; we will call a block horizontal if its unit side is parallel to side $B$, and vertical otherwise. Denote the line parallel to $B$, at a distance $x$ from $B$ and i... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 54,634 |
Folklore
Prove that natural numbers can be placed at the vertices of a polyhedron such that in any two vertices connected by an edge, the numbers are not coprime, and in any two vertices not connected by an edge, the numbers are coprime.
Note: There are infinitely many prime numbers.
# | Place different prime numbers on the edges of a polyhedron. Now, in the vertices, write the products of the numbers standing on the edges that converge at this vertex. The numbers placed satisfy the requirements of the problem.
Author: $\underline{\text { Folklore }}$
In the cells of a $10 \times 10$ square table, no... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 54,635 |
Prove that there does not exist a polynomial $P(x)$ with integer coefficients for which $P(6)=5$ and $P(14)=9$.
# | Use the Bezout's theorem for integer polynomials.
## Solution
Bezout's theorem for integer polynomials. For any polynomial $P(x)$ with integer coefficients and any distinct integers $a$ and $b$, the number
$P(a)-P(b)$ is divisible by $a-b$.
Proof. The difference $P(a)-P(b)$ is a sum of expressions of the form $a^{k... | proof | Algebra | proof | Yes | Yes | olympiads | false | 54,640 |
a) In a square of area 6, three polygons of area 3 are placed. Prove that among them, there will be two polygons
the area of the common part of which is not less than 1.
b) In a square of area 5, nine polygons of area 1 are placed. Prove that among them, there will be two polygons the area of the common part of which... | a) According to problem $\underline{58106}$ a) $6=9-\left(S_{12}+S_{23}+S_{13}\right)+S_{123}$, that is, $S_{12}+S_{23}+S_{13}=3+S_{123} \geq 3$.
Therefore, one of the numbers $S_{12}, S_{23}, S_{13}$ is not less than 1.
b) According to problem $\underline{58106}$ b) $5 \geq 9-M_{2}$, i.e., $M_{2} \geq 4$. Since from... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,641 |
Consider a circle of radius 1. Describe a regular $n$-sided polygon around it and inscribe a regular $n$-sided polygon in it. Denote their perimeters by $P_{n}$ (for the circumscribed) and $p_{n}$ (for the inscribed).
a) Find $P_{4}, p_{4}, P_{6}$, and $p_{6}$.
b) Prove that the following recurrence relations hold: $... | b) Let $a_{n}$ and $b_{n}$ denote the sides of the inscribed and circumscribed regular $n$-gons. The midpoints $K, L, M$ of the sides $A B, B C, C D$ of the circumscribed regular $2 n$-gon are consecutive vertices of the inscribed regular $2 n$-gon. Moreover, $KM$ is a side of the inscribed regular $n$-gon, the point $... | P_{96}\approx6.285429,p_{96}\approx6.282064 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 54,642 |
On the plane, there are four points that do not lie on the same straight line. Prove that there exists a non-acute triangle with vertices at these points.
# | Let's consider various cases of the arrangement of four given points on a plane.
1) The four given points are vertices of a convex quadrilateral. The sum of the angles of this quadrilateral is $360^{\circ}$. Therefore, at least one of its angles is not acute. Let's denote the vertex where the angle is not acute by $A$... | proof | Geometry | proof | Yes | Yes | olympiads | false | 54,643 |
10,11 [ Proof by contradiction $]$
Given a convex quadrilateral $A B C D$. Pedestrian Petya starts from vertex $A$, walks along side $A B$ and continues along the perimeter of the quadrilateral. Pedestrian Vasya starts from vertex $A$ simultaneously with Petya, walks along diagonal $A C$ and arrives at $C$ simultaneou... | Let Petya arrive at vertex $B$ at the same time Vasya passes point $X$ on the diagonal $A C$, and at vertex $C$ when Tolya passes point $Y$ on the diagonal $B D$. Then
$A B: B C = A X: X C, B C: C D = B Y: Y D$. Therefore, $B X$ and $C Y$ are the angle bisectors of angles $B$ and $C$ of the quadrilateral. If Vasya and... | proof | Geometry | proof | Yes | Yes | olympiads | false | 54,646 |
Shapovalov A.V.
At a round table sit ten people, each with several nuts. In total, there are a hundred nuts. On a common signal, each person passes part of their nuts to the person on their right: half if they (the one giving) had an even number, or one nut plus half of the remainder if they had an odd number. This op... | Let's take 10 nuts from each person sitting at the table (for some, the number of nuts may become negative). From each "half" (the part given to the neighbor and the part kept for oneself), 5 nuts will be subtracted. Therefore, the rule of passing nuts will not be violated. Now the total number of nuts at the table is ... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 54,650 |
Shapovalov A.V. Fedia has three sticks. If it is impossible to form a triangle with them, Fedia shortens the longest stick by the sum of the lengths of the other two. If the length of the stick does not become zero and it is still impossible to form a triangle, Fedia repeats the operation, and so on. Can this process ... | The polynomial $P(x)=x^{3}-x^{2}-x-1$ has a root $t>1$, since $P(1)<0$. Then $t^{3}=t^{2}+t+1>t^{2}+t$. Taking the lengths of the sticks to be $t^{3}, t^{2}, t$. After the first cut, we get sticks with lengths $t^{2}, t, 1$. Since the ratio of the lengths has not changed, the process will continue indefinitely.
## Ans... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 54,652 |
Shaovalov A.V.
Let's say a deck of 52 cards is properly arranged if each pair of adjacent cards matches in suit or rank, the same is true for the top and bottom cards, and the ace of spades is on top. Prove that the number of ways to properly arrange the deck
a) is divisible by $12!;$
b) is divisible by $13!$ | Obviously, the correct arrangement of cards in a deck corresponds to a cyclic rook tour of a $4 \times 13$ board (horizontals correspond to suits, and verticals to ranks), starting and ending at the bottom-left corner, corresponding to the ace of spades (mark this cell). Such a tour can be conveniently encoded by numbe... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,654 |
Zaslavsky A.A.
In a single-round football tournament, $n>4$ teams played. For a win, 3 points were awarded, for a draw 1, and for a loss 0. It turned out that all teams scored the same number of points.
a) Prove that there will be four teams with the same number of wins, the same number of draws, and the same number of... | a) If two teams have scored the same number of points, then the difference between the number of draws they have is a multiple of 3.
The number of draws a team has is between 0 and $n-1$. Therefore, the number of groups, each of which consists of teams with the same number of wins, draws, and losses, does not exceed $... | 10 | Combinatorics | proof | Yes | Yes | olympiads | false | 54,655 |
Berpow S.L.
Can the cells of an infinite grid sheet be filled with natural numbers in such a way that for any natural numbers $m, n > 100$ the sum of the numbers in any $m \times n$ rectangle of cells is divisible by $m+n$?
# | Suppose it was possible. Consider any square A of size $200 \times 200$. Let it be a corner square of some square B of size $200 t \times 200 t$, where $t$ is some natural number that does not divide the sum of the numbers in square A.
We will divide the figure $B \backslash A$ into rectangles of size $200 \times 200(... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 54,656 |
Asahanov $H . X$.
On the plane, there are $n>1$ points. Two players take turns to connect an unconnected pair of points with a vector in one of two possible directions. If after a player's move, the sum of all drawn vectors is zero, the second player wins; if the next move is impossible and the sum has never been zero... | The first player wins. We will show that it is sufficient for the first player to draw a vector with the maximum abscissa each turn, and among all vectors with an abscissa equal to the maximum, the vector with the maximum ordinate. Indeed, let's prove that then the sum of all drawn vectors will have either a positive a... | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 54,657 |
Kanunnikov A.L.
For each prime $p$, find the greatest natural power of the number $p$ !, by which the number $(p^2)!$ is divisible. | If $\left(p^{2}\right)!$ is divisible by $(p!)^{n}$, then $n \leq p+1$, since $p$ enters the factorization of the number $p!$ into prime factors with an exponent of 1 (and thus in the factorization of the number $(p!)^{n}$ - with an exponent of $n$), while in the factorization of the number $(p^{2})!$ - with an exponen... | p+1 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 54,659 |
Shapovaoov A.V.
Two chess teams of equal size played a match: each played once with each member of the other team. In each game, 1 point was given for a win, $1 / 2$ for a draw, and 0 for a loss. In the end, the teams scored an equal number of points. Prove that some two participants of the match also scored an equal ... | a) Since 25 games were played, the total number of points scored by the teams is 25, which means each team scored 12.5 points.
Assume that each chess player has a different number of points. Each of the 10 chess players could have scored 0, 0.5, 1, 1.5, ..., 4.5, 5 points - a total of 11 options. Therefore, exactly on... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,660 |
Zaslavsky A.A. Given a triangle $ABC$. A line $l$ is tangent to the inscribed circle of the triangle. Denote by $l_{a}, l_{b}, l_{c}$ the lines symmetric to $l$ with respect to the bisectors of the external angles of the triangle. Prove that the triangle formed by these lines is equal to triangle $ABC$. | Note that the center $I$ of the inscribed circle of triangle $ABC$ is the orthocenter of triangle $KLM$, formed by the external angle bisectors. Draw a line $m$ through $I$, parallel to $l$. The lines $m_a, m_b, m_c$, symmetric to $m$ with respect to these bisectors, intersect at a single point $T$, lying on the circum... | proof | Geometry | proof | Yes | Yes | olympiads | false | 54,661 |
Shapovesov A.V.
Initially, the monomials $1, x, x^{2}, \ldots, x^{n}$ were written on the board. Agreeing in advance, $k$ boys simultaneously calculated each the sum of some two polynomials written on the board every minute, and wrote the result on the board. After $m$ minutes, among others, the polynomials $S_{1}=1+x... | Let's construct a graph corresponding to the final situation on the board: if a polynomial $P$ appears as the sum of polynomials $Q$ and $R$, we will draw arrows from $P$ to $Q$ and $R$. If there is an oriented path from polynomial $F$ to $G$, we will say that $G$ participates in $F$ (in particular, $F$ participates in... | \geq\frac{2n}{k+1} | Algebra | proof | Yes | Yes | olympiads | false | 54,662 |
Berezin V.N.
Find the sums
a) $1 \cdot n+2(n-1)+3(n-2)+\ldots+n \cdot 1$.
b) $S_{n, k}=(1 \cdot 2 \cdot \ldots \cdot k) \cdot(n(n-1) \ldots(n-k+1))+(2 \cdot 3 \cdot \ldots \cdot(k+1)) \cdot((n-1)(n-2) \ldots(n-k))+\ldots+((n-k+1)(n-k+$
2)... $\cdot n) \cdot(k(k-1) \cdot \ldots \cdot 1)$ | a) First, let's find the sum of the form $1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + (n-1) n$.
Notice that $3 k(k+1) = k(k+1)(k+2) - (k-1) k(k+1)$. Adding these equalities for $k$ from 1 to $n-1$, we get:
$3(1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + (n-1) n) = (n-1) n(n+1)$.
Let's return to solving the problem.... | (k!)^2\cdotC_{n+k+1}^{2k+1} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 54,665 |
Authors: Shapovalov A.V., Spivak A.V.
The fortification system consists of dugouts. Some of the dugouts are connected by trenches, and from each dugout, one can run to another. A foot soldier... | a) Let's denote the shelters as shown in the diagram. We will limit the initial positions of the infantryman to shelters $O$, $A_{2}$, $B_{2}$, and $C_{2}$ (we will call these even shelters, and the rest will be odd). We assume that the infantryman is so lucky that he escapes if it is at all possible.
. Therefore, the parity of their number in vertex $A$ is also the same; then all of them are od... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,671 |
In some cells of a $100 \times 100$ board, there is a chip. We call a cell beautiful if there is an even number of chips in the cells adjacent to it by side.
Can exactly one cell of the board be beautiful? | Lemma. For any cell $X$ on the board, there exists a set $S$ consisting of an even number of cells and containing $X$, such that each cell on the board has an even number of neighbors in $S$.
Proof. We color the cells of the board in a checkerboard pattern so that $X$ becomes black. Consider one of the diagonals passi... | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 54,672 |
Berr S.L.
Ten cars are driving in one direction along a highway. The highway passes through several settlements. Each of the cars travels at a certain constant speed in the settlements and at a different constant speed outside the settlements. These speeds may differ for different cars. Along the highway, there are 20... | Let's introduce a coordinate system Oxyt. Denote by $M$ the point on the highway where the first car is located at the initial moment. For each point $A$ on the highway, we associate a point $T_{A}\left(x_{A}, y_{A}\right)$ on the plane $O x y$, where $x_{A}$ is the total length of the segments of the path $A M$ within... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,673 |
Chebotarev A.S.
On a plane, there is a circle. What is the minimum number of lines that need to be drawn so that, by symmetrically reflecting the given circle relative to these lines (in any order a finite number of times), it can cover any given point on the plane? | 1) We will prove that three lines are sufficient. Let the horizontal line $a$ and line $b$, forming a $45^{\circ}$ angle with line $a$, contain the center $O$ of the circle, and let line $c$ be parallel to $a$ and be at a distance of $0.5R$ from it, where $R$ is the radius of the given circle (see figure).
The composi... | 3 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 54,674 |
Kasparov G.A.
When the hockey tournament (in one round) ended, it turned out that for each group of teams, one could find a team (possibly from the same group) that scored an odd number of points in games with teams from this group. Prove that an even number of teams participated in the tournament. (Loss - 0 points, d... | Let the number of teams be $N$. First, let's represent the tournament results in the form of a tournament table $N \times N$. At the intersection of the $i$-th row and the $j$-th column, we place the number $a_{ij}$ of points scored by the $i$-th team in the match against the $j$-th team. We will assume that a team sco... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,675 |
Andzans A.
The number of edges of a polyhedron is 100.
a) What is the maximum number of edges that a plane, not passing through its vertices, can intersect if the polyhedron is convex?
b) Prove that for a non-convex polyhedron, this number can be 96,
c) but cannot be 100. | a) Estimation. No more than $2 / 3$ of the edges of a convex polyhedron can be intersected by one plane.
Indeed, in each face, no more than two sides are intersected, and the number of sides of this face is at least 3, meaning that in each face, no more than $2 / 3$ of its sides are intersected. Summing these inequali... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 54,676 |
Raiigorodskii A.M.
Consider a graph where the vertices correspond to all possible three-element subsets of the set $\left\{1,2,3, \ldots, 2^{k}\right\}$, and edges are drawn between vertices that correspond to subsets intersecting at exactly one element. Find the minimum number of colors needed to color the vertices o... | Let $N=C_{n}^{3}$ be the number of vertices of our graph $G$, that is, the number of triples of elements from the set $\{1, \ldots, n\}$. The minimum number of colors $\chi(G)$, in which the vertices can be properly colored, is called the chromatic number of the graph.
We will show that $\chi(G) \geq 1 / 6(n-1)(n-2)$.... | \frac{1}{6}(2^k-1)(2^k-2) | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 54,678 |
$\underline{\text { Tolpygo A.K. }}$
With a natural number (written in the decimal system), the following operations are allowed:
A) append the digit 4 at the end;
B) append the digit 0 at the end;
C) divide by 2 (if the number is even).
For example, if we perform operations C, C, A, and B sequentially on the numb... | a) Instead of obtaining the number 1972 from the number 4 using operations A, B, and C, we will try to obtain the number 4 from the number 1972 using the inverse operations:
A'
B' ) erasing the digit 0 at the end;
C' ) multiplying the number by 2.
In this process, we will apply operation A' or B' whenever possible ... | proof | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 54,679 |
Ostrovsky $M$.
A number from 1 to 144 is guessed. You are allowed to select one subset of the set of numbers from 1 to 144 and ask whether the guessed number belongs to it. You have to pay 2 rubles for a "yes" answer and 1 ruble for a "no" answer. What is the minimum amount of money needed to surely guess the number? | Let $a_{1}=2, a_{2}=3, a_{i}=a_{i-} 1+a_{i-} 2$ for $i \geq 3$. Then $a 10=144$. We will prove by induction that among not less than $a_{i}$ numbers, the guessed number cannot be found by paying less than $i+1$ rubles.
For $i=1$ and $i=2$, this is true.
Suppose there are not less than $a_{i}$ numbers. Then either the... | 11 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 54,682 |
Toom A.l.
On an infinite grid of white paper, $n$ cells are painted black. At moments $t=1,2,3,...$, all cells on the sheet are simultaneously repainted according to the following rule: each cell $k$ acquires the color that the majority of three cells had in the previous moment: the cell $k$ itself and its neighbors t... | Let $B$ be some set of black cells. Denote by $\Gamma$ (from the word "voting") the recoloring operator, i.e., through $\Gamma(B)$ we denote the set of cells that will result from $B$, through $\Gamma^{2}(B)$ - the set $\Gamma(\Gamma(B))$, into which $B$ will transition in two steps, through $\Gamma^{3}(B)$ - in three ... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,685 |
Galperin G.A.
What is the maximum number of points that can be placed a) on a plane; b)* in space so that no triangle with vertices at these points is obtuse?
(Of course, it is implied in the condition that no three points should lie on the same line - without this restriction, any number of points can be placed.) | Let us prove that a larger number of points cannot be placed.
First of all (this applies to both the planar and spatial problems), if $A_{i}$ and $A_{j}$ are any two of the $n$ points $A_{1}, A_{2}, \ldots, A_{n}$ satisfying the problem's condition, then all these points must belong to the set of points $M$ for which ... | )4points,b)8points | Geometry | math-word-problem | Yes | Yes | olympiads | false | 54,687 |
Zaslavsky A.A.
The graphs of two quadratic trinomials intersect at two points. In both points, the tangents to the graphs are perpendicular.
Is it true that the axes of symmetry of the graphs coincide? | The graphs $y=1 / 8\left(x^{2}+6 x-25\right)$ and $y=1 / 8\left(25+6-x^{2}\right)$ have axes $x= \pm 3$, and intersect at $x= \pm 5$. The product of the tangents of the angles of inclination of the tangents at the points of intersection is $1 / 64(2 \cdot 5+6)(6-2 \cdot 5)=-1$. Therefore, the tangents at these points a... | notfound | Algebra | proof | Yes | Yes | olympiads | false | 54,692 |
Shapovalov A.V.
In a deck, some cards lie face down. From time to time, Petya takes out a stack of one or several consecutive cards from the deck, where the top and bottom cards are face down, flips the entire stack as a whole, and inserts it back into the same place in the deck (if the "stack" consists of only one ca... | Let's assign a digit to each card: one if it is face down, and two if it is face up. Writing these digits from left to right, starting with the digit corresponding to the top card, we get some natural number. Each time the stack of cards is flipped, this number decreases: the two corresponding to the top card in the st... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,694 |
Tamarkin D:
In the class, there are 16 students. Every month, the teacher divides the class into two groups.
What is the minimum number of months that must pass so that every two students end up in different groups at some point? | Example. The figure shows how to divide a class into two groups so that any two students are in different groups in at least one of the four months. Each student corresponds to a column in the table, and each month corresponds to a row. A zero in a cell of the table means that the student is in the first group, and a o... | 4 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 54,695 |
Raskin I.V.
a) A miser knight keeps gold coins in six chests. Once, counting them, he noticed that if any two chests are opened, the coins lying in them can be evenly distributed between these two chests. He also noticed that if any 3, 4, or 5 chests are opened, the coins lying in them can also be rearranged so that t... | a) First method. Divide the chests into three pairs. The total number of coins in each pair of chests is even, so the number of coins in all six chests is also even. Now divide the chests into two triplets. The number of coins in each triplet is a multiple of 3, so the total number of coins in all the chests is also a ... | proof | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 54,696 |
$\underline{\text { Tolpygo A.K. }}$
In space, there are 30 non-zero vectors. Prove that among them, there are two vectors such that the angle between them is less than $45^{\circ}$. | Place the origins of all vectors at point $O$. Surround each vector with a cone with vertex $O$ and an "aperture angle" of $45^{\circ}$ (the vector is directed along the axis of this cone). Each cone cuts out a cap on the unit sphere (the area of which is $4 \pi$). The area of the cap is greater than the area of its ba... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,698 |
Shapovalov A.V.
a) On each of the fields of the top and bottom horizontal lines of an $8 \times 8$ chessboard, there is a chip: white ones at the bottom, black ones at the top. In one move, it is allowed to move any chip to an adjacent free cell vertically or horizontally. What is the minimum number of moves required ... | a) Evaluation. To reach the opposite side of the board, a piece needs to make seven vertical moves. However, at least one of the two pieces standing on the same vertical must make a horizontal move (otherwise, they would not be able to pass each other). Therefore, together these pieces will make no fewer than 15 moves.... | 120 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 54,699 |
Bogdanov I.I.
Does there exist a convex $N$-gon, all sides of which are equal, and all vertices of which lie on the parabola $y=x^{2}$, if
a) $N=2011$;
b) $N=2012 ?$ | a) Let $O$ be the vertex of the parabola. On the right branch, we lay out 1005 equal chords $O A_{1}, A_{1} A_{2}, A_{2} A_{3}, \ldots, A_{1004} A_{1005}$ of length $t$. Consider the broken line $O B_{1} B_{2} \ldots B_{1005}$, symmetric to $O A_{1} A_{2} \ldots A_{1005}$ with respect to the axis of the parabola.
Obvi... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 54,701 |
10,11
[Application of trigonometric formulas (geometry) $]$
[ Identical transformations (trigonometry). ]
In a convex quadrilateral $A B C D: A C \perp B D, \angle B C A=10^{\circ}, \angle B D ... | Let $K$ and $M$ be the points of intersection of the line $CB$ with the line $AD$ and the circumcircle of triangle $ACD$, respectively (see figure). Then
$\angle MDA = \angle MCA = 10^{\circ}$, so $DM$ is the bisector of angle $KDB$. Also note that $\angle ABD = 50^{\circ}$, $\angle CBD = 80^{\circ}$, hence $\angle KB... | 60 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 54,708 |
7,8, |
| | Counting in two ways | |
| | [ Partitions into pairs and groups; bijections | |
| | [ Proof by contradiction | |
| | Extreme principle (other). | |
Author: Bapat R.B.
In each cell of a square table, a number is written. It is known that in each row of the table, the sum of the two largest numbers i... | Let the table have $n$ rows. In each row, we take the two largest numbers and write down these $2n$ numbers in ascending order. Due to the equality of sums, the pairs include the first and last, the second and second-to-last, and so on. We mark the $n+1$ largest of the written numbers in the table. By the Pigeonhole Pr... | b | Combinatorics | proof | Yes | Yes | olympiads | false | 54,709 |
9,10,11 |
| :---: | :---: | :---: |
| | Dirichlet's Principle (miscellaneous) | |
| | Counting in Two Ways | |
| | [Pairing and Grouping; Bijections] | |
| | [Proof by Contradiction] | |
| | Extreme Principle (miscellaneous). | |
Author: Bapat R.B.
In each cell of a square table, a real number is written. I... | a) See problem $\underline{116213}$.
b)
| 2 | 2 | 3 | 0 |
| :--- | :--- | :--- | :--- |
| 2 | 2 | 3 | 0 |
| 2 | 2 | 0 | 3 |
| 2 | 2 | 0 | 3 |
Send a comment | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 54,710 |
Pak I.
Given a pyramid $S A_{1} A_{2} \ldots A_{n}$, the base of which is a convex polygon $A_{1} A_{2} \ldots A_{n}$. For each $i=1,2, \ldots, n$, in the plane of the base, we construct a triangle $X_{i} A_{i} A_{i+1}$, equal to the triangle $S A_{i} A_{i+1}$ and lying on the same side of the line $A_{i} A_{i+1}$ as ... | Consider an arbitrary point $P$ on the base and prove that it is covered by one of the triangles. Take a small sphere lying inside the pyramid and touching the base at point $P$ (such a sphere clearly exists). Start increasing its radius while maintaining the condition of tangency; then at some moment it will first tou... | proof | Geometry | proof | Yes | Yes | olympiads | false | 54,711 |
9,10,11 |
| :---: | :---: | :---: |
| | Sequences (other). | |
| | Rotation helps solve the problem | |
| | Symmetry and involutive transformations | |
Authors: V. A. Vladimirov, R. N. Izmailov.
For each pair of real numbers $a$ and $b$, consider the sequence of numbers $p_{\mathrm{n}}=[2\{a n+b\}]$. Any $k$ co... | Let's represent numbers as points on a unit circle (numbers with the same fractional part correspond to the same point on the circle, see the comment to the solution of problem 6 for 10th grade of the 1997 olympiad). Then the sequence $x_{\mathrm{n}}=\{a n+b\}$ corresponds to a sequence of points on the circle obtained... | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 54,713 |
| [ Isosceles, Inscribed, and Circumscribed Trapezoids |
| :---: | :---: |
| | Inscribed and Circumscribed Circles |
| | Spiral Similarity (other). |
| | Circles Inscribed in Segments |
| | Law of Sines |
Authors: Zaslavsky A.A., Protasov V.Yu.
Trapezoid \(ABCD\) is inscribed in circle \(w\) (\(AD \parallel BC\))... | Note that $X$ and $Y$ are diametrically opposite points, hence $\angle X A Y = \angle X B Y = 90^{\circ}$. Let $I$ and $J$ be the centers of the inscribed circles of triangles $A B C$ and $A B D$, respectively. Then, by the "cloverleaf theorem" (see problem $\underline{\text{53119}}$), $X B = X I$ and $Y A = Y J$. Addi... | proof | Geometry | proof | Yes | Yes | olympiads | false | 54,714 |
Consider the triangle $E$.
On the coordinate plane, a triangle is placed such that its translations by vectors with integer coordinates do not overlap.
a) Can the area of such a triangle be greater than 1/2?
b) Find the greatest possible area of such a triangle. | a) For example, such a triangle has vertices at points $(0,0), (4/3, 2/3)$, and $(2/3, 4/3)$ (see figure). It is easy to see that the sides of the translated triangles pass through the vertices of the original triangle, and that the area of the triangle is $2/3$.
Authors: Shapovalov A.V., Doienko V.V.
From the available sequences $\left\{b_{n}\right\}$ and $\left\{c_{n}\right\}$ (possibly $\left\{b_{n}\right\}$ coincides with $\left\{c_{n}\right\}$), ... | Let's explain how to act in points a) and b).
The allowed operations can transform the sequence $\left\{a_{n}\right\}$ into the sequence $\left\{a_{n+1}-a_{n}\right\}$.
Such a transformation is denoted by $T$, and $m$-fold application of transformation $T$ is denoted by $T^{m}$. Note that if $P$ is a polynomial of de... | ),b)Possible;)Impossible | Algebra | math-word-problem | Yes | Yes | olympiads | false | 54,716 |
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